back to list

Lehman's 'Bach' scale

🔗Carl Lumma <clumma@yahoo.com>

8/17/2005 5:23:57 PM

I don't want to get into an argument, but I thought I'd
make some observations. I've read Brad's web site on a
number of occasions.

I'm not entirely clear if Brad is making a claim that his
scale is *the* scale of Bach, or just if Bach's music
sounds good in it. He hasn't presented any objective way
of testing the latter, so the claim is essentially
meaningless. It certainly can't be taken as evidence for
the former, as much of his text seems to do.

The other evidence for the tuning -- the squiggles on the
title page of the WTC -- am I the only one who thinks this
is a bit of a stretch?

Many of the other arguments on Brad's website focus on
attacking Barnes, Kellner, and other 'Bach scale'
scholars. That certainly isn't hard to do, but again it
isn't evidence in favor of Brad's proposal. As far as
iconography goes, Kellner's page (apparently down at the
moment) on Bach's signet ring was more convincing than
the squiggles.

As for evidence in the first place of there being one
'Bach scale' waiting for us to rediscovered, I haven't
seen much.

-Carl

🔗Brad Lehman <bpl@umich.edu>

8/18/2005 7:49:42 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> I don't want to get into an argument, but I thought I'd
> make some observations. I've read Brad's web site on a
> number of occasions.
>
> I'm not entirely clear if Brad is making a claim that his
> scale is *the* scale of Bach, or just if Bach's music
> sounds good in it. He hasn't presented any objective way
> of testing the latter, so the claim is essentially
> meaningless. It certainly can't be taken as evidence for
> the former, as much of his text seems to do.

Carl, the page of musical examples to "test the latter" empirically
is at
http://www-personal.umich.edu/~bpl/larips/testpieces.html
and a shorter version of that is in the second half of the article.
Reproduce the musical experiment yourself, in practice, to hear what
is being claimed!

Some practical instructions to set up the temperament for this test
are at
http://www-personal.umich.edu/~bpl/larips/practical.html
and, of course, one should also set up any other comparative
temperaments of one's own choice to hear the differences, in the same
music.

The supplementary file for part 2 is available at
http://em.oxfordjournals.org/cgi/content/full/33/2/211/DC1
giving a closer analysis of BWV 622, 591, and 802-5.

Several Bach compositions are demonstrated directly in recorded
examples at
http://www-personal.umich.edu/~bpl/larips/samples.html
with four full-length CDs to be released shortly. Those are played
on organ and harpsichord.

An hour-long interview about this whole thing, including additional
musical examples played by me and Richard Egarr, will be available
within the next few weeks at
http://www.bbc.co.uk/radio3/earlymusicshow/
I have not heard Egarr's recording yet myself, but it is a new disc
of the Goldberg Variations.

Bradley Lehman

🔗Michael Zapf <zapfzapfzapf@yahoo.de>

8/18/2005 7:44:18 AM

<Kellner's page (apparently down at the
moment)>

Kellner died several years ago, he can't defend
himself anymore.
Michael




___________________________________________________________
Yahoo! Messenger - NEW crystal clear PC to PC calling worldwide with voicemail http://uk.messenger.yahoo.com

🔗rumsong <rumsong@telus.net>

8/18/2005 9:26:34 AM

--- In tuning@yahoogroups.com, Michael Zapf
<zapfzapfzapf@y...> wrote:
> <Kellner's page (apparently down at the
> moment)>
>
> Kellner died several years ago, he can't defend
> himself anymore.
> Michael

Greetings,

I had suspected this, but I am very sad to hear of it. I owed him a
letter, but life and work prevented my proper reply.

I studied his theory many years ago, and while I do not hold to it, I
think he worked with honesty towards what he thought was a
solution to a difficult problem. Perhaps this is all we can do
since Truth, Absolute Truth is such a slippery 'thing.'

All best wishes,

Gordon Rumson

🔗Carl Lumma <clumma@yahoo.com>

8/18/2005 6:43:45 PM

> > I don't want to get into an argument, but I thought I'd
> > make some observations. I've read Brad's web site on a
> > number of occasions.
> >
> > I'm not entirely clear if Brad is making a claim that his
> > scale is *the* scale of Bach, or just if Bach's music
> > sounds good in it. He hasn't presented any objective way
> > of testing the latter, so the claim is essentially
> > meaningless. It certainly can't be taken as evidence for
> > the former, as much of his text seems to do.
>
> Carl, the page of musical examples to "test the latter"
> empirically is at
> http://www-personal.umich.edu/~bpl/larips/testpieces.html

That's just a list of pieces. What is the procedure for
testing them? Notice I used the word "objective".

>and a shorter version of that is in the second half of the
>article. Reproduce the musical experiment yourself, in
>practice, to hear what is being claimed!

That sounds like going to a faith healer, and 'feeling the
results for myself'.

> Several Bach compositions are demonstrated directly in
> recorded examples at
> http://www-personal.umich.edu/~bpl/larips/samples.html
> with four full-length CDs to be released shortly. Those are
> played on organ and harpsichord.

These aren't even comparisons -- only your temperament is
demonstrated.

By the way, it seems very unlikely that the Kleines Harmonisches
Labyrinth was written by Bach.

-Carl

🔗monz <monz@tonalsoft.com>

8/18/2005 10:57:06 PM

Hi Brad and Michael,

--- In tuning@yahoogroups.com, Michael Zapf <zapfzapfzapf@y...> wrote:

> > [Brah Lehman:]
> > <Kellner's page (apparently down at the
> > moment)>
>
> Kellner died several years ago, he can't defend
> himself anymore.

But he was a member of this group before his passing,
and he did indeed defend himself and his work on
"Bach's tuning".

... too bad that the Yahoo search feature sucks so much
that i'm not willing to invest the time to look for
his posts ... but hopefully someone else will.
They're in here.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Aaron Krister Johnson <aaron@akjmusic.com>

8/19/2005 9:13:21 AM

On Friday 19 August 2005 12:57 am, monz wrote:
> Hi Brad and Michael,
>
> --- In tuning@yahoogroups.com, Michael Zapf <zapfzapfzapf@y...> wrote:
> > > [Brah Lehman:]
> > > <Kellner's page (apparently down at the
> > > moment)>
> >
> > Kellner died several years ago, he can't defend
> > himself anymore.
>
> But he was a member of this group before his passing,
> and he did indeed defend himself and his work on
> "Bach's tuning".
>
> ... too bad that the Yahoo search feature sucks so much
> that i'm not willing to invest the time to look for
> his posts ... but hopefully someone else will.
> They're in here.

Where where you Monz, when I suggested we move over to Google groups for that
very reason? ;)

-Aaron.

🔗Brad Lehman <bpl@umich.edu>

8/19/2005 10:06:42 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > http://www-personal.umich.edu/~bpl/larips/testpieces.html
>
> That's just a list of pieces. What is the procedure for
> testing them? Notice I used the word "objective".

I have already outlined a suggested procedure, at
http://www-personal.umich.edu/~bpl/larips/faq2.html

- Set up this temperament on a good harpsichord: and preferably by
ear both to understand its structure and because Bach himself had no
electronic aids.

- Play through all the musical examples of the article, footnotes,
and supplementary files--listening carefully to the highlighted
features in the sound. The article is about sound and listening, an
attempt to hear things inside an aesthetic scheme written down by
Bach, testing my hypothesis of its correct interpretation and its
place in Bach's milieu.

- Play Bach's repertoire, and especially the compositions that had
been most problematic in other temperaments, listening closely for
overall balance of musical elements.

- Do similar listening tests with at least Werckmeister III,
Vallotti, regular 1/6 comma, and equal temperament to hear the
differences; and additionally with any other proposed "Bach"
temperaments for further comparison. A point here is to locate the
Bach expectations within musical and historical context, which
presupposes the reader's ability to hear and compare this with other
contemporary layouts (including Neidhardt's and Sorge's, for the
adventurous and thorough!).

Notice that my suggestion *does not* say listen to MIDI files run
through a computer. That in itself might be interesting, but Bach
could not have done so. It's up to hands-on playing on acoustic
instruments, experiencing them by *playing* the repertoire directly
and noting what it does to your own sense of melody/harmony/timing,
as to the tensions and resolutions in the music. Bach wrote the WTC
for musicians to work on in their own keyboard practice and lessons.
If he somehow also envisioned computer-listening or recording-
listening at some point in his distant future, with people having the
music played for them passively by a machine, that would really be
something else!

> > Several Bach compositions are demonstrated directly in
> > recorded examples at
> > http://www-personal.umich.edu/~bpl/larips/samples.html
> > with four full-length CDs to be released shortly. Those are
> > played on organ and harpsichord.
>
> These aren't even comparisons -- only your temperament is
> demonstrated.

And those are for people who don't play keyboards at all, so they can
at least hear the music on appropriate acoustic instruments with the
proposed tuning. Broader comparisons require actually doing some
homework: finding other comparative recordings by other people, and
(better yet) hands-on playing of the repertoire oneself. I'm not
going to have multiple additional recording sessions and production
cost, just to excuse people from going and doing their own homework
and investigations....!

>
> By the way, it seems very unlikely that the Kleines Harmonisches
> Labyrinth was written by Bach.

I have addressed exactly that question of authenticity in the Oxford
supplement to part 2 of the article, as part of discussing that
piece. Some _Bach-Jahrbuch_ articles, and indeed a recent volume of
the _Neue Bach-Ausgabe_ (2003) as well, do include this piece in the
Bach repertoire after all. The books/articles I've cited are from
2000, 2001, and 2003 on this. And yes, I know that Ledbetter in 2002
and Williams in 2003 continue to vote it out, and that the 1998 BWV
still has it in the "questionable" index.

Whether some people would allow this composition as a testing example
or not, based *only* on the question whether Bach himself wrote it
(to everybody's unanimous satisfaction), is to miss the broader
point. The piece, as it stands, is a useful one with which to
explore temperament issues hands-on. And it's a fairly easy one to
play, too, on any keyboard: with only one optional pedal-point in the
last section, or just restrike it a couple of times if there's no
pedalboard available.

As I mentioned here a couple of days ago, the four Duetti BWV 802-5
are even better in that regard as hands-on test pieces, and they are
unquestionably by JSB. There's nowhere for any less-than-worthy
temperament to hide in their brutally exposed texture of two voices,
either melodically or with two-note intervals harmonically.

Bradley Lehman

🔗monz <monz@tonalsoft.com>

8/19/2005 11:55:50 AM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:

> On Friday 19 August 2005 12:57 am, monz wrote:
> >
> > ... too bad that the Yahoo search feature sucks so much
> > that i'm not willing to invest the time to look for
> > his posts ... but hopefully someone else will.
> > They're in here.
>
> Where where you Monz, when I suggested we move over
> to Google groups for that very reason? ;)

I was busy working on the new Tonalsoft website ...
where *we* plan to eventually host the tuning list!

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Carl Lumma <clumma@yahoo.com>

8/19/2005 1:59:50 PM

> > > http://www-personal.umich.edu/~bpl/larips/testpieces.html
> >
> > That's just a list of pieces. What is the procedure for
> > testing them? Notice I used the word "objective".
>
> I have already outlined a suggested procedure, at
> http://www-personal.umich.edu/~bpl/larips/faq2.html
>
> - Set up this temperament on a good harpsichord: and preferably by
> ear both to understand its structure and because Bach himself had
> no electronic aids.
>
> - Play through all the musical examples of the article, footnotes,
> and supplementary files--listening carefully to the highlighted
> features in the sound. The article is about sound and listening, an
> attempt to hear things inside an aesthetic scheme written down by
> Bach, testing my hypothesis of its correct interpretation and its
> place in Bach's milieu.
>
> - Play Bach's repertoire, and especially the compositions that had
> been most problematic in other temperaments, listening closely for
> overall balance of musical elements.
>
> - Do similar listening tests with at least Werckmeister III,
> Vallotti, regular 1/6 comma, and equal temperament to hear the
> differences; and additionally with any other proposed "Bach"
> temperaments for further comparison. A point here is to locate the
> Bach expectations within musical and historical context, which
> presupposes the reader's ability to hear and compare this with
> other contemporary layouts (including Neidhardt's and Sorge's,
> for the adventurous and thorough!).
>
> Notice that my suggestion *does not* say listen to MIDI files run
> through a computer.

This isn't an objective test, it's a subjective one. Who cares
if I do this and think your temperament sounds 'better' in some
way? Who cares if I think Werckmeister III sounds better?

-Carl

🔗Brad Lehman <bpl@umich.edu>

8/19/2005 7:06:17 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > http://www-personal.umich.edu/~bpl/larips/testpieces.html
> > >
> > > That's just a list of pieces. What is the procedure for
> > > testing them? Notice I used the word "objective".
> >
> > I have already outlined a suggested procedure, at
> > http://www-personal.umich.edu/~bpl/larips/faq2.html
> >
> > - Set up this temperament on a good harpsichord: and preferably by
> > ear both to understand its structure and because Bach himself had
> > no electronic aids.
> >
> > - Play through all the musical examples of the article, footnotes,
> > and supplementary files--listening carefully to the highlighted
> > features in the sound. The article is about sound and listening,
an
> > attempt to hear things inside an aesthetic scheme written down by
> > Bach, testing my hypothesis of its correct interpretation and its
> > place in Bach's milieu.
> >
> > - Play Bach's repertoire, and especially the compositions that
had
> > been most problematic in other temperaments, listening closely
for
> > overall balance of musical elements.
> >
> > - Do similar listening tests with at least Werckmeister III,
> > Vallotti, regular 1/6 comma, and equal temperament to hear the
> > differences; and additionally with any other proposed "Bach"
> > temperaments for further comparison. A point here is to locate
the
> > Bach expectations within musical and historical context, which
> > presupposes the reader's ability to hear and compare this with
> > other contemporary layouts (including Neidhardt's and Sorge's,
> > for the adventurous and thorough!).
> >
> > Notice that my suggestion *does not* say listen to MIDI files run
> > through a computer.
>
> This isn't an objective test, it's a subjective one. Who cares
> if I do this and think your temperament sounds 'better' in some
> way? Who cares if I think Werckmeister III sounds better?

The point is not to evaluate things as better/worse/whatever. The
point is to experience them directly in the first place, in the same
way that Bach would have done: sitting and working directly at an
acoustic instrument, hands-on, playing through the music carefully in
different ways to hear what is there, and to experience any effect it
might have on one's own musicality or conception of the piece.

How might one explain the concept of "orange" to a person who has
never troubled to view an orange object directly? Orange isn't
qualitatively better or worse than other colors; it's just something
to experience directly rather than speculating about/against. It's
some measurable range of a wave pattern, objectively. One might
quibble what the boundaries of orange are, on either side toward the
neighboring colors of different frequencies, sure. But the simplest
way to distinguish among orange, green, and violet is to look at them
side by side with a fair and controlled test.... Whether someone
personally prefers the green or the violet ahead of the orange is
beside the point.

Your question, in the first place, was how to try my results
empirically. I recommended some ways to do so, and in Bach
repertoire where differences are especially noticeable (i.e. where it
might most affect someone's choice of a temperament for a Bach
concert). A refusal to go do it, then, is just backpedaling against
an attempt to be empirical, isn't it?.... Empiricism only when it's
not too inconvenient, and won't take time/effort?

Brad Lehman

🔗Carl Lumma <clumma@yahoo.com>

8/21/2005 10:54:13 AM

> Your question, in the first place, was how to try my results
> empirically. I recommended some ways to do so, and in Bach
> repertoire where differences are especially noticeable (i.e.
> where it might most affect someone's choice of a temperament
> for a Bach concert). A refusal to go do it, then, is just
> backpedaling against an attempt to be empirical, isn't it?....
> Empiricism only when it's not too inconvenient, and won't take
> time/effort?

My point is that you've made some very strong claims for
something that you derived through what can only be called
far-fetched means. I don't doubt that your scale is a
fine well temperament, distinguishable (under very careful
listening) from equal or even other well temperaments. I
do doubt that Bach intended a particular scale for the WTC.
I believe he, like most keyboardists, had a favored method
he used to tune his intruments, which he probably taught to
his students. But I am not convinced that your method
corresponds to it. If your point is, "Hey, well temperaments
are great!" then great. But there are lots of people saying
that. You distinguish yourself by making a particular
argument, and I happen to think it's unconvincing. That's
all.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

8/24/2005 12:45:51 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > Your question, in the first place, was how to try my results
> > empirically. I recommended some ways to do so, and in Bach
> > repertoire where differences are especially noticeable (i.e.
> > where it might most affect someone's choice of a temperament
> > for a Bach concert). A refusal to go do it, then, is just
> > backpedaling against an attempt to be empirical, isn't it?....
> > Empiricism only when it's not too inconvenient, and won't take
> > time/effort?
>
> My point is that you've made some very strong claims for
> something that you derived through what can only be called
> far-fetched means. I don't doubt that your scale is a
> fine well temperament,

Not so fast, Carl! You'll find the specifics of the Lehman-"Bach"
temperament here:
http://www-personal.umich.edu/~bpl/larips/math.html

After looking at the numbers, I must say that I have a few doubts.
Here's a table showing the total error in cents of the intervals in
each major triad:

Db Ab Eb Bb F. C. G. D. A. E. B. F#
39 35 31 23 20 20 27 35 43 39 35 31

For comparison, following is the total error of the intervals of a
major triad in each of the "big three" historical regular
temperaments:

Meantone: 11 cents
12-ET: 31 cents
Pythagorean: 43 cents

In the Lehman-"Bach" temperament the numbers progressively decrease
going from Db to F and then level off at C -- so far so good. They
then increase going from C to A; the intonation favors the keys in
the flat direction (a minor flaw IMO, characteristic of many well-
temperaments, including Werckmeister III), so there's nothing
unexpected there. The numbers then *decrease* going from A to F#,
which is exactly the *opposite* of what is expected (and desired) in
a well-temperament -- a very serious flaw indeed. This leads me to
think that either Brad has not properly deciphered some of
Bach's "squiggles", or if he has, then Bach's temperament is not a
very good one. While you might get some interesting results playing
some of Bach's music in it, I would not expect anything in the key of
A major (its worst triad) to fare particularly "well" (pun intended.

Besides this, there's a wide fifth (Bb-F) that, strictly speaking,
would by itself exclude this temperament from the "well-temperament"
category, inasmuch as this causes the total error of the intervals in
the 12 major triads (379.3c) to exceed the minimum theoretically
possible (387.5c). My reason for bringing this point up is not to
nitpick about a categorical technicality, but rather to make the
point that the inclusion of a wide fifth is *totally unnecessary* in
a temperament in which the narrowest fifths are tempered only 1/6 of
a pythagorean comma (see following paragraph).

I found on this website:
http://www.rollingball.com/TemperamentsFrames.htm
a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) that's
a perfect (even if "unauthentic") alternative to Lehman-"Bach". It
has 1/6-p-comma fifths in a chain from C to F# and just fifths from
F#=Gb to C. If you check the numbers in the following table, you'll
see that none of the major triads from Eb to E (corresponding to the
8 usable ones in a standard meantone tuning) has an error
significantly exceeding that of 12-ET:

Major triad:.. Db Ab Eb Bb F. C. G. D. A. E. B. F#
Lehman-"Bach": 39 35 31 23 20 20 27 35 43 39 35 31
Valotti-Young: 43 43 35 26 20 20 20 20 27 35 43 43
Werckmst. III: 43 43 31 20 _8 20 31 31 31 31 43 43

I've also included Werckmeister III for comparison; although it has
room for improvement, at least its numbers demonstrate a reasonable
progression of intonation from the best to worst keys, and we *do
know* that Bach actually played organs that were so tuned. Maybe he
tuned his harpsichord this way (at least sometimes), or maybe he
tuned it differently -- I'll leave that for others to debate. But I
would have a difficult time accepting the notion that Bach intended
his _Well-tempered Clavier_ to be played in a temperament unworthy of
the name "well-tempered".

Brad, it was not my intention to be unkind in my criticism of your
work, and I hope that you'll be open to reviewing your methodology in
light of my observations.

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

8/24/2005 1:10:33 PM

Correction:

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> ... inasmuch as this causes the total error of the intervals in
> the 12 major triads (379.3c) to exceed the minimum theoretically
> possible (387.5c). ...

Sorry! This should have read:

... inasmuch as this causes the total error of the intervals in
the 12 major triads (379.3c) to exceed the minimum theoretically
possible (375.4c). ...

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/24/2005 1:16:37 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

[...]

Hiya George,

Why not show the total absolute errors in the intervals in each of the
minor triads as well? I mean, as long as you're showing it in each of
the major triads already . . .

Cheers,
Paul

🔗Ozan Yarman <ozanyarman@superonline.com>

8/24/2005 1:20:29 PM

George, I'm still quite in the dark as to why you think Bach would not have approved MY well-temperament. It IS very easy to tune by ear too. 8-)

Cordially,
Ozan

12-tone Well Temperament from practically 159tET
|
0: 1/1 C Dbb unison, perfect prime
1: 90.225 cents C# Db
2: 196.090 cents D Ebb
3: 294.135 cents D# Eb
4: 392.180 cents E Fb
5: 498.045 cents F Gbb
6: 588.270 cents F# Gb
7: 701.955 cents G Abb
8: 792.180 cents G# Ab
9: 898.045 cents A Bbb
10: 996.090 cents A# Bb
11: 1086.315 cents B Cb
12: 2/1 C Dbb octave

0: 0.000 cents 0.000 0 0 commas C
7: 701.955 cents -0.000 0 0 commas G
2: 694.135 cents -7.820 -240 -1/3 Pyth. commas D
9: 701.955 cents -7.820 -240 -1/3 Pyth. commas A
4: 694.135 cents -15.640 -480 -2/3 Pyth. commas E
11: 694.135 cents -23.460 -720 -1 Pyth. commas B
6: 701.955 cents -23.460 -720 -1 Pyth. commas F#
1: 701.955 cents -23.460 -720 -1 Pyth. commas C#
8: 701.955 cents -23.460 -720 -1 Pyth. commas G#
3: 701.955 cents -23.460 -720 -1 Pyth. commas Eb
10: 701.955 cents -23.460 -720 -1 Pyth. commas Bb
5: 701.955 cents -23.460 -720 -1 Pyth. commas F
12: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic co C
Average absolute difference: 18.2467 cents
Root mean square difference: 20.8237 cents
Maximum absolute difference: 23.4600 cents
Maximum formal fifth difference: 7.8200 cents

0-4-7: Major Triad diff. 5.866, 0.000
C E G
1-5-8: Major Triad diff. 21.506, 0.000
C# F G#
2-6-9: Major Triad diff. 5.866, 0.000
D F# A
3-7-10: Major Triad diff. 21.506, 0.000
Eb G Bb
4-8-11: Major Triad diff. 13.686,-7.820
E G# B
5-9-12: Major Triad diff. 13.686, 0.000
F A C
6-10-13: Major Triad diff. 21.506, 0.000
F# Bb C#
7-11-14: Major Triad diff. -1.954,-7.820
G B D
8-12-15: Major Triad diff. 21.506, 0.000
G# C Eb
9-13-16: Major Triad diff. 5.866,-7.820
A C# E
10-14-17: Major Triad diff. 13.686, 0.000
Bb D F

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 24 Ağustos 2005 Çarşamba 22:45
Subject: [tuning] Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > Your question, in the first place, was how to try my results
> > empirically. I recommended some ways to do so, and in Bach
> > repertoire where differences are especially noticeable (i.e.
> > where it might most affect someone's choice of a temperament
> > for a Bach concert). A refusal to go do it, then, is just
> > backpedaling against an attempt to be empirical, isn't it?....
> > Empiricism only when it's not too inconvenient, and won't take
> > time/effort?
>
> My point is that you've made some very strong claims for
> something that you derived through what can only be called
> far-fetched means. I don't doubt that your scale is a
> fine well temperament,

Not so fast, Carl! You'll find the specifics of the Lehman-"Bach"
temperament here:
http://www-personal.umich.edu/~bpl/larips/math.html

After looking at the numbers, I must say that I have a few doubts.
Here's a table showing the total error in cents of the intervals in
each major triad:

Db Ab Eb Bb F. C. G. D. A. E. B. F#
39 35 31 23 20 20 27 35 43 39 35 31

For comparison, following is the total error of the intervals of a
major triad in each of the "big three" historical regular
temperaments:

Meantone: 11 cents
12-ET: 31 cents
Pythagorean: 43 cents

In the Lehman-"Bach" temperament the numbers progressively decrease
going from Db to F and then level off at C -- so far so good. They
then increase going from C to A; the intonation favors the keys in
the flat direction (a minor flaw IMO, characteristic of many well-
temperaments, including Werckmeister III), so there's nothing
unexpected there. The numbers then *decrease* going from A to F#,
which is exactly the *opposite* of what is expected (and desired) in
a well-temperament -- a very serious flaw indeed. This leads me to
think that either Brad has not properly deciphered some of
Bach's "squiggles", or if he has, then Bach's temperament is not a
very good one. While you might get some interesting results playing
some of Bach's music in it, I would not expect anything in the key of
A major (its worst triad) to fare particularly "well" (pun intended.

Besides this, there's a wide fifth (Bb-F) that, strictly speaking,
would by itself exclude this temperament from the "well-temperament"
category, inasmuch as this causes the total error of the intervals in
the 12 major triads (379.3c) to exceed the minimum theoretically
possible (387.5c). My reason for bringing this point up is not to
nitpick about a categorical technicality, but rather to make the
point that the inclusion of a wide fifth is *totally unnecessary* in
a temperament in which the narrowest fifths are tempered only 1/6 of
a pythagorean comma (see following paragraph).

I found on this website:
http://www.rollingball.com/TemperamentsFrames.htm
a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) that's
a perfect (even if "unauthentic") alternative to Lehman-"Bach". It
has 1/6-p-comma fifths in a chain from C to F# and just fifths from
F#=Gb to C. If you check the numbers in the following table, you'll
see that none of the major triads from Eb to E (corresponding to the
8 usable ones in a standard meantone tuning) has an error
significantly exceeding that of 12-ET:

Major triad:.. Db Ab Eb Bb F. C. G. D. A. E. B. F#
Lehman-"Bach": 39 35 31 23 20 20 27 35 43 39 35 31
Valotti-Young: 43 43 35 26 20 20 20 20 27 35 43 43
Werckmst. III: 43 43 31 20 _8 20 31 31 31 31 43 43

I've also included Werckmeister III for comparison; although it has
room for improvement, at least its numbers demonstrate a reasonable
progression of intonation from the best to worst keys, and we *do
know* that Bach actually played organs that were so tuned. Maybe he
tuned his harpsichord this way (at least sometimes), or maybe he
tuned it differently -- I'll leave that for others to debate. But I
would have a difficult time accepting the notion that Bach intended
his _Well-tempered Clavier_ to be played in a temperament unworthy of
the name "well-tempered".

Brad, it was not my intention to be unkind in my criticism of your
work, and I hope that you'll be open to reviewing your methodology in
light of my observations.

Best,

--George

🔗Carl Lumma <clumma@yahoo.com>

8/24/2005 2:12:25 PM

> > My point is that you've made some very strong claims for
> > something that you derived through what can only be called
> > far-fetched means. I don't doubt that your scale is a
> > fine well temperament,
>
> Not so fast, Carl! You'll find the specifics of the
> Lehman-"Bach" temperament here:
> http://www-personal.umich.edu/~bpl/larips/math.html
>
> After looking at the numbers, I must say that I have a few
> doubts. Here's a table showing the total error in cents of
> the intervals in each major triad:
>
> Db Ab Eb Bb F. C. G. D. A. E. B. F#
> 39 35 31 23 20 20 27 35 43 39 35 31
>
> For comparison, following is the total error of the intervals
> of a major triad in each of the "big three" historical regular
> temperaments:
>
> Meantone: 11 cents
> 12-ET: 31 cents
> Pythagorean: 43 cents

This makes it look like the average figure for 12-tone scales
can be less than 31. I think a more apt comparison would be to
show all 12 keys of some popular well temperaments.

Personally I'd use only the errors of the major third and fifth.
I don't hear nearly as much of a difference in the consonance of
minor thirds in the range of 290-320 cents as of major thirds in
the 380-410 cents range.

> In the Lehman-"Bach" temperament the numbers progressively
> decrease going from Db to F and then level off at C -- so far
> so good. They then increase going from C to A; the intonation
> favors the keys in the flat direction (a minor flaw IMO,
> characteristic of many well-temperaments, including Werckmeister
> III), so there's nothing unexpected there. The numbers then
> *decrease* going from A to F#, which is exactly the *opposite*
> of what is expected (and desired) in a well-temperament -- a
> very serious flaw indeed. This leads me to think that either
> Brad has not properly deciphered some of Bach's "squiggles", or
> if he has, then Bach's temperament is not a very good one.
> While you might get some interesting results playing some of
> Bach's music in it, I would not expect anything in the key of
> A major (its worst triad) to fare particularly "well" (pun
> intended.

In fact I hadn't noticed this. Good point.

> the total error of the intervals in the 12 major
> triads (379.3c) to exceed the minimum theoretically
> possible (387.5c).

There are maybe some rounding errors going on... above you
say the value for a 12-tET triad is 31 cents; here the minimum
possible is over 32.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/24/2005 2:21:24 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Personally I'd use only the errors of the major third and fifth.
> I don't hear nearly as much of a difference in the consonance of
> minor thirds in the range of 290-320 cents as of major thirds in
> the 380-410 cents range.

But in the context of a major triad, which is what George was talking
about, I find that the tuning of the minor third matters just as much
(particularly if you allow for inversions). A triad like 0-400-720
sounds markedly better to me than 0-380-720, even though though the
latter has a better major third and the same perfect fifth. Similar
conclusions held for me when I compared different triads with narrow
perfect fifths.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/24/2005 2:32:13 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Personally I'd use only the errors of the major third and fifth.
> > I don't hear nearly as much of a difference in the consonance of
> > minor thirds in the range of 290-320 cents as of major thirds in
> > the 380-410 cents range.
>
> But in the context of a major triad, which is what George was talking
> about, I find that the tuning of the minor third matters just as much
> (particularly if you allow for inversions). A triad like 0-400-720
> sounds markedly better to me than 0-380-720,

Oops -- that was supposed to be 390, not 380.

> even though though the
> latter has a better major third and the same perfect fifth. Similar
> conclusions held for me when I compared different triads with narrow
> perfect fifths.

Perhaps a better example given your ranges would be to follow George
Secor and compare the 22-equal (0-382-709) and 19-equal (0-379-695)
major triads. The 19-equal one is a good deal more concordant, and the
explanation clearly can't lie with either the 'fifth' or the 'major
third'. Try this and some other examples out for yourself and see if
you don't agree.

🔗Ozan Yarman <ozanyarman@superonline.com>

8/24/2005 2:53:19 PM

Paul, could the "proportional stretch" factor explain the triadic consonances you mentioned? I noticed similar things myself during my investigations.

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 25 Ağustos 2005 Perşembe 0:32
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Personally I'd use only the errors of the major third and fifth.
> > I don't hear nearly as much of a difference in the consonance of
> > minor thirds in the range of 290-320 cents as of major thirds in
> > the 380-410 cents range.
>
> But in the context of a major triad, which is what George was talking
> about, I find that the tuning of the minor third matters just as much
> (particularly if you allow for inversions). A triad like 0-400-720
> sounds markedly better to me than 0-380-720,

Oops -- that was supposed to be 390, not 380.

> even though though the
> latter has a better major third and the same perfect fifth. Similar
> conclusions held for me when I compared different triads with narrow
> perfect fifths.

Perhaps a better example given your ranges would be to follow George
Secor and compare the 22-equal (0-382-709) and 19-equal (0-379-695)
major triads. The 19-equal one is a good deal more concordant, and the
explanation clearly can't lie with either the 'fifth' or the 'major
third'. Try this and some other examples out for yourself and see if
you don't agree.

🔗Carl Lumma <clumma@yahoo.com>

8/24/2005 5:46:15 PM

> > A triad like 0-400-720
> > sounds markedly better to me than 0-380-720,
>
> Oops -- that was supposed to be 390, not 380.

720 is out of well temperament range for a 5th.

> Perhaps a better example given your ranges would be to follow
> George Secor and compare the 22-equal (0-382-709) and
> 19-equal (0-379-695) major triads. The 19-equal one is a good
> deal more concordant, and the explanation clearly can't lie
> with either the 'fifth' or the 'major third'. Try this and some
> other examples out for yourself and see if you don't agree.

That's not in my ranges, with a 327 minor third in the first case.

I only hear 19's triads (in root position) as a little bit more
concordant than 22's.

This does bear further investigation...

-Carl

🔗Bradley P Lehman <bpl@umich.edu>

8/25/2005 5:48:56 AM

> Subject: Further doubts about Lehman's 'Bach' scale
> (...) While you might get some interesting results playing
> some of Bach's music in it, I would not expect anything in the key of
> A major (its worst triad) to fare particularly "well" (pun intended.
>

Rather than "expecting" some, or speculating, how about tuning a harpsichord and an organ this way and playing some? From more than a year of doing so, A major has become one of my favorite keys to play in and listen to, with its melodic smoothness and the brightness of those sharps. The note C#, as it turns out, is exactly midway between A and F. One could turn your observations right around and say them the other way: in other temperaments, the interval Db-F is so much worse than A-C# that the music played using it (such as quite a bit of Bach's in F minor) "doesn't fare particularly well".

> Besides this, there's a wide fifth (Bb-F) that, strictly speaking,
> would by itself exclude this temperament from the "well-temperament"
> category, inasmuch as this causes the total error of the intervals in
> the 12 major triads (379.3c) to exceed the minimum theoretically
> possible (387.5c).

No, look again: *strictly speaking* that interval is the diminished 6th A#-F. I've explained this thoroughly in my FAQ pages; please take a look.
http://www-personal.umich.edu/~bpl/larips/faq3.html

Better yet, read the whole paper too, and try its musical points hands-on at a harpsichord, to hear the effects in real music. Believe me, I was as skeptical/dubious as anybody BEFORE setting it up on harpsichord and playing it for a bit; it looks ludicrous on paper IN SPECULATION but music isn't speculation.

Also keep in mind that your argument from "the well-temperament category" here is based on a *modern* definition made up by Owen Jorgensen et al; there's nothing historically that would constrain Bach to conform to categories made up a couple hundred years after he was gone.

> My reason for bringing this point up is not to
> nitpick about a categorical technicality, but rather to make the
> point that the inclusion of a wide fifth is *totally unnecessary* in
> a temperament in which the narrowest fifths are tempered only 1/6 of
> a pythagorean comma (see following paragraph).

I see that Jorgensen's speculative notions about the undesirability of "harmonic waste" are alive and well. But within such a paradigm, where the expectations *have been made up in the 20th century*, of course my solution is going to be judged and found wanting. The key here is to come to it with 17th century expectations and habits, and then to see/hear how new it is against that background. Against *that* relief it sounds so smoothly equal and so unremarkable that it doesn't call a huge amount of attention *to itself*, but rather it calls attention to the *the music* and the way tonal progressions behave.

Keep in mind that Bach was as master of harmony, both simple and complex. A couple of days ago I played through the "Schmucke dich" BWV 654 and the "Nun komm der heiden Heiland" BWV 659, marveling at the blend of harmonic/melodic motion in these pieces, in this tuning (there is a pipe organ tuned this way in Indiana, and a second one being completed now in Helsinki). In that same session I played through the "Liebster Jesu" settings BWV 633-634 [in A major!] with their canons, suspended 6ths, D#s, etc...*musically* it's wonderful. And obviously there's no way I can convince anybody who would prefer his/her own speculations on paper ahead of listening to and actually *playing* the music hands-on.

For any who are willing to listen, there is a BBC radio 3 program about this on Sunday, and available on the web all next week:
http://www.bbc.co.uk/radio3/earlymusicshow/pip/z9s45/

>
> I found on this website:
> http://www.rollingball.com/TemperamentsFrames.htm
> a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) that's
> a perfect (even if "unauthentic") alternative to Lehman-"Bach".

What do you mean "unauthentic"? Werckmeister knew the simple layout of six chained tempered 5ths, contrasted against the other six chained pure 5ths, in the 1680s as a typical Venetian style. And his own publications were an attempt to do something different. I've addressed this point in my article. The "Vallotti" style of tuning is *older than* the Neidhardts, Werckmeisters, and Sorges!

> But I
> would have a difficult time accepting the notion that Bach intended
> his _Well-tempered Clavier_ to be played in a temperament unworthy of
> the name "well-tempered".

Again: "unworthy of the name 'well-tempered'" is going by definitions that were made up in the 20th century. Why would Bach care about that? My hypothesis is that Bach used the term _wohltemperirt_ to mean this specific layout that he wrote down himself; and if some people *today* can't accept that it might have a wide "5th" (really a diminished 6th) in it, or the little zigzag where B major is better in tune than E major is, that's a problem of modern expectations, not a problem of Bach's practice. And to see what's in this hypothesis, please read the actual article and try the music as it recommends, rather than relying solely on internet chatter about it.

> Brad, it was not my intention to be unkind in my criticism of your
> work, and I hope that you'll be open to reviewing your methodology in
> light of my observations.

I haven't taken offense at your remarks, George; it's only that I've already covered all that ground myself in the months *before* writing the article, and indeed in the 21 years of working as a harpsichord tuner before that. Your remarks here haven't told me anything yet that I didn't already consider *before* putting my boldly anti-establishment stuff into the article. I know that this article calls for a paradigm shift in the way people think about and listen to temperaments, and it took me some months myself to realize what was necessary to say in it. I had to drop the encrusted habits of 20+ years out the window myself before taking this different harmonic balance seriously myself. And then, having heard it and played in it for a while, I had to figure out the theory as to why it works. The layout of intervals is different enough that it calls for its own methods of analysis, arising from the sounds produced. Its handling of tonics/dominants is simple and logical, *after* one is able to dump old habits and expectations out the window and play directly in the music, to hear what's there.

Bradley Lehman
http://www.larips.com

🔗George D. Secor <gdsecor@yahoo.com>

8/25/2005 9:53:11 AM

--- In tuning@yahoogroups.com, Bradley P Lehman <bpl@u...> wrote:
> > Subject: Further doubts about Lehman's 'Bach' scale
> > (...) While you might get some interesting results playing
> > some of Bach's music in it, I would not expect anything in the
key of
> > A major (its worst triad) to fare particularly "well" (pun
intended.
> >
>
> Rather than "expecting" some, or speculating, how about tuning a
> harpsichord and an organ this way and playing some? From more than
a year
> of doing so, A major has become one of my favorite keys to play in
and
> listen to, with its melodic smoothness and the brightness of those
sharps.
> The note C#, as it turns out, is exactly midway between A and F.
One
> could turn your observations right around and say them the other
way: in
> other temperaments, the interval Db-F is so much worse than A-C#
that the
> music played using it (such as quite a bit of Bach's in F
minor) "doesn't
> fare particularly well".

I agree with much of what you have to say. I had my own piano tuned
to my own #2 well-temperament for about 5 years (did it myself, by
ear) and, as I observed in a previous message (#59697):

<< After a very short time I found that, as I became accustomed to
the differences in intonation in the various keys, I tended to hear
the "remote" pythagorean triads as having a different "mood" rather
than being unacceptably out of tune. I might compare my first
reaction to swimming in cold water: initially shocking, but
ultimately invigorating. >>

Obviously *some* of the triads in a circulating unequal temperament
will be "worse" than others, so the question comes down to
determining which ones. As long as we agree that the temperament
you've presented is at variance with conventional expectations (that
an F# major triad should sound more dissonant than A), then I suppose
I should drop my objections (which were on acoustic grounds) and
leave it to others to debate the other issues.

> > Besides this, there's a wide fifth (Bb-F) that, strictly speaking,
> > would by itself exclude this temperament from the "well-
temperament"
> > category, inasmuch as this causes the total error of the
intervals in
> > the 12 major triads (379.3c) to exceed the minimum theoretically
> > possible (387.5c).
>
> No, look again: *strictly speaking* that interval is the diminished
6th
> A#-F. I've explained this thoroughly in my FAQ pages; please take
a look.
> http://www-personal.umich.edu/~bpl/larips/faq3.html

Yes, it can be a diminished 6th in certain contexts (such as your
tuning routine), but musically those two tones occur much more often
as a 5th, Bb-F.

> Better yet, read the whole paper too, and try its musical points
hands-on
> at a harpsichord, to hear the effects in real music.

If only I had the time. :-(

> Believe me, I was as
> skeptical/dubious as anybody BEFORE setting it up on harpsichord
and
> playing it for a bit; it looks ludicrous on paper IN SPECULATION
but music
> isn't speculation.
>
> Also keep in mind that your argument from "the well-temperament
category"
> here is based on a *modern* definition made up by Owen Jorgensen et
al;
> there's nothing historically that would constrain Bach to conform
to
> categories made up a couple hundred years after he was gone.

Yes, I was aware of that, hence my following disclaimer:

> > My reason for bringing this point up is not to
> > nitpick about a categorical technicality, but rather to make the
> > point that the inclusion of a wide fifth is *totally unnecessary*
in
> > a temperament in which the narrowest fifths are tempered only 1/6
of
> > a pythagorean comma (see following paragraph).
>
> I see that Jorgensen's speculative notions about the undesirability
of
> "harmonic waste" are alive and well.

I aquired that notion independently. I devised my #2 (well-
temperament) and #3 (temperament ordinaire) tunings in 1975, several
years before I learned about Jorgensen's work (from a copy of the
limited-edition _Tuning the Historical Temperaments by Ear_ that I
purchased around 1978), by which I discovered that my #2 tuning met
all of Jorgensen's (rather redundant) conditions for a well-
temperament. My #3 temperament has considerable "harmonic waste",
but that's unavoidable (and therefore acceptable), given the purpose
for which it's intended.

> But within such a paradigm, where
> the expectations *have been made up in the 20th century*, of course
my
> solution is going to be judged and found wanting. The key here is
to come
> to it with 17th century expectations and habits, and then to
see/hear how
> new it is against that background. Against *that* relief it sounds
so
> smoothly equal and so unremarkable that it doesn't call a huge
amount of
> attention *to itself*, but rather it calls attention to the *the
music*
> and the way tonal progressions behave.
>
> Keep in mind that Bach was as master of harmony, both simple and
complex.
> A couple of days ago I played through the "Schmucke dich" BWV 654
and the
> "Nun komm der heiden Heiland" BWV 659, marveling at the blend of
> harmonic/melodic motion in these pieces, in this tuning (there is a
pipe
> organ tuned this way in Indiana, and a second one being completed
now in
> Helsinki). In that same session I played through the "Liebster
Jesu"
> settings BWV 633-634 [in A major!] with their canons, suspended
6ths, D#s,
> etc...*musically* it's wonderful. And obviously there's no way I
can
> convince anybody who would prefer his/her own speculations on paper
ahead
> of listening to and actually *playing* the music hands-on.

It may be argued that if you listen to practically any unfamiliar
tuning enough times, you'll eventually get accustomed to it and might
end up preferring it for that (and no other) reason.

Brad, if you really want to make a convincing demonstration that
you've discovered Bach's keyboard temperament, then I think that you
should also make parallel recordings of some of these selections in
the (theoretically "perfect" alternative) Valotti-Young 1/6-
pythagorean-comma well-temperament (on the same instrument) so that
anyone can make a head-to-head comparison. I realize that it may not
be very practical to retune an entire organ, but I think a
harpsichord would suffice. The listener would be allowed repeated
hearings in order to become accustomed to each tuning and could then
draw conclusions.

> For any who are willing to listen, there is a BBC radio 3 program
about
> this on Sunday, and available on the web all next week:
> http://www.bbc.co.uk/radio3/earlymusicshow/pip/z9s45/
>
> > I found on this website:
> > http://www.rollingball.com/TemperamentsFrames.htm
> > a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799)
that's
> > a perfect (even if "unauthentic") alternative to Lehman-"Bach".
>
> What do you mean "unauthentic"? Werckmeister knew the simple
layout of
> six chained tempered 5ths, contrasted against the other six chained
pure
> 5ths, in the 1680s as a typical Venetian style. And his own
publications
> were an attempt to do something different. I've addressed this
point in
> my article. The "Vallotti" style of tuning is *older than* the
> Neidhardts, Werckmeisters, and Sorges!

Okay, I stand corrected. My chief interest in these temperaments is
in arriving at theoretically optimal or elegant solutions, rather
than involvement with historical details. I first encountered the
tuning as "Young's No. 2" in Barbour's _Tuning and Temperament_ and
had the impression that it was post-Bach. The head-to-head
comparison I suggested above would then be between two
arguably "authentic" tunings.

> > But I
> > would have a difficult time accepting the notion that Bach
intended
> > his _Well-tempered Clavier_ to be played in a temperament
unworthy of
> > the name "well-tempered".
>
> Again: "unworthy of the name 'well-tempered'" is going by
definitions that
> were made up in the 20th century. Why would Bach care about that?
My
> hypothesis is that Bach used the term _wohltemperirt_ to mean this
> specific layout that he wrote down himself; and if some people
*today*
> can't accept that it might have a wide "5th" (really a diminished
6th) in
> it, or the little zigzag where B major is better in tune than E
major is,
> that's a problem of modern expectations, not a problem of Bach's
practice.
> And to see what's in this hypothesis, please read the actual
article and
> try the music as it recommends, rather than relying solely on
internet
> chatter about it.

Point taken (IF your interpretation of Bach's notations are correct).

> > Brad, it was not my intention to be unkind in my criticism of your
> > work, and I hope that you'll be open to reviewing your
methodology in
> > light of my observations.
>
> I haven't taken offense at your remarks, George; it's only that
I've
> already covered all that ground myself in the months *before*
writing the
> article, and indeed in the 21 years of working as a harpsichord
tuner
> before that. Your remarks here haven't told me anything yet that I
didn't
> already consider *before* putting my boldly anti-establishment
stuff into
> the article. I know that this article calls for a paradigm shift
in the
> way people think about and listen to temperaments, and it took me
some
> months myself to realize what was necessary to say in it. I had to
drop
> the encrusted habits of 20+ years out the window myself before
taking this
> different harmonic balance seriously myself.

Okay, I can appreciate that.

> And then, having heard it
> and played in it for a while, I had to figure out the theory as to
why it
> works. The layout of intervals is different enough that it calls
for its
> own methods of analysis, arising from the sounds produced. Its
handling
> of tonics/dominants is simple and logical, *after* one is able to
dump old
> habits and expectations out the window and play directly in the
music, to
> hear what's there.

And even if the rest of us can get past that, it will still take
quite a bit more convincing. You're facing a uphill battle, my
friend, but I wish you the best.

--George

🔗George D. Secor <gdsecor@yahoo.com>

8/25/2005 10:35:53 AM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> George, I'm still quite in the dark as to why you think Bach would
not have approved MY well-temperament. It IS very easy to tune by ear
too. 8-)
>
> Cordially,
> Ozan
>
> 12-tone Well Temperament from practically 159tET
> |
> 0: 1/1 C Dbb unison, perfect prime
> 1: 90.225 cents C# Db
> 2: 196.090 cents D Ebb
> 3: 294.135 cents D# Eb
> 4: 392.180 cents E Fb
> 5: 498.045 cents F Gbb
> 6: 588.270 cents F# Gb
> 7: 701.955 cents G Abb
> 8: 792.180 cents G# Ab
> 9: 898.045 cents A Bbb
> 10: 996.090 cents A# Bb
> 11: 1086.315 cents B Cb
> 12: 2/1 C Dbb octave

The alternation of narrow and just 5ths that Gene noted produces a
zig-zag progression in intonation going (by 5ths) from F to A. You
also have 6 triads with total error (43) equal to pythagorean:

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Ozan Yarman: 43 43 43 27 27 12 16 12 27 43 43 43
Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

As the numbers for my #2 temperament indicate, it's possible to have
3 triads with error in the lower 'teens (1/4-comma meantone error =
11) in combination with 5 ~pythagorean triads. If you want meantone-
like intonation in your 3 best triads in combination with only 4
triads having significantly greater error than 12-ET, you'd have to
have some wide fifths (as in my #3 temperament).

As for whether Bach would approve of your temperament, after hearing
what Brad had to say, perhaps I should not be so prejudicial. ;-)

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

8/25/2005 10:50:11 AM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
>
> [...]
>
> Hiya George,
>
> Why not show the total absolute errors in the intervals in each of
the
> minor triads as well? I mean, as long as you're showing it in each of
> the major triads already . . .

Because I don't have the figures handy. I set up a spreadsheet that
only calculates the individual and total errors in the major triads,
but there's an easy way to determine these for the minor triads. As
long as there are no 5ths wider than just or major 3rds narrower than
just, the major and minor minor triads contained in a major-7th chord
will have the same total error (i.e., twice the error of the minor 3rd
of the triad). Thus the F major and A minor triads in a well-
temperament will have the same total error.

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

8/25/2005 11:43:02 AM

So, did I finally made it into the "temperament league"? Does this well temperament of mine (12 out of practically 159-tET) deserve to be called "Yarman #5" ?

Then the others would be:

Yarman #1 : 12-tone 79 MOS 159-tET
Yarman #2: 12-tone 67 MOS 135-tET
Yarman #3: 12-tone 55 MOS 277-tET
Yarman #4: 12-tone pure fourths (inverse of `Cordier pure fifths`)

Cordially,
Ozan

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 25 Ağustos 2005 Perşembe 20:35
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> George, I'm still quite in the dark as to why you think Bach would
not have approved MY well-temperament. It IS very easy to tune by ear
too. 8-)
>
> Cordially,
> Ozan
>
> 12-tone Well Temperament from practically 159tET
> |
> 0: 1/1 C Dbb unison, perfect prime
> 1: 90.225 cents C# Db
> 2: 196.090 cents D Ebb
> 3: 294.135 cents D# Eb
> 4: 392.180 cents E Fb
> 5: 498.045 cents F Gbb
> 6: 588.270 cents F# Gb
> 7: 701.955 cents G Abb
> 8: 792.180 cents G# Ab
> 9: 898.045 cents A Bbb
> 10: 996.090 cents A# Bb
> 11: 1086.315 cents B Cb
> 12: 2/1 C Dbb octave

The alternation of narrow and just 5ths that Gene noted produces a
zig-zag progression in intonation going (by 5ths) from F to A. You
also have 6 triads with total error (43) equal to pythagorean:

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Ozan Yarman: 43 43 43 27 27 12 16 12 27 43 43 43
Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

As the numbers for my #2 temperament indicate, it's possible to have
3 triads with error in the lower 'teens (1/4-comma meantone error =
11) in combination with 5 ~pythagorean triads. If you want meantone-
like intonation in your 3 best triads in combination with only 4
triads having significantly greater error than 12-ET, you'd have to
have some wide fifths (as in my #3 temperament).

As for whether Bach would approve of your temperament, after hearing
what Brad had to say, perhaps I should not be so prejudicial. ;-)

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

8/25/2005 2:38:10 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> So, did I finally made it into the "temperament league"?

Yep!

> Does this well temperament of mine (12 out of practically 159-tET)
deserve to be called "Yarman #5" ?

It's your privilege to number them as you wish.

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

8/26/2005 6:25:39 AM

Great!! but dear George, can you show me how to practically tune the cent values in a modified 1/4 comma meantone:

0: 1/1 C Dbb unison, perfect prime
1: 76.049 cents C# Db
2: 193.157 cents D Ebb
3: 279.471 cents D# Eb
4: 386.314 cents E Fb
5: 493.157 cents F Gbb
6: 579.471 cents F# Gb
7: 696.578 cents G Abb
8: 772.627 cents G# Ab
9: 889.735 cents A Bbb
10: 986.314 cents A# Bb
11: 1082.892 cents B Cb
12: 1200.000 cents C Dbb

0: 0.000 cents 0.000 0 0 commas C
7: 696.578 cents -5.377 -165 -1/4 synt. commas G
2: 696.578 cents -10.753 -330 -1/2 synt. commas D
9: 696.578 cents -16.130 -495 -3/4 synt. commas A
4: 696.578 cents -21.506 -660 -1 synt. commas E
11: 696.578 cents -26.883 -825 -5/4 synt. commas B
6: 696.578 cents -32.259 -990 -3/2 synt. commas F#
1: 696.578 cents -37.636 -1155 -7/4 synt. commas C#
8: 696.578 cents -43.013 -1320 -2 synt. commas G#
3: 706.843 cents -38.124 -1170 -39/22 synt. commas Eb
10: 706.843 cents -33.236 -1020 -17/11 synt. commas Bb
5: 706.843 cents -28.348 -870 -29/24 Pyth. commas F
12: 706.843 cents -23.460 -720 -1 Pyth. commas C
Average absolute difference: 26.3938 cents
Root mean square difference: 29.8622 cents
Maximum absolute difference: 43.0126 cents
Maximum formal fifth difference: 5.3766 cents

so that I can achieve equal beatings for the major triads? What is the most feasible formula that calibrates the beats in a tempered 4:5:6 to equal? Specifically, how can I distrubute the `wolf-fifth remnant` to the tones in an optimal manner so as to do minimal damage in consonant interval approximations?

Cordially,
Ozan

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 26 Ağustos 2005 Cuma 0:38
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> So, did I finally made it into the "temperament league"?

Yep!

> Does this well temperament of mine (12 out of practically 159-tET)
deserve to be called "Yarman #5" ?

It's your privilege to number them as you wish.

--George

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

SPONSORED LINKS Music education grants Music business education Music industry education
Music production education Music education software Music theory education

------------------------------------------------------------------------------
YAHOO! GROUPS LINKS

a.. Visit your group "tuning" on the web.

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

------------------------------------------------------------------------------

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/26/2005 11:56:37 AM

Hi Ozan,

I have no idea what you mean by 'the "proportional stretch" factor'.
Would you explain?

Thanks,
Paul

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, could the "proportional stretch" factor explain the triadic
consonances you mentioned? I noticed similar things myself during my
investigations.
>
> Cordially,
> Ozan
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 25 Aðustos 2005 Perþembe 0:32
> Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > > Personally I'd use only the errors of the major third and
fifth.
> > > I don't hear nearly as much of a difference in the consonance
of
> > > minor thirds in the range of 290-320 cents as of major thirds
in
> > > the 380-410 cents range.
> >
> > But in the context of a major triad, which is what George was
talking
> > about, I find that the tuning of the minor third matters just
as much
> > (particularly if you allow for inversions). A triad like 0-400-
720
> > sounds markedly better to me than 0-380-720,
>
> Oops -- that was supposed to be 390, not 380.
>
> > even though though the
> > latter has a better major third and the same perfect fifth.
Similar
> > conclusions held for me when I compared different triads with
narrow
> > perfect fifths.
>
> Perhaps a better example given your ranges would be to follow
George
> Secor and compare the 22-equal (0-382-709) and 19-equal (0-379-
695)
> major triads. The 19-equal one is a good deal more concordant,
and the
> explanation clearly can't lie with either the 'fifth' or
the 'major
> third'. Try this and some other examples out for yourself and see
if
> you don't agree.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/26/2005 12:05:37 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> > Perhaps a better example given your ranges would be to follow
> > George Secor and compare the 22-equal (0-382-709) and
> > 19-equal (0-379-695) major triads. The 19-equal one is a good
> > deal more concordant, and the explanation clearly can't lie
> > with either the 'fifth' or the 'major third'. Try this and some
> > other examples out for yourself and see if you don't agree.
>
> That's not in my ranges, with a 327 minor third in the first case.

Did you try other examples? How about 0-385-705 vs. 0-382-698?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/26/2005 12:30:53 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> >
> > [...]
> >
> > Hiya George,
> >
> > Why not show the total absolute errors in the intervals in each
of
> the
> > minor triads as well? I mean, as long as you're showing it in
each of
> > the major triads already . . .
>
> Because I don't have the figures handy. I set up a spreadsheet
that
> only calculates the individual and total errors in the major
triads,
> but there's an easy way to determine these for the minor triads.
As
> long as there are no 5ths wider than just or major 3rds narrower
than
> just, the major and minor minor triads contained in a major-7th
chord
> will have the same total error (i.e., twice the error of the minor
3rd
> of the triad). Thus the F major and A minor triads in a well-
> temperament will have the same total error.
>
> --George

Egads, you're right! So for some of the tunings, you really didn't
need the figures to be 'handy' -- just rotate the figures for the
major triads four positions and you get the figures for the minor
triads. But that wouldn't have worked for the other tunings, because
some of the fifths were wider than pure?

P.S. I can't see what sense it makes to talk about A#-F as a
diminished sixth, while not also talking about B-Gb, etc., as
diminished sixths, in the context of a circulating 12-note fixed
tuning system.

🔗George D. Secor <gdsecor@yahoo.com>

8/26/2005 1:34:50 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Great!! but dear George, can you show me how to practically tune
the cent values in a modified 1/4 comma meantone:
>
>
> 0: 1/1 C Dbb unison, perfect prime
> 1: 76.049 cents C# Db
> 2: 193.157 cents D Ebb
> 3: 279.471 cents D# Eb
> 4: 386.314 cents E Fb
> 5: 493.157 cents F Gbb
> 6: 579.471 cents F# Gb
> 7: 696.578 cents G Abb
> 8: 772.627 cents G# Ab
> 9: 889.735 cents A Bbb
> 10: 986.314 cents A# Bb
> 11: 1082.892 cents B Cb
> 12: 1200.000 cents C Dbb

Sorry, Ozan. I know how to calculate beat rates, but tuning routines
aren't one of my areas of expertise.

Could someone else please help?

--George

🔗George D. Secor <gdsecor@yahoo.com>

8/26/2005 2:19:13 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Great!! but dear George, can you show me how to practically tune
the cent values in a modified 1/4 comma meantone:
> ...
> so that I can achieve equal beatings for the major triads? What is
the most feasible formula that calibrates the beats in a tempered
4:5:6 to equal? Specifically, how can I distrubute the `wolf-fifth
remnant` to the tones in an optimal manner so as to do minimal damage
in consonant interval approximations?

Wait a minute -- I think I may have partially misread your first
question (which had a couple of tables inserted), but in any case,
there needs to be a little clarification.

By "equal beatings", do you mean:
1) Equal beat-rates for intervals *within* a single triad, or
2) Equal beat rates for the same interval in two different triads?

Whatever the case, this is something that I haven't thought about
until quite recently, so I'm not the best person to answer this.

> What is the most feasible formula that calibrates the beats in a
tempered 4:5:6 to equal?

Do you mean "calculates the beats ... "? If I needed to do that, I'd
set it up in a spreadsheet, but there must be a resource somewhere
out on the Internet that will do it for you.

> Specifically, how can I distrubute the `wolf-fifth remnant` to the
tones in an optimal manner so as to do minimal damage in consonant
interval approximations?

I can't find any wolf fifth here:

> 0: 1/1 C Dbb unison, perfect prime
> 1: 76.049 cents C# Db
> 2: 193.157 cents D Ebb
> 3: 279.471 cents D# Eb
> 4: 386.314 cents E Fb
> 5: 493.157 cents F Gbb
> 6: 579.471 cents F# Gb
> 7: 696.578 cents G Abb
> 8: 772.627 cents G# Ab
> 9: 889.735 cents A Bbb
> 10: 986.314 cents A# Bb
> 11: 1082.892 cents B Cb
> 12: 1200.000 cents C Dbb

Are you referring to the wolf fifth in a strict 12-tone 1/4-comma
meantone temperament? It's not too hard to eliminate the wolf fifth,
but the answer to your question depends on how much you're willing to
temper the thirds. For some examples of modified meantone
temperaments, check out:

http://www.rollingball.com/TemperamentsFrames.htm

--George

🔗George D. Secor <gdsecor@yahoo.com>

8/26/2005 2:43:55 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > ... there's an easy way to determine these for the minor triads.
As
> > long as there are no 5ths wider than just or major 3rds narrower
than
> > just, the major and minor minor triads contained in a major-7th
chord
> > will have the same total error (i.e., twice the error of the
minor 3rd
> > of the triad). Thus the F major and A minor triads in a well-
> > temperament will have the same total error.
> >
> > --George
>
> Egads, you're right! So for some of the tunings, you really didn't
> need the figures to be 'handy' -- just rotate the figures for the
> major triads four positions and you get the figures for the minor
> triads.

Yep!

> But that wouldn't have worked for the other tunings, because
> some of the fifths were wider than pure?

Yes, in triads with wide fifths the total error is twice the absolute
error of the third having the larger absolute error (generally the
major 3rd). In the Lehman-Bach temperament Bb-F is a wide fifth, and
Bb-D has 2 cents greater absolute error than D-F.

--George

> P.S. I can't see what sense it makes to talk about A#-F as a
> diminished sixth, while not also talking about B-Gb, etc., as
> diminished sixths, in the context of a circulating 12-note fixed
> tuning system.

Yeah, I thought the same thing, even after reading the link to Brad's
FAQ.

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

8/26/2005 3:06:31 PM

Simple! If you stretch the fifth by 20 cents, you should stretch the major third as much as 1 over 702/386=0.54985754985754985754985754985766, which makes it 10.1 cents to achieve harmonic consonance.

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 26 Ağustos 2005 Cuma 21:56
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Hi Ozan,

I have no idea what you mean by 'the "proportional stretch" factor'.
Would you explain?

Thanks,
Paul

🔗Carl Lumma <clumma@yahoo.com>

8/26/2005 3:26:34 PM

> > > Perhaps a better example given your ranges would be to follow
> > > George Secor and compare the 22-equal (0-382-709) and
> > > 19-equal (0-379-695) major triads. The 19-equal one is a good
> > > deal more concordant, and the explanation clearly can't lie
> > > with either the 'fifth' or the 'major third'. Try this and some
> > > other examples out for yourself and see if you don't agree.
> >
> > That's not in my ranges, with a 327 minor third in the first case.
>
> Did you try other examples? How about 0-385-705 vs. 0-382-698?

I clearly prefer the Maj3rd and 5th of the first chord, and
the min3rd of the second, in two different timbres. With
a clarinet, both triads sound the same except the first one
beats more. With an english horn, both sound the same except
the second one beats more.

What do you hear?

Anybody else?

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/26/2005 3:39:55 PM

I get 11 cents, not 10.1 cents, according to your own calculation.
Sure, this is one way of looking at it, but it seems less mysterious
simply (or firstly) to acknowledge that the mistuning of the minor
third matters too (as well as the mistunings of the fifth and major
third), especially if we are looking for an 'explanation'. You wrote:

'Paul, could the "proportional stretch" factor explain the triadic
consonances you mentioned? I noticed similar things myself during my
investigations.'

Although trying to stretch things proportionally may make sense, and
one may be able to derive it as a solution in certain circumstances
under certain assumptions, it doesn't seem to serve as
an 'explanation' for anything. Instead, it's something that might
present itself in someone's investigations and brainstormings and
then cry out for an explanation itself. In the case of a triad with
given fifth, the explanation seems to be that the mistuning of the
minor third matters too.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Simple! If you stretch the fifth by 20 cents, you should stretch
the major third as much as 1 over
702/386=0.54985754985754985754985754985766, which makes it 10.1 cents
to achieve harmonic consonance.
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 26 Aðustos 2005 Cuma 21:56
> Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
>
>
> Hi Ozan,
>
> I have no idea what you mean by 'the "proportional stretch"
factor'.
> Would you explain?
>
> Thanks,
> Paul

🔗Ozan Yarman <ozanyarman@superonline.com>

8/26/2005 5:03:38 PM

Yes! sorry, you are right, it was 11 cents. My mistake there. But Paul, the mistuning of the minor third is - obviously - directly correlated to the mistuning of the major third in a consonant triad.

I have merely proposed a model to explain the phenomenon observed. By no means do I make any ontological claims in that respect. If the model works, then a triad indeed should require the "proportional stretch" formula to sound consonant. Do you have any relevant evidence to the contrary?

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 27 Ağustos 2005 Cumartesi 1:39
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

I get 11 cents, not 10.1 cents, according to your own calculation.
Sure, this is one way of looking at it, but it seems less mysterious
simply (or firstly) to acknowledge that the mistuning of the minor
third matters too (as well as the mistunings of the fifth and major
third), especially if we are looking for an 'explanation'. You wrote:

'Paul, could the "proportional stretch" factor explain the triadic
consonances you mentioned? I noticed similar things myself during my
investigations.'

Although trying to stretch things proportionally may make sense, and
one may be able to derive it as a solution in certain circumstances
under certain assumptions, it doesn't seem to serve as
an 'explanation' for anything. Instead, it's something that might
present itself in someone's investigations and brainstormings and
then cry out for an explanation itself. In the case of a triad with
given fifth, the explanation seems to be that the mistuning of the
minor third matters too.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Simple! If you stretch the fifth by 20 cents, you should stretch
the major third as much as 1 over
702/386=0.54985754985754985754985754985766, which makes it 10.1 cents
to achieve harmonic consonance.
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 26 Ağustos 2005 Cuma 21:56
> Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
>
>
> Hi Ozan,
>
> I have no idea what you mean by 'the "proportional stretch"
factor'.
> Would you explain?
>
> Thanks,
> Paul

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

------------------------------------------------------------------------------
YAHOO! GROUPS LINKS

a.. Visit your group "tuning" on the web.

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

------------------------------------------------------------------------------

🔗Ozan Yarman <ozanyarman@superonline.com>

8/27/2005 8:21:08 AM

Dear George, by "equal beating" I mean firstly:

1) Equal beat-rates for intervals *within* a single triad,

and secondly:

2) Equal beat rates for the same interval in two different triads.

if you are in the know as to how to convert pitch information to vibrations per second, I may perhaps demonstrate whether or not I understand how to tune to `equal (or rather, even) beating`.

I am indeed referring to the wolf of 1/4 comma meantone which I `tempered` - so to speak - in such a way as to make sure the fifths do not howl anymore. But I am not sure how to distribute the remnant optimally so as to preserve the `equal-beating` property.

Does your modified meantone contain any `equal-beating` intervals?

Cordially,
Ozan

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 27 Ağustos 2005 Cumartesi 0:19
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Great!! but dear George, can you show me how to practically tune
the cent values in a modified 1/4 comma meantone:
> ...
> so that I can achieve equal beatings for the major triads? What is
the most feasible formula that calibrates the beats in a tempered
4:5:6 to equal? Specifically, how can I distrubute the `wolf-fifth
remnant` to the tones in an optimal manner so as to do minimal damage
in consonant interval approximations?

Wait a minute -- I think I may have partially misread your first
question (which had a couple of tables inserted), but in any case,
there needs to be a little clarification.

By "equal beatings", do you mean:
1) Equal beat-rates for intervals *within* a single triad, or
2) Equal beat rates for the same interval in two different triads?

Whatever the case, this is something that I haven't thought about
until quite recently, so I'm not the best person to answer this.

> What is the most feasible formula that calibrates the beats in a
tempered 4:5:6 to equal?

Do you mean "calculates the beats ... "? If I needed to do that, I'd
set it up in a spreadsheet, but there must be a resource somewhere
out on the Internet that will do it for you.

> Specifically, how can I distrubute the `wolf-fifth remnant` to the
tones in an optimal manner so as to do minimal damage in consonant
interval approximations?

I can't find any wolf fifth here:

> 0: 1/1 C Dbb unison, perfect prime
> 1: 76.049 cents C# Db
> 2: 193.157 cents D Ebb
> 3: 279.471 cents D# Eb
> 4: 386.314 cents E Fb
> 5: 493.157 cents F Gbb
> 6: 579.471 cents F# Gb
> 7: 696.578 cents G Abb
> 8: 772.627 cents G# Ab
> 9: 889.735 cents A Bbb
> 10: 986.314 cents A# Bb
> 11: 1082.892 cents B Cb
> 12: 1200.000 cents C Dbb

Are you referring to the wolf fifth in a strict 12-tone 1/4-comma
meantone temperament? It's not too hard to eliminate the wolf fifth,
but the answer to your question depends on how much you're willing to
temper the thirds. For some examples of modified meantone
temperaments, check out:

http://www.rollingball.com/TemperamentsFrames.htm

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 2:03:35 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Yes! sorry, you are right, it was 11 cents. My mistake there. But
>Paul, the mistuning of the minor third is - obviously - directly
>correlated to the mistuning of the major third in a consonant triad.

Sometimes it's independent, for example if you're only specifying
absolute deviations from JI.

> I have merely proposed a model to explain the phenomenon observed.
>By no means do I make any ontological claims in that respect. If the
>model works, then a triad indeed should require the "proportional
>stretch" formula to sound consonant. Do you have any relevant
>evidence to the contrary?

Not necessarily evidence, but I can propose a test. I believe that if
we're talking about a root-position close-voiced major triad,
approximating 4:5:6, and the 4:6 (or 2:3) is stretched by 20 cents,
then the triad is likely to sound more concordant when the 4:5 is
stretched by only 9 1/3 cents (and hence the 5:6 is stretched by 10
2/3 cents). The reasoning here is that 4:5 is a slightly stronger
concordance than 5:6; stronger concordances are slightly more
sensitive to mistuning than weaker ones; thus the 4:5 should be
mistuned just a bit less than the 5:6 in order to prevent one or the
other interval from being too damaged by mistuning. So the test would
be to compare your proposal,

0 - 397 1/3 - 720

against mine,

0 - 395 2/3 - 720

with various timbres and in various registers, and try to determine
which one sounds concordant. Probably a tough test to get significant
results with, due to the closeness of the two triads to one another.

> Cordially,
> Ozan

Best,
Paul

🔗George D. Secor <gdsecor@yahoo.com>

8/29/2005 2:43:10 PM

HEY EVERYONE!

Having been unable to ignore the discussion about equal-beating
temperaments lately, I have some startling news about a "temperament
(extra)ordinaire!"

I say "startling" because I never expected to be thinking about
dumping a couple of "optimal" tunings I devised 30 years ago in favor
of something that it took me less than a couple of hours to arrive at
this past Saturday.

But first I need to answer Ozan's message:

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George, by "equal beating" I mean firstly:
>
> 1) Equal beat-rates for intervals *within* a single triad,
>
> and secondly:
>
> 2) Equal beat rates for the same interval in two different triads.
>
> if you are in the know as to how to convert pitch information to
vibrations per second, I may perhaps demonstrate whether or not I
understand how to tune to `equal (or rather, even) beating`.

Well, whichever you mean, why don't I provide you with a spreadsheet
I was already working on to calculate frequencies (cols. W-Y), beat
rates (cols. S-U), beat ratios (cols. O-Q), and errors from just
(cols. G,I,K,M) from cents (cols.E-F):

/tuning-math/files/secor/WT-wksht.xls

You can input the cents directly into cells E8-E19, or you can
specify the departure of each tone from 12-ET, expressed as a
fraction of a comma (either didymus, pythagorean, archytas/septimal,
or skhisma) and let the spreadsheet calculate the cents. Put your
base frequency for A in K2, the type of comma (d/p/a/s) in G3, the
fraction of that comma to use as your unit of departure in G4, and
the numerator of the fraction in B15. I presently have figures
entered for a regular 5/17-didymus-comma temperament with A=220, for
which the intervals in a root-position major triad are all equal-
beating. If you change G4 to 7 and b15 to -1 (negative number for a
*narrow* 5th), you'll see that the 1/7th-comma temperament is very
close to having proportional-beating triads.

Now change G4 to 23 and b15 to -5, and you'll see that the 5/23rd-
comma temperament is very close to having equal-beating 5ths and
major 3rds (with minor 3rds beating 4X -- more about this below).
I'm a bit perplexed as to why the ratios between the beat-rates are
*almost* exact integers (imprecision in the calculations perhaps, but
I'm not so sure).

Observe in all of the above regular temperaments that F# and Gb are
different pitches. The numbers to the right of column F are all
calculated using Gb (not F#).

If you want to see the Valotti-Young 1/6-pythagorean-comma well-
temperament, put "p" in G3, 6 in G4, -1 in B15, and 0 in B8 through
B13. (Note that cents for F# and Gb are now the same.)

You can construct your own temperament either by changing the numbers
in the light-cyan-colored cells, or you can enter cents directly in
cells E8-E19. If your temperament does not conform to Owen
Jorgensen's strict definition of a well-temperament, this will be
indicated by one or more asterisks appearing in columns H, J, L, or N.

> I am indeed referring to the wolf of 1/4 comma meantone which I
`tempered` - so to speak - in such a way as to make sure the fifths
do not howl anymore. But I am not sure how to distribute the remnant
optimally so as to preserve the `equal-beating` property.

Now you can experiment using the spreadsheet. Besides the fractions
of a didymus comma I already mentioned above, you may find 5/14,
5/19, and 5/21 to be of interest.

> Does your modified meantone contain any `equal-beating` intervals?

Nope, but my brand-new temperament (extra)ordinare does:

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
12
!
86.53330
194.55680
294.12876
389.11361
499.91792
585.54105
697.27840
789.37483
891.83521
997.96292
1086.39201

2/1

It's in the 'temperament ordinaire' category, but after trying it out
on my Scalatron, I'd have to consider it a bit 'extradordinaire'.

I've made available a temperament worksheet with all sorts of
figures, should anyone be interested:

/tuning-math/files/secor/GS_1.xls

Observe that all of the major triads from Eb to E are proportional-
beating (i.e., the beat rates are related by rational integers or
fractions -- I had to tweak the numbers in column B slightly to get
the beat-ratios to come out exactly proportional). I also inserted
an extra column (G) so that Kraig Grady (and myself) could try this
on a Scalatron. (I'd be interested in your opinion, Kraig!)

Here's a summary of how the triad errors compare to my other two
temperaments:

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
Secor #1:... 54 49 34 21 15 15 15 19 26 37 48 54
Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

Again, for comparison, following is the total error of the intervals
of a major triad in each of the "big three" historical regular
temperaments:

1/4-comma meantone: 11 cents
12-ET: 31 cents
Pythagorean: 43 cents

However, the numbers don't tell the whole story (see below).

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote: (#59901)
> IMO, when a chord is played in which two or more of
> the intervals beat at the same rate, it sets up an
> unwanted pulse which will probably interfere with the
> rhythmic structure of the piece.

I've observed that this is also occurs to some extent in a triad with
a just interval, such as 1/3 or 1/4-comma meantone temperament, in
which the beat rates are proportional. However, I don't find it
particularly objectionable -- rather like a good vibrato.

Anyway, I've had a chance to play around with this on my Scalatron
the past couple of days, and I've decided to call it "#1", because I
think it tops both my #2 (well-temperament) and (in all respects) my
#3 (temperament ordinaire) tunings. (I devised my old #1 temperament
in 1964, and #2 and #3 eventually replaced it. Now, IMO, this new #1
could replace all of them.)

After listening and comparing, I've concluded that the 8 equal- and
proportional-beating major triads in this temperament sound as if
they have less total error than the numbers would indicate, with the
faster beats tending to be masked by the slower ones. In conducting
some head-to-head comparisons of the C and G major triads with those
in 31-ET (12c total error), I found that they sound at least as good;
likewise the E and Eb major triads as compared with 12-ET. (The
Scalatron permits me to jump back and forth instantly from one tuning
to another, which easily allows me to make comparisons using the same
timbres.) I also found, for some reason, that the B and Ab major
triads sound only about as dissonant as pythagorean triads, even
though they have about 5 cents greater total error.

I was frankly a bit surprised at how quickly I was able to come up
with proportional beat rates for so many triads, thinking that it
should have been a lot harder to do this and beginning to experience
some feelings of doubt that there must be some room for improvement --
at this point I don't know. (But the more I play around with it,
the less my doubt.)

Anyway, I hope that some others will try this tuning and give me your
reactions -- at least go through the triads in Scala using the chord
playing function in the keyboard clavier. Convince me that this
isn't all a dream!

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

8/29/2005 2:51:37 PM

Paul, absolute deviations from JI is not independent of consonance in my opinion. However, I understand and approve of your harmonic entropy model, and agree that 5/4 is more sensitive to mistuning as compared to 6/5.

Therefore, I am eager to hear of the results of the test you proposed. If the good folk of tuning list is willing to participate in it, we can load the MIDI or WAV files up on my website for auditory comparison.

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 0:03
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Yes! sorry, you are right, it was 11 cents. My mistake there. But
>Paul, the mistuning of the minor third is - obviously - directly
>correlated to the mistuning of the major third in a consonant triad.

Sometimes it's independent, for example if you're only specifying
absolute deviations from JI.

> I have merely proposed a model to explain the phenomenon observed.
>By no means do I make any ontological claims in that respect. If the
>model works, then a triad indeed should require the "proportional
>stretch" formula to sound consonant. Do you have any relevant
>evidence to the contrary?

Not necessarily evidence, but I can propose a test. I believe that if
we're talking about a root-position close-voiced major triad,
approximating 4:5:6, and the 4:6 (or 2:3) is stretched by 20 cents,
then the triad is likely to sound more concordant when the 4:5 is
stretched by only 9 1/3 cents (and hence the 5:6 is stretched by 10
2/3 cents). The reasoning here is that 4:5 is a slightly stronger
concordance than 5:6; stronger concordances are slightly more
sensitive to mistuning than weaker ones; thus the 4:5 should be
mistuned just a bit less than the 5:6 in order to prevent one or the
other interval from being too damaged by mistuning. So the test would
be to compare your proposal,

0 - 397 1/3 - 720

against mine,

0 - 395 2/3 - 720

with various timbres and in various registers, and try to determine
which one sounds concordant. Probably a tough test to get significant
results with, due to the closeness of the two triads to one another.

> Cordially,
> Ozan

Best,
Paul

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

------------------------------------------------------------------------------
YAHOO! GROUPS LINKS

a.. Visit your group "tuning" on the web.

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

------------------------------------------------------------------------------

🔗Carl Lumma <clumma@yahoo.com>

8/29/2005 3:01:11 PM

> ! secor12_1.scl
> !
> George Secor's 12-tone temperament ordinaire #1
> 12
> !
> 86.53330
> 194.55680
> 294.12876
> 389.11361
> 499.91792
> 585.54105
> 697.27840
> 789.37483
> 891.83521
> 997.96292
> 1086.39201
> 2/1
>
> It's in the 'temperament ordinaire' category, but after trying
> it out on my Scalatron, I'd have to consider it a
> bit 'extradordinaire'.
>
> I've made available a temperament worksheet with all sorts of
> figures, should anyone be interested:
>
> /tuning-math/files/secor/GS_1.xls

Hiya George (and Ozan),

Do you find you like the triads where all the intervals beat
at the same rate? I would think it would be better to stagger
the beats, so that they do not result in large amplitude
changes. Or does this not happen?

-Carl

🔗Jon Szanto <jszanto@cox.net>

8/29/2005 3:06:36 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> HEY EVERYONE!

Look, George is shouting with GLEE! :)

> Nope, but my brand-new temperament (extra)ordinare does:
>
> ! secor12_1.scl

Copied to a file...

> It's in the 'temperament ordinaire' category, but after trying
> it out on my Scalatron...

George, you *do* realize just how extraordinary the last part of that
sentence is, at least to 99.99999% of the people out here? (the one's
without Scalatrons, that is)

> Anyway, I hope that some others will try this tuning and give me your
> reactions -- at least go through the triads in Scala using the chord
> playing function in the keyboard clavier. Convince me that this
> isn't all a dream!

Some of the best stuff in the world is in dreams. But I've put it on
my "must check out" pile.

Cheers,
Jon

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 3:41:02 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> I'm a bit perplexed as to why the ratios between the beat-rates are
> *almost* exact integers (imprecision in the calculations perhaps,
but
> I'm not so sure).

No, your calculations are quite correct. This observation of *almost*
exact-integer beat rates has come up a few times here and
particularly on the tuning-math list. Gene in particular has posted a
great deal about it -- I hope he'll point you to some relevant
postings and/or articles.

> I was frankly a bit surprised at how quickly I was able to come up
> with proportional beat rates for so many triads, thinking that it
> should have been a lot harder to do this

Bob Wendell's posts on the subject might lead one to believe that
this is extremely difficult, almost an intractable problem, while
Gene's paint a different picture.

> and beginning to experience
> some feelings of doubt that there must be some room for
improvement --
> at this point I don't know. (But the more I play around with it,
> the less my doubt.)

Again, since Gene has worked a lot on this subject, I'm hoping to see
him provide you with an illuminating, well-rounded reply and
assessment of your new tuning systems alongside relevant alternatives.

Gene?

🔗Carl Lumma <clumma@yahoo.com>

8/29/2005 3:48:40 PM

> > I was frankly a bit surprised at how quickly I was able to
> > come up with proportional beat rates for so many triads,
> > thinking that it should have been a lot harder to do this
>
> Bob Wendell's posts on the subject might lead one to believe that
> this is extremely difficult, almost an intractable problem, while
> Gene's paint a different picture.

However, didn't Gene agree that his main tuning, the "Natural
Synchronous Well" was indeed a unique point in the space of
such tunings?

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 3:58:10 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > I was frankly a bit surprised at how quickly I was able to
> > > come up with proportional beat rates for so many triads,
> > > thinking that it should have been a lot harder to do this
> >
> > Bob Wendell's posts on the subject might lead one to believe that
> > this is extremely difficult, almost an intractable problem, while
> > Gene's paint a different picture.
>
> However, didn't Gene agree that his main tuning, the "Natural
> Synchronous Well" was indeed a unique point in the space of
> such tunings?

To the best of my recollection, yes, but really Gene should answer
this. And I'd love to see a comparison of George's new proposal against
this and other relevant alternatives.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 4:02:56 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Paul, absolute deviations from JI is not independent of consonance
>in my opinion.

If you thought I stated something to the contrary, we must have
misunderstood one another.

> Therefore, I am eager to hear of the results of the test you
>proposed. If the good folk of tuning list is willing to participate
>in it, we can load the MIDI or WAV files up on my website for
>auditory comparison.

I would never trust MIDI files for this purpose, especially as the
difference between the two triads is of the same order of magnitude as
the resolution of most sound cards! But I suppose I could prepare a
series of .wav files for this purpose -- we need a variety of timbres
and registers to test -- any suggestions? Also, we would need to
instruct the listener to try several different volume levels for each
example, or something similar . . .

🔗Carl Lumma <clumma@yahoo.com>

8/29/2005 4:25:37 PM

> > > > I was frankly a bit surprised at how quickly I was able to
> > > > come up with proportional beat rates for so many triads,
> > > > thinking that it should have been a lot harder to do this
> > >
> > > Bob Wendell's posts on the subject might lead one to believe
> > > that this is extremely difficult, almost an intractable
> > > problem, while Gene's paint a different picture.
> >
> > However, didn't Gene agree that his main tuning, the "Natural
> > Synchronous Well" was indeed a unique point in the space of
> > such tunings?
>
> To the best of my recollection, yes, but really Gene should answer
> this. And I'd love to see a comparison of George's new proposal
> against this and other relevant alternatives.

Bob's scale has brats of 2 and 3:2 on all triads. George's has
brats of 1 on most triads, 6:5 on one or two, and some more
complex ones (if memory (from 20 min ago) serves).

-Carl

🔗Ozan Yarman <ozanyarman@superonline.com>

8/29/2005 4:26:38 PM

Hi again George!
----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 0:43
Subject: [tuning] Temperament (Extra)ordinare! (Was: Further doubts ...)

HEY EVERYONE!

Having been unable to ignore the discussion about equal-beating
temperaments lately, I have some startling news about a "temperament
(extra)ordinaire!"

Wowsers!

I say "startling" because I never expected to be thinking about
dumping a couple of "optimal" tunings I devised 30 years ago in favor
of something that it took me less than a couple of hours to arrive at
this past Saturday.

But first I need to answer Ozan's message:

Let's get to it!

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George, by "equal beating" I mean firstly:
>
> 1) Equal beat-rates for intervals *within* a single triad,
>
> and secondly:
>
> 2) Equal beat rates for the same interval in two different triads.
>
> if you are in the know as to how to convert pitch information to
vibrations per second, I may perhaps demonstrate whether or not I
understand how to tune to `equal (or rather, even) beating`.

Well, whichever you mean, why don't I provide you with a spreadsheet
I was already working on to calculate frequencies (cols. W-Y), beat
rates (cols. S-U), beat ratios (cols. O-Q), and errors from just
(cols. G,I,K,M) from cents (cols.E-F):

/tuning-math/files/secor/WT-wksht.xls

SNIP!

You can construct your own temperament either by changing the numbers
in the light-cyan-colored cells, or you can enter cents directly in
cells E8-E19. If your temperament does not conform to Owen
Jorgensen's strict definition of a well-temperament, this will be
indicated by one or more asterisks appearing in columns H, J, L, or N.

Goodness gracious! It was just the kind of thing I was thinking about yesterday! Dear George, you are truly blessed. :) I congratulate you for providing such a resourceful little gadget. It's the dream of any tuning enthusiast!

> I am indeed referring to the wolf of 1/4 comma meantone which I
`tempered` - so to speak - in such a way as to make sure the fifths
do not howl anymore. But I am not sure how to distribute the remnant
optimally so as to preserve the `equal-beating` property.

Now you can experiment using the spreadsheet. Besides the fractions
of a didymus comma I already mentioned above, you may find 5/14,
5/19, and 5/21 to be of interest.

I am eager to start my experiments, but how do I input these fractions you mentioned?

> Does your modified meantone contain any `equal-beating` intervals?

Nope, but my brand-new temperament (extra)ordinare does:

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
12
!
86.53330
194.55680
294.12876
389.11361
499.91792
585.54105
697.27840
789.37483
891.83521
997.96292
1086.39201

2/1

It sounds terrific on my organ! Why, this is the best temperament I have ever heard so far and it should obviously qualify to replace the horrible 12-tET default of all the synths currently under production.

It's in the 'temperament ordinaire' category, but after trying it out
on my Scalatron, I'd have to consider it a bit 'extradordinaire'.

I've made available a temperament worksheet with all sorts of
figures, should anyone be interested:

/tuning-math/files/secor/GS_1.xls

Observe that all of the major triads from Eb to E are proportional-
beating (i.e., the beat rates are related by rational integers or
fractions -- I had to tweak the numbers in column B slightly to get
the beat-ratios to come out exactly proportional).

Uh, slow down please! And tell me what beat-rates, or BRATS are and what their significance is in your own words.

I also inserted
an extra column (G) so that Kraig Grady (and myself) could try this
on a Scalatron. (I'd be interested in your opinion, Kraig!)

Here's a summary of how the triad errors compare to my other two
temperaments:

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
Secor #1:... 54 49 34 21 15 15 15 19 26 37 48 54
Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

Again, for comparison, following is the total error of the intervals
of a major triad in each of the "big three" historical regular
temperaments:

1/4-comma meantone: 11 cents
12-ET: 31 cents
Pythagorean: 43 cents

And what is the total error for this extra-ordinary temperament of yours?

However, the numbers don't tell the whole story (see below).

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote: (#59901)
> IMO, when a chord is played in which two or more of
> the intervals beat at the same rate, it sets up an
> unwanted pulse which will probably interfere with the
> rhythmic structure of the piece.

I've observed that this is also occurs to some extent in a triad with
a just interval, such as 1/3 or 1/4-comma meantone temperament, in
which the beat rates are proportional. However, I don't find it
particularly objectionable -- rather like a good vibrato.

Playing with dozens of 12-tone systems just showed me that beats should be calibrated in only a special way, or otherwise the temperament in question becomes intolerable.

Anyway, I've had a chance to play around with this on my Scalatron
the past couple of days, and I've decided to call it "#1", because I
think it tops both my #2 (well-temperament) and (in all respects) my
#3 (temperament ordinaire) tunings. (I devised my old #1 temperament
in 1964, and #2 and #3 eventually replaced it. Now, IMO, this new #1
could replace all of them.)

After listening and comparing, I've concluded that the 8 equal- and
proportional-beating major triads in this temperament sound as if
they have less total error than the numbers would indicate, with the
faster beats tending to be masked by the slower ones. In conducting
some head-to-head comparisons of the C and G major triads with those
in 31-ET (12c total error), I found that they sound at least as good;
likewise the E and Eb major triads as compared with 12-ET. (The
Scalatron permits me to jump back and forth instantly from one tuning
to another, which easily allows me to make comparisons using the same
timbres.) I also found, for some reason, that the B and Ab major
triads sound only about as dissonant as pythagorean triads, even
though they have about 5 cents greater total error.

I can personally say that it is by far the most optimal solution in my opinion that deserves to replace 12-tET utilized in contemporary music education and practice worldwide.

I was frankly a bit surprised at how quickly I was able to come up
with proportional beat rates for so many triads, thinking that it
should have been a lot harder to do this and beginning to experience
some feelings of doubt that there must be some room for improvement --
at this point I don't know. (But the more I play around with it,
the less my doubt.)

Anyway, I hope that some others will try this tuning and give me your
reactions -- at least go through the triads in Scala using the chord
playing function in the keyboard clavier. Convince me that this
isn't all a dream!

--George

It is a dream come true!

Cordially,
Ozan

🔗Ozan Yarman <ozanyarman@superonline.com>

8/29/2005 4:29:42 PM

Carl, do not underestimate the power of the beat! Or in other words, may the force beat with you! 8=)

Cordially,
Ozan

----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 1:01
Subject: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> ! secor12_1.scl
> !
> George Secor's 12-tone temperament ordinaire #1
> 12
> !
> 86.53330
> 194.55680
> 294.12876
> 389.11361
> 499.91792
> 585.54105
> 697.27840
> 789.37483
> 891.83521
> 997.96292
> 1086.39201
> 2/1
>
> It's in the 'temperament ordinaire' category, but after trying
> it out on my Scalatron, I'd have to consider it a
> bit 'extradordinaire'.
>
> I've made available a temperament worksheet with all sorts of
> figures, should anyone be interested:
>
> /tuning-math/files/secor/GS_1.xls

Hiya George (and Ozan),

Do you find you like the triads where all the intervals beat
at the same rate? I would think it would be better to stagger
the beats, so that they do not result in large amplitude
changes. Or does this not happen?

-Carl

🔗Ozan Yarman <ozanyarman@superonline.com>

8/29/2005 4:34:09 PM

Dear Paul,
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 2:02
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Paul, absolute deviations from JI is not independent of consonance
>in my opinion.

If you thought I stated something to the contrary, we must have
misunderstood one another.

Yes, once more it seems! Hopefully there will be no end to it. ;)

> Therefore, I am eager to hear of the results of the test you
>proposed. If the good folk of tuning list is willing to participate
>in it, we can load the MIDI or WAV files up on my website for
>auditory comparison.

I would never trust MIDI files for this purpose, especially as the
difference between the two triads is of the same order of magnitude as
the resolution of most sound cards! But I suppose I could prepare a
series of .wav files for this purpose -- we need a variety of timbres
and registers to test -- any suggestions? Also, we would need to
instruct the listener to try several different volume levels for each
example, or something similar . . .

Timbres:
Piano, Trumpet, Clarinet, Celesta and Flutes

Common Register:
G3 to C5

We may safely disregard volume at this time I believe.

Cordially,
Ozan

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 4:40:40 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> ! secor12_1.scl
> !
> George Secor's 12-tone temperament ordinaire #1, proportional beating

I get 4, ~640/441, 3, 4/3, 2, 2, 14/9, 4, 4/3, 7/3, 3/2, ~795/469
for the brats. It does look interesting.

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 4:57:05 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> No, your calculations are quite correct. This observation of *almost*
> exact-integer beat rates has come up a few times here and
> particularly on the tuning-math list. Gene in particular has posted a
> great deal about it -- I hope he'll point you to some relevant
> postings and/or articles.

I gave a table of meantone fractional comma tunings which had near
low-integer-ratio brats to go with them a few days back.

> Again, since Gene has worked a lot on this subject, I'm hoping to see
> him provide you with an illuminating, well-rounded reply and
> assessment of your new tuning systems alongside relevant alternatives.
>
> Gene?

So far I haven't had a chance to test it, but I wonder what sort of
reply you are looking for. Cooking up alternatives already?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 5:00:01 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> It sounds terrific on my organ! Why, this is the best temperament I
>have ever heard so far and it should obviously qualify to replace >the
horrible 12-tET default of all the synths currently under >production.

Unless you plan to force the musical world to suddenly revert to
spending most of its time in the key of C major and nearby keys, and/or
to playing mostly early 18th century music, or foresee that such
regressive changes are for some reason in our future, such a
replacement would seem extremely inappropriate. As an additional
transposable tuning *option* (or preset) for synths, along with Robert
Wendell's and other equal-beating schemes, any of which could be
employed at the discretion of the performer, I could certainly endorse
its adoption, though! Perhaps I just took your enthusiasm too
literally, my dear Ozan.

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 5:13:11 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> > However, didn't Gene agree that his main tuning, the "Natural
> > Synchronous Well" was indeed a unique point in the space of
> > such tunings?
>
> To the best of my recollection, yes, but really Gene should answer
> this. And I'd love to see a comparison of George's new proposal against
> this and other relevant alternatives.

I'm not sure what you mean by a "unique point", but certainly it
didn;t have the barqoue complexity of George's system, and stuck with
brats that clearly seemed to be worthwhile.

One possibility would be to start with a circulating temperament, and
see how well one could mutate it into something with more of the
interesting beat ratios. Starting with bifrost, for instance, we have
brats of infinity, ~9/7, ~3, 3/2, ~17/9, infinity, 3/2, ~59/50, ~9/4,
3/2, 3/2. If we kept the infinites and 3/2s, and made the 3 an exact
3, how close can we get the others to good ratios?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 5:23:13 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Timbres:
> Piano, Trumpet, Clarinet, Celesta and Flutes

Interesting. Piano of course gets you into issues of inharmonic
timbres, and the Celesta into other issues (I really doubt I could hear
the difference between those two chords on the Celesta). I can
create .wav files with Matlab but I certainly doubt my ability to
emulate these instruments with mathematical functions! Maybe someone
else can come to the rescue here . . .

> Common Register:
> G3 to C5

This would be a range of pitch for the lowest note? Highest note?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 5:33:42 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > No, your calculations are quite correct. This observation of
*almost*
> > exact-integer beat rates has come up a few times here and
> > particularly on the tuning-math list. Gene in particular has
posted a
> > great deal about it -- I hope he'll point you to some relevant
> > postings and/or articles.
>
> I gave a table of meantone fractional comma tunings which had near
> low-integer-ratio brats to go with them a few days back.

Interestingly, I just referenced that table in a post here, but I was
thinking more about your well-temperament stuff.

> > Again, since Gene has worked a lot on this subject, I'm hoping to
see
> > him provide you with an illuminating, well-rounded reply and
> > assessment of your new tuning systems alongside relevant
alternatives.
> >
> > Gene?
>
> So far I haven't had a chance to test it, but I wonder what sort of
> reply you are looking for. Cooking up alternatives already?

I'm under the impression that George is unfamiliar with a lot of the
work on equal-beating well-temperaments that Robert Wendell, you, and
others have done. So someone should start introducing him to it. Of
course, there are two sometimes different meanings of "equal-beating
well-temperament" -- well-temperaments in which some of the different
occurences of like intervals have synchronized beatings, making
tuning by ear easier (Jorgensen is big on these); and well-
temperaments in which unlike intervals cohabitating in a given
consonant chord (for all or many occurences of that chord) have
synchronized beatings, giving the chord a sort of "second-order
coherence" that may somewhat mitigate its impurity. We seem to be
discussing the second meaning here.

🔗Ozan Yarman <ozanyarman@superonline.com>

8/29/2005 5:34:55 PM

Huh, I hate enforcing ideas on people. In fact, that is one of the reasons why I stick to this forum given its open-mindedness and all that.

Still, you do a great deal of injustice to George's calculations. One can very well play any kind of 12-tone music in his temperament on any key one likes. The tone-color is a bonus and it is only a matter of which frequency you want to take as basis in order to start the cycle of fifths. I played it, did you?

Maybe it escaped you, but I simply imagined 12-tET to be too crude to be accepted as default. I didn't mean to dump the availabilities out of the window! Simply put, non-microtonal synths can perform much better with George's tuning as default instead of the borin' ol' 12-tET in my opinion.

I for one wish to tune my piano to this temperament straight away.

Great work George!

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 3:00
Subject: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> It sounds terrific on my organ! Why, this is the best temperament I
>have ever heard so far and it should obviously qualify to replace >the
horrible 12-tET default of all the synths currently under >production.

Unless you plan to force the musical world to suddenly revert to
spending most of its time in the key of C major and nearby keys, and/or
to playing mostly early 18th century music, or foresee that such
regressive changes are for some reason in our future, such a
replacement would seem extremely inappropriate. As an additional
transposable tuning *option* (or preset) for synths, along with Robert
Wendell's and other equal-beating schemes, any of which could be
employed at the discretion of the performer, I could certainly endorse
its adoption, though! Perhaps I just took your enthusiasm too
literally, my dear Ozan.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 5:39:27 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > However, didn't Gene agree that his main tuning, the "Natural
> > > Synchronous Well" was indeed a unique point in the space of
> > > such tunings?
> >
> > To the best of my recollection, yes, but really Gene should
answer
> > this. And I'd love to see a comparison of George's new proposal
against
> > this and other relevant alternatives.
>
> I'm not sure what you mean by a "unique point", but certainly it
> didn;t have the barqoue complexity of George's system, and stuck
with
> brats that clearly seemed to be worthwhile.

Now wouldn't this be a worthwhile fact, if accompanied with plenty of
supporting data on "Natural Synchronous Well" and other systems of
Bob's and your devising, and other helpful explanatory writing, to
point out to George in this thread?

I'm just trying to get some *real* communication going here.

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 5:51:33 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I'm under the impression that George is unfamiliar with a lot of the
> work on equal-beating well-temperaments that Robert Wendell, you, and
> others have done. So someone should start introducing him to it.

Wendell came up with some well-temperaments, not ordinaire, which had
some very nice beat ratios. I pointed out that you can set up systems
of equations and solve for specific arrangements of beat ratios as a
way to construct these sorts of temperaments, and that therefore the
problem may not be as intractible as suggested. One question that you
need to answer for yourself at the outset is which brats are
interesting enough to be worth bothering with, and therefore of
solving for. I'd say 3/2, 2, 3, 4, infinity and -1 are all worthwhile,
but I'm not sure about some of the stuff George seems to be using.

🔗Ozan Yarman <ozanyarman@superonline.com>

8/29/2005 6:07:36 PM

Paul,
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 3:23
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Timbres:
> Piano, Trumpet, Clarinet, Celesta and Flutes

Interesting. Piano of course gets you into issues of inharmonic
timbres, and the Celesta into other issues (I really doubt I could hear
the difference between those two chords on the Celesta).

I trust you will.

I can
create .wav files with Matlab but I certainly doubt my ability to
emulate these instruments with mathematical functions! Maybe someone
else can come to the rescue here . . .

Gene perhaps? Or Charles?

> Common Register:
> G3 to C5

This would be a range of pitch for the lowest note? Highest note?

If you are familiar with the note-naming conventions, you will recall that C4 is the middle C of the piano. I am thinking of a common range for all of the timbres above. 1.5 octaves is plenty room for chordal modulation. But the last note should have been C#5.

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 6:14:43 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

>> > Common Register:
>> > G3 to C5
>
>> This would be a range of pitch for the lowest note? Highest note?
>
>
> If you are familiar with the note-naming conventions, you will recall
>that C4 is the middle C of the piano.

Of course.

>I am thinking of a common range for all of the timbres above. 1.5
>octaves is plenty room for chordal modulation. But the last note
>should have been C#5.

You don't seem to have answered my question. Whether you meant the
lowest note or the highest note, you're still talking about a range of
1.5 octave within which to transpose the chord.

🔗Aaron Krister Johnson <aaron@akjmusic.com>

8/29/2005 6:22:06 PM

On Monday 29 August 2005 7:34 pm, Ozan Yarman wrote:
> Huh, I hate enforcing ideas on people. In fact, that is one of the reasons
> why I stick to this forum given its open-mindedness and all that.
>
> Still, you do a great deal of injustice to George's calculations. One can
> very well play any kind of 12-tone music in his temperament on any key one
> likes. The tone-color is a bonus and it is only a matter of which frequency
> you want to take as basis in order to start the cycle of fifths. I played
> it, did you?
>
> Maybe it escaped you, but I simply imagined 12-tET to be too crude to be
> accepted as default. I didn't mean to dump the availabilities out of the
> window! Simply put, non-microtonal synths can perform much better with
> George's tuning as default instead of the borin' ol' 12-tET in my opinion.
>
> I for one wish to tune my piano to this temperament straight away.

Ozan,

It sounds to me like you would also rather like 1/7-comma meantone. It has
some very sweet builtin beat ratios, and is at a kind of 'sweet spot' in the
space of meantones: the fifths are large enough to make it a circulating
temperament, the largest fifth is a tiny bit larger than Pythagorean, and
very close to being 19/15, and it's just subversive enough in sound to be not
the same old 12-equal, but you can play music of any Western era in it
without fundamentally 'breaking it'!

What I do is also distribute the wolf a bit between Ab-Eb and Eb-Bb. The wolf
in the case of 1/7-comma is really samll enough that putting it between 2
fifths makes it sound like a very very mild distortion of a well-temperament,
or rather, a very very mild temperament ordinaire.

Margo and I were talking about 1/7-comma for about an hour on the phone a few
days ago....check it out!

In the meantime, George's tuning does seem intriguing...

How is Istanbul these days?

Warmly,
Aaron.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/29/2005 6:47:15 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > I'm under the impression that George is unfamiliar with a lot of
the
> > work on equal-beating well-temperaments that Robert Wendell, you,
and
> > others have done. So someone should start introducing him to it.
>
> Wendell came up with some well-temperaments, not ordinaire, which had
> some very nice beat ratios.

A well-temperament, not ordinaire, would seem to be a better candidate
for the type of replacement of 12-equal Ozan is talking about. I hope
Ozan will give Wendell's tunings the kind of hearing he gave one of
Secor's.

🔗Herman Miller <hmiller@IO.COM>

8/29/2005 7:04:34 PM

George D. Secor wrote:

> Anyway, I hope that some others will try this tuning and give me your > reactions -- at least go through the triads in Scala using the chord > playing function in the keyboard clavier. Convince me that this > isn't all a dream!

I tried it a little with the keyboard hooked up to Scala, and retuned some MIDI files, and I have to say that it's a very colorful tuning. The synchronized beating creates an interesting effect in the remote keys, and the keys around C are very pleasant. I'm not sure how it compares with similar tunings; it's been a while since I was looking at tunings like this, but it seems to hold up pretty well with a range of different styles.

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 9:21:52 PM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> It sounds to me like you would also rather like 1/7-comma meantone.

Has anyone ever sung the praises of 91-equal meantone, I wonder? It's
got a brat of 1.99, which ought to do for it in the beat department.
It's wolf fifth is 712 cents, which as Aaron notes you can pretty well
get away with.

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 9:24:34 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> A well-temperament, not ordinaire, would seem to be a better candidate
> for the type of replacement of 12-equal Ozan is talking about

Especially since I meant to say "not extraordinaire". Mixing pure
fifths with the 1/7 comma meantone fifths Aaron is advocating leads to
a mild tempering, with beat ratios to smooth out the pain of the
mistuning for you if you feel they help.

🔗Carl Lumma <clumma@yahoo.com>

8/29/2005 10:34:02 PM

> > > However, didn't Gene agree that his main tuning, the "Natural
> > > Synchronous Well" was indeed a unique point in the space of
> > > such tunings?
> >
> > To the best of my recollection, yes, but really Gene should
> > answer this. And I'd love to see a comparison of George's new
> > proposal against this and other relevant alternatives.
>
> I'm not sure what you mean by a "unique point", but certainly it
> didn;t have the barqoue complexity of George's system, and stuck
> with brats that clearly seemed to be worthwhile.

I seem to remember the claim being that it is the only well
temperament with all triads having brats of 3:2 or 2, and that
you agreed. Now let's check that...

...crud. That must have been when I was reading the list on the
web.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

8/29/2005 10:40:51 PM

> Wendell came up with some well-temperaments, not ordinaire, which had
> some very nice beat ratios. I pointed out that you can set up systems
> of equations and solve for specific arrangements of beat ratios as a
> way to construct these sorts of temperaments, and that therefore the
> problem may not be as intractible as suggested. One question that you
> need to answer for yourself at the outset is which brats are
> interesting enough to be worth bothering with, and therefore of
> solving for. I'd say 3/2, 2, 3, 4, infinity and -1 are all
> worthwhile, but I'm not sure about some of the stuff George seems
> to be using.

Hi Gene,

Wondering if you could help clear up the importance of sign with
brats. With a brat of 1, the major and minor thirds beat together,
right? How is -1 different?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

8/29/2005 11:24:05 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Wondering if you could help clear up the importance of sign with
> brats. With a brat of 1, the major and minor thirds beat together,
> right? How is -1 different?

It would differ in intonation, in ways that depend on the tuning. For
instance, a meantone fifth with a brat of +1 is a useless 679.56
cents, whereas -1 gives 695.63 cents. In general, if we stick to pure
meantone (no circulating temperaments) a brat of b goes with a fifth
which is a root of (3-2b)x^4 - 10x + 10b = 0. Other linear
temperaments, or nonregular temperaments, would be quite different.

🔗Kraig Grady <kraiggrady@anaphoria.com>

8/30/2005 1:46:23 AM

BTW i also find George's tuning is very impressive and sounds good in all keys

>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Ozan Yarman <ozanyarman@superonline.com>

8/30/2005 7:05:10 AM

>I am thinking of a common range for all of the timbres above. 1.5
>octaves is plenty room for chordal modulation. But the last note
>should have been C#5.

You don't seem to have answered my question. Whether you meant the
lowest note or the highest note, you're still talking about a range of 1.5 octave within which to transpose the chord.

-------

Paul, if you mean a base frequency to commence the test, that would be a 260Hz C4.

🔗Ozan Yarman <ozanyarman@superonline.com>

8/30/2005 7:33:58 AM

>I am thinking of a common range for all of the timbres above. 1.5
>octaves is plenty room for chordal modulation. But the last note
>should have been C#5.

You don't seem to have answered my question. Whether you meant the
lowest note or the highest note, you're still talking about a range of 1.5
octave within which to transpose the chord.

-------

Paul, if you mean a base frequency to commence the test, that would be a
260Hz C4.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/30/2005 9:17:54 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > A well-temperament, not ordinaire, would seem to be a better
candidate
> > for the type of replacement of 12-equal Ozan is talking about
>
> Especially since I meant to say "not extraordinaire". Mixing pure
> fifths with the 1/7 comma meantone fifths Aaron is advocating leads to
> a mild tempering, with beat ratios to smooth out the pain of the
> mistuning for you if you feel they help.

And Wendell's well-temperaments?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/30/2005 9:18:35 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > However, didn't Gene agree that his main tuning, the "Natural
> > > > Synchronous Well" was indeed a unique point in the space of
> > > > such tunings?
> > >
> > > To the best of my recollection, yes, but really Gene should
> > > answer this. And I'd love to see a comparison of George's new
> > > proposal against this and other relevant alternatives.
> >
> > I'm not sure what you mean by a "unique point", but certainly it
> > didn;t have the barqoue complexity of George's system, and stuck
> > with brats that clearly seemed to be worthwhile.
>
> I seem to remember the claim being that it is the only well
> temperament with all triads having brats of 3:2 or 2, and that
> you agreed. Now let's check that...
>
> ...crud. That must have been when I was reading the list on the
> web.
>
> -Carl

I'd be interested to see Gene's reply.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/30/2005 9:25:35 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Wondering if you could help clear up the importance of sign with
> > brats. With a brat of 1, the major and minor thirds beat
together,
> > right? How is -1 different?
>
> It would differ in intonation, in ways that depend on the tuning.
For
> instance, a meantone fifth with a brat of +1 is a useless 679.56
> cents,

Far from useless, I've been extolling the virtues of such Mavila
tunings here and on MMM, but mostly to deaf ears. Another Mavila
tuning with synchronous beat rates is Wilson's meta-mavila:

http://www.anaphoria.com/meantone-mavila.PDF

whose fifth is 1200*(1-.436385705396) = 676.34 cents.

> whereas -1 gives 695.63 cents.

That's Wilson's meta-meantone (see the article above);

1200*.579692031034 = 695.63 cents.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/30/2005 9:36:31 AM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> >I am thinking of a common range for all of the timbres above. 1.5
> >octaves is plenty room for chordal modulation. But the last note
> >should have been C#5.
>
> You don't seem to have answered my question. Whether you meant the
> lowest note or the highest note, you're still talking about a range
of 1.5 octave within which to transpose the chord.
>
> -------
>
> Paul, if you mean a base frequency to commence the test, that would
be a 260Hz C4.

Oddly, the text below the dashes doesn't show up until I click
on "Reply"! This is crazy! What on earth is Yahoo doing here?

But you still didn't answer my question. You gave the range G3 to
C#5, did you not? I still don't know whether that refers to the range
for lowest note or the highest note (or something else).

🔗Ozan Yarman <ozanyarman@superonline.com>

8/30/2005 9:51:09 AM

Oh I see what you mean! C4 should be the lowest note of the triad in root position.

Oz.
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 19:36
Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> >I am thinking of a common range for all of the timbres above. 1.5
> >octaves is plenty room for chordal modulation. But the last note
> >should have been C#5.
>
> You don't seem to have answered my question. Whether you meant the
> lowest note or the highest note, you're still talking about a range
of 1.5 octave within which to transpose the chord.
>
> -------
>
> Paul, if you mean a base frequency to commence the test, that would
be a 260Hz C4.

Oddly, the text below the dashes doesn't show up until I click
on "Reply"! This is crazy! What on earth is Yahoo doing here?

But you still didn't answer my question. You gave the range G3 to
C#5, did you not? I still don't know whether that refers to the range
for lowest note or the highest note (or something else).

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

8/30/2005 10:05:46 AM

Ok, I hope someone can prepare some examples.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Oh I see what you mean! C4 should be the lowest note of the triad
in root position.
>
> Oz.
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 30 Aðustos 2005 Salý 19:36
> Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> >
> > >I am thinking of a common range for all of the timbres above.
1.5
> > >octaves is plenty room for chordal modulation. But the last
note
> > >should have been C#5.
> >
> > You don't seem to have answered my question. Whether you meant
the
> > lowest note or the highest note, you're still talking about a
range
> of 1.5 octave within which to transpose the chord.
> >
> > -------
> >
> > Paul, if you mean a base frequency to commence the test, that
would
> be a 260Hz C4.
>
> Oddly, the text below the dashes doesn't show up until I click
> on "Reply"! This is crazy! What on earth is Yahoo doing here?
>
> But you still didn't answer my question. You gave the range G3 to
> C#5, did you not? I still don't know whether that refers to the
range
> for lowest note or the highest note (or something else).

🔗Gene Ward Smith <gwsmith@svpal.org>

8/30/2005 12:23:52 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> And Wendell's well-temperaments?

That's what I was describing--his mixture of pure fifths (brat 3/2)
with 1/7 comma fifths (brat 2.)

🔗Gene Ward Smith <gwsmith@svpal.org>

8/30/2005 12:34:52 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I'd be interested to see Gene's reply.

I don't recall proving such a thing, and one of the brats is going to
be only almost exactly a pure 2 or 3/2, instead of exactly. However,
if you arrange the fifths in a non-waste way, there's really only one
way to do this, modulo differences which don't actually matter.

🔗George D. Secor <gdsecor@yahoo.com>

8/30/2005 2:46:32 PM

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> > HEY EVERYONE!
>
> Look, George is shouting with GLEE! :)

And I'm (gleefully) overwhelmed just trying to read the replies!!!
(Still haven't gotten thru all of them -- please be patient -- I'll
answer soon!

--George

🔗Carl Lumma <clumma@yahoo.com>

8/30/2005 9:06:10 PM

> Oddly, the text below the dashes doesn't show up until I click
> on "Reply"! This is crazy! What on earth is Yahoo doing here?

They think it's a signature, probably. Google started filtering
those out in gmail, and yahoo's probably trying to copy them.
The whole idea is stupid.

-Carl

🔗terry0051 <terry0051@yahoo.com>

8/31/2005 1:00:00 AM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> HEY EVERYONE!
>
> Having been unable to ignore the discussion about equal-beating
> temperaments lately, I have some startling news about a
"temperament
> (extra)ordinaire!"

That's really interesting: and it put me in mind of
a temp.ordinaire of interestingly similar type that was
discussed in one of the piano tuning groups about four
years ago -- (apologies for the repetition if you are
already familiar with this one).

According to the reports of the time, it was tried out and
much liked, and there, too, some of its benefits were being
tentatively attributed to "close attention to internal
agreements of beat speeds and not just purity of intervals".

The starting message of the older thread is at
http://www.ptg.org/pipermail/caut/2001-May/004075.html

and the temperament used was defined by cent deviations from
unstretched equal temperament 100 cents/semitone:
A, Bb, B, C ,C#, D, Eb, E, F, F#, G, G#, A
0, +7, -6,+9,-2, +3, +2,-3, +12, -4, +6, -1, 0
[possibly "G +6" may have been a typo for "G +5"].

The source was cited as "The Well Tempered Organ", Charles A
Padgham (Oxford 1986).

I confess that I haven't yet managed to have a listen-in
(I haven't yet succeeded in getting my computer to do
this scale stuff, even with 'Scala', but maybe I will
soon crack this thing!), but if I may add a question:-

I notice that your temperament is defined with
precision down to 10^-5 cents, while the discussion four
years ago was based on trial of a temperament defined only
to the nearest cent. How much precision of implementation
do you think is needed in practice to realise these
internal agreements of beat-rate?

Kind regards
Terry Stancliffe
Cambridge

🔗terry0051 <terry0051@yahoo.com>

8/31/2005 1:05:46 AM

Correction to previous message:
Sorry -- the possible typo seemed to be in g# (-2?), not in g.
(would remove a seeming +/- anomaly in tempering of fifths).
Terry

--- In tuning@yahoogroups.com, "terry0051" <terry0051@y...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

🔗pgreenhaw@nypl.org

8/31/2005 10:18:11 AM
Attachments

I was pushing around numbers last night and decided to see what would be
generated when I used 5^(1/2) as a generator and 3:1 folding point (see
attachment) -- as was expected, MOSs happened at various steps of the
procedure -- but when reduced back into the 2:1 space, the MOS evaporated.

Has anyone toyed around with this notion of folding around other points
besides the 2:1? If so, anything curious come out of it?

Paul

🔗George D. Secor <gdsecor@yahoo.com>

9/1/2005 1:18:22 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Hi again George!

Hello, again! Sorry for my delay in replying -- I had to take some
time off yesterday due to an illness in my family. (And I have time
today to reply to only a few things.)

> ----- Original Message -----
> From: George D. Secor
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> > ...
> > > if you are in the know as to how to convert pitch information
to
> vibrations per second, I may perhaps demonstrate whether or not I
> understand how to tune to `equal (or rather, even) beating`.
>
> Well, whichever you mean, why don't I provide you with a
spreadsheet
> I was already working on to calculate frequencies (cols. W-Y),
beat
> rates (cols. S-U), beat ratios (cols. O-Q), and errors from just
> (cols. G,I,K,M) from cents (cols.E-F):
>
> /tuning-math/files/secor/WT-wksht.xls
> ...
> > Now you can experiment using the spreadsheet. Besides the
fractions
> > of a didymus comma I already mentioned above, you may find
5/14,
> > 5/19, and 5/21 to be of interest.
>
> I am eager to start my experiments, but how do I input these
fractions you mentioned?

For n/d-comma, put the number d in cell G4 and -n in B15. Observe
that all of the major triads from Gb thru G have the same beat ratios.

> > Does your modified meantone contain any `equal-beating`
intervals?
>
> Nope, but my brand-new temperament (extra)ordinare does:
> ...
> I've made available a temperament worksheet with all sorts of
> figures, should anyone be interested:
>
> /tuning-math/files/secor/GS_1.xls
>
> Observe that all of the major triads from Eb to E are
proportional-
> beating (i.e., the beat rates are related by rational integers or
> fractions -- I had to tweak the numbers in column B slightly to
get
> the beat-ratios to come out exactly proportional).
>
> Uh, slow down please! And tell me what beat-rates, or BRATS are and
what their significance is in your own words.

A beat-rate is simply the number of beats per second that occur in a
tempered consonance. A triad has synchronous beating when the ratios
between the beat rates for its constituent intervals can be expressed
with small integers.

> ...
> And what is the total error for this extra-ordinary temperament of
yours?

For the 12 major triads: ~386.5 cents (in cell H21); since this isn't
a well-temperament, that figure exceeds the theoretical minimum of
~375.4 cents for a curculating 12-tone temperament.

> Anyway, I hope that some others will try this tuning and give me
your
> reactions -- at least go through the triads in Scala using the
chord
> playing function in the keyboard clavier. Convince me that this
> isn't all a dream!

And for those others who did: thank you for your feedback -- much
appreciated.

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

9/1/2005 1:44:55 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > ! secor12_1.scl
> > ...
> > /tuning-math/files/secor/GS_1.xls
>
> Hiya George (and Ozan),
>
> Do you find you like the triads where all the intervals beat
> at the same rate? I would think it would be better to stagger
> the beats, so that they do not result in large amplitude
> changes. Or does this not happen?
>
> -Carl

It seems to give the consonant chords a more "unified"
or "integrated" sound, as if the instrument is "singing" with a
vibrato.

Only the 5th & major 3rd of the C and G major triads beat at a 1:1
ratio in this temperament, but the overwhelming majority of the other
beat ratios in the best triads (from Eb to E major) are also
integers, so those triads also have a "singing" quality.

I don't think that the amplitude changes are objectionably large --
even in the 5/17-comma temperament with its 1:1:1 beat ratio (which I
previously incorporated in the best keys of my 19-tone well-
temperament -- a tuning I currently have stored in my Scalatron). I
expect that one would have to deliberately adjust the phase
relationships between the tones in order for the beats to reinforce
one another so as to produce an amplitude large enough to be judged
objectionable.

--George

🔗Carl Lumma <clumma@yahoo.com>

9/1/2005 4:05:59 PM

> > Hiya George (and Ozan),
> >
> > Do you find you like the triads where all the intervals beat
> > at the same rate? I would think it would be better to stagger
> > the beats, so that they do not result in large amplitude
> > changes. Or does this not happen?
> >
> > -Carl
>
> It seems to give the consonant chords a more "unified"
> or "integrated" sound, as if the instrument is "singing" with a
> vibrato.
>
> Only the 5th & major 3rd of the C and G major triads beat at a 1:1
> ratio in this temperament, but the overwhelming majority of the
> other beat ratios in the best triads (from Eb to E major) are also
> integers, so those triads also have a "singing" quality.
>
> I don't think that the amplitude changes are objectionably large --
> even in the 5/17-comma temperament with its 1:1:1 beat ratio (which
> I previously incorporated in the best keys of my 19-tone well-
> temperament -- a tuning I currently have stored in my Scalatron).
> I expect that one would have to deliberately adjust the phase
> relationships between the tones in order for the beats to reinforce
> one another so as to produce an amplitude large enough to be judged
> objectionable.

Interesting.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/1/2005 6:12:10 PM

--- In tuning@yahoogroups.com, "terry0051" <terry0051@y...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> How much precision of implementation
> do you think is needed in practice to realise these
> internal agreements of beat-rate?
>
> Kind regards
> Terry Stancliffe
> Cambridge

I don't see that George replied; but one often needs far better than 1
cent precision to get the beating ratios to sound audibly right. So
most electronic synths are out of the question for hearing the effects.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/1/2005 6:14:42 PM

--- In tuning@yahoogroups.com, pgreenhaw@n... wrote:

> I was pushing around numbers last night and decided to see what would
be
> generated when I used 5^(1/2) as a generator and 3:1 folding point
(see
> attachment) -- as was expected, MOSs happened at various steps of the
> procedure -- but when reduced back into the 2:1 space, the MOS
evaporated.
>
>
> Has anyone toyed around with this notion of folding around other
points
> besides the 2:1?

Yes.

> If so, anything curious come out of it?

Are you familiar with the Bohlen-Pierce scale?

I'd like to snail-mail you my 'Middle Path' paper, if I may. The new
version is over a year old now, but I'd love your comments. May I have
your address?

>
> Paul

-Another Paul

🔗monz <monz@tonalsoft.com>

9/1/2005 11:47:03 PM

Hi Paul and Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning@yahoogroups.com, pgreenhaw@n... wrote:
>
> > Has anyone toyed around with this notion of folding
> > around other points besides the 2:1?
>
> Yes.
>
> > If so, anything curious come out of it?
>
> Are you familiar with the Bohlen-Pierce scale?

Bohlen-Pierce uses 3:1 as the identity interval.

Tonescape allows you to use any interval as the
identity interval, and also allows the creation of
tunings which have no identity interval.

In addition, the generators of the tuning can be
anything.

Both the identity interval and generators can be
specified as either ratios or cents values (to
3 decimal places).

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Ozan Yarman <ozanyarman@superonline.com>

9/2/2005 4:43:38 AM

Identity interval??? What happened to generator and period?

----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 02 Eylül 2005 Cuma 9:47
Subject: [tuning] Re: MOS at folding points other than the 2:1

Hi Paul and Paul,

SNIP!

Both the identity interval and generators can be
specified as either ratios or cents values (to
3 decimal places).

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗pgreenhaw@nypl.org

9/2/2005 7:43:31 AM

What about other MOSs at places other than the 2:1? Does finding MOSs at
a 3:1 folding-point even matter in the grand scheme of things? Is there
anything "coherent" with that? Or any integrity to be gained?

P

___________________________________________
Paul Greenhaw
Music Specialist II
The New York Public Library for the Performing Arts
40 Lincoln Center Plaza
New York, NY 10023
(212) 870-1892
__________________________________________

"monz" <monz@tonalsoft.com>
Sent by: tuning@yahoogroups.com
09/02/2005 02:47 AM
Please respond to tuning

To: tuning@yahoogroups.com
cc:
Subject: [tuning] Re: MOS at folding points other than the
2:1

Hi Paul and Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning@yahoogroups.com, pgreenhaw@n... wrote:
>
> > Has anyone toyed around with this notion of folding
> > around other points besides the 2:1?
>
> Yes.
>
> > If so, anything curious come out of it?
>
> Are you familiar with the Bohlen-Pierce scale?

Bohlen-Pierce uses 3:1 as the identity interval.

Tonescape allows you to use any interval as the
identity interval, and also allows the creation of
tunings which have no identity interval.

In addition, the generators of the tuning can be
anything.

Both the identity interval and generators can be
specified as either ratios or cents values (to
3 decimal places).

-monz
http://tonalsoft.com
Tonescape microtonal music software

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

SPONSORED LINKS
Music education
Music production education
Music education degree
Degree education music online
Music business education
Music education online

YAHOO! GROUPS LINKS

Visit your group "tuning" on the web.

To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

🔗Carl Lumma <clumma@yahoo.com>

9/2/2005 9:31:02 AM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Identity interval??? What happened to generator and period?

Period = identity interval.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

9/2/2005 9:37:58 AM

--- In tuning@yahoogroups.com, pgreenhaw@n... wrote:
> What about other MOSs at places other than the 2:1? Does finding
> MOSs at a 3:1 folding-point even matter in the grand scheme of
> things? Is there anything "coherent" with that? Or any
> integrity to be gained?

I don't think there are universal answers to these questions at
this point. Listen, make some music, and if you go platinum
go back and say it was coherent. :)

-Carl

PS- Charles Carpenter's band made 2 CDs of Bohlen-Pierce music
in a rather aggressive fusion style. If you're interested in
scales like this, you might want to get these recordings. But
even if you don't like them, it isn't conclusive evidence against
the 3:1-equivalence idea.

🔗Graham Breed <gbreed@gmail.com>

9/2/2005 9:53:24 AM

Ozan Yarman wrote:
> Identity interval??? What happened to generator and period?

The period is an equal division of the identity interval. Usually the identity interval is an octave. The period is the interval by which the tuning repeats, and the identity interval is a higher level concept.

Graham

🔗Carl Lumma <clumma@yahoo.com>

9/2/2005 10:34:35 AM

> > Identity interval??? What happened to generator and period?
>
> The period is an equal division of the identity interval. Usually
> the identity interval is an octave. The period is the interval by
> which the tuning repeats, and the identity interval is a higher
> level concept.
> Graham

That's an alternate usage. But in this usage, the identity
interval has nothing to do with the pitches -- it's just a
statement that you do not intend the period to sound like an
equivalence interval.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/2/2005 4:30:04 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> > Identity interval??? What happened to generator and period?
>
> Period = identity interval.

No, it isn't necessarily. For example, most composers who use the
octatonic (diminished) scale in 12-equal still consider the identity
interval to be the octave, even though the period of the scale is 1/4
octave.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/2/2005 4:32:55 PM

The identity interval, or interval of equivalence, is usually taken
to be the octave (2:1 or thereabouts), while a scale may repeat more
than once per octave and thus have a period that is 1/2, 1/3, 1/4,
1/5, . . . octave. If you're talking about 2D tuning systems, then
you just need to then specify a generator, and you can make that
specification unique if you insist that it's less than 1/2 the size
of the period.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Identity interval??? What happened to generator and period?
>
> ----- Original Message -----
> From: monz
> To: tuning@yahoogroups.com
> Sent: 02 Eylül 2005 Cuma 9:47
> Subject: [tuning] Re: MOS at folding points other than the 2:1
>
>
> Hi Paul and Paul,
>
> SNIP!
>
> Both the identity interval and generators can be
> specified as either ratios or cents values (to
> 3 decimal places).
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/2/2005 4:36:19 PM

--- In tuning@yahoogroups.com, pgreenhaw@n... wrote:

> What about other MOSs at places other than the 2:1? Does finding
MOSs at
> a 3:1 folding-point even matter in the grand scheme of things?

Well, what does matter?

> Is there
> anything "coherent" with that? Or any integrity to be gained?

The Bohlen-Pierce scale is quite unique and interesting. Some great
music has been made in it, for example by Charles Carpenter. Can we
hear 3:1 as an interval of equivalence? I'm still skeptical. But
sometimes the music is *right* even if the theory behind it is wrong.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/2/2005 4:41:31 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > Identity interval??? What happened to generator and period?
> >
> > The period is an equal division of the identity interval. Usually
> > the identity interval is an octave. The period is the interval by
> > which the tuning repeats, and the identity interval is a higher
> > level concept.
> > Graham
>
> That's an alternate usage.

It seems to be the more meaningful one, by far. Why do you say
it's "alternate"?

> But in this usage, the identity
> interval has nothing to do with the pitches -- it's just a
> statement that you do not intend the period to sound like an
> equivalence interval.

When you start naming notes, notating things, and even simply saying
how many notes there are in the scale, knowing the identity interval
(or interval of equivalence) is necessary.

🔗monz <monz@tonalsoft.com>

9/2/2005 9:08:04 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> > >
> > > Identity interval??? What happened to generator and period?
> >
> > Period = identity interval.
>
> No, it isn't necessarily. For example, most composers
> who use the octatonic (diminished) scale in 12-equal
> still consider the identity interval to be the octave,
> even though the period of the scale is 1/4 octave.

It should also be noted that, at least concerning the
way Tonescape works, the period and identity-interval
are also generators.

They're just considered differently from what's called
"generators", because of the equivalence aspect.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

9/2/2005 9:25:13 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> When you start naming notes, notating things, and even
> simply saying how many notes there are in the scale,
> knowing the identity interval (or interval of equivalence)
> is necessary.

*If* there is an identity-interval.

As i pointed out (and as you well know, of course, Paul),
scales don't necessarily have to have an identity-interval.

The ancient Greek tunings, based on similar tetrachords,
had a "sort-of" identity-interval of 4/3 ratio ...
however, because of the "tone of disjunction" which
occurs above _mese_ in the Greater Perfect System and
above _proslambanomenos_, the identity aspect isn't
totally consistent. Thus, the Greek tunings are best
designed in Tonescape without an identity-interval.

(BTW, word is that John Chalmers is currently composing
some sample pieces in these Greek scales with Tonescape.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Ozan Yarman <ozanyarman@superonline.com>

9/6/2005 7:12:37 AM

Equal beating 6/5 = 8/5 same. Almost 1/7-comma
|
0: 1/1 C unison, perfect prime
1: 92.146 cents
2: 197.756 cents D
3: 303.366 cents Dx Eb
4: 395.512 cents E
5: 501.122 cents F
6: 593.268 cents
7: 698.878 cents G
8: 804.488 cents Gx Ab
9: 896.634 cents A
10: 1002.244 cents Ax Bb
11: 1094.390 cents B
12: 2/1 C octave

> I for one wish to tune my piano to this temperament straight away.

Ozan,

It sounds to me like you would also rather like 1/7-comma meantone. It has some very sweet builtin beat ratios, and is at a kind of 'sweet spot' in the space of meantones: the fifths are large enough to make it a circulating temperament, the largest fifth is a tiny bit larger than Pythagorean, and very close to being 19/15, and it's just subversive enough in sound to be not the same old 12-equal, but you can play music of any Western era in it without fundamentally 'breaking it'!

---------

Aaron, 1/7-comma meantone does indeed sound sweet to my ears when I play a first inversion dominant seventh. However, when one modulates to major or minor from there, the triad becomes pretty ugly. Moreover, I hardly think that 712 cents to be `tiny bit larger than Pythagorean`. This is a good temperament, but it can be improved in my opinion.

Now, George has formulated an excellent temperament which I have just yesterday implemented on my Bechstein Imperial Grand, and it sounds absolutely terrific.

-----------

What I do is also distribute the wolf a bit between Ab-Eb and Eb-Bb. The wolf in the case of 1/7-comma is really samll enough that putting it between 2 fifths makes it sound like a very very mild distortion of a well-temperament, or rather, a very very mild temperament ordinaire.

Margo and I were talking about 1/7-comma for about an hour on the phone a few days ago....check it out!

-------------

Your modification would require both Ab-Eb and Eb-Bb to be about 706 cents wide. I'm not sure if that would bode well with the equal-beating nature of 1/7-comma meantone.

-------------

In the meantime, George's tuning does seem intriguing...

Yes, indeed! I'm the honored musician who first implemented George's brand new tuning to an acoustic grand piano. I'm delighted with the results!

How is Istanbul these days?

------------

Mildly warm, windy and humid due to the Marmara climate, but not intolerably so. It's good though for my hypoglycaemic diet.

Warmly,
Aaron.

Cordially,
Ozan

🔗Ozan Yarman <ozanyarman@superonline.com>

9/6/2005 7:26:43 AM

Paul, Wendell's temperaments are also very elegant, however, my ears prefer Secor temperament ordinaire #1. It seems to be more balanced on the white keys and flat tonalities in comparison.

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 4:47
Subject: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

A well-temperament, not ordinaire, would seem to be a better candidate for the type of replacement of 12-equal Ozan is talking about. I hope Ozan will give Wendell's tunings the kind of hearing he gave one of Secor's.

🔗Ozan Yarman <ozanyarman@superonline.com>

9/6/2005 7:44:56 AM

Hello, again! Sorry for my delay in replying -- I had to take some
time off yesterday due to an illness in my family. (And I have time today to reply to only a few things.)

It's quite alright dear George. I have delayed to reply myself.

For n/d-comma, put the number d in cell G4 and -n in B15. Observe
that all of the major triads from Gb thru G have the same beat ratios.

Now that is fascinating! But how does one go to a temperament where the octave closes with equal-beating triads after, say, 30-40 tones?

>
> Uh, slow down please! And tell me what beat-rates, or BRATS are and what their significance is in your own words.

A beat-rate is simply the number of beats per second that occur in a tempered consonance. A triad has synchronous beating when the ratios between the beat rates for its constituent intervals can be expressed with small integers.

Okay, ratios between the beat-rates... as in the ratio of the beat rate of 5/4 to the beat rate of 3/2? In the case of 5/17-comma temperament, the major third beats 4.14 times per second and the fifth the same, so the ratio is 1/1. In the case of the 5/14-comma temperament, the BRAT of the consonant major triad is 10/5=2. I think I got it! The comma-fractions you gave yield multiples of beat-ratios. Right?

> ...
> And what is the total error for this extra-ordinary temperament of yours?

For the 12 major triads: ~386.5 cents (in cell H21); since this isn't a well-temperament, that figure exceeds the theoretical minimum of ~375.4 cents for a curculating 12-tone temperament.

That is because you have chosen to increase the error in the remote keys in favor of the more accustomed keys. Correct?

> Anyway, I hope that some others will try this tuning and give me
your reactions -- at least go through the triads in Scala using the
chord playing function in the keyboard clavier. Convince me that this isn't all a dream!

And for those others who did: thank you for your feedback -- much
appreciated.

Best,

--George

I have implemented your tuning on my Bechstein Imperial Grand just yesterday. Being the first to tune an acoustic instrument to your new temperament (my tuners almost demanded double the price for the extra effort!), let me also be the first to say that I simply loved the results. Good Work dear George!

Cordially,
Ozan

🔗George D. Secor <gdsecor@yahoo.com>

9/6/2005 12:02:37 PM

--- In tuning@yahoogroups.com, "terry0051" <terry0051@y...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > HEY EVERYONE!
> >
> > Having been unable to ignore the discussion about equal-beating
> > temperaments lately, I have some startling news about
a "temperament
> > (extra)ordinaire!"
>
> That's really interesting: and it put me in mind of
> a temp.ordinaire of interestingly similar type that was
> discussed in one of the piano tuning groups about four
> years ago -- (apologies for the repetition if you are
> already familiar with this one).

No, I wasn't aware of it. My main interest is in non-12 tunings, so
I haven't made it a point to keep fully informed about the more
recent developments concerning well temperaments or _temperaments
ordinaires_.

> According to the reports of the time, it was tried out and
> much liked, and there, too, some of its benefits were being
> tentatively attributed to "close attention to internal
> agreements of beat speeds and not just purity of intervals".
>
> The starting message of the older thread is at
> http://www.ptg.org/pipermail/caut/2001-May/004075.html
>
> and the temperament used was defined by cent deviations from
> unstretched equal temperament 100 cents/semitone:
> A, Bb, B, C ,C#, D, Eb, E, F, F#, G, G#, A
> 0, +7, -6,+9,-2, +3, +2,-3, +12, -4, +6, -1, 0
> [possibly "G +6" may have been a typo for "G +5"].
>
> [Correction:
> Sorry -- the possible typo seemed to be in g# (-2?), not in g.
> (would remove a seeming +/- anomaly in tempering of fifths).
> Terry]

I agree that this is probably a typo and have taken this into account
in my attempts to reconstruct the temperament below.

> The source was cited as "The Well Tempered Organ", Charles A
> Padgham (Oxford 1986).

Not having that at hand, it's not clear to me whether the table of
offsets defines a temperament that was devised in the 20th century or
is an attempt to enumerate (in cents) a description of a temperment
dating from a much earlier time. (Should the latter be the case,
then I might question whether the figures are even correct to the
nearest cent.)

> I confess that I haven't yet managed to have a listen-in
> (I haven't yet succeeded in getting my computer to do
> this scale stuff, even with 'Scala', but maybe I will
> soon crack this thing!), but if I may add a question:-
>
> I notice that your temperament is defined with
> precision down to 10^-5 cents, while the discussion four
> years ago was based on trial of a temperament defined only
> to the nearest cent. How much precision of implementation
> do you think is needed in practice to realise these
> internal agreements of beat-rate?
>
> Kind regards
> Terry Stancliffe
> Cambridge

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> I don't see that George replied; but one often needs far better
than 1
> cent precision to get the beating ratios to sound audibly right. So
> most electronic synths are out of the question for hearing the
effects.

I'm inclined to agree with Paul about this. Since the pitch
increments on my Scalatron vary (from 0.8 to 1.6 cents) according to
the interval the pitch makes with "A", I would also have to
investigate how many of the triads have the pitches close enough to
exhibit the desired effect on my instrument. (I do remember as I
consulted a lookup table to set the Scalatron switches for each note,
that many of the pitches were within 0.2 cents or so -- I'll have to
listen closely to the beating of the individual intervals in each
triad and do a qualitative comparison. In any case, I'm *extremely*
happy with the way my temperament sounds.)

In attempting to analyze the temperament described by whole-cent
offsets, I found that this precision is barely adequate for
*determining* the intended beat rates. For example, C-G is 3 cents
narrower than in 12-ET, which would be 5 cents narrower than just,
but this could apply to a 5th ranging anywhere from a 1/4-comma (5.4
cents) narrow to the 5/23-comma (4.7 cents narrow) fifth that I have
in my #1 temperament. If one (reasonably) assumes that the chain of
fifths from F to B are the same size and that all of the pitches in
the table are indeed to the nearest cent, then the fifth must be
rather closer to 5.0 cents. But it's difficult to say for sure,
since the offsets may be, at best, a second-hand description.

The text in the above link provides another clue:

<< In this tempering in the central keys of 1 or 2 sharps and flats,
for example, beats in thirds or fifths agree with beats in sixths
which creates a wonderful clarity of texture to chords and harmonies.
>>

If "agree" is interpreted as a 1:1 beat-ratio, this description could
apply to the F, C, and G major triads in a temperament with 5ths 5/23-
comma narrow from F to B (1:1 beat ratio between 5th & M3), as long
as we allow a little leeway with the cents. Using this as a starting
point, I've constructed a temperament here:

/tuning-math/files/secor/TO-523.xls

! TO-523.scl
!
Possible 5/23-comma temperament ordinaire
12
!
86.53508
194.55680
294.12920
389.11361
502.72160
585.53449
697.27840
789.38184
891.83521
997.01621
1086.39201
2/1

The description could also apply to a temperament with 10/43-comma
(5.002 cents narrow) fifths from F to B. Although the M3:5th ratio
(for F, C, and G major triads) is 1:2, the m6:5th ratio (in a
tempered 4:5:6:8) would be 1:1. If Bb-F is then made ~6.5/43-comma
wide (which gives the correct number of cents for Bb), then the m3:M3
ratio in the Bb major triad (root position) will be 1:1. If B-F# is
made 1/43-comma narrow, the M3:5th ratio in the D major triad (root
position) would be 2:1, which could still be interpreted
as "agreeable"; this gives F# an offset of -4.7 cents instead of -4,
which is close. Following this, I've attempted to construct a
temperament reasonably close to the offsets given in the table:

/tuning-math/files/secor/TO-1043.xls

! TO-1043.scl
!
Possible 10/43-comma temperament ordinaire
12
!
87.62999
193.90577
292.61883
387.81153
503.04712
586.20299
696.95288
789.04548
890.85865
997.83297
1084.76441
2/1

Both of the above constructions conform fairly well to figures
derived from the cents-offset table, particularly when the total
absolute error of the major triads are calculated (and then compared
with my #1 temperament):

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
From table:. 59 49 35 19 13 13 13 23 33 39 43 49
TO-1043.xls: 59 49 36 20 13 13 13 22 31 40 44 52
TO-523.xls:. 60 49 34 22 15 15 15 19 26 37 48 52
Secor #1:... 54 49 34 21 15 15 15 19 26 37 48 54

The most important difference between those three and my #1
temperament (extra)ordinaire is that mine has the narrowest 5ths in a
chain only from C-B rather than F-B, which allows Db-F to have
significantly less error (without increasing the total absolute error
in the F major triad). This, in turn, significantly reduces the
total absolute error of the Db major triad -- the characteristic
essential to making it "extraordinaire" (and still #1 in my book,
BTW!).

I've tightened up the calculations for #1 and have updated the
spreadsheet accordingly:

/tuning-math/files/secor/GS_1.xls

This is the updated .scl file:

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
12
!
86.53508
194.55680
294.12834
389.11361
499.91715
585.53449
697.27840
789.38184
891.83521
997.96215
1086.39201
2/1

Best,

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/6/2005 12:29:38 PM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
> wrote:
> > > >
> > > > Identity interval??? What happened to generator and period?
> > >
> > > Period = identity interval.
> >
> > No, it isn't necessarily. For example, most composers
> > who use the octatonic (diminished) scale in 12-equal
> > still consider the identity interval to be the octave,
> > even though the period of the scale is 1/4 octave.
>
>
> It should also be noted that, at least concerning the
> way Tonescape works, the period and identity-interval
> are also generators.

That's unfortunate, because for plenty of important scales and
tunings, such as diminished, the identity-interval (the octave) is
*not* one of the generators.

> They're just considered differently from what's called
> "generators", because of the equivalence aspect.

For the diminished example, one of the two generators is 1/4-octave.
Neither of the generators is the octave or identity-interval.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/6/2005 12:32:37 PM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > When you start naming notes, notating things, and even
> > simply saying how many notes there are in the scale,
> > knowing the identity interval (or interval of equivalence)
> > is necessary.
>
>
> *If* there is an identity-interval.
>
> As i pointed out (and as you well know, of course, Paul),
> scales don't necessarily have to have an identity-interval.

The pitches of scales are unaffected whether there is an identity
interval or not. It's simply how we *name* and *notate* the scales
that is affected.

> The ancient Greek tunings, based on similar tetrachords,
> had a "sort-of" identity-interval of 4/3 ratio ...
> however, because of the "tone of disjunction" which
> occurs above _mese_ in the Greater Perfect System and
> above _proslambanomenos_, the identity aspect isn't
> totally consistent.

Monz, you're confusing "identity" with "periodicity".

> Thus, the Greek tunings are best
> designed in Tonescape without an identity-interval.

I think this is just silly because the Greeks used similar
nomenclature for notes an octave apart, just as we do today.

🔗Gene Ward Smith <gwsmith@svpal.org>

9/6/2005 1:11:20 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Okay, ratios between the beat-rates... as in the ratio of the beat
rate of 5/4 to the beat rate of 3/2? In the case of 5/17-comma
temperament, the major third beats 4.14 times per second and the fifth
the same, so the ratio is 1/1. In the case of the 5/14-comma
temperament, the BRAT of the consonant major triad is 10/5=2. I think
I got it! The comma-fractions you gave yield multiples of beat-ratios.
Right?

A brat of 2 goes with 1/7 comma. 5/14 comma is a brat of 1/4, and is
for a fifth (a root of f^4-4f+1) even flatter than 19-et. However,
there *is* a 2 in there; in fact, two 2s, if we look at the associated
beat ratios for all three chord consituants taken in pairs, we get two
2 beat ratios.

If someone were looking for a 19-tone circulating temperament, this
fifth would be a good one to keep in mind.

🔗George D. Secor <gdsecor@yahoo.com>

9/6/2005 1:26:48 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> [gs:]
> Hello, again! Sorry for my delay in replying -- I had to take some
> time off yesterday due to an illness in my family. (And I have
time today to reply to only a few things.)
> [oy:]
> It's quite alright dear George. I have delayed to reply myself.

This has turned into even more of a delay -- I learned last Friday
afternoon that my mother-in-law (87 years old) died the previous
night, and I have had almost no time until today to catch up with the
discussion.

Tom Dent has contributed some valuable observations (which will take
me some time to analyze and/or digest), to which I hope to make a
detailed reply in 2 or 3 days (hang in there, Tom!).

> [gs:]
> For n/d-comma, put the number d in cell G4 and -n in B15. Observe
> that all of the major triads from Gb thru G have the same beat
ratios.
>
> [oy:]
> Now that is fascinating! But how does one go to a temperament where
the octave closes with equal-beating triads after, say, 30-40 tones?

That's something I've never tried. The first thing I would do is add
more rows to my spreadsheet, then take it from there. You would have
to change some of the formulas for a 41-tone circulating system,
since you'd no longer be dealing mainly with narrow 5ths.

However, I have done it with part of a circle of 19. The 5/17-comma
5ths are not exactly 1:1:1 (5th:M3:m3) equal-beating (would need to
be 695.6304 cents), so I'll have to recalculate these figures when I
have more time:

! secor_19wt.scl
!
George Secor's 19-tone well temperament with ten 5/17-comma fifths
19
!
69.40735
131.54493
191.25924
260.66659
317.95765
382.51849
451.92584
504.37038
573.77773
638.33856
695.62962
765.03697
824.57129
886.88886
956.29622
1011.16402
1078.14811
1145.13220
2/1

With C as 1/1, the 6:7:9 triads on F, C, G, and D are greatly
improved over 19-ET, due to 7:9 having very low error. I came up
with the essential idea for this in 1978.

> > [oy:]
> > Uh, slow down please! And tell me what beat-rates, or BRATS are
and what their significance is in your own words.
> [gs:]
> A beat-rate is simply the number of beats per second that occur in
a tempered consonance. A triad has synchronous beating when the
ratios between the beat rates for its constituent intervals can be
expressed with small integers.
>
> [oy:]
> Okay, ratios between the beat-rates... as in the ratio of the beat
rate of 5/4 to the beat rate of 3/2? In the case of 5/17-comma
temperament, the major third beats 4.14 times per second and the
fifth the same, so the ratio is 1/1. In the case of the 5/14-comma
temperament, the BRAT of the consonant major triad is 10/5=2. I think
I got it! The comma-fractions you gave yield multiples of beat-
ratios. Right?

Yep!

> [oy:]
> > ...
> > And what is the total error for this extra-ordinary temperament
of yours?
> [gs:]
> For the 12 major triads: ~386.5 cents (in cell H21); since this
isn't a well-temperament, that figure exceeds the theoretical minimum
of ~375.4 cents for a curculating 12-tone temperament.
>
> [oy:]
> That is because you have chosen to increase the error in the remote
keys in favor of the more accustomed keys. Correct?

It's because I've chosen to increase the total absolute error in
those triads over and above that of a pythagorean triad by using some
fifths wider than just.

> [gs:]
> > Anyway, I hope that some others will try this tuning and give me
> your reactions -- at least go through the triads in Scala using the
> chord playing function in the keyboard clavier. Convince me that
this isn't all a dream!
>
> And for those others who did: thank you for your feedback -- much
> appreciated.
>
> Best,
>
> --George
>
> [oy:]
> I have implemented your tuning on my Bechstein Imperial Grand just
yesterday. Being the first to tune an acoustic instrument to your new
temperament (my tuners almost demanded double the price for the extra
effort!), let me also be the first to say that I simply loved the
results. Good Work dear George!

I guess that's the ultimate compliment -- you've made my day!

Best,

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

9/6/2005 1:32:56 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> That's unfortunate, because for plenty of important scales and
> tunings, such as diminished, the identity-interval (the octave) is
> *not* one of the generators.

It should be kept in mind anyway that the whole question of generators
is really distinct from the question of identity intevals. It makes
perfectly good sense to use 25/24 and 128/125 as generators for
meantone, for instance.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/6/2005 2:00:41 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> > [gs:]
> > Hello, again! Sorry for my delay in replying -- I had to take some
> > time off yesterday due to an illness in my family. (And I have
> time today to reply to only a few things.)
> > [oy:]
> > It's quite alright dear George. I have delayed to reply myself.
>
> This has turned into even more of a delay -- I learned last Friday
> afternoon that my mother-in-law (87 years old) died the previous
> night,

My deepest condolences, George, and my best wishes to your family.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/6/2005 2:04:50 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > That's unfortunate, because for plenty of important scales and
> > tunings, such as diminished, the identity-interval (the octave) is
> > *not* one of the generators.
>
> It should be kept in mind anyway that the whole question of generators

Or periods!

> is really distinct from the question of identity intevals. It makes
> perfectly good sense to use 25/24 and 128/125 as generators for
> meantone, for instance.

Or, more immediately relevant to Western notation, 16;15 and 9;8 (or
identically, 16;15 and 10;9) -- this tells you that the minor second
and major second suffice as a basis from which one can construct any
meantone interval. I use semicolons because these ratios are not to be
taken in their JI sense but rather their meantone-tempered incarnations
are meant, something you might have wanted to make clear in your post.

🔗Ozan Yarman <ozanyarman@superonline.com>

9/6/2005 2:13:36 PM

Yes I have, but I'm not sure anymore about the accuracy of my synth after what George said.
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 06 Eylül 2005 Salı 22:46
Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone

>
> Now, George has formulated an excellent temperament which I have
>just yesterday implemented on my Bechstein Imperial Grand, and it
>sounds absolutely terrific.

Ozan, have you tried either of Bob Wendell's Synchronous Well
Temperaments?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/6/2005 2:35:57 PM

Well I'm glad this point is finally making an impact on you. You can
check your synth's accuracy at:

http://www.microtonal-synthesis.com/

Anyone -- Is Kyma listed here? If so, under what brand?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Yes I have, but I'm not sure anymore about the accuracy of my synth
after what George said.
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 06 Eylül 2005 Salý 22:46
> Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone
>
>
>
> >
> > Now, George has formulated an excellent temperament which I
have
> >just yesterday implemented on my Bechstein Imperial Grand, and
it
> >sounds absolutely terrific.
>
> Ozan, have you tried either of Bob Wendell's Synchronous Well
> Temperaments?

🔗Gene Ward Smith <gwsmith@svpal.org>

9/6/2005 3:40:53 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Or, more immediately relevant to Western notation, 16;15 and 9;8 (or
> identically, 16;15 and 10;9) -- this tells you that the minor second
> and major second suffice as a basis from which one can construct any
> meantone interval.

I was thinking of Eytan Agmon when I picked that pair, since 25;24 is
one step of 12-et and 0 steps of 7 et, and 128;125 is 0 steps of 12 et
and one step of 7-et. Your pair leads to the two vals <2 3 4| and
<5 8 12| (5-et.) 5 and 7 et together are then associated to 16;15 and
25;24, another reasonable pair of generators.

🔗Gene Ward Smith <gwsmith@svpal.org>

9/6/2005 6:42:48 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> I was thinking of Eytan Agmon when I picked that pair, since 25;24 is
> one step of 12-et and 0 steps of 7 et, and 128;125 is 0 steps of 12 et
> and one step of 7-et. Your pair leads to the two vals <2 3 4| and
> <5 8 12| (5-et.) 5 and 7 et together are then associated to 16;15 and
> 25;24, another reasonable pair of generators.

I've posted more on this here:

/tuning-math/message/12632

🔗monz <monz@tonalsoft.com>

9/6/2005 10:24:08 PM

Hi Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Thus, the Greek tunings are best
> > designed in Tonescape without an identity-interval.
>
> I think this is just silly because the Greeks used similar
> nomenclature for notes an octave apart, just as we do today.

Only for certain notes. For many others, the two different
octaves of what for us is one pitch-class, each had its
own totally unique symbol in both the "vocal" and the
"instrumental" notation.

As far as the "regular" nomenclature (_proslambanomenos_,
_mese_, etc.) -- no, there was no indication of octave
equivalence at all. The notes were named strictly
according to the periodicity of the 4/3 ratio.
Take a look at my page about the PIS:

http://tonalsoft.com/enc/p/pis.aspx

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Ozan Yarman <ozanyarman@superonline.com>

9/7/2005 9:22:51 AM

Graham, can you give me some concrete examples from some famous recent
tunings? Like, Blackjack, Mavila, Metameantone...

Cordially,
Ozan

----- Original Message -----
From: "Graham Breed" <gbreed@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 02 Eyl�l 2005 Cuma 19:53
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

> Ozan Yarman wrote:
> > Identity interval??? What happened to generator and period?
>
> The period is an equal division of the identity interval. Usually the
> identity interval is an octave. The period is the interval by which the
> tuning repeats, and the identity interval is a higher level concept.
>
>
> Graham
>
>

🔗Ozan Yarman <ozanyarman@superonline.com>

9/7/2005 9:33:08 AM

It is indeed making an impact on me since the beginning Paul, but I don't see Yamaha SW1000XG listed anywhere.
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 07 Eylül 2005 Çarşamba 0:35
Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone

Well I'm glad this point is finally making an impact on you. You can check your synth's accuracy at:

http://www.microtonal-synthesis.com/

Anyone -- Is Kyma listed here? If so, under what brand?

🔗Graham Breed <gbreed@gmail.com>

9/7/2005 1:22:12 PM

Ozan Yarman wrote:
> Graham, can you give me some concrete examples from some famous recent
> tunings? Like, Blackjack, Mavila, Metameantone...

Hello! Yes, well, all the examples in your motley collection have the octave as the period. An important one that doesn't is the diaschismic family, including that explained in Paul Erlich's 22 tone temperament paper. I explain them in my website, I think at

http://x31eq.com/diaschis.htm

The 10 note MOS is of the form

s s s s L s s s s L

in terms of small and large intervals. In 22-equal, the small interval is 2 steps and the large one is 3 steps. There's no way of defining this scale with the octave as a generator, because you lose half the notes. Hence the period has to be set at a half octave, but for convenience you can still think of the scale as being 10 notes to an octave.

Another example is the octatonic scale in 12-equal:

1 2 1 2 1 2 1 2

This repeats 4 times in an octave, and can be generalized to give a linear temperament with a period of a quarter octave.

My linear temperament searches threw up a lot of examples where the period is quite small. They are most useful in the higher limits, and so less famous because of the difficulties involved in making music with such beasties. My favourite is the one I call "mystery" which does 15-limit JI by dividing the octave into 29 equal parts, and using either 0 or 1 of the other generator (around 16 cents) to get to each 15-limit interval.

Looked at from another direction, meantone always divides the major third into two equal parts. If, for some strange reason, you decided that the major third was to be your identity interval, the whole tone would have to be your period. Some regular temperaments happen to equally divide an interval that wouldn't be so divisible in JI. If one of those intervals happens to be the octave, then the octave can't be the period.

Graham

🔗George D. Secor <gdsecor@yahoo.com>

9/7/2005 1:27:44 PM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
> On Monday 29 August 2005 7:34 pm, Ozan Yarman wrote:
> > ... Simply put, non-microtonal synths can perform much better with
> > George's tuning as default instead of the borin' ol' 12-tET in my
opinion.
> >
> > I for one wish to tune my piano to this temperament straight away.
>
> Ozan,
>
> It sounds to me like you would also rather like 1/7-comma meantone.
It has
> some very sweet builtin beat ratios, and is at a kind of 'sweet
spot' in the
> space of meantones: the fifths are large enough to make it a
circulating
> temperament, the largest fifth is a tiny bit larger than
Pythagorean, and
> very close to being 19/15, and it's just subversive enough in sound
to be not
> the same old 12-equal, but you can play music of any Western era in
it
> without fundamentally 'breaking it'!
>
> What I do is also distribute the wolf a bit between Ab-Eb and Eb-
Bb. The wolf
> in the case of 1/7-comma is really samll enough that putting it
between 2
> fifths makes it sound like a very very mild distortion of a well-
temperament,
> or rather, a very very mild temperament ordinaire.
>
> Margo and I were talking about 1/7-comma for about an hour on the
phone a few
> days ago....check it out!
>
> In the meantime, George's tuning does seem intriguing...
> ...
> Warmly,
> Aaron.

Aaron,

Curiousity got the better of me, so I decided to try my hand at a 1/7-
comma well-temperament (no wide 5ths) and came up with the following:

! Secor_WT1-7.scl
!
George Secor's 1/7-comma well-temperament
12
!
93.97984
197.75601
297.88984
395.51202
501.12200
593.26802
698.87800
795.93484
869.63401
999.88484
1094.39002
2/1

Here are some numbers to compare with Tom Dent't 5/32-comma well-
temperament (ref. msgs. 60165, 60248):

Total absolute error (cents):
Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Secor WT1-7: 42 36 29 25 25 25 25 25 28 34 41 43
Dent WT5-32: 36 35 31 27 23 23 27 31 35 35 37 37

Both temperaments have only 5 proportional-beating triads (F thru A
in mine, C thru E in Tom's). I haven't heard either of these yet,
but I intend to try them out in Scala (even if the accuracy isn't up
to exhibiting the proportional beating).

Considering that I was able to get 8 proportional-beating triads in
my #1 temperament (extra)ordinaire and 7 in each of two other (5/23
and 10/43-comma) temperaments ordinaire, I'm now more inclined to
believe that Robert Wendell's equal-beating well-temperaments were
indeed more difficult to construct (than temperaments ordinaires).

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> I'm under the impression that George is unfamiliar with a lot of
the
> work on equal-beating well-temperaments that Robert Wendell, you,
and
> others have done. So someone should start introducing him to it.

You're right about that, Paul. Now that I've had a chance to
experiment with a less-deviant-from-12-equal temperament, I'll have a
greater appreciation for them.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

9/7/2005 3:42:17 PM

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Ozan Yarman wrote:

> My linear temperament searches threw up a lot of examples where the
> period is quite small. They are most useful in the higher limits, and
> so less famous because of the difficulties involved in making music
with
> such beasties. My favourite is the one I call "mystery" which does
> 15-limit JI by dividing the octave into 29 equal parts, and using
either
> 0 or 1 of the other generator (around 16 cents) to get to each 15-limit
> interval.

Mystery is quite interesting if you are interested in higher limits,
and might be looked at from the point of view of someone interested in
maqams. I'll put in a plug for another temperament, ennealimmal, which
divides the octave into nine parts, and which by the time you get up
to mystery size, meaning 54 or 63 notes, is able to give a large
amount of harmony which has the interesting property of being sensibly
JI. Splitting the period in half, to 18 rather than 9 parts to an
octave allows something similar for the 11 limit rather than the 7 limit.

🔗George D. Secor <gdsecor@yahoo.com>

9/8/2005 11:26:00 AM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> ...
> Curiousity got the better of me, so I decided to try my hand at a
1/7-
> comma well-temperament (no wide 5ths) and came up with the
following:
> ...

Oops! I tried it last night in Scala and found that the "A" was way
off (reversed a couple of digits) -- here's the corrected file:

! Secor_WT1-7.scl
!
George Secor's 1/7-comma well-temperament
12
!
93.97984
197.75601
297.88984
395.51202
501.12200
593.26802
698.87800
795.93484
896.63401
999.88484
1094.39002
2/1

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/8/2005 2:03:05 PM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> Hi Paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > Thus, the Greek tunings are best
> > > designed in Tonescape without an identity-interval.
> >
> > I think this is just silly because the Greeks used similar
> > nomenclature for notes an octave apart, just as we do today.
>
>
>
> Only for certain notes. For many others, the two different
> octaves of what for us is one pitch-class, each had its
> own totally unique symbol in both the "vocal" and the
> "instrumental" notation.
>
> As far as the "regular" nomenclature (_proslambanomenos_,
> _mese_, etc.) -- no, there was no indication of octave
> equivalence at all.

Oops! Thanks for correcting me, then.

> The notes were named strictly
> according to the periodicity of the 4/3 ratio.
> Take a look at my page about the PIS:
>
> http://tonalsoft.com/enc/p/pis.aspx

I don't see it. How does this nomenclature fall "strictly according
to the periodicity of the 4/3 ratio?" It seems that for the two upper
tetrachords, the interval at which the names repeat can be either 4:3
or 3:2, but neither of these intervals is repeated in such a
capacity; and across the 'mese', there doesn't seem to be any
indication of equivalence at all. Am I missing something?

🔗Richard Eldon Barber <bassooner42@yahoo.com>

9/8/2005 3:02:25 PM

Kyma is under Symbolic Sound. Although I wouldn't call Kyma the
synth, rather its external engine, the Capybara 320.

Anyone-- feel free to buy me one.

-Rick

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> Well I'm glad this point is finally making an impact on you. You can
> check your synth's accuracy at:
>
> http://www.microtonal-synthesis.com/
>
> Anyone -- Is Kyma listed here? If so, under what brand?
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> > Yes I have, but I'm not sure anymore about the accuracy of my synth
> after what George said.
> > ----- Original Message -----
> > From: wallyesterpaulrus
> > To: tuning@yahoogroups.com
> > Sent: 06 Eylül 2005 Salý 22:46
> > Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone
> >
> >
> >
> > >
> > > Now, George has formulated an excellent temperament which I
> have
> > >just yesterday implemented on my Bechstein Imperial Grand, and
> it
> > >sounds absolutely terrific.
> >
> > Ozan, have you tried either of Bob Wendell's Synchronous Well
> > Temperaments?

🔗Ozan Yarman <ozanyarman@superonline.com>

9/10/2005 3:02:44 AM

George,

This has turned into even more of a delay -- I learned last Friday
afternoon that my mother-in-law (87 years old) died the previous
night, and I have had almost no time until today to catch up with the
discussion.

My condolences. May she rest in peace.

>
> [oy:]
> Now that is fascinating! But how does one go to a temperament where
the octave closes with equal-beating triads after, say, 30-40 tones?

That's something I've never tried. The first thing I would do is add
more rows to my spreadsheet, then take it from there. You would have
to change some of the formulas for a 41-tone circulating system,
since you'd no longer be dealing mainly with narrow 5ths.

One could require as much as 70-80 rows there.

However, I have done it with part of a circle of 19. The 5/17-comma
5ths are not exactly 1:1:1 (5th:M3:m3) equal-beating (would need to
be 695.6304 cents), so I'll have to recalculate these figures when I
have more time:

! secor_19wt.scl
!
George Secor's 19-tone well temperament with ten 5/17-comma fifths
19
!
69.40735
131.54493
191.25924
260.66659
317.95765
382.51849
451.92584
504.37038
573.77773
638.33856
695.62962
765.03697
824.57129
886.88886
956.29622
1011.16402
1078.14811
1145.13220
2/1

Neat! Many maqams fit into this scheme.

With C as 1/1, the 6:7:9 triads on F, C, G, and D are greatly
improved over 19-ET, due to 7:9 having very low error. I came up
with the essential idea for this in 1978.

Eventually, I have myself seen that equal temperaments provide only close approximations to the intended to tuning.

> [oy:]
> > ...
> > And what is the total error for this extra-ordinary temperament
of yours?
> [gs:]
> For the 12 major triads: ~386.5 cents (in cell H21); since this
isn't a well-temperament, that figure exceeds the theoretical minimum
of ~375.4 cents for a curculating 12-tone temperament.

How exactly does this 375 cents accumulate? And how does it become a theoretical minimum?

>
> [oy:]
> I have implemented your tuning on my Bechstein Imperial Grand just
yesterday. Being the first to tune an acoustic instrument to your new
temperament (my tuners almost demanded double the price for the extra
effort!), let me also be the first to say that I simply loved the
results. Good Work dear George!

I guess that's the ultimate compliment -- you've made my day!

Best,

--George

The pleasure is all mine.

Cordially,
Ozan

🔗Ozan Yarman <ozanyarman@superonline.com>

9/10/2005 6:30:09 AM

Hello Graham!

----- Original Message -----
From: "Graham Breed" <gbreed@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 07 Eyl�l 2005 �ar�amba 23:22
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1
>
> Hello! Yes, well, all the examples in your motley collection have the
> octave as the period. An important one that doesn't is the diaschismic
> family, including that explained in Paul Erlich's 22 tone temperament
> paper. I explain them in my website, I think at
>
> http://x31eq.com/diaschis.htm
>
> The 10 note MOS is of the form
>
> s s s s L s s s s L
>
> in terms of small and large intervals. In 22-equal, the small interval
> is 2 steps and the large one is 3 steps. There's no way of defining
> this scale with the octave as a generator, because you lose half the
> notes. Hence the period has to be set at a half octave, but for
> convenience you can still think of the scale as being 10 notes to an
octave.
>

So the identity interval is the octave as you said previously. Accordinly,
the generator interval is the period?

> Another example is the octatonic scale in 12-equal:
>
> 1 2 1 2 1 2 1 2
>
> This repeats 4 times in an octave, and can be generalized to give a
> linear temperament with a period of a quarter octave.
>

I remember a famous Lex Luther melody from Superman series based on this
very mode.

>
> My linear temperament searches threw up a lot of examples where the
> period is quite small. They are most useful in the higher limits, and
> so less famous because of the difficulties involved in making music with
> such beasties. My favourite is the one I call "mystery" which does
> 15-limit JI by dividing the octave into 29 equal parts, and using either
> 0 or 1 of the other generator (around 16 cents) to get to each 15-limit
> interval.
>

I was once interested in 29-EDO. But the more I investigated, the more I
discovered the need for higher cardinalities in Maqam Music.

>
> Looked at from another direction, meantone always divides the major
> third into two equal parts.

That had escaped me since now. Interesting!

> If, for some strange reason, you decided
> that the major third was to be your identity interval, the whole tone
> would have to be your period.

It's beginning to sink in ever so slowly.

> Some regular temperaments happen to
> equally divide an interval that wouldn't be so divisible in JI. If one
> of those intervals happens to be the octave, then the octave can't be
> the period.

Unless the identity interval is 4/1?

>
>
> Graham
>
>
>

Cordially,
Ozan

🔗Graham Breed <gbreed@gmail.com>

9/10/2005 7:28:04 AM

> Hello Graham!

Hiya Ozan!

> So the identity interval is the octave as you said previously. Accordinly,
> the generator interval is the period?

The generator interval is the generator. The period is a special kind of generator. You need two generators to define a two-dimensional scale (or linear temperament, where the period-generator becomes invisible)

>>Another example is the octatonic scale in 12-equal:
>>
>>1 2 1 2 1 2 1 2
>>
>>This repeats 4 times in an octave, and can be generalized to give a
>>linear temperament with a period of a quarter octave.
> > I remember a famous Lex Luther melody from Superman series based on this
> very mode.

Could be, it gets around!

> I was once interested in 29-EDO. But the more I investigated, the more I
> discovered the need for higher cardinalities in Maqam Music.

Did you check 58? It's the next step in the mystery chain. Good for 15-limit harmony, may or may not be what you want for Maqam music.

>>Some regular temperaments happen to
>>equally divide an interval that wouldn't be so divisible in JI. If one
>>of those intervals happens to be the octave, then the octave can't be
>>the period.
> > Unless the identity interval is 4/1?

Yes, you can always define it that way, but most people would be confused by an identity interval that isn't the octave. If a scale doesn't have an octave, this is unavoidable. Otherwise, we usually divide the octave to get a new period. It's a matter of convenience. That is, it's *supposed* to make it easier to understand.

Graham

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/12/2005 1:36:31 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Hello Graham!
>
> ----- Original Message -----
> From: "Graham Breed" <gbreed@g...>
> To: <tuning@yahoogroups.com>
> Sent: 07 Eylül 2005 Çarþamba 23:22
> Subject: Re: [tuning] Re: MOS at folding points other than the 2:1
> >
> > Hello! Yes, well, all the examples in your motley collection
have the
> > octave as the period. An important one that doesn't is the
diaschismic
> > family, including that explained in Paul Erlich's 22 tone
temperament
> > paper. I explain them in my website, I think at
> >
> > http://x31eq.com/diaschis.htm
> >
> > The 10 note MOS is of the form
> >
> > s s s s L s s s s L
> >
> > in terms of small and large intervals. In 22-equal, the small
interval
> > is 2 steps and the large one is 3 steps. There's no way of
defining
> > this scale with the octave as a generator, because you lose half
the
> > notes. Hence the period has to be set at a half octave, but for
> > convenience you can still think of the scale as being 10 notes to
an
> octave.
> >
>
>
>
> So the identity interval is the octave as you said previously.
Accordinly,
> the generator interval is the period?

If I may jump in . . .

The generator is not the period. In the case of the diatonic scale,
the generator is the fifth (or fourth) and the period is the octave.
In the case of the scale Graham refers to above:

s s s s L s s s s L

the period is a half-octave, or 4*s+L; the generator can be expressed
as s, or as 4s (or as 5s+L or as 8s+L). Start with 1 note per period
and then construct a chain of generators from each starting note.
After iterating the generation four times, you will obtain the above
as the arrangement of notes within each octave span.

> > Another example is the octatonic scale in 12-equal:
> >
> > 1 2 1 2 1 2 1 2
> >
> > This repeats 4 times in an octave, and can be generalized to give
a
> > linear temperament with a period of a quarter octave.
> >
>
>
> I remember a famous Lex Luther melody from Superman series based on
this
> very mode.

The generator here is 1 or 2 (or equivalently, any integer not
divisible by 3) in steps of 12-equal.

> > My linear temperament searches threw up a lot of examples where
the
> > period is quite small. They are most useful in the higher
limits, and
> > so less famous because of the difficulties involved in making
music with
> > such beasties. My favourite is the one I call "mystery" which
does
> > 15-limit JI by dividing the octave into 29 equal parts, and using
either
> > 0 or 1 of the other generator (around 16 cents) to get to each 15-
limit
> > interval.
> >
>
>
> I was once interested in 29-EDO. But the more I investigated, the
more I
> discovered the need for higher cardinalities in Maqam Music.

Have you considered this system of Graham's though, with a generator
of around 16 cents?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/12/2005 1:40:01 PM

Jumping in again (sorry . . .)

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> >>Some regular temperaments happen to
> >>equally divide an interval that wouldn't be so divisible in JI. If
one
> >>of those intervals happens to be the octave, then the octave can't
be
> >>the period.
> >
> > Unless the identity interval is 4/1?
>
> Yes, you can always define it that way,

It doesn't seem to me that that would help -- the octave still can't be
the period, no matter what the identity interval, if one of the
tempered JI intervals happens to be the octave.

> Otherwise, we usually
> divide the octave to get a new period.

Right -- which wouldn't be 4/1, but rather would occur *twice as many
times* in the 4/1 as it does in the octave.

🔗Graham Breed <gbreed@gmail.com>

9/13/2005 2:34:22 AM

wallyesterpaulrus wrote:

> It doesn't seem to me that that would help -- the octave still can't be > the period, no matter what the identity interval, if one of the > tempered JI intervals happens to be the octave.

Oh, yes, I clearly didn't understand the question there...

Graham

🔗George D. Secor <gdsecor@yahoo.com>

9/13/2005 11:58:38 AM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> [gs:]
> However, I have done it with part of a circle of 19. The 5/17-
comma
> 5ths are not exactly 1:1:1 (5th:M3:m3) equal-beating (would need
to
> be 695.6304 cents), so I'll have to recalculate these figures
when I
> have more time:
>
> ! secor_19wt.scl
> !
> George Secor's 19-tone well temperament with ten 5/17-comma fifths
> 19
> !
> 69.40735
> 131.54493
> 191.25924
> 260.66659
> 317.95765
> 382.51849
> 451.92584
> 504.37038
> 573.77773
> 638.33856
> 695.62962
> 765.03697
> 824.57129
> 886.88886
> 956.29622
> 1011.16402
> 1078.14811
> 1145.13220
> 2/1

> [oy:]
> Neat! Many maqams fit into this scheme.

> With C as 1/1, the 6:7:9 triads on F, C, G, and D are greatly
> improved over 19-ET, due to 7:9 having very low error. I came up
> with the essential idea for this in 1978.
>
> Eventually, I have myself seen that equal temperaments provide only
close approximations to the intended to tuning.

I find that 19-ET leaves much to be desired when it comes to the 7ths
(all types). The diatonic major 7th is rather low, so is not the
most effective leading tone. The minor 7th is rather high, so does
not contribute to a very effective (diatonic) dominant 7th chord.
The augmented 6th is rather low, so does not contribute to a very
effective harmonic 7th chord. This well-temperament does much to
alleviate these problems, in a few keys at least.

Breaking news: I've not only recalculated the 5/17-comma fifths --
I've also developed a 19-tone well-temperament with 15 proportional-
beating major triads.

There are, at the moment, 2 versions. The first one has the
proportional-beating triads in the sharp direction:

! secor_19wt1.scl
!
George Secor's 19-tone proportional-beating (5/17-comma) well
temperament (v.1)
19
!
69.41306
131.21719
191.26088
260.67394
318.03803
382.52175
451.93481
504.36956
573.78263
638.12678
695.63044
765.04350
824.94573
886.89131
956.30438
1011.12652
1078.15219
1145.03265
2/1

Rather than put out an entire spreadsheet with all the figures, I'll
summarize with the following:

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
Db... 2.6627 0.6693 1.7822 (not proportional)
Ab... 2.1159 1.4839 3.1399 (not proportional)
Eb... 1.6394 24.486 40.143 (not proportional)
Bb... 1.1830 1.3784 1.6307 (not proportional)
F.... 5.0000 10.000 2.0000
C.... 1.0000 1.0000 1.0000
G.... 1.0000 1.0000 1.0000
D.... 1.0000 1.0000 1.0000
A.... 1.0000 1.0000 1.0000
E.... 1.0000 1.0000 1.0000
B.... 1.0000 1.0000 1.0000
F#... 1.0000 1.0000 1.0000
C#... 1.0000 1.0000 1.0000
G#... 1.6667 ------ ------
D#... 2.3333 1.0000 2.3333
A#... 3.0000 0.5000 1.5000
E#/Fb 2.5000 0.8000 2.0000
B#/Cb 2.5000 0.8000 2.0000
Fx/Gb 2.5000 0.8000 2.0000
Total triad error: 330.864 cents

The second version was a little trickier to construct; it has 3 of
the propotional-beating triads in the flat direction:

! secor_19wt2.scl
!
George Secor's 19-tone proportional-beating (5/17-comma) well
temperament (v.2)
19
!
69.41306
131.21719
191.26088
260.67394
317.43304
382.52175
451.93481
504.36956
573.78263
638.12678
695.63044
765.04350
824.40909
886.89131
956.30438
1010.63200
1078.15219
1145.03265
2/1

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
Db... 2.5000 0.8000 2.0000
Ab... 2.0000 2.0000 4.0000
Eb... 1.5452 5.4877 8.4796 (not proportional)
Bb... 1.1538 1.3000 1.5000
F.... 5.0000 10.000 2.0000
C.... 1.0000 1.0000 1.0000
G.... 1.0000 1.0000 1.0000
D.... 1.0000 1.0000 1.0000
A.... 1.0000 1.0000 1.0000
E.... 1.0000 1.0000 1.0000
B.... 1.0000 1.0000 1.0000
F#... 1.0000 1.0000 1.0000
C#... 1.0000 1.0000 1.0000
G#... 1.6667 ------ ------
D#... 2.3333 1.0000 2.3333
A#... 3.0000 0.5000 1.5000
E#/Fb 2.6004 0.7139 1.8566 (not proportional)
B#/Cb 2.6132 0.7043 1.8406 (not proportional)
Fx/Gb 2.5925 0.7201 1.8668 (not proportional)
Total triad error: 331.853 cents

The two versions differ in the tuning of only 3 tones, with the
differences on the order of ~1/2-cent each. I haven't listened to
them yet to see if I can tell any difference -- or whether either of
these is really worth the trouble vs. the original temperament quoted
at the top of this message (which has 5ths of only 2 different
sizes). :-)

The total triad error, BTW, is the sum of the total absolute error of
all 19 major triads, which is smaller than that of the 12 major
triads of 12-ET.

The rest of this discussion pertains to my 12-tone temperament (extra)
ordinaire:

> ...
> > [oy:]
> > > ...
> > > And what is the total error for this extra-ordinary
temperament of yours?
> > [gs:]
> > For the 12 major triads: ~386.5 cents (in cell H21); since this
> isn't a well-temperament, that figure exceeds the theoretical
minimum
> of ~375.4 cents for a curculating 12-tone temperament.
>
> How exactly does this 375 cents accumulate? And how does it become
a theoretical minimum?

Take the sum of the total absolute errors of all 12 triads. The
theoretical minimum is the same as the sum-total for 12-ET, and you
won't exceed this figure if you have no 5ths or minor 3rds wider than
just and no major 3rds narrower than just.

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

9/13/2005 12:16:06 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

Oops!!! The beat ratios for F major triad in the tables for both
versions should be:

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
F.... 1.0000 1.0000 1.0000

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

9/16/2005 2:22:00 PM

> Hiya Ozan!
>

Hello again.

> The generator interval is the generator. The period is a special kind
> of generator. You need two generators to define a two-dimensional scale
> (or linear temperament, where the period-generator becomes invisible)
>

And it becomes visible in rank 3 temperaments?

> Did you check 58? It's the next step in the mystery chain. Good for
> 15-limit harmony, may or may not be what you want for Maqam music.
>

In 58, the cycle of fifths produce super-pythagorean intervals, which do not
very well suit my needs. My default is a meantone-pythagorean hybrid that
closes the cycle in between 31 or 55 jumps.

> Unless the identity interval is 4/1?
>
> Yes, you can always define it that way, but most people would be
> confused by an identity interval that isn't the octave. If a scale
> doesn't have an octave, this is unavoidable. Otherwise, we usually
> divide the octave to get a new period. It's a matter of convenience.
> That is, it's *supposed* to make it easier to understand.
>

So, the `period` AKA the `interval of periodicity` is the interval where the
scale repeats itself using the exact same intervals. This generates the
`interval of equivalance`, as in the `octave`, or else, is squeezed inside
it any number of times.

>
> Graham
>
>

Cordially,
Ozan

🔗Ozan Yarman <ozanyarman@superonline.com>

9/17/2005 3:23:09 PM

Paul, I did not find the oppurtunity to thank you for your informative post. Can you point to me a scala file for Graham's tuning where 16 cents is the generator?

Cordially,
Ozan

------------------------

> So the identity interval is the octave as you said previously.Accordinly,the generator interval is the period?

If I may jump in . . .

The generator is not the period. In the case of the diatonic scale,
the generator is the fifth (or fourth) and the period is the octave.
In the case of the scale Graham refers to above:

s s s s L s s s s L

the period is a half-octave, or 4*s+L; the generator can be expressed as s, or as 4s (or as 5s+L or as 8s+L). Start with 1 note per period and then construct a chain of generators from each starting note. After iterating the generation four times, you will obtain the above as the arrangement of notes within each octave span.

> I was once interested in 29-EDO. But the more I investigated, the
more I discovered the need for higher cardinalities in Maqam Music.

Have you considered this system of Graham's though, with a generator
of around 16 cents?

🔗Ozan Yarman <ozanyarman@superonline.com>

9/17/2005 3:39:00 PM

Dear George,
----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 13 Eylül 2005 Salı 21:58
Subject: [tuning] 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)
SNIP

I find that 19-ET leaves much to be desired when it comes to the 7ths (all types). The diatonic major 7th is rather low, so is not the most effective leading tone. The minor 7th is rather high, so does not contribute to a very effective (diatonic) dominant 7th chord. The augmented 6th is rather low, so does not contribute to a very effective harmonic 7th chord. This well-temperament does much to alleviate these problems, in a few keys at least.

Your recent endeavour in equal-beating temperaments is truly commendable!

Breaking news: I've not only recalculated the 5/17-comma fifths --
I've also developed a 19-tone well-temperament with 15 proportional-
beating major triads.

There are, at the moment, 2 versions. The first one has the
proportional-beating triads in the sharp direction:

! secor_19wt1.scl
!
George Secor's 19-tone proportional-beating (5/17-comma) well
temperament (v.1)
19
!
69.41306
131.21719
191.26088
260.67394
318.03803
382.52175
451.93481
504.36956
573.78263
638.12678
695.63044
765.04350
824.94573
886.89131
956.30438
1011.12652
1078.15219
1145.03265
2/1

Rather than put out an entire spreadsheet with all the figures, I'll summarize with the following:

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
Db... 2.6627 0.6693 1.7822 (not proportional)
Ab... 2.1159 1.4839 3.1399 (not proportional)
Eb... 1.6394 24.486 40.143 (not proportional)
Bb... 1.1830 1.3784 1.6307 (not proportional)
F.... 1.0000 1.0000 1.0000
C.... 1.0000 1.0000 1.0000
G.... 1.0000 1.0000 1.0000
D.... 1.0000 1.0000 1.0000
A.... 1.0000 1.0000 1.0000
E.... 1.0000 1.0000 1.0000
B.... 1.0000 1.0000 1.0000
F#... 1.0000 1.0000 1.0000
C#... 1.0000 1.0000 1.0000
G#... 1.6667 ------ ------
D#... 2.3333 1.0000 2.3333
A#... 3.0000 0.5000 1.5000
E#/Fb 2.5000 0.8000 2.0000
B#/Cb 2.5000 0.8000 2.0000
Fx/Gb 2.5000 0.8000 2.0000
Total triad error: 330.864 cents

This is pretty good.

The second version was a little trickier to construct; it has 3 of
the propotional-beating triads in the flat direction:

! secor_19wt2.scl
!
George Secor's 19-tone proportional-beating (5/17-comma) well
temperament (v.2)
19
!
69.41306
131.21719
191.26088
260.67394
317.43304
382.52175
451.93481
504.36956
573.78263
638.12678
695.63044
765.04350
824.40909
886.89131
956.30438
1010.63200
1078.15219
1145.03265
2/1

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
Db... 2.5000 0.8000 2.0000
Ab... 2.0000 2.0000 4.0000
Eb... 1.5452 5.4877 8.4796 (not proportional)
Bb... 1.1538 1.3000 1.5000
F.... 1.0000 1.0000 1.0000
C.... 1.0000 1.0000 1.0000
G.... 1.0000 1.0000 1.0000
D.... 1.0000 1.0000 1.0000
A.... 1.0000 1.0000 1.0000
E.... 1.0000 1.0000 1.0000
B.... 1.0000 1.0000 1.0000
F#... 1.0000 1.0000 1.0000
C#... 1.0000 1.0000 1.0000
G#... 1.6667 ------ ------
D#... 2.3333 1.0000 2.3333
A#... 3.0000 0.5000 1.5000
E#/Fb 2.6004 0.7139 1.8566 (not proportional)
B#/Cb 2.6132 0.7043 1.8406 (not proportional)
Fx/Gb 2.5925 0.7201 1.8668 (not proportional)
Total triad error: 331.853 cents

Ditto.

The two versions differ in the tuning of only 3 tones, with the
differences on the order of ~1/2-cent each. I haven't listened to
them yet to see if I can tell any difference -- or whether either of
these is really worth the trouble vs. the original temperament quoted
at the top of this message (which has 5ths of only 2 different
sizes). :-)

The total triad error, BTW, is the sum of the total absolute error of
all 19 major triads, which is smaller than that of the 12 major
triads of 12-ET.

Exceptionally so! I assume that the total error in 12-tET is 12*(400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the minor thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum total error for the triads. Correct?

The rest of this discussion pertains to my 12-tone temperament (extra)ordinaire:

> How exactly does this 375 cents accumulate? And how does it become a theoretical minimum?

Take the sum of the total absolute errors of all 12 triads. The
theoretical minimum is the same as the sum-total for 12-ET, and you won't exceed this figure if you have no 5ths or minor 3rds wider than just and no major 3rds narrower than just.

Best,

--George

Cordially,
Ozan

🔗George D. Secor <gdsecor@yahoo.com>

9/19/2005 10:57:18 AM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
> ----- Original Message -----
> From: George D. Secor
> To: tuning@yahoogroups.com
> Sent: 13 Eylül 2005 Salý 21:58
> Subject: [tuning] 19-tone Well-temperament (Was: Temperament
(Extra)Ordinaire ...)
> SNIP
> [gs:]
> I find that 19-ET leaves much to be desired when it comes to the
7ths (all types). The diatonic major 7th is rather low, so is not
the most effective leading tone. The minor 7th is rather high, so
does not contribute to a very effective (diatonic) dominant 7th
chord. The augmented 6th is rather low, so does not contribute to a
very effective harmonic 7th chord. This well-temperament does much
to alleviate these problems, in a few keys at least.
>
> [oy:]
> Your recent endeavour in equal-beating temperaments is truly
commendable!
>
> [gs:]
> Breaking news: I've not only recalculated the 5/17-comma fifths -
-
> I've also developed a 19-tone well-temperament with 15
proportional-
> beating major triads.
>
> There are, at the moment, 2 versions. The first one has the
> proportional-beating triads in the sharp direction:
>
> ! secor_19wt1.scl
> !
> George Secor's 19-tone proportional-beating (5/17-comma) well
> temperament (v.1)
> 19
> !
> 69.41306
> 131.21719
> 191.26088
> 260.67394
> 318.03803
> 382.52175
> 451.93481
> 504.36956
> 573.78263
> 638.12678
> 695.63044
> 765.04350
> 824.94573
> 886.89131
> 956.30438
> 1011.12652
> 1078.15219
> 1145.03265
> 2/1
>
> Rather than put out an entire spreadsheet with all the figures,
I'll summarize with the following:
>
> Major ----Beat Ratios----
> Triad M3/5th 5th/m3 M3/m3
> ----- ------ ------ ------
> Db... 2.6627 0.6693 1.7822 (not proportional)
> Ab... 2.1159 1.4839 3.1399 (not proportional)
> Eb... 1.6394 24.486 40.143 (not proportional)
> Bb... 1.1830 1.3784 1.6307 (not proportional)
> F.... 1.0000 1.0000 1.0000
> C.... 1.0000 1.0000 1.0000
> G.... 1.0000 1.0000 1.0000
> D.... 1.0000 1.0000 1.0000
> A.... 1.0000 1.0000 1.0000
> E.... 1.0000 1.0000 1.0000
> B.... 1.0000 1.0000 1.0000
> F#... 1.0000 1.0000 1.0000
> C#... 1.0000 1.0000 1.0000
> G#... 1.6667 ------ ------
> D#... 2.3333 1.0000 2.3333
> A#... 3.0000 0.5000 1.5000
> E#/Fb 2.5000 0.8000 2.0000
> B#/Cb 2.5000 0.8000 2.0000
> Fx/Gb 2.5000 0.8000 2.0000
> Total triad error: 330.864 cents
>
>
> [oy:]
> This is pretty good.
>
> [gs:]
> The second version was a little trickier to construct; it has 3
of
> the propotional-beating triads in the flat direction:
>
> ! secor_19wt2.scl
>
> (SNIP!)
>
> Total triad error: 331.853 cents
>
> [oy:]
> Ditto.
>
> [gs:]
> The two versions differ in the tuning of only 3 tones, with the
> differences on the order of ~1/2-cent each. I haven't listened
to
> them yet to see if I can tell any difference -- or whether either
of
> these is really worth the trouble vs. the original temperament
quoted
> at the top of this message (which has 5ths of only 2 different
> sizes). :-)

Now that I've listened to them, I find that I really can't hear any
appreciable difference. On the basis of the numbers I've decided
that I'll advocate version 1 (for several reasons, but chiefly
because it has the lowest total error in the 19 triads).

> The total triad error, BTW, is the sum of the total absolute
error of
> all 19 major triads, which is smaller than that of the 12 major
> triads of 12-ET.
>
> [oy:]
> Exceptionally so! I assume that the total error in 12-tET is 12*
(400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds
PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the minor
thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum total
error for the triads. Correct?

You forgot to add in the total error for the 5ths: 12*(701.9550 -
700) = 23.4600, which gives a total absolute error of 375.3909 cents
for 12 triads.

Best,

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

9/19/2005 11:40:54 AM

> Exceptionally so! I assume that the total error in 12-tET is 12*
(400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds
PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the minor thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum total
error for the triads. Correct?

(GS)
You forgot to add in the total error for the 5ths: 12*(701.9550 -
700) = 23.4600, which gives a total absolute error of 375.3909 cents for 12 triads.

-------

Ouch! You are right. But I did figure out the correct calculation, did I not?

--------

Best,

--George

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/19/2005 12:23:04 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> So, the `period` AKA the `interval of periodicity` is the interval
where the
> scale repeats itself using the exact same intervals.

Yes.

> This generates the
> `interval of equivalance`, as in the `octave`, or else, is squeezed
inside
> it any number of times.

The period normally generates the interval of equivalence, whether in
1 step (when the two are the same) or in any whole number N steps
(when one has to iterate the period N times in order to generate the
interval of equivalence. And the interval of equivalence is normally
taken as pretty much given a priori (some say pan-culturally) and
understood to be the "octave" or (~)2:1 ratio, though experiments
with 3:1 as the interval of equivalence certainly yield musically
intriguing, if not convincing, results.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/19/2005 1:02:58 PM

I don't know of such a Scala file -- and Graham should provide one if
he hasn't already -- but the basic idea is to take 29-equal, and
superimpose another 29-equal set about 16 cents higher (and in theory
another 29-equal set about 16 cents higher than that, and so on) --
this 58-note scale, I believe, is one of the most efficient 2D
systems for acheiving lots of higher-limit harmonies.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, I did not find the oppurtunity to thank you for your
informative post. Can you point to me a scala file for Graham's
tuning where 16 cents is the generator?
>
> Cordially,
> Ozan
>
> ------------------------
>
>
> > So the identity interval is the octave as you said
previously.Accordinly,the generator interval is the period?
>
>
> If I may jump in . . .
>
> The generator is not the period. In the case of the diatonic scale,
> the generator is the fifth (or fourth) and the period is the
octave.
> In the case of the scale Graham refers to above:
>
> s s s s L s s s s L
>
> the period is a half-octave, or 4*s+L; the generator can be
expressed as s, or as 4s (or as 5s+L or as 8s+L). Start with 1 note
per period and then construct a chain of generators from each
starting note. After iterating the generation four times, you will
obtain the above as the arrangement of notes within each octave span.
>
>
> > I was once interested in 29-EDO. But the more I investigated, the
> more I discovered the need for higher cardinalities in Maqam Music.
>
> Have you considered this system of Graham's though, with a
generator
> of around 16 cents?

🔗Graham Breed <gbreed@gmail.com>

9/19/2005 2:10:22 PM

wallyesterpaulrus wrote:
> I don't know of such a Scala file -- and Graham should provide one if > he hasn't already -- but the basic idea is to take 29-equal, and > superimpose another 29-equal set about 16 cents higher (and in theory > another 29-equal set about 16 cents higher than that, and so on) -- > this 58-note scale, I believe, is one of the most efficient 2D > systems for acheiving lots of higher-limit harmonies.

This is completely untested...

!mystery58.scl
Mystery temperament with 16 cent generator
58
!
16.000
41.379
57.379
82.759
98.759
124.138
140.138
165.517
181.517
206.897
222.897
248.276
264.276
289.655
305.655
331.034
347.034
372.414
388.414
413.793
429.793
455.172
471.172
496.552
512.552
537.931
553.931
579.310
595.310
620.690
636.690
662.069
678.069
703.448
719.448
744.828
760.828
786.207
802.207
827.586
843.586
868.966
884.966
910.345
926.345
951.724
967.724
993.103
1009.103
1034.483
1050.483
1075.862
1091.862
1117.241
1133.241
1158.621
1174.621
1200.000

Graham

🔗George D. Secor <gdsecor@yahoo.com>

9/19/2005 2:36:26 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> > Exceptionally so! I assume that the total error in 12-tET is 12*
> (400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds
> PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the
minor thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum
total
> error for the triads. Correct?
>
> (GS)
> You forgot to add in the total error for the 5ths: 12*(701.9550 -
> 700) = 23.4600, which gives a total absolute error of 375.3909
cents for 12 triads.
>
> -------
> (OY)
> Ouch! You are right. But I did figure out the correct calculation,
did I not?

Apart from the omission of the 5th, your calculation was correct.

Best,

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

9/20/2005 9:09:33 AM

So the interval of equivalence differing from the period is only the result of the tuning jargon developed by the pioneers of microtonality and the efforts of the explorers on this list?

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 19 Eylül 2005 Pazartesi 22:23
Subject: [tuning] Re: MOS at folding points other than the 2:1

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> So, the `period` AKA the `interval of periodicity` is the interval
where the
> scale repeats itself using the exact same intervals.

Yes.

> This generates the
> `interval of equivalance`, as in the `octave`, or else, is squeezed
inside
> it any number of times.

The period normally generates the interval of equivalence, whether in
1 step (when the two are the same) or in any whole number N steps
(when one has to iterate the period N times in order to generate the
interval of equivalence. And the interval of equivalence is normally
taken as pretty much given a priori (some say pan-culturally) and
understood to be the "octave" or (~)2:1 ratio, though experiments
with 3:1 as the interval of equivalence certainly yield musically
intriguing, if not convincing, results.

🔗Ozan Yarman <ozanyarman@superonline.com>

9/20/2005 12:17:26 PM

Paul, I thought you had said to me that 29-tET was unsuitable for Western polyphony. What's so special about this particular 58?
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 19 Eylül 2005 Pazartesi 23:02
Subject: [tuning] Re: MOS at folding points other than the 2:1

I don't know of such a Scala file -- and Graham should provide one if he hasn't already -- but the basic idea is to take 29-equal, and superimpose another 29-equal set about 16 cents higher (and in theory another 29-equal set about 16 cents higher than that, and so on) -- this 58-note scale, I believe, is one of the most efficient 2D systems for acheiving lots of higher-limit harmonies.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/21/2005 12:28:44 PM

I wouldn't say that. There are plenty of examples in Western music,
from Rimsky-Korsakov to Miles Davis, of scales like 3-1-3-1-3-1 and 1-
2-1-2-1-2-1-2, where the interval of equivalence is 2 or 3 or 4 times
the period.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> So the interval of equivalence differing from the period is only
the result of the tuning jargon developed by the pioneers of
microtonality and the efforts of the explorers on this list?
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 19 Eylül 2005 Pazartesi 22:23
> Subject: [tuning] Re: MOS at folding points other than the 2:1
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
>
> > So, the `period` AKA the `interval of periodicity` is the
interval
> where the
> > scale repeats itself using the exact same intervals.
>
> Yes.
>
> > This generates the
> > `interval of equivalance`, as in the `octave`, or else, is
squeezed
> inside
> > it any number of times.
>
> The period normally generates the interval of equivalence,
whether in
> 1 step (when the two are the same) or in any whole number N steps
> (when one has to iterate the period N times in order to generate
the
> interval of equivalence. And the interval of equivalence is
normally
> taken as pretty much given a priori (some say pan-culturally) and
> understood to be the "octave" or (~)2:1 ratio, though experiments
> with 3:1 as the interval of equivalence certainly yield musically
> intriguing, if not convincing, results.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

9/21/2005 1:18:48 PM

It's still unsuitable for Western polyphony, but extremely efficient
for delivering lots of complete 13-limit and 15-limit near-JI
harmonies. To my ear, these kind of harmonies (including variants where
some notes are omitted) are the only places where near-JI tuning of
ratios of 13 and 15 really matters aurally and is noticeably different
from not-near-JI tuning of such intervals. So if you're interested in
such all intervals and in polyphony, such a system seems like an
excellent choice. So easy to notate too! Looks like Gene suggested a
generator of 15.82578774 cents in one 13-odd-limit (so not considering
ratios of 15) context, in case 16 cents wasn't exact enough for you :)

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, I thought you had said to me that 29-tET was unsuitable for
Western polyphony. What's so special about this particular 58?
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 19 Eylül 2005 Pazartesi 23:02
> Subject: [tuning] Re: MOS at folding points other than the 2:1
>
>
> I don't know of such a Scala file -- and Graham should provide one
if he hasn't already -- but the basic idea is to take 29-equal, and
superimpose another 29-equal set about 16 cents higher (and in theory
another 29-equal set about 16 cents higher than that, and so on) --
this 58-note scale, I believe, is one of the most efficient 2D systems
for acheiving lots of higher-limit harmonies.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/15/2005 6:37:02 AM

Hi George,

On Mon, 29 Aug 2005, George D. Secor wrote:
>
> HEY EVERYONE!
>
> Having been unable to ignore the discussion about equal-beating
> temperaments lately, I have some startling news about a "temperament
> (extra)ordinaire!"
... [snipt]

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
12
!
86.53330
194.55680
294.12876
389.11361
499.91792
585.54105
697.27840
789.37483
891.83521
997.96292
1086.39201

2/1

... [snipt]

> After listening and comparing, I've concluded that the 8 equal- and
> proportional-beating major triads in this temperament sound as if
> they have less total error than the numbers would indicate, with the
> faster beats tending to be masked by the slower ones. In conducting
> some head-to-head comparisons of the C and G major triads with those
> in 31-ET (12c total error), I found that they sound at least as good;
> likewise the E and Eb major triads as compared with 12-ET. (The
> Scalatron permits me to jump back and forth instantly from one tuning
> to another, which easily allows me to make comparisons using the same
> timbres.) I also found, for some reason, that the B and Ab major
> triads sound only about as dissonant as pythagorean triads, even
> though they have about 5 cents greater total error.
>
> I was frankly a bit surprised at how quickly I was able to come up
> with proportional beat rates for so many triads, thinking that it
> should have been a lot harder to do this and beginning to experience
> some feelings of doubt that there must be some room for improvement --
> at this point I don't know. (But the more I play around with it,
> the less my doubt.)
>
> Anyway, I hope that some others will try this tuning and give me your
> reactions -- at least go through the triads in Scala using the chord
> playing function in the keyboard clavier. Convince me that this
> isn't all a dream!

I expect you've had plenty of feedback by now?
Anyway, FWIW, here are my reactions to tuning up your
"extraordinary" temperament, then playing with it for
an hour or so using both piano and harpsichord samples
on my Roland keyboard.

Everything is fine - EXCEPT the sounds of the bare Db,
Eb, E, F#, Ab and B major triads in root position with
root in the octave above middle C. Other inversions of
these triads are (more or less) acceptable, while more
open chords using these harmonies are fine. Playing an
accompanied melody, or improvised two- and three-part
counterpoint, all sounds pleasant enough to me. The only
problem I have is with these triads in that range.

I checked out the actual sizes of the fifths and thirds
in all major chords. The fifths are not a problem - by
construction, almost. (I don't really like fifths as
narrow as 697 cents, but that's the price we pay for
being able to play in all keys, I guess.) The major thirds
in these chords, without exception, are a little wide to
very wide, and I think this explains the worst of my
difficulty with them. Still, the E-G# is only 400.26122,
hardly different from the 12EDO third of 400 cents.
I can't really explain why the E major triad on E4 sounds
so wrong to me. Is it the minor third of ~297 cents? It
can't be the fifth - the D, G and A major chords, which
have the same size fifth, are all quite good; tho the D
chord seems noticeably flat, the A sounds a little better
than in 12EDO and the G major chord sounds superb -
much better than in 12EDO.

Perhaps all the beat frequencies for all my problem
triads lie in a range that I react badly to. Anyway,
something about them sets my teeth on edge!

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.0/132 - Release Date: 13/10/05

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/17/2005 12:48:50 PM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> I can't really explain why the E major triad on E4 sounds
> so wrong to me. Is it the minor third of ~297 cents?

I think that may very well be your explanation.

> Perhaps all the beat frequencies for all my problem
> triads lie in a range that I react badly to.

How did you implement this tuning, exactly? Did you check the tuning
independently after you set it?

🔗George D. Secor <gdsecor@yahoo.com>

10/17/2005 2:44:28 PM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Hi George,
>
> On Mon, 29 Aug 2005, George D. Secor wrote:
> >
> > HEY EVERYONE!
> >
> > Having been unable to ignore the discussion about equal-beating
> > temperaments lately, I have some startling news about
a "temperament
> > (extra)ordinaire!"
> ... [snipt]
> >
> > I was frankly a bit surprised at how quickly I was able to come
up
> > with proportional beat rates for so many triads, thinking that it
> > should have been a lot harder to do this and beginning to
experience
> > some feelings of doubt that there must be some room for
improvement --
> > at this point I don't know. (But the more I play around with
it,
> > the less my doubt.)

A couple of months have passed since I posted that, and I'm happy to
say that I've not only improved it (proportional beating in the
*minor* as well as the major triads), but after extensive
experimentation involving dozens of tuning attempts, I've selected
the five best tunings to be members of an entire suite of
proportional-beating well-temperaments (WT's) and _temperaments
ordinaires_ (TO's) -- details to be given at a later date (once I'm
sure that I've settled on my final selection).

> > Anyway, I hope that some others will try this tuning and give me
your
> > reactions -- at least go through the triads in Scala using the
chord
> > playing function in the keyboard clavier. Convince me that this
> > isn't all a dream!
>
> I expect you've had plenty of feedback by now?

Yes, but more is welcome.

> Anyway, FWIW, here are my reactions to tuning up your
> "extraordinary" temperament, then playing with it for
> an hour or so using both piano and harpsichord samples
> on my Roland keyboard.
>
> Everything is fine - EXCEPT the sounds of the bare Db,
> Eb, E, F#, Ab and B major triads in root position with
> root in the octave above middle C. Other inversions of
> these triads are (more or less) acceptable, while more
> open chords using these harmonies are fine. Playing an
> accompanied melody, or improvised two- and three-part
> counterpoint, all sounds pleasant enough to me. The only
> problem I have is with these triads in that range.

They all have total absolute error greater than the triads of 12-ET,
and the problem is mostly with the thirds. You'll need to understand
that the purpose of a _temperament ordinaire_ is to have high
(meantone-like) consonance in as many triads as possible after taming
the "wolf". In my temperament (extra)ordinaire wolf fifths are
completely eliminated. The 3rds will of necessity be worse than in
12-ET, but the objective is to have as few "worst" triads as
possible. I'm a bit surprised that you had problems with E and
especially Eb -- I would judge them to be not significantly worse
than 12-ET. (However, see my comment about becoming "spoiled",
below.)

> I checked out the actual sizes of the fifths and thirds
> in all major chords. The fifths are not a problem - by
> construction, almost. (I don't really like fifths as
> narrow as 697 cents, but that's the price we pay for
> being able to play in all keys, I guess.)

No, narrow fifths are the price we pay for being able to get the
thirds closer to just in the best triads.

> The major thirds
> in these chords, without exception, are a little wide to
> very wide, and I think this explains the worst of my
> difficulty with them.

Yes.

> Still, the E-G# is only 400.26122,
> hardly different from the 12EDO third of 400 cents.
> I can't really explain why the E major triad on E4 sounds
> so wrong to me. Is it the minor third of ~297 cents?

Probably. (Those who think that the beating of minor 3rds aren't
nearly as important as beating major 3rds should take note.)

Another possibility is that once you've gotten "spoiled" hearing how
nice the triads from Bb thru A sound, you're jarred by the relative
dissonance of triads only slightly worse than 12-ET. In my latest
experimentation with WT's and TO's I've listened and relistened (and
compared and recompared) tunings with one another and have been
occasionally surprised to find that I may experience different
reactions at different times -- sometimes only a matter of 10 or 15
minutes apart. At one time a Pythagorean triad may sound only
marginally acceptable and at another time a triad with even more
total absolute error than that (such as the worst ones in the
temperament extraordinaire) will sound more acceptable. Likewise,
sometimes a low-contrast modern well-temperament (with a relatively
small difference in intonation between the best and worst triads)
will sound as if all the triads are very close to 12-ET (and
therefore probably not worth the effort), whereas at other times the
difference between the best and worst will seem quite effective.

There seems to be some sort of "hearing fatigue" in operation here,
and it has made me look at Aaron Johnson's "Killing the Buddha of
well-temperament" message
/tuning/topicId_60103.html#60103
with a new perspective.

> It
> can't be the fifth - the D, G and A major chords, which
> have the same size fifth, are all quite good; tho the D
> chord seems noticeably flat,

I assume you're referring primarily to the 3rd as "flat" -- that's
what make the triad more consonant.

> the A sounds a little better
> than in 12EDO and the G major chord sounds superb -
> much better than in 12EDO.

The G, D, and A major triads are significantly better than what
you'll hear in most other circulating 12-tone temperaments.

> Perhaps all the beat frequencies for all my problem
> triads lie in a range that I react badly to. Anyway,
> something about them sets my teeth on edge!

Yes, that's understandable.

You'll have to try out the other temperaments in my suite and let me
know what you think. All of the others will have less error in the
worst triads, and one of them is a well-temperament that has
proportional beating in *all 24* major & minor triads (a feat to
behold!).

Best,

--George

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/18/2005 8:30:54 PM

Hi Paul,

On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > I can't really explain why the E major triad on E4 sounds
> > so wrong to me. Is it the minor third of ~297 cents?
>
> I think that may very well be your explanation.
>
> > Perhaps all the beat frequencies for all my problem
> > triads lie in a range that I react badly to.
>
> How did you implement this tuning, exactly?

Exactly: I used the Keyboard Scale, + and - buttons of the
Roland E-28 keyboard. Holding the Keyboard Scale button
down for two seconds changes the tuning mode to one in
which you can adjust the tuning of each note name (octave
equivalent pitch class) independently by from -64 to +63
cents, in steps of 1 cent, away from 12-EDO.

It's also possible to toggle between the altered tuning and
the standard 12-EDO tuning by tapping the Keyboard Scale
button.

>... Did you check the tuning
> independently after you set it?

No. I have no specialised tools for doing so. But my
impression is that the frequency change per tap of the +
and - buttons is [log] linear.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.2/139 - Release Date: 17/10/05

🔗klaus schmirler <KSchmir@online.de>

10/19/2005 4:38:39 AM

Yahya Abdal-Aziz wrote:

> Exactly: I used the Keyboard Scale, + and - buttons of the > Roland E-28 keyboard. Holding the Keyboard Scale button > down for two seconds changes the tuning mode to one in > which you can adjust the tuning of each note name (octave
> equivalent pitch class) independently by from -64 to +63
> cents, in steps of 1 cent, away from 12-EDO.
> > It's also possible to toggle between the altered tuning and > the standard 12-EDO tuning by tapping the Keyboard Scale
> button.
> > >>... Did you check the tuning >>independently after you set it?
> > > No. I have no specialised tools for doing so. But my > impression is that the frequency change per tap of the + > and - buttons is [log] linear.

I'd be suspicious when the settings are from -64 to 63 - I have a cheap old Kawai keyboard with these settings, and it doesn't work that way. My C+50 was way higher than the C#-50. Nor did I find any logic in the step sizes except that they seemed to increase towards the edges.

klaus

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/20/2005 3:21:27 PM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Hi Paul,
>
> On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> >
> > > I can't really explain why the E major triad on E4 sounds
> > > so wrong to me. Is it the minor third of ~297 cents?
> >
> > I think that may very well be your explanation.
> >
> > > Perhaps all the beat frequencies for all my problem
> > > triads lie in a range that I react badly to.
> >
> > How did you implement this tuning, exactly?
>
> Exactly: I used the Keyboard Scale, + and - buttons of the
> Roland E-28 keyboard. Holding the Keyboard Scale button
> down for two seconds changes the tuning mode to one in
> which you can adjust the tuning of each note name (octave
> equivalent pitch class) independently by from -64 to +63
> cents, in steps of 1 cent, away from 12-EDO.
>
> It's also possible to toggle between the altered tuning and
> the standard 12-EDO tuning by tapping the Keyboard Scale
> button.
>
> >... Did you check the tuning
> > independently after you set it?
>
> No. I have no specialised tools for doing so. But my
> impression is that the frequency change per tap of the +
> and - buttons is [log] linear.

Be suspicious. Monz did a test where he found that, for at least one
synth model, the increments were not at all equal (in his case, some
were one cent and some were two cents), even though nominally, if you
trusted the manufacturer, they should have been. The same is most
definitely true of another synth, my Ensoniq. Synth manufacturers
have had very little market pressure to implement accurate
microtuning offsets. So you should *always* do an independent check
when you care about 1 or 2 cents accuracy.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/21/2005 1:28:05 AM

On Wed, 19 Oct 2005, klaus schmirler wrote:
>
> Yahya Abdal-Aziz wrote:
>
> > Exactly: I used the Keyboard Scale, + and - buttons of the
> > Roland E-28 keyboard. Holding the Keyboard Scale button
> > down for two seconds changes the tuning mode to one in
> > which you can adjust the tuning of each note name (octave
> > equivalent pitch class) independently by from -64 to +63
> > cents, in steps of 1 cent, away from 12-EDO.
> >
> > It's also possible to toggle between the altered tuning and
> > the standard 12-EDO tuning by tapping the Keyboard Scale
> > button.
> >
> >
> >>... Did you check the tuning
> >>independently after you set it?
> >
> >
> > No. I have no specialised tools for doing so. But my
> > impression is that the frequency change per tap of the +
> > and - buttons is [log] linear.
>
> I'd be suspicious when the settings are from -64 to 63 - I have a
> cheap old Kawai keyboard with these settings, and it doesn't work that
> way. My C+50 was way higher than the C#-50. Nor did I find any logic
> in the step sizes except that they seemed to increase towards the edges.

Klaus,

I've just checked. My C+50 and C#-50 are identical.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.4/143 - Release Date: 19/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/21/2005 1:28:17 AM

On Thu, 20 Oct 2005, "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > Hi Paul,
> >
> > On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > > <yahya@m...> wrote:
> > >
> > > > I can't really explain why the E major triad on E4 sounds
> > > > so wrong to me. Is it the minor third of ~297 cents?
> > >
> > > I think that may very well be your explanation.
> > >
> > > > Perhaps all the beat frequencies for all my problem
> > > > triads lie in a range that I react badly to.
> > >
> > > How did you implement this tuning, exactly?
> >
> > Exactly: I used the Keyboard Scale, + and - buttons of the
> > Roland E-28 keyboard. Holding the Keyboard Scale button
> > down for two seconds changes the tuning mode to one in
> > which you can adjust the tuning of each note name (octave
> > equivalent pitch class) independently by from -64 to +63
> > cents, in steps of 1 cent, away from 12-EDO.
> >
> > It's also possible to toggle between the altered tuning and
> > the standard 12-EDO tuning by tapping the Keyboard Scale
> > button.
> >
> > >... Did you check the tuning
> > > independently after you set it?
> >
> > No. I have no specialised tools for doing so. But my
> > impression is that the frequency change per tap of the +
> > and - buttons is [log] linear.
>
> Be suspicious. Monz did a test where he found that, for at least one
> synth model, the increments were not at all equal (in his case, some
> were one cent and some were two cents), even though nominally, if you
> trusted the manufacturer, they should have been. The same is most
> definitely true of another synth, my Ensoniq. Synth manufacturers
> have had very little market pressure to implement accurate
> microtuning offsets. So you should *always* do an independent check
> when you care about 1 or 2 cents accuracy.

Paul,

I've answered (elsewhere) the suggestion that the C +50c might not
equal C# -50c on some keyboards. I've demonstrated clearly on my
Roland E-28 that the two are identical.

To be doubly sure, I've just now checked that C +63c equals C# -37c.

So I'm content to follow my ear on this one.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.4/143 - Release Date: 19/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/21/2005 1:27:53 AM

Hi George,

On Mon, 17 Oct 2005, "George D. Secor" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> >
> > Hi George,
> >
> > On Mon, 29 Aug 2005, George D. Secor wrote:
> > >
> > > HEY EVERYONE!
> > >
> > > Having been unable to ignore the discussion about equal-beating
> > > temperaments lately, I have some startling news about
> > > a "temperament (extra)ordinaire!"
... [ much snipt]

> A couple of months have passed since I posted that, and I'm happy to
> say that I've not only improved it (proportional beating in the
> *minor* as well as the major triads), but after extensive
> experimentation involving dozens of tuning attempts, I've selected
> the five best tunings to be members of an entire suite of
> proportional-beating well-temperaments (WT's) and _temperaments
> ordinaires_ (TO's) -- details to be given at a later date (once I'm
> sure that I've settled on my final selection).

I look forward to reading the details.

> > > Anyway, I hope that some others will try this tuning and give me
> > > your reactions -- at least go through the triads in Scala using
> > > the chord playing function in the keyboard clavier. Convince me
> > > that this isn't all a dream!
> >
> > I expect you've had plenty of feedback by now?
>
> Yes, but more is welcome.
>
> > Anyway, FWIW, here are my reactions to tuning up your
> > "extraordinary" temperament, then playing with it for
> > an hour or so using both piano and harpsichord samples
> > on my Roland keyboard.
> >
> > Everything is fine - EXCEPT the sounds of the bare Db,
> > Eb, E, F#, Ab and B major triads in root position with
> > root in the octave above middle C. Other inversions of
> > these triads are (more or less) acceptable, while more
> > open chords using these harmonies are fine. Playing an
> > accompanied melody, or improvised two- and three-part
> > counterpoint, all sounds pleasant enough to me. The only
> > problem I have is with these triads in that range.
>
> They all have total absolute error greater than the triads of 12-ET,
> and the problem is mostly with the thirds. You'll need to understand
> that the purpose of a _temperament ordinaire_ is to have high
> (meantone-like) consonance in as many triads as possible after taming
> the "wolf". In my temperament (extra)ordinaire wolf fifths are
> completely eliminated. The 3rds will of necessity be worse than in
> 12-ET, but the objective is to have as few "worst" triads as
> possible.

Thanks for that explanation; until now, I had no idea what
the goals of a "temp�rament ordinaire" were.

> ... I'm a bit surprised that you had problems with E and
> especially Eb -- I would judge them to be not significantly worse
> than 12-ET. (However, see my comment about becoming "spoiled",
> below.)

I can rule this out; I toggled between the two tunings several times,
playing identical passages in each.

> > I checked out the actual sizes of the fifths and thirds
> > in all major chords. The fifths are not a problem - by
> > construction, almost. (I don't really like fifths as
> > narrow as 697 cents, but that's the price we pay for
> > being able to play in all keys, I guess.)
>
> No, narrow fifths are the price we pay for being able to get the
> thirds closer to just in the best triads.

Hmmm. So the nature of the compromise in these tunings
is that (unlike any EDO) some keys MUST be worse than
others.

> > The major thirds
> > in these chords, without exception, are a little wide to
> > very wide, and I think this explains the worst of my
> > difficulty with them.
>
> Yes.

> > Still, the E-G# is only 400.26122,
> > hardly different from the 12EDO third of 400 cents.
> > I can't really explain why the E major triad on E4 sounds
> > so wrong to me. Is it the minor third of ~297 cents?
>
> Probably. (Those who think that the beating of minor 3rds aren't
> nearly as important as beating major 3rds should take note.)

> Another possibility is that once you've gotten "spoiled" hearing how
> nice the triads from Bb thru A sound, you're jarred by the relative
> dissonance of triads only slightly worse than 12-ET. In my latest
> experimentation with WT's and TO's I've listened and relistened (and
> compared and recompared) tunings with one another and have been
> occasionally surprised to find that I may experience different
> reactions at different times -- sometimes only a matter of 10 or 15
> minutes apart.

I noticed the same thing, but over much shorter time intervals, mostly
when I was contrasting the effect of the two tunings (your extra-TO
and 12-EDO) by playing a short phrase of perhaps two bars, first in
one and then the other, then repeating both - in the pattern 1 2 1 2.

> ... At one time a Pythagorean triad may sound only
> marginally acceptable and at another time a triad with even more
> total absolute error than that (such as the worst ones in the
> temperament extraordinaire) will sound more acceptable. Likewise,
> sometimes a low-contrast modern well-temperament (with a relatively
> small difference in intonation between the best and worst triads)
> will sound as if all the triads are very close to 12-ET (and
> therefore probably not worth the effort), whereas at other times the
> difference between the best and worst will seem quite effective.
>
> There seems to be some sort of "hearing fatigue" in operation here,

Perhaps my even shorter-term "hearing fatigue" arose from the
fact I have Chronic Fatigue Syndrome ... ? :-) Yes, I thought much
the same thing, so I took a rest break and came back to it. The first
time I repeated the fourfold pattern 1 2 1 2 as above, I still heard
the representations of the first tuning differently. So the effect I
heard probably had more to do with the different contrasts, as
follows:

Tuning 1 2 1 2
Contrast 0-1 1-2 2-1 1-2

All of which makes me even more certain that I will never listen
to an adaptive tuning without a distinct feeling of nausea! (Slight
OT: Is barbershop tuning necessarily adaptive? It certainly sounds
so to me. And yes, although I do love the sweet harmonies I hear,
that style of singing usually does leave me feeling queasy.)

> ... and it has made me look at Aaron Johnson's "Killing the Buddha of
> well-temperament" message
> /tuning/topicId_60103.html#60103
> with a new perspective.

I'll look it up.

> > It
> > can't be the fifth - the D, G and A major chords, which
> > have the same size fifth, are all quite good; tho the D
> > chord seems noticeably flat,
>
> I assume you're referring primarily to the 3rd as "flat" -- that's
> what make the triad more consonant.

Yes, but the whole chord is noticeably "squashed" when compared
to the 12-EDO.

> > > the A sounds a little better
> > than in 12EDO and the G major chord sounds superb -
> > much better than in 12EDO.
>
> The G, D, and A major triads are significantly better than what
> you'll hear in most other circulating 12-tone temperaments.
>
> > Perhaps all the beat frequencies for all my problem
> > triads lie in a range that I react badly to. Anyway,
> > something about them sets my teeth on edge!
>
> Yes, that's understandable.
>
> You'll have to try out the other temperaments in my suite and let me
> know what you think. All of the others will have less error in the
> worst triads, and one of them is a well-temperament that has
> proportional beating in *all 24* major & minor triads (a feat to
> behold!).

I await with breath abated ... :-)

>
> Best,
> --George

Likewise,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.2/139 - Release Date: 17/10/05

🔗monz <monz@tonalsoft.com>

10/21/2005 11:15:30 AM

Hi Yahya and Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> >
> > > ... Did you check the tuning independently after you set it?
> >
> > No. I have no specialised tools for doing so. But my
> > impression is that the frequency change per tap of the +
> > and - buttons is [log] linear.
>
> Be suspicious. Monz did a test where he found that, for
> at least one synth model, the increments were not at all
> equal (in his case, some were one cent and some were two
> cents), even though nominally, if you trusted the
> manufacturer, they should have been. The same is most
> definitely true of another synth, my Ensoniq. Synth
> manufacturers have had very little market pressure to
> implement accurate microtuning offsets. So you should
> *always* do an independent check when you care about
> 1 or 2 cents accuracy.

Those tests were done on my previous and current computer
soundcards. Their tuning resolutions, as usual for MIDI
hardware, are quantized to 6 bits, which gives 64 divisions
per semitone and thus 768 per octave.

*But*, while my old soundcard did give 768-edo resolution,
my current one doesn't. The current one is quantized to a
768-tone subset of 1200-edo, so that each tuning "zone"
encompasses either 1 or 2 cents. Obviously, this makes it
fiendishly difficult to determine the precise tuning of
any MIDI i do on my computer.

Here are the old posts:

/tuning/topicId_45522.html#45522

/tuning/topicId_45522.html#45533

Thankfully, Tonescape (which is now the only software i
use to compose music) has now entered the phase of
development where we are able to create mp3's without
using MIDI, and our tuning resolution uses floating point
cents values with so many decimal places that i can't
remember how many there are. If we get 8 decimal places,
then it's (1.2 * 10^11)-edo, which is precise enough
for me. ;-)

The question which still remains for me is: is the
tuning resolution i discovered for my soundcard relevant
only to MIDI, or to any sound it produces? I did the
test using MIDI software, so i have no idea what resolution
the soundcard gives for non-MIDI audio.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗George D. Secor <gdsecor@yahoo.com>

10/21/2005 1:02:52 PM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Hi George,
>
> On Mon, 17 Oct 2005, "George D. Secor" wrote:
> >...
> > You'll have to try out the other temperaments in my suite and let
me
> > know what you think. All of the others will have less error in
the
> > worst triads, and one of them is a well-temperament that has
> > proportional beating in *all 24* major & minor triads (a feat to
> > behold!).
>
> I await with breath abated ... :-)

Please don't hold it for more than a few seconds at a time, because
I'm not able to reply at length right now. ;-)

When I do (hopefully by the end of next week), I think I should make
a separate file describing the 5 temperaments in the suite (including
the purpose of each) and include a link to it so it will be easier to
locate later.

Best,

--George

🔗Aaron Krister Johnson <aaron@akjmusic.com>

10/21/2005 1:34:56 PM

it seems that all this exploration into equal-beating well temperaments has
given me a great question:

what is the simplest possible 12-note temperament where all 24 major and minor
triads have rationally proportional beating? here 'simplest' means that the
brats (beat ratios for the un-initiated) are the lowest numbers in the
numerator and denominator that they can be.....

anyone? this is ripe for george or gene to explore.

-aaron.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/21/2005 2:59:23 PM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
>
>
> it seems that all this exploration into equal-beating well
temperaments has
> given me a great question:
>
> what is the simplest possible 12-note temperament where all 24 major
and minor
> triads have rationally proportional beating?

If you are asking for a tuning where the brats are all rational, any
tuning with rational values, producing rational values of fifths and
thirds, will do.

🔗Carl Lumma <clumma@yahoo.com>

10/21/2005 3:53:03 PM

> > it seems that all this exploration into equal-beating well
> temperaments has
> > given me a great question:
> >
> > what is the simplest possible 12-note temperament where all
> > 24 major and minor
> > triads have rationally proportional beating?
>
> If you are asking for a tuning where the brats are all rational,
> any tuning with rational values, producing rational values of
> fifths and thirds, will do.

Bob Wendell's Natural Synchronous Well claimed to be this tuning.
Brats of 2 or 1.5 or all 24 triads (except a few of the values are
only very close). Bob made great claims for this scale, and it
apparently got the attention of some piano tuners, and I've
tried it on my MIDI keyboard with Scala. I wasn't immediately
bowled over.

Gene seemed to agree that this was the simplest tuning in this
sense at the time, but more recently he seemed to weaken those
claims...

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

10/21/2005 4:24:03 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Gene seemed to agree that this was the simplest tuning in this
> sense at the time, but more recently he seemed to weaken those
> claims...

Bob's scale does not precisely fit Aaron's requirements; I first need
to get clear if all the beat ratios actually need to be exact. The 12
note Pythagorean scale, for instance, has all brats of exactly 3/2,
except for the triad with the wolf fifth, which is rational but hardly
simple.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/21/2005 4:56:59 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Bob's scale does not precisely fit Aaron's requirements; I first need
> to get clear if all the beat ratios actually need to be exact. The 12
> note Pythagorean scale, for instance, has all brats of exactly 3/2,
> except for the triad with the wolf fifth, which is rational but hardly
> simple.

From out of the Scala archives, here is my rational number version of
Wendell Well. It has all-rational beat ratios. Ten of the major brats
are simple, being either 3/2 or 2. The other two are close to 2.

! wendell1r.scl
!
Rational version of wendell1.scl by Gene Ward Smith
12
!
24000/22729
76460/68187
27000/22729
85600/68187
4/3
32100/22729
102140/68187
36000/22729
114400/68187
40500/22729
42800/22729
2/1

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/23/2005 1:19:52 AM

Hi Monz,

On Fri, 21 Oct 2005, "monz" wrote:
>
> Hi Yahya and Paul,
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> > >
> > > > ... Did you check the tuning independently after you set it?
> > >
> > > No. I have no specialised tools for doing so. But my
> > > impression is that the frequency change per tap of the +
> > > and - buttons is [log] linear.
> >
> > Be suspicious. Monz did a test where he found that, for
> > at least one synth model, the increments were not at all
> > equal (in his case, some were one cent and some were two
> > cents), even though nominally, if you trusted the
> > manufacturer, they should have been. The same is most
> > definitely true of another synth, my Ensoniq. Synth
> > manufacturers have had very little market pressure to
> > implement accurate microtuning offsets. So you should
> > *always* do an independent check when you care about
> > 1 or 2 cents accuracy.
>
>
> Those tests were done on my previous and current computer
> soundcards. Their tuning resolutions, as usual for MIDI
> hardware, are quantized to 6 bits, which gives 64 divisions
> per semitone and thus 768 per octave.
>
> *But*, while my old soundcard did give 768-edo resolution,
> my current one doesn't. The current one is quantized to a
> 768-tone subset of 1200-edo, so that each tuning "zone"
> encompasses either 1 or 2 cents. Obviously, this makes it
> fiendishly difficult to determine the precise tuning of
> any MIDI i do on my computer.
>
> Here are the old posts:
>
> /tuning/topicId_45522.html#45522
>
> /tuning/topicId_45522.html#45533

I'm having no luck retrieving either message;
I tried yesterday and have retried today.

Would you please briefly describe your test method?

> Thankfully, Tonescape (which is now the only software i
> use to compose music) has now entered the phase of
> development where we are able to create mp3's without
> using MIDI, and our tuning resolution uses floating point
> cents values with so many decimal places that i can't
> remember how many there are. If we get 8 decimal places,
> then it's (1.2 * 10^11)-edo, which is precise enough
> for me. ;-)

A resolution that probably exceeds the capabilities of any
(sentient) creature on Earth!

> The question which still remains for me is: is the
> tuning resolution i discovered for my soundcard relevant
> only to MIDI, or to any sound it produces? I did the
> test using MIDI software, so i have no idea what resolution
> the soundcard gives for non-MIDI audio.

Sorry, I can't help you with an answer to this.

The only thoughts I have on the matter are, that since
the soundcard is a digital device, it necessarily DOES
have a fixed resolution that will depend on the chip and
logic design. Thus, the manufacturer would probably be
best able to advise you on the resolution they intended
the card to have.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/23/2005 1:27:08 AM

George,

On Fri, 21 Oct 2005 "George D. Secor" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > Hi George,
> >
> > On Mon, 17 Oct 2005, "George D. Secor" wrote:
> > >...
> > > You'll have to try out the other temperaments in my suite and let
> > > me
> > > know what you think. All of the others will have less error in
> > > the
> > > worst triads, and one of them is a well-temperament that has
> > > proportional beating in *all 24* major & minor triads (a feat to
> > > behold!).
> >
> > I await with breath abated ... :-)
>
> Please don't hold it for more than a few seconds at a time, because
> I'm not able to reply at length right now. ;-)

Just when I was "Blue turning gray" ...

> When I do (hopefully by the end of next week), I think I should make
> a separate file describing the 5 temperaments in the suite (including
> the purpose of each) and include a link to it so it will be easier to
> locate later.
>
> Best,
>
> --George

Looking forward to it.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗monz <monz@tonalsoft.com>

10/23/2005 7:58:06 AM

Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> On Fri, 21 Oct 2005, "monz" wrote:

> > Here are the old posts:
> >
> > /tuning/topicId_45522.html#45522
> >
> > /tuning/topicId_45522.html#45533
>
> I'm having no luck retrieving either message;
> I tried yesterday and have retried today.

They are there. Just keep trying.

> Would you please briefly describe your test method?

Very briefly ...

Using Cakewalk 9.03, i probably used the "ocarina"
general-MIDI sound because it's the closest thing MIDI
gives to a sine wave. I set two identical tracks to
a long sustained note at MIDI-note 69 (A-440 Hz).

Then, while keeping the pitch of Track 1 constant,
and using the finest pitch-bend resolution in Cakewalk --
the 12mu (dodekamu) level, 4096 steps per semitone
-- i incremented the pitch-bend values of Track 2 until
i heard the notes go out of phase, indicating that
the pitch of Track 2 had changed minutely.

On my old soundcard, the pitch change occurred each
time i incremented the 12mu values by 64, which indicates
that the soundcard was giving 6mu (hexamu) resolution,
that is, 4096/64 = 64 steps per semitone, which equals
768-edo.

But on my new soundcard, the pitch change occurred at
unequally-spaced intervals which incremented by either
40 or 41 (12mu) steps, or 80 or 81 steps, which means
that each pitch-change "zone" is quantized to either
1 or 2 cents:

4096/40 or 4096/41 = ~1 cent
4096/80 or 4096/81 = ~2 cents

So in effect, what i get is a 768-tone subset of 1200-edo.

> The only thoughts I have on the matter are, that since
> the soundcard is a digital device, it necessarily DOES
> have a fixed resolution that will depend on the chip and
> logic design. Thus, the manufacturer would probably be
> best able to advise you on the resolution they intended
> the card to have.

Without bothering to get any more scientific about it
than i already have, i just think it's easiest and safest
to assume that digital audio hardware is not going to
give resolution better than 2 cents.

So essentially, while it's interesting to do tuning theory
to any level of resolution you like, in practice, doing
music on a computer, it's probably useless to even
contemplate anything finer than 600-edo.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Jon Szanto <jszanto@cox.net>

10/23/2005 1:02:37 PM

Monz,

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> Without bothering to get any more scientific about it
> than i already have, i just think it's easiest and safest
> to assume that digital audio hardware is not going to
> give resolution better than 2 cents.
>
> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing
> music on a computer, it's probably useless to even
> contemplate anything finer than 600-edo.

I think you need to really re-think a statement like the above. While
that may be true for some audio cards, and mainly if you are staying
within the limitations of midi (and pitch bends), which these days I
don't know why people would, it doesn't apply at all if you want to
utilize other digital sound/music tools. For instance, I think you can
get very accurate tunings using Csound, and while I don't know how
many people have done extremely careful testing, my suspicion is that
softsynths with built-in tuning capacity will get better pitch
accuracy than relying on pitch bends.

I bet a Csound guru would know for sure.

Cheers,
Jon

🔗Gene Ward Smith <gwsmith@svpal.org>

10/23/2005 2:19:16 PM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing
> music on a computer, it's probably useless to even
> contemplate anything finer than 600-edo.

That assumes music on a computer must use a soundcard.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/23/2005 2:24:15 PM

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

> I bet a Csound guru would know for sure.

Csound tuning has nothing to do with either pitch bends or soundcards.
The waveform is computed, and the tuning is given in frequency terms.

🔗Jon Szanto <jszanto@cox.net>

10/23/2005 5:11:49 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
>
> > I bet a Csound guru would know for sure.
>
> Csound tuning has nothing to do with either pitch bends or soundcards.
> The waveform is computed, and the tuning is given in frequency terms.

Ah, yes, that was my point. By rendering a file you aren't locked into
any hardware paradigm at all. I'm just not exactly sure of Csound's
tuning resolution, but I'd bet the farm it is a lot better than pitch
bending.

Of course, you do give up the fun of interacting with the music in a
live manner, but for some people that isn't an issue. BTW, Gene, are
there currently any really good midi file -> Csound .sco utilities?

Cheers,
Jon

🔗Gene Ward Smith <gwsmith@svpal.org>

10/23/2005 7:00:59 PM

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

> Of course, you do give up the fun of interacting with the music in a
> live manner, but for some people that isn't an issue. BTW, Gene, are
> there currently any really good midi file -> Csound .sco utilities?

There's a utility which has been around for years, but it isn't much
use. What is needed is a good suite of GM instruments for Csound.

🔗monz <monz@tonalsoft.com>

10/24/2005 2:58:30 AM

Hi Jon,

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
>
> Monz,
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > Without bothering to get any more scientific about it
> > than i already have, i just think it's easiest and safest
> > to assume that digital audio hardware is not going to
> > give resolution better than 2 cents.
> >
> > So essentially, while it's interesting to do tuning theory
> > to any level of resolution you like, in practice, doing
> > music on a computer, it's probably useless to even
> > contemplate anything finer than 600-edo.
>
> I think you need to really re-think a statement like the
> above. While that may be true for some audio cards, and
> mainly if you are staying within the limitations of midi
> (and pitch bends), which these days I don't know why
> people would, it doesn't apply at all if you want to
> utilize other digital sound/music tools. For instance,
> I think you can get very accurate tunings using Csound,
> and while I don't know how many people have done extremely
> careful testing, my suspicion is that softsynths with
> built-in tuning capacity will get better pitch accuracy
> than relying on pitch bends.
>
> I bet a Csound guru would know for sure.

Agreed, and i'm glad you wrote this ... i'm not sure why
i didn't qualify what i wrote myself, because i had already
written in a previous post that this resolution applies
to MIDI, but that i have no idea of the resolution of any
soundcards when doing non-MIDI digital audio. In fact,
i'm immersed in a study of Csound myself right now. I'll
see if i can find that answer.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

10/24/2005 3:48:52 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
>
> > Of course, you do give up the fun of interacting with
> > the music in a live manner, but for some people that
> > isn't an issue. BTW, Gene, are there currently any really
> > good midi file -> Csound .sco utilities?
>
> There's a utility which has been around for years, but it
> isn't much use. What is needed is a good suite of GM
> instruments for Csound.

And amazingly, after all these years, it seems that still
it doesn't exist. That's exactly what i'm deep into right
now: studying how Csound .orc files would do GM.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/24/2005 8:21:50 AM

Hi Monz,

On Sun, 23 Oct 2005, "monz" wrote:
> Hi Yahya,
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > On Fri, 21 Oct 2005, "monz" wrote:
> > > Here are the old posts:
> > >
> > > /tuning/topicId_45522.html#45522
> > >
> > > /tuning/topicId_45522.html#45533
> >
> > I'm having no luck retrieving either message;
> > I tried yesterday and have retried today.
>
> They are there. Just keep trying.

Success!

> > Would you please briefly describe your test method?
>
> Very briefly ...
>
> Using Cakewalk 9.03, i probably used the "ocarina"
> general-MIDI sound because it's the closest thing MIDI
> gives to a sine wave. I set two identical tracks to
> a long sustained note at MIDI-note 69 (A-440 Hz).
>
> Then, while keeping the pitch of Track 1 constant,
> and using the finest pitch-bend resolution in Cakewalk --
> the 12mu (dodekamu) level, 4096 steps per semitone
> -- i incremented the pitch-bend values of Track 2 until
> i heard the notes go out of phase, indicating that
> the pitch of Track 2 had changed minutely.
>
> On my old soundcard, the pitch change occurred each
> time i incremented the 12mu values by 64, which indicates
> that the soundcard was giving 6mu (hexamu) resolution,
> that is, 4096/64 = 64 steps per semitone, which equals
> 768-edo.
>
> But on my new soundcard, the pitch change occurred at
> unequally-spaced intervals which incremented by either
> 40 or 41 (12mu) steps, or 80 or 81 steps, which means
> that each pitch-change "zone" is quantized to either
> 1 or 2 cents:
>
> 4096/40 or 4096/41 = ~1 cent
> 4096/80 or 4096/81 = ~2 cents
>
> So in effect, what i get is a 768-tone subset of 1200-edo.

Thank you, that is quite clear. So to test the MIDI
frequency resolution of our sound sources (card or external
MIDI source), using whatever MIDI software we have to
hand, we could also try to find the smallest MIDI Pitch Bend
that audibly changes the simplest timbre. Obviously, only
software with fine PB resolution would be useful in such an
exercise.

> > The only thoughts I have on the matter are, that since
> > the soundcard is a digital device, it necessarily DOES
> > have a fixed resolution that will depend on the chip and
> > logic design. Thus, the manufacturer would probably be
> > best able to advise you on the resolution they intended
> > the card to have.
>
> Without bothering to get any more scientific about it
> than i already have, i just think it's easiest and safest
> to assume that digital audio hardware is not going to
> give resolution better than 2 cents.

I don't know about that ... I suspect the actual limits
to the frequency resolution of digital audio hardware
will be much better than that. In practice, this will
probably only matter in music with slow sustained sounds
or very slow changes in pitch. For "most" music, your
resolution of 2 cents is pretty much the limit of human
pitch discrimination, isn't it?

> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing
> music on a computer, it's probably useless to even
> contemplate anything finer than 600-edo.

Even if we do slowly evolving sine textures in a La Monte
Young kind of way?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Jon Szanto <jszanto@cox.net>

10/24/2005 8:26:23 AM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> And amazingly, after all these years, it seems that still
> it doesn't exist. That's exactly what i'm deep into right
> now: studying how Csound .orc files would do GM.

I don't think it is that amazing, really, as most Csounders don't seem
to be working with midi (or GM, for that matter). Why don't you post
an open question, and see if Dave Seidel or Prent Rodgers or Bill
Sethares or ??? can help you?

Cheers,
Jon

P.S. I'm assuming you have "The Csound Book"...

🔗Carl Lumma <clumma@yahoo.com>

10/24/2005 12:38:03 PM

> Monz,
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > Without bothering to get any more scientific about it
> > than i already have, i just think it's easiest and safest
> > to assume that digital audio hardware is not going to
> > give resolution better than 2 cents.
> >
> > So essentially, while it's interesting to do tuning theory
> > to any level of resolution you like, in practice, doing
> > music on a computer, it's probably useless to even
> > contemplate anything finer than 600-edo.
>
> I think you need to really re-think a statement like the above.

Indeed. Today's softsynths write audio data directly to
audio devices, and thus can be as accurate as their authors
cared to make them.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

10/24/2005 12:44:20 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > Bob's scale does not precisely fit Aaron's requirements; I first
need
> > to get clear if all the beat ratios actually need to be exact.
The 12
> > note Pythagorean scale, for instance, has all brats of exactly
3/2,
> > except for the triad with the wolf fifth, which is rational but
hardly
> > simple.
>
> From out of the Scala archives, here is my rational number version
of
> Wendell Well. It has all-rational beat ratios. Ten of the major
brats
> are simple, being either 3/2 or 2. The other two are close to 2.
>
> ! wendell1r.scl
> !
> Rational version of wendell1.scl by Gene Ward Smith
> 12
> !
> 24000/22729
> 76460/68187
> 27000/22729
> 85600/68187
> 4/3
> 32100/22729
> 102140/68187
> 36000/22729
> 114400/68187
> 40500/22729
> 42800/22729
> 2/1

Gene, I'm impressed!

But, OTOH, I hope you realize that there are reasonably simple brats
in only 5 minor triads (the ones with just 5ths), and the rest are
pretty much hit-or-miss (mostly the latter) -- much like what I got
in my _temperament (extra)ordinaire_ a couple months back.

In attempting to construct synchronous temperaments (with irrational
intervals) with both proportional *major and minor* triads, I found
that, unless the 5ths are exact, the minor-triad brats will at best
only approximate simple ratios when the majors are exact (or vice
versa), or you can compromise and make both inexact by about half and
thereby get most of the brats to within 0.02 of simple numbers --
close enough, I would say, considering that the required adjustment
is very small. In the following example, each of the tempered fifths
is modified by less than 0.002 cents (i.e., -3.9114 vs. -3.9132 cents
for an exact 2/11-comma narrow fifth).

Here's a sneak preview of my 2/11-comma well-temperament (with beat
synchrony in virtually all 24 major & minor triads). Bear in mind
that this is an honest-to-goodness well-temperament that meets *all*
of Jorgensen's requirements, with key contrasts that are very similar
to the Valotti-Young (1/6-pythagorean-comma) temperament:

! secor_WT2-11.scl
!
George Secor's 2/11-comma well-temperament, proportional beating
12
!
92.13884
196.08721
296.01562
392.17442
499.92562
590.20835
698.04361
794.06062
894.13082
997.97062
1090.21803
2/1

As to how close it comes to perfection, judge for yourself:

Major -----Beat Ratios-----
Triad M3/5th m3/5th m3/M3
----- ------- ------- -----
Ab... ------- ------- 1.50
Eb... ------- ------- 1.50
Bb... ------- ------- 1.50
F.... 7.01 13.02 1.86
C.... 2.50 6.26 2.50
G.... 2.50 6.26 2.50
D.... 3.34 7.51 2.25
A.... 5.01 10.01 5.00
E.... 6.67 12.51 1.87
B…... 16.63 27.45 1.65
F#... 1467.47 2203.71 1.50
C#... 1084.06 1628.58 1.50

Minor -----Beat Ratios-----
Triad M3/5th m3/5th m3/M3
----- ------- ------- -----
Ab... ------- ------- 1.00
Eb... ------- ------- 1.00
Bb... ------- ------- 1.00
F.... 20.74 22.74 1.10
C.... 7.99 9.99 1.25
G.... 6.01 8.01 1.33
D.... 4.02 6.02 1.50
A.... 2.99 4.99 1.67
E.... 2.99 4.99 1.67
B…... 7.93 9.93 1.25
F#... 950.96 952.96 1.00
C#... 933.34 935.34 1.00

The F# and C# triads have 5ths that are nearly just, so you can
consider those 3- and 4-digit numbers as approaching infinity. As
for B major, it's your call as to whether 1.65 is close enough to 5/3
to qualify as proportional; or for F minor, whether 11/10 is simple
enough.

Anyway, it turns out that I came up with at least two other
synchonous temperaments (one well, the other _ordinaire_) that I like
better than this one. I'll share them once I've settled on the best
way to categorize the whole collection.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

10/24/2005 7:22:56 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> Gene, I'm impressed!
>
> But, OTOH, I hope you realize that there are reasonably simple brats
> in only 5 minor triads (the ones with just 5ths), and the rest are
> pretty much hit-or-miss (mostly the latter) -- much like what I got
> in my _temperament (extra)ordinaire_ a couple months back.

The beat ratios are Wendell's targets; what I did was precisely solve
for them. The brat for 12-et is 1.7133, so a mixture of 1.5 and 2.0
brats makes sense. The brats for 1.5 are 3/2, 0, infinity and their
inverses, which is nice. The 2 brats are not so good, being 2, -5,
-1/10 and their inverses. However, you need to go up to a brat of 4 to
get something simpler, giving 4, -1, -1/4, and you then need to
balance off the flatter fifths with either more pure fifths, or even
fifths with brats less than 3/2. Wendell liked the sound of these brat
2 triads, and they represent the easiest way to tackle the problem.

> In attempting to construct synchronous temperaments (with irrational
> intervals) with both proportional *major and minor* triads, I found
> that, unless the 5ths are exact, the minor-triad brats will at best
> only approximate simple ratios when the majors are exact (or vice
> versa), or you can compromise and make both inexact by about half and
> thereby get most of the brats to within 0.02 of simple numbers --
> close enough, I would say, considering that the required adjustment
> is very small.

That's correct. I tend to think making the major triad brats exact and
letting the minor triad brats be a little bit off makes the most sense.

> As to how close it comes to perfection, judge for yourself:
>
> Major -----Beat Ratios-----
> Triad M3/5th m3/5th m3/M3
> ----- ------- ------- -----
> Ab... ------- ------- 1.50
> Eb... ------- ------- 1.50
> Bb... ------- ------- 1.50
> F.... 7.01 13.02 1.86
> C.... 2.50 6.26 2.50
> G.... 2.50 6.26 2.50
> D.... 3.34 7.51 2.25
> A.... 5.01 10.01 5.00
> E.... 6.67 12.51 1.87
> B…... 16.63 27.45 1.65
> F#... 1467.47 2203.71 1.50
> C#... 1084.06 1628.58 1.50

You didn't like Wendell's brats of 2, but I can't see this is an
improvement. A 5/2 brat has a brat orbit of 5/2, -5/2, -4/25, which
doesn't seem as good as 2. The 5 brat has an orbit 5, -5/7, -7/25, and
I would urge you to dump it in favor of one or more brats of 4
instead. I don't know where the 1.86 and 1.85 come from, but a brat
of 13/7 has an orbit 13/7, -7, -1/13. 9/4 is another not very terrific
looking brat, with an orbit of 9/4, -10/3, -2/15.

I recommend using as many of the "magic" brats as possible; load up on
3/2 and 4, and fill in a gap with 2 or 3.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/25/2005 9:54:43 AM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> On Thu, 20 Oct 2005, "wallyesterpaulrus" wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> > >
> > > Hi Paul,
> > >
> > > On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote:
> > > >
> > > > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > > > <yahya@m...> wrote:
> > > >
> > > > > I can't really explain why the E major triad on E4 sounds
> > > > > so wrong to me. Is it the minor third of ~297 cents?
> > > >
> > > > I think that may very well be your explanation.
> > > >
> > > > > Perhaps all the beat frequencies for all my problem
> > > > > triads lie in a range that I react badly to.
> > > >
> > > > How did you implement this tuning, exactly?
> > >
> > > Exactly: I used the Keyboard Scale, + and - buttons of the
> > > Roland E-28 keyboard. Holding the Keyboard Scale button
> > > down for two seconds changes the tuning mode to one in
> > > which you can adjust the tuning of each note name (octave
> > > equivalent pitch class) independently by from -64 to +63
> > > cents, in steps of 1 cent, away from 12-EDO.
> > >
> > > It's also possible to toggle between the altered tuning and
> > > the standard 12-EDO tuning by tapping the Keyboard Scale
> > > button.
> > >
> > > >... Did you check the tuning
> > > > independently after you set it?
> > >
> > > No. I have no specialised tools for doing so. But my
> > > impression is that the frequency change per tap of the +
> > > and - buttons is [log] linear.
> >
> > Be suspicious. Monz did a test where he found that, for at least
one
> > synth model, the increments were not at all equal (in his case,
some
> > were one cent and some were two cents), even though nominally, if
you
> > trusted the manufacturer, they should have been. The same is most
> > definitely true of another synth, my Ensoniq. Synth manufacturers
> > have had very little market pressure to implement accurate
> > microtuning offsets. So you should *always* do an independent
check
> > when you care about 1 or 2 cents accuracy.
>
> Paul,
>
> I've answered (elsewhere) the suggestion that the C +50c might not
> equal C# -50c on some keyboards. I've demonstrated clearly on my
> Roland E-28 that the two are identical.
>
> To be doubly sure, I've just now checked that C +63c equals C# -37c.
>
> So I'm content to follow my ear on this one.
>
> Regards,
> Yahya

Neither of your checks does anything to ensure that the 100 steps per
semitone are equal -- and as Monz found with another synth, they may
very well be unequal.

🔗Graham Breed <gbreed@gmail.com>

10/25/2005 10:11:27 AM

Jon Szanto wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >>And amazingly, after all these years, it seems that still
>>it doesn't exist. That's exactly what i'm deep into right
>>now: studying how Csound .orc files would do GM.
> > I don't think it is that amazing, really, as most Csounders don't seem
> to be working with midi (or GM, for that matter). Why don't you post
> an open question, and see if Dave Seidel or Prent Rodgers or Bill
> Sethares or ??? can help you?

Enough Csounders are working with MIDI to get the MIDI support working and do a lot of work on optimizing the real-time support. Besides, a standard instrument library would be useful even if you don't use MIDI. I think the problem here arises from the academic nature of the project. Programming a real-time synthesis engine is an interesting project. Writing a user-friendly front-end is interesting. Implementing a novel form of synthesis is interesting. But getting some boring, general purpose instruments for newbies isn't interesting, and so it gets neglected.

It's a worthy project, certainly. I don't know how I could help with it now as I'm still a relative newcomer. I have some legato insruments with custom timbres that work well, but they're not very generic. If anybody has Sound Fonts that they like, you can try duplicating the envelopes and whatever else gets left out of Csound. The trouble is, all you will have is some wrapped up Sound Fonts and so the results won't be much different from any other Sound Font based synthesizer. Except for the tuning, which makes it a niche application.

I looked at the Fluid opcodes. They wrap up a MIDI-based synthesizer. So you get full Sound Font support, but you still have to address it with MIDI note numbers, not frequencies :( Retuning it will be as difficult as any MIDI synth.

There's also a Moog emulator in there.

If you post an open question, you should probably attract Stephen Yi's attention as well. He may have a way of abstracting instruments without tuning for Blue. You should also check Blue's feature set before claiming Csound doesn't do something ;)

> Cheers,
> Jon
> > P.S. I'm assuming you have "The Csound Book"...

Not that useful here, because it's also academically oriented, and so about interesting projects rather than quickly adapting your MIDI skills. Also a lot of it doesn't consider real-time use.

There's also a CD of instruments you can order, and some of them are good, but they're all instruments that somebody else cooked up for some reason; they're either academically interesting or specifically designed for some piece of music. It also takes a heck of a long time to go through them all to see if one of them is what you want.

Graham

🔗Jon Szanto <jszanto@cox.net>

10/25/2005 10:20:24 AM

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Enough Csounders are working with MIDI to get the MIDI support working
> and do a lot of work on optimizing the real-time support.

I'm always willing to stand corrected; I guess it is just that
virtually all of the Csound "end results" I have heard weren't
anything related to MIDI, but more custom crafted scores and
freer-form experimental pieces (say, for instance, like Prent for the
former and Dave Seidel for the latter). I know of the midi support,
but it just didn't seem that many were working in that direction.

> If you post an open question, you should probably attract Stephen Yi's
> attention as well. He may have a way of abstracting instruments
without
> tuning for Blue. You should also check Blue's feature set before
> claiming Csound doesn't do something ;)

Oh, Blue is veeery cool, one of the few things that *might* get me
into Csound some day. :)

Cheers,
Jon

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/25/2005 10:19:50 AM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> All of which makes me even more certain that I will never listen
> to an adaptive tuning without a distinct feeling of nausea!

Have you listened to any of John deLaubenfels's adaptive tuning
examples? How about ones that use the Vicentino approach? I found that
retunings of 11 cents give me a queasy feeling, but these approaches
typically, for pre-Beethoven music, don't exceed 6 cent retuning
motions, and my tummy is not disturbed by those. And you?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/25/2005 10:52:45 AM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> The question which still remains for me is: is the
> tuning resolution i discovered for my soundcard relevant
> only to MIDI, or to any sound it produces?

Only to MIDI -- it obviously wouldn't apply to a .wav file!

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/25/2005 11:03:44 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > it seems that all this exploration into equal-beating well
> > temperaments has
> > > given me a great question:
> > >
> > > what is the simplest possible 12-note temperament where all
> > > 24 major and minor
> > > triads have rationally proportional beating?
> >
> > If you are asking for a tuning where the brats are all rational,
> > any tuning with rational values, producing rational values of
> > fifths and thirds, will do.
>
> Bob Wendell's Natural Synchronous Well claimed to be this tuning.
> Brats of 2 or 1.5 or all 24 triads (except a few of the values are
> only very close). Bob made great claims for this scale, and it
> apparently got the attention of some piano tuners, and I've
> tried it on my MIDI keyboard with Scala. I wasn't immediately
> bowled over.

Few MIDI keyboards provide the necessary resolution to acheive aural
synchrony between the beat rates.

🔗Dave Seidel <dave@superluminal.com>

10/25/2005 11:09:01 AM

My use of MIDI with Csound has so far been limited to using hardware MIDI controllers to manipulate Csound in real time (i.e., for performance). Csound does have plenty of other MIDI capabilities, but I haven't explored them yet myself. I know it is capable of emitting MIDI events as well as consuming them. I've also seen, but not tried, .orc files that take realtime input from a MIDI source and output a new Csound .orc file.

I'm not sure what Monz means by "doing GM". If he means replicating the full set of instruments represented in GM, that seems like an enormous amount of work, not sure if it's worth the trouble. OTOH, if he means producing GM-compatible MIDI messages, that's probably a piece of cake, relatively speaking.

One combination I'm interested in exploring (I understand others have already done stuff like this) is a combination of Csound and Keykit, where KeyKit streams MIDI to Csound for rendering. With Csound's essentially infinite tuning capabilities, could be very powerful.

- Dave

Jon Szanto wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >>And amazingly, after all these years, it seems that still
>>it doesn't exist. That's exactly what i'm deep into right
>>now: studying how Csound .orc files would do GM.
> > > I don't think it is that amazing, really, as most Csounders don't seem
> to be working with midi (or GM, for that matter). Why don't you post
> an open question, and see if Dave Seidel or Prent Rodgers or Bill
> Sethares or ??? can help you?
> > Cheers,
> Jon
> > P.S. I'm assuming you have "The Csound Book"...

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/25/2005 11:22:48 AM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Without bothering to get any more scientific about it
> than i already have, i just think it's easiest and safest
> to assume that digital audio hardware is not going to
> give resolution better than 2 cents.
>
> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing
> music on a computer, it's probably useless to even
> contemplate anything finer than 600-edo.

Monz, you're equating computer music with MIDI or keyboard synths. Bad
equation. I can create .wav files for you with as much resolution as
mathematics (and a sampling rate of 44.1 KHz or whatever) will allow.

🔗Carl Lumma <clumma@yahoo.com>

10/25/2005 11:39:37 AM

> > > If you are asking for a tuning where the brats are all rational,
> > > any tuning with rational values, producing rational values of
> > > fifths and thirds, will do.
> >
> > Bob Wendell's Natural Synchronous Well claimed to be this tuning.
> > Brats of 2 or 1.5 or all 24 triads (except a few of the values
> > are only very close). Bob made great claims for this scale, and
> > it apparently got the attention of some piano tuners, and I've
> > tried it on my MIDI keyboard with Scala. I wasn't immediately
> > bowled over.
>
> Few MIDI keyboards provide the necessary resolution to acheive aural
> synchrony between the beat rates.

The pitch bend method I used gives, in theory, 4096 steps per
semitone. Many synths discard least significant bits of bend,
I'm told, and I haven't measure the performance of mine. But
since it's a digital additive synth, there's no reason for it
not to be accurate. I haven't tried to measure its accuracy,
but I suspect it's very accurate.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/25/2005 11:46:44 AM

> I'm not sure what Monz means by "doing GM". If he means
> replicating the full set of instruments represented in GM,

That's what we mean.

> that seems like an enormous amount of work, not sure if
> it's worth the trouble.

???

It's absolutely fundamental to have some sort of standard
orchestra.

How hard could it be? Even if they were just bad wavetables,
you've got to have some form of standard orchestra. What a
joke.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

10/25/2005 12:19:29 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > Gene, I'm impressed!
> >
> > But, OTOH, I hope you realize that there are reasonably simple
brats
> > in only 5 minor triads (the ones with just 5ths), and the rest
are
> > pretty much hit-or-miss (mostly the latter) -- much like what I
got
> > in my _temperament (extra)ordinaire_ a couple months back.
>
> The beat ratios are Wendell's targets; what I did was precisely
solve
> for them. The brat for 12-et is 1.7133, so a mixture of 1.5 and 2.0
> brats makes sense. The brats for 1.5 are 3/2, 0, infinity and their
> inverses, which is nice. The 2 brats are not so good, being 2, -5,
> -1/10 and their inverses. However, you need to go up to a brat of 4
to
> get something simpler, giving 4, -1, -1/4, and you then need to
> balance off the flatter fifths with either more pure fifths, or even
> fifths with brats less than 3/2. Wendell liked the sound of these
brat
> 2 triads, and they represent the easiest way to tackle the problem.

I didn't take a really close look at his temperaments until after I
did most of the work on mine, and I see that my approach is quite
different.

> > In attempting to construct synchronous temperaments (with
irrational
> > intervals) with both proportional *major and minor* triads, I
found
> > that, unless the 5ths are exact, the minor-triad brats will at
best
> > only approximate simple ratios when the majors are exact (or vice
> > versa), or you can compromise and make both inexact by about half
and
> > thereby get most of the brats to within 0.02 of simple numbers --
> > close enough, I would say, considering that the required
adjustment
> > is very small.
>
> That's correct. I tend to think making the major triad brats exact
and
> letting the minor triad brats be a little bit off makes the most
sense.

I agree, and that's exactly what I did with all of the other
synchronous temperaments I worked on over the past couple of months.
With this one there was actually a completely different reason for
making both the major and minor brat ratios approximate: if the tones
in a chain from Ab thru G# are pitched so that that brat ratios for
major triads Ab thru E are exact then, G# is 0.01661 cents lower in
pitch than Ab; widening all of the tempered fifths (spanning F thru
G#) by 1/9 of that miniscule amount makes them the same pitch.
(Hmmm, I just noticed that this improves some of the minor 3rd brat
ratios but worsens others, but fortunately it's only a slight change -
- oh, well!)

> > As to how close it comes to perfection, judge for yourself:
> >
> > Major -----Beat Ratios-----
> > Triad M3/5th m3/5th m3/M3
> > ----- ------- ------- -----
> > Ab... ------- ------- 1.50
> > Eb... ------- ------- 1.50
> > Bb... ------- ------- 1.50
> > F.... 7.01 13.02 1.86
> > C.... 2.50 6.26 2.50
> > G.... 2.50 6.26 2.50
> > D.... 3.34 7.51 2.25
> > A.... 5.01 10.01 2.00 [corrected from 5.00]
> > E.... 6.67 12.51 1.87
> > B…... 16.63 27.45 1.65
> > F#... 1467.47 2203.71 1.50
> > C#... 1084.06 1628.58 1.50
>
> You didn't like Wendell's brats of 2, but I can't see this is an
> improvement.

Oh, I liked it okay, because I used it in my A major triad (note that
I had the wrong figure in the 3rd column of the above table, which
I've now noted as corrected).

> A 5/2 brat has a brat orbit of 5/2, -5/2, -4/25, which
> doesn't seem as good as 2. The 5 brat has an orbit 5, -5/7, -7/25,
and
> I would urge you to dump it in favor of one or more brats of 4
> instead.

My primary objective was to keep the total absolute error of the
major triads for F, C, and G under 20 cents and Bb and D under 24
cents (which is to say, 5 triads significantly better than 12-ET;
Wendell has only 3). I therefore selected brat ratios that would not
only accomplish this, but also create a smooth change in intonation
as one moves about the circle of fifths (whereas Wendell has all the
major triads from D to B slightly worse than 12-ET).

In other words, my priorities were different. Synchronized beating
is a very subtle effect, and it requires considerable tuning
precision. The error of the individual intervals from just
intonation is much more obvious, so I have selected brats that will
produce a decent well-temperament.

> I don't know where the 1.86 and 1.85 come from, but a brat
> of 13/7 has an orbit 13/7, -7, -1/13.

The untweaked value for 1.86 is 1.85714285714..., or 13/7.

I didn't have 1.85 anywhere; did you mean 1.87? Its untweaked value
is 1.875.

> 9/4 is another not very terrific
> looking brat, with an orbit of 9/4, -10/3, -2/15.
>
> I recommend using as many of the "magic" brats as possible; load up
on
> 3/2 and 4, and fill in a gap with 2 or 3.

Okay, I'll take another look at this. I just think that a lot of the
minor 3rd brats won't be as simple as we would like.

--George

🔗Jon Szanto <jszanto@cox.net>

10/25/2005 12:23:22 PM

--- In tuning@yahoogroups.com, Dave Seidel <dave@s...> wrote:
> One combination I'm interested in exploring (I understand others have
> already done stuff like this) is a combination of Csound and Keykit,
> where KeyKit streams MIDI to Csound for rendering. With Csound's
> essentially infinite tuning capabilities, could be very powerful.

Cool. I'm using KeyKit and routing it to VST instruments that already
are microtuneable. I haven't done much new coding in KeyKit, but it
seems like a good environment to be doing some basic gymnastics.
Please report if you get your implementation working (I'm guessing it
could be done with the CsoundVST?).

Cheers,
Jon

🔗Jon Szanto <jszanto@cox.net>

10/25/2005 12:28:38 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> It's absolutely fundamental to have some sort of standard orchestra.

Fundamental only if you want to make orchestral mockups. When I
suggested that velocity sensitivity was fundamental in a keyboard, you
pointed out a couple areas where it wasn't necessary; I'd say that
there are a few areas of music where an orchestra isn't necessary. :)

> How hard could it be?

Maybe not hard, but time consuming? The question then becomes, who
will sit down and do it?

> Even if they were just bad wavetables,
> you've got to have some form of standard orchestra. What a
> joke.

When I hear another 'orchestral' piece with bad sounds, that is when
the comedy starts!

Cheers,
Jon

🔗Dave Seidel <dave@superluminal.com>

10/25/2005 12:45:13 PM

I understand what you're saying, but you need to keep in mind the history of Csound, and what it is (and has been) very often used for, which is instrument design from scratch and experimentation with new forms of synthesis.

Does Max/MSP supply a standard GM instrument library? Does PD? What makes you think Csound is or should be a general purpose synth?

Here's the first sentence from "What is Csound?" (http://www.csounds.com/whatis/index.html): "Csound is a programming language designed and optimized for sound rendering and signal processing."

- Dave

Carl Lumma wrote:
>>I'm not sure what Monz means by "doing GM". If he means
>>replicating the full set of instruments represented in GM,
> > > That's what we mean.
> > >>that seems like an enormous amount of work, not sure if
>>it's worth the trouble.
> > > ???
> > It's absolutely fundamental to have some sort of standard
> orchestra.
> > How hard could it be? Even if they were just bad wavetables,
> you've got to have some form of standard orchestra. What a
> joke.
> > -Carl

🔗Carl Lumma <clumma@yahoo.com>

10/25/2005 12:53:12 PM

> > It's absolutely fundamental to have some sort of standard
> > orchestra.
>
> Fundamental only if you want to make orchestral mockups.

I menat "orchestra" in a general sense here. Note that GM
contains things that aren't traditional elements of an
orchestra, like funk bass.

> > Even if they were just bad wavetables,
> > you've got to have some form of standard orchestra. What a
> > joke.
>
> When I hear another 'orchestral' piece with bad sounds, that is
> when the comedy starts!

At least it would (presumably) have the tunability of csound.
Bad orchestral sounds are still great for mock-ups.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/25/2005 12:55:55 PM

> I understand what you're saying, but you need to keep in mind the
> history of Csound, and what it is (and has been) very often used
> for, which is instrument design from scratch and experimentation
> with new forms of synthesis.
>
> Does Max/MSP supply a standard GM instrument library? Does PD?
> What makes you think Csound is or should be a general purpose
> synth?

I guess I'm just shocked that nobody's done it.

-Carl

🔗Dave Seidel <dave@superluminal.com>

10/25/2005 1:19:19 PM

I don't do any VST, so I'm not sure yet how it will work. I think that the command-line version of Csound has a way of receiving events on a pipe or something. Let you know if and when I get there. :-)

- Dave

Jon Szanto wrote:
> Cool. I'm using KeyKit and routing it to VST instruments that already
> are microtuneable. I haven't done much new coding in KeyKit, but it
> seems like a good environment to be doing some basic gymnastics.
> Please report if you get your implementation working (I'm guessing it
> could be done with the CsoundVST?).
> > Cheers,
> Jon

🔗Gene Ward Smith <gwsmith@svpal.org>

10/25/2005 3:12:29 PM

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> I think the problem here arises from the academic nature of the
> project. Programming a real-time synthesis engine is an interesting
> project. Writing a user-friendly front-end is interesting.
> Implementing a novel form of synthesis is interesting. But getting
some
> boring, general purpose instruments for newbies isn't interesting, and
> so it gets neglected.

In other words, most Csound people are interested in making weird
sounds, not music. Music mostly requires normal sounds, which
apparently are boring. However, there *is* interest in emulating
acoustic instruments, and I find it surprising this hasn't led to a
suite of GM instruments by now.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/25/2005 3:30:59 PM

--- In tuning@yahoogroups.com, Dave Seidel <dave@s...> wrote:

> I'm not sure what Monz means by "doing GM". If he means replicating
the
> full set of instruments represented in GM, that seems like an enormous
> amount of work, not sure if it's worth the trouble.

It depends on whether Csound is supposed to be used for music or not,
I suppose.

🔗Jon Szanto <jszanto@cox.net>

10/25/2005 4:05:20 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> In other words, most Csound people are interested in making weird
> sounds, not music. Music mostly requires normal sounds, which
> apparently are boring.

Another quote to add to my collection. Amazing.

Cheers,
Jon

🔗Dave Seidel <dave@superluminal.com>

10/25/2005 7:23:26 PM

Are you serious, Gene? Are you really that close-minded, or are these gratuitous remarks just a grumpy reflex?

You're welcome to think what you wish of my music, of course, although I don't assume that you've heard any of it (if you're curious, which I doubt, go to http://mysterybear.net). In point of fact, I use Csound when I want to make certain kinds of music, and I use MIDI-driven emulated instruments when I want to make other kinds of music. Different tools are suited for different jobs, and it's ridiculous to try to make a square peg fit in a round hole.

You might be (genuinely) interested in the music of Prent Rodgers (http://prodgers13.home.comcast.net/), who uses Csound to manipulate sampled instruments, with marvelous results. No crappy emulation, either.

- Dave

Gene Ward Smith wrote:
> In other words, most Csound people are interested in making weird
> sounds, not music. Music mostly requires normal sounds, which
> apparently are boring. However, there *is* interest in emulating
> acoustic instruments, and I find it surprising this hasn't led to a
> suite of GM instruments by now.

Gene Ward Smith wrote:
> It depends on whether Csound is supposed to be used for music or not,
> I suppose.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/25/2005 8:37:03 PM

Hi Paul,

On Tue, 25 Oct 2005, "wallyesterpaulrus" wrote:
[much snipt]
> > I've answered (elsewhere) the suggestion that the C +50c might not
> > equal C# -50c on some keyboards. I've demonstrated clearly on my
> > Roland E-28 that the two are identical.
> >
> > To be doubly sure, I've just now checked that C +63c equals C# -37c.
> > So I'm content to follow my ear on this one.
>
> Neither of your checks does anything to ensure that the 100 steps per
> semitone are equal -- and as Monz found with another synth, they may
> very well be unequal.

By doing a few more of these checks (say about 24 more), I could
at best show that the steps on the "down" side equal those on the
"up", at least in the region of overlap; that is, that each of the notes
C+37c to C+63c match the notes C#-63c to C#-37c.

So you're right, of course; I could easily have alternating steps of
sizes, say 2/3 and 4/3 of a cent. But I repeat my assertion that
the steps seem equal to me. I have no measurements to back this
up, just my ear.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/25/2005 8:37:07 PM

On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > All of which makes me even more certain that I will never listen
> > to an adaptive tuning without a distinct feeling of nausea!
>
> Have you listened to any of John deLaubenfels's adaptive tuning
> examples? How about ones that use the Vicentino approach? I found that
> retunings of 11 cents give me a queasy feeling, but these approaches
> typically, for pre-Beethoven music, don't exceed 6 cent retuning
> motions, and my tummy is not disturbed by those. And you?

Those delights yet await me. I should do so. Have you links?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Gene Ward Smith <gwsmith@svpal.org>

10/25/2005 9:03:10 PM

--- In tuning@yahoogroups.com, Dave Seidel <dave@s...> wrote:
>
> Are you serious, Gene? Are you really that close-minded, or are these
> gratuitous remarks just a grumpy reflex?

I'm pretty much of a grumpy reflex guy, but the only real music I know
of produced with Csound is by Prent Rodgers, and he uses it in a
highly idiosyncratic way. Then again, I haven't heard *your* music,
which I will immediately listen to now.

However, it's clear historically that Csound started out as an MIT
engineering project, and it seems to me that judging from things scuh
as the Csound book and the Csound mailing list, sound engineering and
not music is the real interest of most of the people involved.

> You might be (genuinely) interested in the music of Prent Rodgers
> (http://prodgers13.home.comcast.net/), who uses Csound to manipulate
> sampled instruments, with marvelous results. No crappy emulation,
either.

I love Prent's music. We've exchanged emails and CDs, and one thing is
clear--Prent is interested in Csound as a means of making *music*. I
don't see a lot of that in the Csound world. Or at least, that is my
grumpy reflex at the moment. It seems to me that if there was more
interest in music anong the Soundrs there would obviously be more
Csound instruments with musical qualities publically available.

🔗Aaron Krister Johnson <aaron@akjmusic.com>

10/25/2005 9:44:49 PM

On Tuesday 25 October 2005 5:12 pm, Gene Ward Smith wrote:

> In other words, most Csound people are interested in making weird
> sounds, not music. Music mostly requires normal sounds, which
> apparently are boring. However, there *is* interest in emulating
> acoustic instruments, and I find it surprising this hasn't led to a
> suite of GM instruments by now.

should a the output moog modular synth not be considered 'music'?

-aaron.

🔗monz <monz@tonalsoft.com>

10/25/2005 11:33:05 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> > I think the problem here arises from the academic nature
> > of the project. Programming a real-time synthesis engine
> > is an interesting project. Writing a user-friendly
> > front-end is interesting. Implementing a novel form of
> > synthesis is interesting. But getting some boring,
> > general purpose instruments for newbies isn't interesting,
> > and so it gets neglected.
>
> In other words, most Csound people are interested in
> making weird sounds, not music. Music mostly requires
> normal sounds, which apparently are boring. However,
> there *is* interest in emulating acoustic instruments,
> and I find it surprising this hasn't led to a suite
> of GM instruments by now.

Unfortunately, my perceptions agree exactly with what
both of you wrote here. I was amazed, after doing a *lot*
of searching, that there is no catalog of GM instruments
for Csound ... but you can easily find *thousands* of
weird "experiemental" instruments.

It's taken me several weeks to create a decent Csound
nylon-string guitar instrument, which i *finally*
accomplished today to my satisfaction.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

10/25/2005 11:41:16 PM

Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> >
> > > All of which makes me even more certain that I will never listen
> > > to an adaptive tuning without a distinct feeling of nausea!
> >
> > Have you listened to any of John deLaubenfels's
> > adaptive tuning examples? How about ones that use the
> > Vicentino approach? I found that retunings of 11 cents
> > give me a queasy feeling, but these approaches typically,
> > for pre-Beethoven music, don't exceed 6 cent retuning
> > motions, and my tummy is not disturbed by those. And you?
>
> Those delights yet await me. I should do so. Have you links?

John deLaubenfels's music page is here:

http://personalpages.bellsouth.net/j/d/jdelaub/jstudio.htm

Note also that he has retuned a lot of my own 12-edo
stuff, and my MIDI files of pieces such as Schoenberg's
_Verklaerte Nacht_, into adaptive-tuning.

For Vicentino's adaptive-JI, i provide this:

http://tonalsoft.com/enc/a/adaptive-ji.aspx

If you click on the first graphic, it takes you to a
MIDI of a Lasso piece which i specifically chose because
Easley Blackwood uses it to demonstrate the unsuitability
of strict-JI for Renaissance music.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/26/2005 1:09:06 AM

On Tue, 25 Oct 2005, "Carl Lumma" wrote:
>
> > I'm not sure what Monz means by "doing GM". If he means
> > replicating the full set of instruments represented in GM,
>
> That's what we mean.
>
> > that seems like an enormous amount of work, not sure if
> > it's worth the trouble.
>
> ???
>
> It's absolutely fundamental to have some sort of standard
> orchestra.
>
> How hard could it be? Even if they were just bad wavetables,
> you've got to have some form of standard orchestra. What a
> joke.

Hi Carl,

That's only true, of course, if you're trying to make orchestral
music. Personally, I detest the sound of large numbers of
strings played together, and find very few uses for them in
my music. I much prefer the texture of, say, a string quartet,
or other small ensemble.

But perhaps you would regard a collection of the usual chamber
music instruments as "some form of standard orchestra".

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/26/2005 6:08:50 AM

Hi George and Gene,

On Tue, 25 Oct 2005 "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor"
> <gdsecor@y...> wrote:
>
> > Gene, I'm impressed!
> >
> > But, OTOH, I hope you realize that there are reasonably simple brats
> > in only 5 minor triads (the ones with just 5ths), and the rest are
> > pretty much hit-or-miss (mostly the latter) -- much like what I got
> > in my _temperament (extra)ordinaire_ a couple months back.
>
> The beat ratios are Wendell's targets; what I did was precisely solve
> for them. The brat for 12-et is 1.7133, so a mixture of 1.5 and 2.0
> brats makes sense. The brats for 1.5 are 3/2, 0, infinity and their
> inverses, which is nice. The 2 brats are not so good, being 2, -5,
> -1/10 and their inverses. However, you need to go up to a brat of 4 to
> get something simpler, giving 4, -1, -1/4, and you then need to
> balance off the flatter fifths with either more pure fifths, or even
> fifths with brats less than 3/2. Wendell liked the sound of these brat
> 2 triads, and they represent the easiest way to tackle the problem.
>
> > In attempting to construct synchronous temperaments (with irrational
> > intervals) with both proportional *major and minor* triads, I found
> > that, unless the 5ths are exact, the minor-triad brats will at best
> > only approximate simple ratios when the majors are exact (or vice
> > versa), or you can compromise and make both inexact by about half and
> > thereby get most of the brats to within 0.02 of simple numbers --
> > close enough, I would say, considering that the required adjustment
> > is very small.
>
> That's correct. I tend to think making the major triad brats exact and
> letting the minor triad brats be a little bit off makes the most sense.
>
> > As to how close it comes to perfection, judge for yourself:
> >

[snipt]

I've tried to follow this thread, but without much success.
I see you both doing some maths to make the ratios of beats
("brats") proportional - but still don't understand why. So
please tell me -

What is the *musical* purpose of having proportional brats?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/26/2005 6:43:24 AM

Hi Monz,

On Wed, 26 Oct 2005 "monz" wrote:
>
> Hi Yahya,
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > > <yahya@m...> wrote:
> > >
> > > > All of which makes me even more certain that I will never listen
> > > > to an adaptive tuning without a distinct feeling of nausea!
> > >
> > > Have you listened to any of John deLaubenfels's
> > > adaptive tuning examples? How about ones that use the
> > > Vicentino approach? I found that retunings of 11 cents
> > > give me a queasy feeling, but these approaches typically,
> > > for pre-Beethoven music, don't exceed 6 cent retuning
> > > motions, and my tummy is not disturbed by those. And you?
> >
> > Those delights yet await me. I should do so. Have you links?
>
>
> John deLaubenfels's music page is here:
>
> http://personalpages.bellsouth.net/j/d/jdelaub/jstudio.htm
>
>
> Note also that he has retuned a lot of my own 12-edo
> stuff, and my MIDI files of pieces such as Schoenberg's
> _Verklaerte Nacht_, into adaptive-tuning.
>
>
> For Vicentino's adaptive-JI, i provide this:
>
> http://tonalsoft.com/enc/a/adaptive-ji.aspx
>
> If you click on the first graphic, it takes you to a
> MIDI of a Lasso piece which i specifically chose because
> Easley Blackwood uses it to demonstrate the unsuitability
> of strict-JI for Renaissance music.

Thank you very much for these links!

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/26/2005 6:43:23 AM

Monz,

On Wed, 26 Oct 2005, "monz" wrote:
>
> [snip] ... I was amazed, after doing a *lot*
> of searching, that there is no catalog of GM instruments
> for Csound ... but you can easily find *thousands* of
> weird "experiemental" instruments.
>
> It's taken me several weeks to create a decent Csound
> nylon-string guitar instrument, which i *finally*
> accomplished today to my satisfaction.

Now *there's* an achievement! Well done ...
Is this perhaps the first in a usable "orchestra" of
CSound emulations of traditional instruments?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Aaron Krister Johnson <aaron@akjmusic.com>

10/26/2005 8:22:07 AM

On Wednesday 26 October 2005 8:43 am, Yahya Abdal-Aziz wrote:
> Monz,
>
> On Wed, 26 Oct 2005, "monz" wrote:
> > [snip] ... I was amazed, after doing a *lot*
> > of searching, that there is no catalog of GM instruments
> > for Csound ... but you can easily find *thousands* of
> > weird "experiemental" instruments.
> >
> > It's taken me several weeks to create a decent Csound
> > nylon-string guitar instrument, which i *finally*
> > accomplished today to my satisfaction.
>
> Now *there's* an achievement! Well done ...
> Is this perhaps the first in a usable "orchestra" of
> CSound emulations of traditional instruments?

it appears that Prent Rodgers uses samples of trad instruments to good effect
in csound.

-aaron.

🔗Jon Szanto <jszanto@cox.net>

10/26/2005 9:04:03 AM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> it appears that Prent Rodgers uses samples of trad instruments to
good effect
> in csound.

Yes. Prent has done a lot of work on them over the years, so it
definitely wasn't "plug and play". I believe many of his initial
samples came from the McGill University sample collection. Lots of
people have worked with samples for years (obviously); for a GM set
for use in Csound, it would take someone some time and initiative to
get a set ready. There is a fairly complete set of orchestral
instrument samples available one could start with from the University
of Iowa:

http://theremin.music.uiowa.edu/MIS.html

Cheers,
Jon

🔗Dave Seidel <dave@superluminal.com>

10/26/2005 11:17:15 AM

BTW, Csound includes SoundFont support as well.

- Dave

Jon Szanto wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> >>it appears that Prent Rodgers uses samples of trad instruments to
> > good effect > >>in csound.
> > > Yes. Prent has done a lot of work on them over the years, so it
> definitely wasn't "plug and play". I believe many of his initial
> samples came from the McGill University sample collection. Lots of
> people have worked with samples for years (obviously); for a GM set
> for use in Csound, it would take someone some time and initiative to
> get a set ready. There is a fairly complete set of orchestral
> instrument samples available one could start with from the University
> of Iowa:
> > http://theremin.music.uiowa.edu/MIS.html
> > Cheers,
> Jon
> > > > > > > You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> > > > > > >

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 11:27:57 AM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> should a the output moog modular synth not be considered 'music'?

It's exactly like the output of a cello--it would depend on how it was
composed.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 11:31:29 AM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Unfortunately, my perceptions agree exactly with what
> both of you wrote here. I was amazed, after doing a *lot*
> of searching, that there is no catalog of GM instruments
> for Csound ... but you can easily find *thousands* of
> weird "experiemental" instruments.

I did warn you.

> It's taken me several weeks to create a decent Csound
> nylon-string guitar instrument, which i *finally*
> accomplished today to my satisfaction.

What's Tonalsoft going to do with these? Anything independently of
Tonescape?

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 11:48:01 AM

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> What is the *musical* purpose of having proportional brats?

It's a very subtle effect which requires correct tuning, but Wendell
believes it amelorates the mistuning of triads, and in some cases I
think that is true. Unfortunately, the most interesting of these brats
is -1, what I've taken to calling in a more general context (higher
prime limits) "synch tuning", and that isn't well adapated to
circulating temperaments of twelve notes based on meantone. Of course,
Kraig Grady can tell you about Wilson and his Mt Maru scales, and
there's twelve notes of Wilson meantone ("wilwolf") etc to consider.

I think proportional beating is pretty interesting in the ill-tuned,
low complexity temperaments, and Herman has made a few experiments
along those lines.

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 12:26:56 PM

> > It's absolutely fundamental to have some sort of standard
> > orchestra.
> >
> > How hard could it be? Even if they were just bad wavetables,
> > you've got to have some form of standard orchestra. What a
> > joke.
>
> Hi Carl,
>
> That's only true, of course, if you're trying to make orchestral
> music. Personally, I detest the sound of large numbers of
> strings played together, and find very few uses for them in
> my music. I much prefer the texture of, say, a string quartet,
> or other small ensemble.

I happen to agree totally, and have written only chamber and
keyboard music. I used GM to specify instrumentation anyway.

> But perhaps you would regard a collection of the usual chamber
> music instruments as "some form of standard orchestra".

As I said in a later message: yes.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 1:03:10 PM

> > What is the *musical* purpose of having proportional brats?
>
> It's a very subtle effect which requires correct tuning, but
> Wendell believes it amelorates the mistuning of triads, and in
> some cases I think that is true.

I've done some brat comparisons between similarly-mistuned
chords with Cool Edit, and I can't agree. It's a very subtle
effect, quite different from overall mistuning.

> Unfortunately, the most interesting of these brats is -1,

Why is it most interesting? Did you determine this by listening?

> Kraig Grady can tell you about Wilson and his Mt Maru
> scales, and there's twelve notes of Wilson meantone ("wilwolf")
> etc to consider.

I believe those scales also are intended to work with
combination tones.

> I think proportional beating is pretty interesting in the
> ill-tuned, low complexity temperaments, and Herman has made
> a few experiments along those lines.

These were not controlled for key. I'd be shocked if the
improvement noticed in the porcupine overture could be
attributed to brats. Herman, can you tell us what key you
used in blackbeat15.scl, and if it was one of its better
ones?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 1:12:40 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> > Unfortunately, the most interesting of these brats is -1,
>
> Why is it most interesting? Did you determine this by listening?

I do find the sound of it interesting, but this is really a
theory-based claim. Synch beating is the special case of everything
beating at the same rate.

🔗monz <monz@tonalsoft.com>

10/26/2005 3:15:23 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > It's taken me several weeks to create a decent Csound
> > nylon-string guitar instrument, which i *finally*
> > accomplished today to my satisfaction.
>
> What's Tonalsoft going to do with these? Anything
> independently of Tonescape?

Tonescape has taken the big leap from MIDI to self-produced
WAV files. We're writing a synthesis engine which does
direct digital synthesis. So i'm using Csound as a
learning tool right now.

We do have plans (for pretty far in the future, i guess)
to have other products besides Tonescape, and so this
synthesis engine is an important part of several of them.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/26/2005 4:17:30 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Unfortunately, the most interesting of these brats is -1,
> >
> > Why is it most interesting? Did you determine this by listening?
>
> I do find the sound of it interesting, but this is really a
> theory-based claim. Synch beating is the special case of everything
> beating at the same rate.

Specifically, when and only when the "brat" is -1, the three intervals
in the close-voiced root-position major triad (major third, minor
third, perfect fifth) all produce the same rate of beating. All major
triads in Wilson's metameantone have this property . . .

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 4:30:26 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Specifically, when and only when the "brat" is -1, the three intervals
> in the close-voiced root-position major triad (major third, minor
> third, perfect fifth) all produce the same rate of beating. All major
> triads in Wilson's metameantone have this property . . .

And it isn't just the closed-voice root position triads which do this,
either.

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 5:31:11 PM

> > Specifically, when and only when the "brat" is -1, the three
> > intervals in the close-voiced root-position major triad (major
> > third, minor third, perfect fifth) all produce the same rate
> > of beating. All major triads in Wilson's metameantone have this
> > property . . .
>
> And it isn't just the closed-voice root position triads which do
> this, either.

I'm still not sure why this is supposed to be desirable, other
than it makes creating bearing plans much easier.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/26/2005 5:34:39 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > Specifically, when and only when the "brat" is -1, the three
intervals
> > in the close-voiced root-position major triad (major third, minor
> > third, perfect fifth) all produce the same rate of beating. All
major
> > triads in Wilson's metameantone have this property . . .
>
> And it isn't just the closed-voice root position triads which do this,
> either.

Could you list all that do?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/26/2005 5:44:32 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Specifically, when and only when the "brat" is -1, the three
> > > intervals in the close-voiced root-position major triad (major
> > > third, minor third, perfect fifth) all produce the same rate
> > > of beating. All major triads in Wilson's metameantone have this
> > > property . . .
> >
> > And it isn't just the closed-voice root position triads which do
> > this, either.
>
> I'm still not sure why this is supposed to be desirable, other
> than it makes creating bearing plans much easier.
>
> -Carl

I'm not sure how you'd use this for a bearing plan -- Jorgenson likes
tunings where different instances of an interval across the keyboard
have the same rate of beating, but this is not at all the case for
regular metameantone, where the beat rates are (only?) synchronized
within a single major triad at a time.

But some have said that when all the beatings in each chord are
synchronized, the effect is more coherent and less chaotic than when
there are multiple, unrelated rates of beating all occuring at the
same time.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/26/2005 6:05:20 PM

Aaron,

On Wed, 26 Oct 2005 you wrote:
[edited]

[monz]
> > > It's taken me several weeks to create a decent Csound
> > > nylon-string guitar instrument, which i *finally*
> > > accomplished today to my satisfaction.

[Yahya]
> > Now *there's* an achievement! Well done ...
> > Is this perhaps the first in a usable "orchestra" of
> > CSound emulations of traditional instruments?

[aaron]
> it appears that Prent Rodgers uses samples of trad
> instruments to good effect in csound.

My bad. Prent was kind enough to send me a CD of
his music, and I must agree that the sounds are good.
I'd forgotten he's a CSound user.

What I had in mind was that perhaps CSound users
might begin to *share* the instruments that they've
created, so that other users might get a leg up.

At what price? Since it obviously takes quite a deal
of effort to create them, they've got to be worth
something to new users. Still, the community of
CSound users might think it worthwhile to give some
instruments away, as an inducement to get others
using it, thus swelling their ranks ...

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

10/26/2005 6:48:32 PM

On Wed, 26 Oct 2005 "Jon Szanto" wrote:
>
> --- In tuning@yahoogroups.com, Aaron Krister Johnson
> <aaron@a...> wrote:
> > it appears that Prent Rodgers uses samples of trad
> > instruments to good effect in csound.
>
> Yes. Prent has done a lot of work on them over the years, so it
> definitely wasn't "plug and play". I believe many of his initial
> samples came from the McGill University sample collection. Lots of
> people have worked with samples for years (obviously); for a GM set
> for use in Csound, it would take someone some time and initiative to
> get a set ready. There is a fairly complete set of orchestral
> instrument samples available one could start with from the University
> of Iowa:
>
> http://theremin.music.uiowa.edu/MIS.html

Jon,
This looks like it has good coverage of the commoner
instruments. One thing that concerns me is that the
samples are in .aiff format. Didn't someone recently
(you, perhaps) comment on list that AIF files don't
have the best resolution available, and that some other
format would be preferable? From the web page you
linked to:

"All samples are in mono, 16 bit, 44.1 kHz, AIFF format."

According to the Nyquist limit theorem, 44.1 kHz would
give adequate sampling of all frequencies up to 22.05
kHz, right? Which pretty much covers the musical
spectrum. Am I missing something here?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗Herman Miller <hmiller@IO.COM>

10/26/2005 7:12:01 PM

Carl Lumma wrote:
> > These were not controlled for key. I'd be shocked if the
> improvement noticed in the porcupine overture could be
> attributed to brats. Herman, can you tell us what key you
> used in blackbeat15.scl, and if it was one of its better
> ones?

Since there wasn't any indication of what was supposed to be the "better" key of the scale, I just tuned C (MIDI note 60) to the usual 261.63 Hz and used that as the first note of the scale. The music actually starts out in the key of "MIDI note 64" (more or less "E flat") minor, but it happens to be in C major from around 1:55 to around 2:10 and then again briefly around 2:30. It keeps getting back around to "Eb" minor.

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 10:53:42 PM

> > And it isn't just the closed-voice root position triads which
> > do this, either.
>
> Could you list all that do?

With pure octaves, all inversions/voicings will, right?

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 10:56:38 PM

> > I'm still not sure why this is supposed to be desirable, other
> > than it makes creating bearing plans much easier.
> >
> > -Carl
>
> I'm not sure how you'd use this for a bearing plan -- Jorgenson
> likes tunings where different instances of an interval across
> the keyboard have the same rate of beating, but this is not at
> all the case for regular metameantone, where the beat rates are
> (only?) synchronized within a single major triad at a time.

Bearing plans often include things like, 'tune the E until
the major third C-E beats at the same rate as the minor third
E-G'.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 11:02:14 PM

> > http://theremin.music.uiowa.edu/MIS.html
>
> Jon,
> This looks like it has good coverage of the commoner
> instruments. One thing that concerns me is that the
> samples are in .aiff format. Didn't someone recently
> (you, perhaps) comment on list that AIF files don't
> have the best resolution available, and that some other
> format would be preferable? From the web page you
> linked to:
>
> "All samples are in mono, 16 bit, 44.1 kHz, AIFF format."
>
> According to the Nyquist limit theorem, 44.1 kHz would
> give adequate sampling of all frequencies up to 22.05
> kHz, right? Which pretty much covers the musical
> spectrum. Am I missing something here?

Nope, AIFF is just Apple's version of WAV (some Mac
person will probably claim it was the other way around,
but whatever). It's uncompressed audio. The resolution
depends on the resolution chosen when recording it.

For samples, though, there are a ton available for
free in SoundFont format...

http://www.hammersound.net/
http://www.personalcopy.com/

I believe there are free utilities that can extract
individual samples from SoundFont files.

But no need to do so -- Dave says Csound has
SoundFont support (I wonder if they remain as tunable
as a normal Csound instrument?).

Is there anything like the above sites for Csound
patches??

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/26/2005 11:15:21 PM

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> Carl Lumma wrote:
> >
> > These were not controlled for key. I'd be shocked if the
> > improvement noticed in the porcupine overture could be
> > attributed to brats. Herman, can you tell us what key you
> > used in blackbeat15.scl, and if it was one of its better
> > ones?
>
> Since there wasn't any indication of what was supposed to be the
> "better" key of the scale, I just tuned C (MIDI note 60) to the
> usual 261.63 Hz and used that as the first note of the scale.
> The music actually starts out in the key of "MIDI note 64" (more
> or less "E flat") minor, but it happens to be in C major from
> around 1:55 to around 2:10 and then again briefly around 2:30.
> It keeps getting back around to "Eb" minor.

Is Eb the 4th line of the Scala file? And C the first?

If so, the C major 3rd is 3 cents better than in 15-equal.
A small difference, but one audible to me in my well
temperament adventures (12-equal also has 400-cent 3rds).

The 5ths are all the same as 15-equal, but C and Eb are
2 of the 5 keys whose minor 3rds are 6 cents better, and
very near just. This is especially big because the zone
above 6/5 gets dissonant very rapidly (it's why 15-equal
doesn't really have better minor 3rds than 12).

The 7:4s are all the same as in 15-equal.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 11:51:56 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Could you list all that do?

"All" would require a lemma, I suppose, but start from

2:3:5
4:5:6
3:4:10
3:8:20
3:16:40
...
5:8:12
5:16:24
5:32:48
...

At least two infinite families.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/26/2005 11:57:36 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > And it isn't just the closed-voice root position triads which
> > > do this, either.
> >
> > Could you list all that do?
>
> With pure octaves, all inversions/voicings will, right?

Up to powers of two. Instead of getting 1,1,1 you might get 1,1,2, etc.

🔗Graham Breed <gbreed@gmail.com>

10/27/2005 5:02:20 AM

Carl Lumma wrote:

> Nope, AIFF is just Apple's version of WAV (some Mac
> person will probably claim it was the other way around,
> but whatever). It's uncompressed audio. The resolution
> depends on the resolution chosen when recording it.

I'm not sure what exactly they are. Wav files are a generic wrapper for any encoding that's registered with Windows, but many applications that work with Wav files aren't this flexible. AIFF files, as usually understood, allow you to set loop points while Wav files don't. So AIFF is more appropriate for sample libraries.

> For samples, though, there are a ton available for
> free in SoundFont format...
> > http://www.hammersound.net/
> http://www.personalcopy.com/
> > I believe there are free utilities that can extract
> individual samples from SoundFont files.
> > But no need to do so -- Dave says Csound has
> SoundFont support (I wonder if they remain as tunable
> as a normal Csound instrument?).

Yes. SoundFont support is at the opcode level, so you write your own instrument to use them. From my copy of the documentation (online somewhere, try Google):

"""
When iflag = 0, inotenum sets the frequency of the output according to the MIDI note number used, and xfreq is used as a multiplier. When iflag = 1, the frequency of the output, is set directly by xfreq. This allows the user to use any kind of micro-tuning based scales. However, this method is designed to work correctly only with presets tuned to the default equal temperament. Attempts to use this method with a preset already having non-standard tunings, or with drum-kit-based presets, could give unexpected results.
"""

But it also says this, which is scarier:

"""
These opcodes only support the sample structure of SF2 files. The modulator structure of the SoundFont2 format is not supported in Csound. Any modulation or processing to the sample data is left to the Csound user, bypassing all restrictions forced by the SF2 standard.
"""

You can also play Sound Fonts using FluidSynth. I still don't know exactly what it is, but it's set up as a multi-channel MIDI synth. So the only way to tune it is to send pitch bend messages to the relevant channel. Which pretty much misses the point. I suppose you could run multiple instances of the Fluid engine to get around the 16-channel limit.

> Is there anything like the above sites for Csound
> patches??

I don't think so. You can order a CD-ROM which has lots of instruments and musical examples on it. There's also a User Defined Opcode (UDO) database online.

Graham

🔗Carl Lumma <clumma@yahoo.com>

10/27/2005 11:00:37 AM

> > Nope, AIFF is just Apple's version of WAV (some Mac
> > person will probably claim it was the other way around,
> > but whatever). It's uncompressed audio. The resolution
> > depends on the resolution chosen when recording it.
>
> I'm not sure what exactly they are. Wav files are a generic
> wrapper for any encoding that's registered with Windows, but
> many applications that work with Wav files aren't this flexible.
> AIFF files, as usually understood, allow you to set loop points
> while Wav files don't. So AIFF is more appropriate for sample
> libraries.

"Acidized WAVs" let you set loop points, and are back-compat.
with WAV.

> > But no need to do so -- Dave says Csound has
> > SoundFont support (I wonder if they remain as tunable
> > as a normal Csound instrument?).
>
> Yes. SoundFont support is at the opcode level, so you write
> your own instrument to use them. From my copy of the
> documentation (online somewhere, try Google):
>
> """
> When iflag = 0, inotenum sets the frequency of the output
> according to the MIDI note number used, and xfreq is used as
> a multiplier. When iflag = 1, the frequency of the output,
> is set directly by xfreq. This allows the user to use any
> kind of micro-tuning based scales. However, this method is
> designed to work correctly only with presets tuned to the
> default equal temperament. Attempts to use this method with
> a preset already having non-standard tunings, or with
> drum-kit-based presets, could give unexpected results.
> """
>
> But it also says this, which is scarier:
>
> """
> These opcodes only support the sample structure of SF2 files.
> The modulator structure of the SoundFont2 format is not
> supported in Csound. Any modulation or processing to the
> sample data is left to the Csound user, bypassing all
> restrictions forced by the SF2 standard.
> """

Good work.

-Carl

🔗Aaron Krister Johnson <aaron@akjmusic.com>

10/27/2005 11:44:15 AM

On Thursday 27 October 2005 7:02 am, Graham Breed wrote:

> You can also play Sound Fonts using FluidSynth.

and let's not forget timidity, which plays them, without distorting (unlike
fluidsynth, which distorts terribly)

-aaron.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/27/2005 1:04:14 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > I'm still not sure why this is supposed to be desirable, other
> > > than it makes creating bearing plans much easier.
> > >
> > > -Carl
> >
> > I'm not sure how you'd use this for a bearing plan -- Jorgenson
> > likes tunings where different instances of an interval across
> > the keyboard have the same rate of beating, but this is not at
> > all the case for regular metameantone, where the beat rates are
> > (only?) synchronized within a single major triad at a time.
>
> Bearing plans often include things like, 'tune the E until
> the major third C-E beats at the same rate as the minor third
> E-G'.

But in this case, how would you

() get the minor third E-G in the first place

() get beyond the C major triad and tune the rest of the notes

?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/27/2005 1:08:39 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> The 5ths are all the same as 15-equal, but C and Eb are
> 2 of the 5 keys whose minor 3rds are 6 cents better,

How is that possible? The "minor 3rd" or approximate 6:5 is only 4
cents off JI in 15-equal!

> and
> very near just. This is especially big because the zone
> above 6/5 gets dissonant very rapidly (it's why 15-equal
> doesn't really have better minor 3rds than 12).

It doesn't? I guess I better get my ears checked . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/27/2005 1:11:21 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > Could you list all that do?
>
> "All" would require a lemma, I suppose, but start from
>
> 2:3:5
> 4:5:6
> 3:4:10
> 3:8:20
> 3:16:40
> ...
> 5:8:12
> 5:16:24
> 5:32:48
> ...
>
> At least two infinite families.

Cool.

🔗Graham Breed <gbreed@gmail.com>

10/27/2005 1:52:04 PM

Aaron Krister Johnson wrote:

> and let's not forget timidity, which plays them, without distorting (unlike > fluidsynth, which distorts terribly)

But you can't host Timidity in Csound, can you?

Graham

🔗Herman Miller <hmiller@IO.COM>

10/27/2005 6:20:07 PM

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>>Since there wasn't any indication of what was supposed to be the >>"better" key of the scale, I just tuned C (MIDI note 60) to the
>>usual 261.63 Hz and used that as the first note of the scale.
>>The music actually starts out in the key of "MIDI note 64" (more
>>or less "E flat") minor, but it happens to be in C major from
>>around 1:55 to around 2:10 and then again briefly around 2:30.
>>It keeps getting back around to "Eb" minor.
> > > Is Eb the 4th line of the Scala file? And C the first?

The 5th line (the first is MIDI note 60, the note I'm calling "Eb" Is MIDI note 64).

🔗Gene Ward Smith <gwsmith@svpal.org>

10/27/2005 7:52:05 PM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
>
> On Thursday 27 October 2005 7:02 am, Graham Breed wrote:
>
> > You can also play Sound Fonts using FluidSynth.
>
> and let's not forget timidity, which plays them, without distorting
(unlike
> fluidsynth, which distorts terribly)

SynthFont is another program which has been improving.

🔗Dave Keenan <d.keenan@bigpond.net.au>

10/28/2005 1:41:42 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> I'm still not sure why this is supposed to be desirable, other
> than it makes creating bearing plans much easier.
>
> -Carl

Small whole-number ratios between the beat rates of the different
intervals making up a chord? I think it has a special sound. I call
it "second-order JI". La Monte Young's Dream House depends on it.
That's the way to think about the Dream House -- not as two-hundred-
and-something limit (first-order) JI.

-- Dave

🔗Ozan Yarman <ozanyarman@superonline.com>

10/28/2005 12:30:57 PM

George, I liked your previous extraordinary temperament better. Maybe it's because of better 5-limit approximation of intervals?

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 24 Ekim 2005 Pazartesi 22:44
Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

SNIP

> ! wendell1r.scl
> !
> Rational version of wendell1.scl by Gene Ward Smith
> 12
> !
> 24000/22729
> 76460/68187
> 27000/22729
> 85600/68187
> 4/3
> 32100/22729
> 102140/68187
> 36000/22729
> 114400/68187
> 40500/22729
> 42800/22729
> 2/1

Gene, I'm impressed!

But, OTOH, I hope you realize that there are reasonably simple brats
in only 5 minor triads (the ones with just 5ths), and the rest are
pretty much hit-or-miss (mostly the latter) -- much like what I got
in my _temperament (extra)ordinaire_ a couple months back.

In attempting to construct synchronous temperaments (with irrational
intervals) with both proportional *major and minor* triads, I found
that, unless the 5ths are exact, the minor-triad brats will at best
only approximate simple ratios when the majors are exact (or vice
versa), or you can compromise and make both inexact by about half and
thereby get most of the brats to within 0.02 of simple numbers --
close enough, I would say, considering that the required adjustment
is very small. In the following example, each of the tempered fifths
is modified by less than 0.002 cents (i.e., -3.9114 vs. -3.9132 cents
for an exact 2/11-comma narrow fifth).

Here's a sneak preview of my 2/11-comma well-temperament (with beat
synchrony in virtually all 24 major & minor triads). Bear in mind
that this is an honest-to-goodness well-temperament that meets *all*
of Jorgensen's requirements, with key contrasts that are very similar
to the Valotti-Young (1/6-pythagorean-comma) temperament:

! secor_WT2-11.scl
!
George Secor's 2/11-comma well-temperament, proportional beating
12
!
92.13884
196.08721
296.01562
392.17442
499.92562
590.20835
698.04361
794.06062
894.13082
997.97062
1090.21803
2/1

As to how close it comes to perfection, judge for yourself:

Major -----Beat Ratios-----
Triad M3/5th m3/5th m3/M3
----- ------- ------- -----
Ab... ------- ------- 1.50
Eb... ------- ------- 1.50
Bb... ------- ------- 1.50
F.... 7.01 13.02 1.86
C.... 2.50 6.26 2.50
G.... 2.50 6.26 2.50
D.... 3.34 7.51 2.25
A.... 5.01 10.01 5.00
E.... 6.67 12.51 1.87
B…... 16.63 27.45 1.65
F#... 1467.47 2203.71 1.50
C#... 1084.06 1628.58 1.50

Minor -----Beat Ratios-----
Triad M3/5th m3/5th m3/M3
----- ------- ------- -----
Ab... ------- ------- 1.00
Eb... ------- ------- 1.00
Bb... ------- ------- 1.00
F.... 20.74 22.74 1.10
C.... 7.99 9.99 1.25
G.... 6.01 8.01 1.33
D.... 4.02 6.02 1.50
A.... 2.99 4.99 1.67
E.... 2.99 4.99 1.67
B…... 7.93 9.93 1.25
F#... 950.96 952.96 1.00
C#... 933.34 935.34 1.00

The F# and C# triads have 5ths that are nearly just, so you can
consider those 3- and 4-digit numbers as approaching infinity. As
for B major, it's your call as to whether 1.65 is close enough to 5/3
to qualify as proportional; or for F minor, whether 11/10 is simple
enough.

Anyway, it turns out that I came up with at least two other
synchonous temperaments (one well, the other _ordinaire_) that I like
better than this one. I'll share them once I've settled on the best
way to categorize the whole collection.

--George

🔗Carl Lumma <clumma@yahoo.com>

10/28/2005 3:30:22 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > > > I'm still not sure why this is supposed to be desirable, other
> > > > than it makes creating bearing plans much easier.
> > > >
> > > > -Carl
> > >
> > > I'm not sure how you'd use this for a bearing plan -- Jorgenson
> > > likes tunings where different instances of an interval across
> > > the keyboard have the same rate of beating, but this is not at
> > > all the case for regular metameantone, where the beat rates are
> > > (only?) synchronized within a single major triad at a time.
> >
> > Bearing plans often include things like, 'tune the E until
> > the major third C-E beats at the same rate as the minor third
> > E-G'.
>
> But in this case, how would you
>
> () get the minor third E-G in the first place
>
> () get beyond the C major triad and tune the rest of the notes
>
> ?

Usually using other devices. What is the point of this line
of questioning?

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/28/2005 3:37:44 PM

> > The 5ths are all the same as 15-equal, but C and Eb are
> > 2 of the 5 keys whose minor 3rds are 6 cents better,
>
> How is that possible? The "minor 3rd" or approximate 6:5 is
> only 4 cents off JI in 15-equal!

That's a good point. It's 2 cents better.

> > and
> > very near just. This is especially big because the zone
> > above 6/5 gets dissonant very rapidly (it's why 15-equal
> > doesn't really have better minor 3rds than 12).
>
> It doesn't? I guess I better get my ears checked . . .

Maybe you should, Paul, maybe you should.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/28/2005 3:42:14 PM

> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>Since there wasn't any indication of what was supposed to be the
> >>"better" key of the scale, I just tuned C (MIDI note 60) to the
> >>usual 261.63 Hz and used that as the first note of the scale.
> >>The music actually starts out in the key of "MIDI note 64" (more
> >>or less "E flat") minor, but it happens to be in C major from
> >>around 1:55 to around 2:10 and then again briefly around 2:30.
> >>It keeps getting back around to "Eb" minor.
> >
> >
> > Is Eb the 4th line of the Scala file? And C the first?
>
> The 5th line (the first is MIDI note 60, the note I'm calling
> "Eb" Is MIDI note 64).

397cents is Eb?

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/28/2005 3:45:35 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > I'm still not sure why this is supposed to be desirable, other
> > than it makes creating bearing plans much easier.
> >
> > -Carl
>
> Small whole-number ratios between the beat rates of the different
> intervals making up a chord? I think it has a special sound.

Can you give me some chord (cents values) to compare?

> I call it "second-order JI". La Monte Young's Dream House
> depends on it. That's the way to think about the Dream
> House -- not as two-hundred-and-something limit (first-order) JI.

While the beat rates in the Dream House may be rational, are
they simple?

When I was at the dream house I didn't notice any special
synchrony. I wasn't listening for it either, but I heard lots
of different things as I moved throughout the space.

-Carl

🔗Herman Miller <hmiller@IO.COM>

10/28/2005 8:14:48 PM

Carl Lumma wrote:
>>>Is Eb the 4th line of the Scala file? And C the first?
>>
>>The 5th line (the first is MIDI note 60, the note I'm calling
>>"Eb" Is MIDI note 64).
> > > 397cents is Eb?

Hmm; you confused me when you asked if C is the first line (Scala files don't include the base note of the scale, since it's the reference for the other notes). With C tuned as 0.0 (which isn't in the Scala file), then "Eb" is the 4th line, 322.836732 cents.

🔗Carl Lumma <clumma@yahoo.com>

10/29/2005 1:35:26 AM

> Carl Lumma wrote:
> >>>Is Eb the 4th line of the Scala file? And C the first?
> >>
> >>The 5th line (the first is MIDI note 60, the note I'm calling
> >>"Eb" Is MIDI note 64).
> >
> >
> > 397cents is Eb?
>
> Hmm; you confused me when you asked if C is the first line (Scala
> files don't include the base note of the scale, since it's the
> reference for the other notes). With C tuned as 0.0 (which isn't
> in the Scala file), then "Eb" is the 4th line, 322.836732 cents.

So I think my analysis was right. If you render a performance with
this, we can see, though:

!
Blackwood[15] with brats of -1, bad C and Eb keys.
15
!
82.83673
165.67346
240.00000
322.83673
405.67346
480.00000
562.83673
645.67346
720.00000
802.83673
885.67346
960.00000
1042.83673
1125.67346
1200.00000
!

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

10/29/2005 1:43:25 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> So I think my analysis was right. If you render a performance with
> this, we can see, though:

This makes the C major chord not synch beating. What does that test?

🔗Carl Lumma <clumma@yahoo.com>

10/29/2005 11:26:07 AM

> > So I think my analysis was right. If you render a performance
> > with this, we can see, though:
>
> This makes the C major chord not synch beating. What does that
> test?

The file comments say "with brats of -1". Maybe they
should say "major triads in 2 out of 3 keys have -1 brats".

-Carl

🔗Carl Lumma <clumma@yahoo.com>

10/29/2005 11:37:40 AM

> > > So I think my analysis was right. If you render a performance
> > > with this, we can see, though:
> >
> > This makes the C major chord not synch beating. What does that
> > test?
>
> The file comments say "with brats of -1". Maybe they
> should say "major triads in 2 out of 3 keys have -1 brats".

This raises the question of how many synch triads are available.
It'd be cool to see a graph with third size from say 200 - 500
cents on the y axis and fifth size from 650 - 750 cents on the
x axis, with synch-tuned triads plotted as dots.

-Carl

🔗Herman Miller <hmiller@IO.COM>

10/29/2005 8:31:05 PM

Carl Lumma wrote:
> So I think my analysis was right. If you render a performance with
> this, we can see, though:

http://home.comcast.net/~teamouse/porcupine-absynth-blackwood-worse.mp3

Compare with:
http://home.comcast.net/~teamouse/porcupine-absynth-blackwood.mp3
http://home.comcast.net/~teamouse/porcupine-absynth.mp3 (15-ET)

> !
> Blackwood[15] with brats of -1, bad C and Eb keys.
> 15
> !
> 82.83673
> 165.67346
> 240.00000
> 322.83673
> 405.67346
> 480.00000
> 562.83673
> 645.67346
> 720.00000
> 802.83673
> 885.67346
> 960.00000
> 1042.83673
> 1125.67346
> 1200.00000
> !

The effect is somewhat interesting in the familiar "comma pump" progression at the end. The one chord in the sequence that's actually closer to being in tune sounds out of place and quickly "resolves" to a more dissonant chord.

🔗Dave Keenan <d.keenan@bigpond.net.au>

10/30/2005 5:41:10 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > Small whole-number ratios between the beat rates of the
different
> > intervals making up a chord? I think it has a special sound.
>
> Can you give me some chord (cents values) to compare?

Hi Carl,

Ratios describe it more precisely than cents. And it has to be tuned
very precisely to get the effect. You'd probably need something
based on digital-counter frequency dividers or phase-locked-loop
frequency multipliers (like the Dreamhouse or a Scalatron). Your
standard MIDI synth isn't likely to be able to do it.

Consider an approximate 4:5:6 chord. Multiply each of those numbers
by 128 to get 512:640:768 then temper it to 512:639:766 (i.e. 512 :
640-1 : 768-2). If those numbers were hertz we'd have a 4 Hz beat
between the lowest relevant harmonics in each of the three
intervals. And if it's tuned precisely enough those beats will be
synchronised so there will be only one simple beat.

This example results in narrowing of the fifth and major and minor
thirds by approximately 4.5, 2.7 and 1.8 cents respectively.

Compare this to 512 : 640-1/phi : 768-phi which should have no
discernable beat synchrony but similar cent errors of approximately
3.7, 1.7 and 2.0 cents. phi = (sqrt(5)+1)/2

I don't have the equipment to generate these accurately enough, so
I'm running on pure theory here in suggesting that these will sound
more different from each other than you'd expect just based on the
cent errors. Maybe someone out there can generate these for us all
to listen to.

> > I call it "second-order JI". La Monte Young's Dream House
> > depends on it. That's the way to think about the Dream
> > House -- not as two-hundred-and-something limit (first-order) JI.
>
> While the beat rates in the Dream House may be rational, are
> they simple?

I haven't analysed the numbers in depth, but my impression is that
they would often be so.

> When I was at the dream house I didn't notice any special
> synchrony. I wasn't listening for it either, but I heard lots
> of different things as I moved throughout the space.

I'm only referring to the temporal aspects not the spatial. I've
only ever heard that software simulation of it that someone kindly
posted a few months back.

If you've got lots of simultaneous frequencies at about the same
amplitude, then if you can hear any kind of rhythmic beating, as
opposed to a complete mess, there's got to be some beat synchrony
going on, doesn't there?

-- Dave

🔗George D. Secor <gdsecor@yahoo.com>

10/31/2005 12:10:16 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> ...
> > As to how close it [secor_WT2-11.scl] comes to perfection, judge
for yourself:
> >
> > Major -----Beat Ratios-----
> > Triad M3/5th m3/5th m3/M3
> > ----- ------- ------- -----
> > Ab... ------- ------- 1.50
> > Eb... ------- ------- 1.50
> > Bb... ------- ------- 1.50
> > F.... 7.01 13.02 1.86
> > C.... 2.50 6.26 2.50
> > G.... 2.50 6.26 2.50
> > D.... 3.34 7.51 2.25
> > A.... 5.01 10.01 5.00
> > E.... 6.67 12.51 1.87
> > B…... 16.63 27.45 1.65
> > F#... 1467.47 2203.71 1.50
> > C#... 1084.06 1628.58 1.50
>
> You didn't like Wendell's brats of 2, but I can't see this is an
> improvement. A 5/2 brat has a brat orbit of 5/2, -5/2, -4/25, which
> doesn't seem as good as 2. The 5 brat has an orbit 5, -5/7, -7/25,
and
> I would urge you to dump it in favor of one or more brats of 4
> instead. I don't know where the 1.86 and 1.85 come from, but a brat
> of 13/7 has an orbit 13/7, -7, -1/13. 9/4 is another not very
terrific
> looking brat, with an orbit of 9/4, -10/3, -2/15.
>
> I recommend using as many of the "magic" brats as possible; load up
on
> 3/2 and 4, and fill in a gap with 2 or 3.

So what do you think of this one?

! secor_TEO4.scl
!
George Secor's 12-tone temperament (extra)ordinaire, proportional
beating (attempt #4)
12
!
87.83846
194.22760
294.41956
389.19724
499.66357
585.88345
697.13843
789.79346
890.82392
997.70857
1086.36510
2/1

Here are the brats and total absolute error for each triad:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... 20.5856 28.3784 1.3786 32.81
Bb... ------- ------- 1.5000 20.41
F.... 5.0000 10.0000 2.0000 12.93
C.... 1.0000 4.0000 4.0000 15.40
G.... 1.0000 4.0000 4.0000 15.56
D.... 1.6667 5.0000 3.0000 21.40
A.... 5.0000 10.0000 2.0000 28.57
E.... 5.0000 10.0000 2.0000 38.14
B.... 14.9750 24.9625 1.6669 48.35
F#... ------- ------- 1.5000 51.02
C#... ------- ------- 1.5000 51.02
G#... 15.0000 20.0000 1.3333 47.79

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... 37.9820 35.9820 0.9473 51.02
Bb... ------- ------- 1.0000 51.02
F.... 29.3069 31.3069 1.0682 51.02
C.... 6.7704 8.7704 1.2954 42.44
G.... 4.1765 6.1765 1.4789 30.14
D.... 1.8036 3.8036 2.1089 20.41
A.... 1.6071 3.6071 2.2444 12.93
E.... 1.2143 3.2143 2.6471 15.40
B.... 4.3750 6.3750 1.4571 15.56
F#... ------- ------- 1.0000 21.40
C#... ------- ------- 1.0000 28.57
G#... 16.1892 14.1892 0.8765 43.48

I found it *extremely* difficult to get reasonably simple brats for
*all* 12 major triads (the Eb and B brats are very close to 11/8 and
5/3, but not exact), and if you do, then it's not too likely that the
brats for the minor triads with tempered fifths will be simple (as
you'll see above). I'll have to work on this some more, if that's
what's required for a "simplest" temperament. (It looks as if 3-
brats on the C and G major triads will give you 2-brats on A and E
minor, so there's hope.)

In order to get the 4-brats (in the majors) you need triads with
relatively low total absolute error, something on the order of
meantone temperament. I used my 5/23-comma temperament (extra)
ordinaire as a guide for arriving at the above, since it already had
some excellent brats in about half the major triads.

/tuning/topicId_59689.html#59999

If you compare the two in a listening test, you'll be hard pressed to
hear any difference.

--George

🔗George D. Secor <gdsecor@yahoo.com>

10/31/2005 12:16:33 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> George, I liked your previous extraordinary temperament better. Maybe
it's because of better 5-limit approximation of intervals?

Yes, the best triads of the (extra)ordinaire are much better. Please
note the latest improvement on the (extra)ordinaire, which has
proportional beating on all 12 major triads:

/tuning/topicId_59689.html#62004

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

10/31/2005 3:05:30 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> So what do you think of this one?

At first glance, it looks better.

> I found it *extremely* difficult to get reasonably simple brats for
> *all* 12 major triads (the Eb and B brats are very close to 11/8 and
> 5/3, but not exact), and if you do, then it's not too likely that the
> brats for the minor triads with tempered fifths will be simple (as
> you'll see above).

Here's a try of mine, which I think has been posted before. The beat
ratios are simple, but one of the fifths is sharp. Nonetheless, it
could be considered circulating.

! brac.scl
circulating temperament with simple beat ratios
! beat ratios 4 3/2 4 3/2 2 2 177/176 4 3/2 2 3/2 2
12
!
56640/53701
60008/53701
63720/53701
67264/53701
71685/53701
75056/53701
80276/53701
84960/53701
89920/53701
95580/53701
100544/53701
2

I'll have to work on this some more, if that's
> what's required for a "simplest" temperament. (It looks as if 3-
> brats on the C and G major triads will give you 2-brats on A and E
> minor, so there's hope.)
>
> In order to get the 4-brats (in the majors) you need triads with
> relatively low total absolute error, something on the order of
> meantone temperament. I used my 5/23-comma temperament (extra)
> ordinaire as a guide for arriving at the above, since it already had
> some excellent brats in about half the major triads.
>
> /tuning/topicId_59689.html#59999
>
> If you compare the two in a listening test, you'll be hard pressed to
> hear any difference.
>
> --George
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/31/2005 4:12:28 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > I'm still not sure why this is supposed to be desirable, other
> > than it makes creating bearing plans much easier.
> >
> > -Carl
>
> Small whole-number ratios between the beat rates of the different
> intervals making up a chord? I think it has a special sound. I call
> it "second-order JI". La Monte Young's Dream House depends on it.

Is that true? Why isn't that first-order JI, and why is the lack of
any upper partials that might beat not a problem?

> That's the way to think about the Dream House -- not as two-hundred-
> and-something limit (first-order) JI.

What about the combinational tones, which I thought were supposer to
be more significant? And what about the fact that a constant Hz shift
of the whole sound ruined the effect according to Daniel Wolf?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

10/31/2005 4:25:22 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > >
> > > > > I'm still not sure why this is supposed to be desirable,
other
> > > > > than it makes creating bearing plans much easier.
> > > > >
> > > > > -Carl
> > > >
> > > > I'm not sure how you'd use this for a bearing plan --
Jorgenson
> > > > likes tunings where different instances of an interval across
> > > > the keyboard have the same rate of beating, but this is not at
> > > > all the case for regular metameantone, where the beat rates
are
> > > > (only?) synchronized within a single major triad at a time.
> > >
> > > Bearing plans often include things like, 'tune the E until
> > > the major third C-E beats at the same rate as the minor third
> > > E-G'.
> >
> > But in this case, how would you
> >
> > () get the minor third E-G in the first place
> >
> > () get beyond the C major triad and tune the rest of the notes
> >
> > ?
>
> Usually using other devices. What is the point of this line
> of questioning?

To find out if indeed "it makes creating bearing plans much easier",
as you claim. If so, I'd be interested to learn how.

🔗Gene Ward Smith <gwsmith@svpal.org>

10/31/2005 6:55:15 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> So what do you think of this one?

Here is a rational-number version of it.

! secorteo4.scl
rational version of secor_TEO4
12
!
17848/16965
10544/9425
4022/3393
106208/84825
566029/424125
71392/50895
126884/84825
8924/5655
141904/84825
2264116/1272375
158872/84825
2

🔗Dave Keenan <d.keenan@bigpond.net.au>

10/31/2005 9:44:17 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > I'm still not sure why this is supposed to be desirable, other
> > > than it makes creating bearing plans much easier.
> > >
> > > -Carl
> >
> > Small whole-number ratios between the beat rates of the
different
> > intervals making up a chord? I think it has a special sound. I
call
> > it "second-order JI". La Monte Young's Dream House depends on
it.
>
> Is that true?

Maybe not. But it seemed to be of major significance in the software
simulation. I realise there's all kinds of other stuff happening in
the _real_ Dream House.

> Why isn't that first-order JI,

Because (if you'll allow me a little hyperbole) beating is what it's
all about.

> and why is the lack of
> any upper partials that might beat not a problem?

When you have multiple sine-waves phase-locked to exact ratios you
can think of the whole thing as a single note of a strange timbre or
you can group the frequencies in various ways and think of each
group as a note with partials (and possibly a missing fundamentals).

So they are _all_ partials, and some of them beat.

> > That's the way to think about the Dream House -- not as two-
hundred-
> > and-something limit (first-order) JI.
>
> What about the combinational tones, which I thought were supposer
to
> be more significant?

OK. It's _one_ way to think about the Dream House, or at least the
software simulation which, as I said, is all I've ever heard.

> And what about the fact that a constant Hz shift
> of the whole sound ruined the effect according to Daniel Wolf?

Why should that affect my opinion. I never said that synched beats
is the _only_ thing happening.

-- Dave Keenan

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 1:43:45 AM

Hi Dave,

> > > Small whole-number ratios between the beat rates of the
> > > different intervals making up a chord? I think it has a
> > > special sound.
> >
> > Can you give me some chord (cents values) to compare?
>
> Hi Carl,
>
> Ratios describe it more precisely than cents. And it has to be
> tuned very precisely to get the effect. You'd probably need
> something based on digital-counter frequency dividers or
> phase-locked-loop frequency multipliers (like the Dreamhouse
> or a Scalatron). Your standard MIDI synth isn't likely to be
> able to do it.

I can render directly to WAV from Cool Edit.

> Consider an approximate 4:5:6 chord. Multiply each of those
> numbers by 128 to get 512:640:768 then temper it
> to 512:639:766 (i.e. 512 : 640-1 : 768-2). If those numbers
> were hertz we'd have a 4 Hz beat between the lowest relevant
> harmonics in each of the three intervals. And if it's tuned
> precisely enough those beats will be synchronised so there
> will be only one simple beat.
>
> This example results in narrowing of the fifth and major and
> minor thirds by approximately 4.5, 2.7 and 1.8 cents
> respectively.
>
> Compare this to 512 : 640-1/phi : 768-phi which should have no
> discernable beat synchrony but similar cent errors of
> approximately 3.7, 1.7 and 2.0 cents. phi = (sqrt(5)+1)/2

This is exactly the kind of comparison I've been asking for
the data to do...

/tuning-math/message/6339

I get...

766.38196601125010515179541316563
639.38196601125010515179541316563
512.0

...Hz. for the messy chord. But hrm, I don't see any brats
of phi with "show /relative beats" in Scala. Isn't that what
you intended?

> I don't have the equipment to generate these accurately enough,
> so I'm running on pure theory here in suggesting that these
> will sound more different from each other than you'd expect
> just based on the cent errors. Maybe someone out there can
> generate these for us all to listen to.

Here are the files:

http://lumma.org/tuning/A.wav
http://lumma.org/tuning/B.wav
(1MB total)

Anybody care to guess which is which? Anybody feel one
sounds better than the other?

> > > I call it "second-order JI". La Monte Young's Dream House
> > > depends on it. That's the way to think about the Dream
> > > House -- not as two-hundred-and-something limit (first-order)
> > > JI.
> >
> > While the beat rates in the Dream House may be rational, are
> > they simple?
>
> I haven't analysed the numbers in depth, but my impression is
> that they would often be so.

I haven't analysed either, but I thought La Monte basically
came up with ratios of large primes in a rather off-the-cuff
manner.

> > When I was at the dream house I didn't notice any special
> > synchrony. I wasn't listening for it either, but I heard lots
> > of different things as I moved throughout the space.
>
> I'm only referring to the temporal aspects not the spatial.

I just mean I heard a lot of stuff, some of it simple,
some of it complex.

> I've only ever heard that software simulation of it that
> someone kindly posted a few months back.

Dave Seidel? I thought I remembered this too, but on his
site I find several pieces inspired by La Monte installations,
but none by the Dream House. Dave?

> If you've got lots of simultaneous frequencies at about the
> same amplitude, then if you can hear any kind of rhythmic
> beating, as opposed to a complete mess, there's got to be
> some beat synchrony going on, doesn't there?

I don't recall hearing recognizable rhythms in the beats.

-Carl

🔗Magnus Jonsson <magnus@smartelectronix.com>

11/1/2005 5:22:53 AM

On Tue, 1 Nov 2005, Carl Lumma wrote:

> http://lumma.org/tuning/A.wav
> http://lumma.org/tuning/B.wav
> (1MB total)
>
> Anybody care to guess which is which? Anybody feel one
> sounds better than the other?

I prefer B.

🔗Jon Szanto <jszanto@cox.net>

11/1/2005 7:23:41 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> Dave Seidel? I thought I remembered this too, but on his
> site I find several pieces inspired by La Monte installations,
> but none by the Dream House. Dave?

IIRC, it was posted by someone but actually created by an Italian
musician, and then the software on his site was removed due to
complaints from LMY himself. David Beardsly will probably remember
this better than I.

Cheers,
Jon

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 11:05:29 AM

> > http://lumma.org/tuning/A.wav
> > http://lumma.org/tuning/B.wav
> > (1MB total)
> >
> > Anybody care to guess which is which? Anybody feel one
> > sounds better than the other?
>
> I prefer B.

Thanks for weighing in!

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

11/1/2005 11:12:12 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > So what do you think of this one?
>
> At first glance, it looks better.

And it sounds as good as it looks.

> > I found it *extremely* difficult to get reasonably simple brats
for
> > *all* 12 major triads (the Eb and B brats are very close to 11/8
and
> > 5/3, but not exact), and if you do, then it's not too likely that
the
> > brats for the minor triads with tempered fifths will be simple
(as
> > you'll see above).
>
> Here's a try of mine, which I think has been posted before. The beat
> ratios are simple,

An understatement :-) -- they're about as simple as they can get
(ignoring the minor triads, of course), at least for something
reasonably useful.

> but one of the fifths is sharp. Nonetheless, it
> could be considered circulating.

Yes, but I would transpose it up a fifth, so the present F would
become C. That would put your two worst major triads on F# and C#
(instead of B and F#) and put the wide fifth between F# and C#.

> ! brac.scl
> circulating temperament with simple beat ratios
> ! beat ratios 4 3/2 4 3/2 2 2 177/176 4 3/2 2 3/2 2
> 12
> !
> 56640/53701
> 60008/53701
> 63720/53701
> 67264/53701
> 71685/53701
> 75056/53701
> 80276/53701
> 84960/53701
> 89920/53701
> 95580/53701
> 100544/53701
> 2

But I still don't think either one of us has answered Aaron's
original question:
/tuning/topicId_59689.html#61731
which I repeat here:

<< what is the simplest possible 12-note temperament where all 24
major and minor triads have rationally proportional beating?
here 'simplest' means that the brats (beat ratios for the un-
initiated) are the lowest numbers in the numerator and denominator
that they can be..... >>

I also assume the following additional requirements:

That it be a *circulating* temperament, and
That it should, at the very least, be reasonably useful from a
musical standpoint -- or should I go beyond that by insisting that it
should sound good (just so we don't overlook the most important point
of this exercise)?

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/1/2005 12:37:51 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > I'm still not sure why this is supposed to be desirable, other
> > > > than it makes creating bearing plans much easier.
> > > >
> > > > -Carl
> > >
> > > Small whole-number ratios between the beat rates of the
> different
> > > intervals making up a chord? I think it has a special sound. I
> call
> > > it "second-order JI". La Monte Young's Dream House depends on
> it.
> >
> > Is that true?
>
> Maybe not. But it seemed to be of major significance in the
software
> simulation. I realise there's all kinds of other stuff happening in
> the _real_ Dream House.
>
> > Why isn't that first-order JI,
>
> Because (if you'll allow me a little hyperbole) beating is what
it's
> all about.
>
> > and why is the lack of
> > any upper partials that might beat not a problem?
>
> When you have multiple sine-waves phase-locked to exact ratios you
> can think of the whole thing as a single note of a strange timbre

Sounds like an aspect of first-order JI, or maybe "zeroth order JI"
where there's only one note with its harmonics. Isn't that very
different from what "second-order JI" is supposed to be about --
slightly mistuned intervals with equal rates of beating?

> or
> you can group the frequencies in various ways and think of each
> group as a note with partials (and possibly a missing fundamentals).

Can you show me these groups please?

🔗Ozan Yarman <ozanyarman@superonline.com>

11/1/2005 1:00:19 PM

They both sound delightful, because of proportional beating. A is more energetic, B is calmer.

----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 01 Kasım 2005 Salı 21:05
Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

> > http://lumma.org/tuning/A.wav
> > http://lumma.org/tuning/B.wav
> > (1MB total)
> >
> > Anybody care to guess which is which? Anybody feel one
> > sounds better than the other?
>
> I prefer B.

Thanks for weighing in!

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 4:20:09 PM

> > > http://lumma.org/tuning/A.wav
> > > http://lumma.org/tuning/B.wav
> > > (1MB total)
> > >
> > > Anybody care to guess which is which? Anybody feel one
> > > sounds better than the other?
> >
> > I prefer B.
>>
>> Thanks for weighing in!
>
> They both sound delightful, because of proportional beating.
> A is more energetic, B is calmer.

Thanks Ozan. I'll wait for Dave and others to reply before
discussing it further.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

11/1/2005 4:41:14 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> Thanks Ozan. I'll wait for Dave and others to reply before
> discussing it further.

You know, if these weren't wav files and it was clear what you wanted
more people might play.

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 4:47:04 PM

> > Thanks Ozan. I'll wait for Dave and others to reply before
> > discussing it further.
>
> You know, if these weren't wav files and it was clear what you
> wanted more people might play.

Why are WAVs bad? They're fairly small.

I don't want anything other than comments. How do they sound?
Can you identify which one is which?

-Carl

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

11/1/2005 5:17:06 PM

On Wed, 02 Nov 2005, "Carl Lumma" wrote:
>
> > > > http://lumma.org/tuning/A.wav
> > > > http://lumma.org/tuning/B.wav
> > > > (1MB total)
> > > >
> > > > Anybody care to guess which is which? Anybody feel one
> > > > sounds better than the other?
> > >
> > > I prefer B.
> > >
> > > Thanks for weighing in!
> >
> > They both sound delightful, because of proportional beating.
> > A is more energetic, B is calmer.
>
> Thanks Ozan. I'll wait for Dave and others to reply before
> discussing it further.

Hi Carl,

I prefer B. A has a much steadier beat, and I find it annoying.
B has a sound that seems to continue evolving throughout, much
like that of a natural instrument, or a group of them.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.362 / Virus Database: 267.12.7/154 - Release Date: 1/11/05

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/1/2005 5:57:21 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> I get...
>
> 766.38196601125010515179541316563
> 639.38196601125010515179541316563
> 512.0
>
> ...Hz. for the messy chord.

Correct.

> But hrm, I don't see any brats
> of phi with "show /relative beats" in Scala. Isn't that what
> you intended?

That would have been nice, but I couldn't figure out how to get it
so I was satisfied with them being other noble numbers.

The beat ratios are somewhere between 3:4:5 and 4:5:7 -- more like
7:9:12. I think these are complex enough to not to count as "nearly
sychronised".

> Here are the files:
>
> http://lumma.org/tuning/A.wav
> http://lumma.org/tuning/B.wav
> (1MB total)
>
> Anybody care to guess which is which? Anybody feel one
> sounds better than the other?

Thanks for doing this, Carl. I now think my earlier theory on beat
rates is crap, as I would have realised if I'd (before now) set up a
spreadsheet to look at the waveforms and their envelopes, even
without listening to it.

There is a lot more to it. The waveform used for the single notes
matters a lot (I used sawtooths, what did you use Carl?). And the
initial phases of the three notes matters. If you choose the right
initial phases and relative amplitudes for the three notes you may
actually be able to get the beats to cancel each other out, or
nearly so.

I think A has the 1:1:1 rates and B has those with no simple ratio.
But I think B sound more restful. It simply sounds like it has
slower beats (maybe 1.5 times slower) and a lower beat amplitude.

-- Dave Keenan

🔗Herman Miller <hmiller@IO.COM>

11/1/2005 6:38:03 PM

Carl Lumma wrote:

> http://lumma.org/tuning/A.wav
> http://lumma.org/tuning/B.wav
> (1MB total)
> > Anybody care to guess which is which? Anybody feel one
> sounds better than the other?

I prefer B with these isolated chords; the beating in A is distracting. It could be an interesting effect in a different context, though. But I think that B sounds better to me for much the same reason that I prefer tempered octaves for electronic sounds: the varied beating adds an extra richness to tones that would otherwise sound static.

🔗Magnus Jonsson <magnus@smartelectronix.com>

11/1/2005 7:35:53 PM

This is exactly the same reason I had for preferring B.

On Wed, 2 Nov 2005, Yahya Abdal-Aziz wrote:

> Hi Carl,
>
> I prefer B. A has a much steadier beat, and I find it annoying.
>
> B has a sound that seems to continue evolving throughout, much
> like that of a natural instrument, or a group of them.
>
> Regards,
> Yahya
>
>

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 10:39:45 PM

> Carl Lumma wrote:
> > So I think my analysis was right. If you render a performance
> > with this, we can see, though:
>
> http://home.comcast.net/~teamouse/porcupine-absynth-blackwood-
> worse.mp3
>
> Compare with:
> http://home.comcast.net/~teamouse/porcupine-absynth-blackwood.mp3
> http://home.comcast.net/~teamouse/porcupine-absynth.mp3 (15-ET)

It sounds worse alright, but see Gene's post that these off
keys are beat-synched. So we have to turn somewhere else to
see if that's really helping.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 10:45:35 PM

> > > So I think my analysis was right. If you render a performance
> > > with this, we can see, though:
> >
> > http://home.comcast.net/~teamouse/porcupine-absynth-blackwood-
> > worse.mp3
> >
> > Compare with:
> > http://home.comcast.net/~teamouse/porcupine-absynth-blackwood.mp3
> > http://home.comcast.net/~teamouse/porcupine-absynth.mp3 (15-ET)
>
> It sounds worse alright, but see Gene's post that these off
> keys are beat-synched. So we have to turn somewhere else to
> see if that's really helping.

You'd think I'd be in danger of wearing out the piece with all
this testing. But it's such a damn fine piece of music, it
hasn't happened yet!

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/1/2005 11:38:44 PM

> > But hrm, I don't see any brats of phi with "show /relative
> > beats" in Scala. Isn't that what you intended?
>
> That would have been nice, but I couldn't figure out how to
> get it so I was satisfied with them being other noble numbers.

According to Gene, the M3/m3 brat is

brat = (6t - 5f)/(4t - 5)

And Phi is

(sqrt(5) + 1)/2

Using your M3

t = 639/512 = 383.60655 cents

That gives me a 5th of

f = sqrt(5)/1280 - 479/320 = 696.32288 cents

ignoring a sign (which I think is ok). Is this right?

> The beat ratios are somewhere between 3:4:5 and 4:5:7 -- more
> like 7:9:12. I think these are complex enough to not to count
> as "nearly sychronised".

Agreed.

> > Here are the files:
> >
> > http://lumma.org/tuning/A.wav
> > http://lumma.org/tuning/B.wav
> > (1MB total)
> >
> > Anybody care to guess which is which? Anybody feel one
> > sounds better than the other?
>
> Thanks for doing this, Carl. I now think my earlier theory on
> beat rates is crap, as I would have realised if I'd (before now)
> set up a spreadsheet to look at the waveforms and their
> envelopes, even without listening to it.
>
> There is a lot more to it. The waveform used for the single
> notes matters a lot (I used sawtooths, what did you use Carl?).

The least-jarring thing with decent harmonic content that
I could coax out of Cool Edit. Which turned out to be the
"inverse squared sine" waveform, at, I believe, equal power
and phase for all three notes, with short fades in and out
added.

> I think A has the 1:1:1 rates and B has those with no simple
> ratio.

That's right.

> But I think B sound more restful. It simply sounds like
> it has slower beats (maybe 1.5 times slower) and a lower beat
> amplitude.

I'm not sure about amplitude, but slower beat rate is the
major difference to my ear. Respondants so far included
Magnus, Herman, Yahya, and yourself. All preferred B for
similar reasons. But my wife prefers A. Go figure. :)

-Carl

🔗Mark <mark@equiton.waitrose.com>

11/2/2005 12:58:52 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > http://lumma.org/tuning/A.wav
> > > > http://lumma.org/tuning/B.wav
> > > > (1MB total)
> > > >
I prefer B too. The beating of A is more active and unsettling. If used
with more movement perhaps it could be got away with, but the sensation
of B is much more calm.

I always like these interesting little tests - it just shows how
sensitive hearing really is.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/2/2005 12:28:21 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> t = 639/512 = 383.60655 cents
>
> That gives me a 5th of
>
> f = sqrt(5)/1280 - 479/320 = 696.32288 cents
>
> ignoring a sign (which I think is ok). Is this right?

No, the fifth has to be a positive number! Change the sign, and you
get a brat of -phi.

🔗Carl Lumma <clumma@yahoo.com>

11/2/2005 1:35:39 PM

> > t = 639/512 = 383.60655 cents
> >
> > That gives me a 5th of
> >
> > f = sqrt(5)/1280 - 479/320 = 696.32288 cents
> >
> > ignoring a sign (which I think is ok). Is this right?
>
> No, the fifth has to be a positive number! Change the sign,
> and you get a brat of -phi.

Maybe this question about negative brats will help:
Does a brat of -1.5 still means one third beats thrice
for every two beatings of the other?

And, what's the fifth that gives a +phi brat with
the 639/512 Maj 3rd?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

11/2/2005 2:41:16 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Maybe this question about negative brats will help:
> Does a brat of -1.5 still means one third beats thrice
> for every two beatings of the other?

Right.

> And, what's the fifth that gives a +phi brat with
> the 639/512 Maj 3rd?

(1918+sqrt(5))/1280, about a fifth of a cent sharp.

🔗Carl Lumma <clumma@yahoo.com>

11/2/2005 3:10:57 PM

> > Maybe this question about negative brats will help:
> > Does a brat of -1.5 still means one third beats thrice
> > for every two beatings of the other?
>
> Right.

Then I think my -phi triad will have the desired effect,
and my statement about it being ok to ignore the sign
was right.

> > And, what's the fifth that gives a +phi brat with
> > the 639/512 Maj 3rd?
>
> (1918+sqrt(5))/1280, about a fifth of a cent sharp.

Thanks. I'll try both. But my triad is closer tuning-wise
to Dave's synch triad, and so it might make a better
comparison.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/2/2005 4:26:06 PM

Right, so Dave's original triad of
512:639:766 Hz.
is still at
http://lumma.org/tuning/A.wav
Gene, can you confirm if this is what you
would call "synch tuned"?

Now we can compare with this triad
512:639:765.505572809 Hz.
at
http://lumma.org/tuning/B2.wav
whose major and minor third beat rates are
related by the golden mean, phi. If the
simple-ratio beating theory were right, we'd
expect this triad to sound very bad indeed,
since in some sense phi is the least rational
number.

Last but not least, check out
512:639:768.094427191 Hz.
at
http://lumma.org/tuning/B3.wav
which is the other phi-beating triad based
on Dave's major 3rd. It's fifth is
farther from the Dave's than B2.wav's.

Comments?

For Gene: all four chords as mp3s, in a zip
file (295KB)...
http://lumma.org/tuning/brats3.zip

Interestingly, the quality loss of A.mp3, even
with LAME's --preset extreme encoding, is
immediately noticeable to me (with headphones).

I played with a parameter in Cool Edit called
"Start Phase" (in degrees). It didn't have any
audible effect.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/2/2005 4:34:57 PM

> Comments?
//
> I played with a parameter in Cool Edit called
> "Start Phase" (in degrees). It didn't have any
> audible effect.

For me, the most important difference between these
chords is beat rate (slower is better).

I know how to calculate beat rates for dyads (assuming
harmonic timbres). But how to calculate them
for triads....

Is the fastest beat the thing we hear? Or is it
the most prominent beat that sticks out the most?

I need to do more listening, but it seems like Dave's
"messy" triad (B.wav) is my favorite.

My guess is that brats are not the right formulation.
The beat rate of the 5th also needs to be considered.
And the desideratum seems to be to stagger them as
much as possible (in other words, to *de*rationalize
them).

I worried earlier that the intervals in synch triads
would beat all together, and thus make the beats
louder. Paul E. said that in real life, phase
differences would make this unlikely.

Cool Edit doesn't seem to be giving me very good
control over phase. Dave, care to make your spreadsheet
public?

-Carl

🔗Magnus Jonsson <magnus@smartelectronix.com>

11/2/2005 5:04:09 PM

In order from best to worst sounding:

* B3 (nice, emphasis moves between harmonics in a lovely way. Slightly weird attack, i guess some cancellation happened at just the wrong moment.)
* B from the original question (similar to B2, but slower beating)
* B2 (wawawawawa, static.)
* A (too rhythmic, as if trying to pull me here and there. Sounds forced...)

On Thu, 3 Nov 2005, Carl Lumma wrote:

> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav
> Gene, can you confirm if this is what you
> would call "synch tuned"?
>
> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi. If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.
>
> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd. It's fifth is
> farther from the Dave's than B2.wav's.
>
> Comments?
>
> For Gene: all four chords as mp3s, in a zip
> file (295KB)...
> http://lumma.org/tuning/brats3.zip
>
> Interestingly, the quality loss of A.mp3, even
> with LAME's --preset extreme encoding, is
> immediately noticeable to me (with headphones).
>
> I played with a parameter in Cool Edit called
> "Start Phase" (in degrees). It didn't have any
> audible effect.
>
> -Carl
>
>
>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>
>
>
>
>

🔗Herman Miller <hmiller@IO.COM>

11/2/2005 6:43:55 PM

Carl Lumma wrote:

> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi. If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.

I think this one sounds worse than the others, but mainly because the beats seem faster.

> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd. It's fifth is
> farther from the Dave's than B2.wav's.

This one's nice and smooth. The faster beats are in a higher register where they don't stand out so badly.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/3/2005 1:19:08 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Gene, can you confirm if this is what you
> would call "synch tuned"?

Sounds that way, and it ought to be.

If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.

It seems to me that is a ridiculous conclusion. Certainly, I can't
recall anyone claiming such a thing. Clearly however the kind of
beating sounds different in the two cases.

Is this what all this has been in aid of? Why did you pick these
particlar chords?

> For Gene: all four chords as mp3s, in a zip
> file (295KB)...
> http://lumma.org/tuning/brats3.zip

I only see three chords.

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 3:19:41 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Comments?
> //
> > I played with a parameter in Cool Edit called
> > "Start Phase" (in degrees). It didn't have any
> > audible effect.

There should be separate start phases for the three notes. It should
only make a difference when the beats are in simple rational ratios.
In that case some phase combinations will give shallower beats as
beats partly cancel each other.

> For me, the most important difference between these
> chords is beat rate (slower is better).

Right, but beat depth matters too.

> I know how to calculate beat rates for dyads (assuming
> harmonic timbres). But how to calculate them
> for triads....

It seems the best way is just to simulate them.

> Is the fastest beat the thing we hear? Or is it
> the most prominent beat that sticks out the most?

It doesn't seem to be any of those. It doesn't seem to be simple to
predict at all.

> Cool Edit doesn't seem to be giving me very good
> control over phase. Dave, care to make your spreadsheet
> public?

Hopefully, by the time you read this I will have uploaded it to:
/tuning/files/Keenan/beats.
xls.zip

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 5:20:57 AM

I wrote:
> Hopefully, by the time you read this I will have uploaded it to:
> /tuning/files/Keenan/beats.
> xls.zip

Nope. It seems tuning_files is too full. I deleted some of my old
stuff but still couldn't upload this (1.9 MB). Maybe someone who has
_lots_ of files stored there could delete some old stuff.

But in any case, thanks to my friend Andrew Smith, I've put it up here:
http://asmith.id.au/~dkeenan/beats.xls.zip

-- Dave Keenan

🔗monz <monz@tonalsoft.com>

11/3/2005 8:17:47 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav
> Gene, can you confirm if this is what you
> would call "synch tuned"?
>
> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi. If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.
>
> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd. It's fifth is
> farther from the Dave's than B2.wav's.
>
> Comments?

I don't understand how any of you guys can objectively
say that any of these are "better" or "worse". ?

To my ears, they all exhibit different beat patterns,
which are simply different rhythms, none especially
better or worse than the others.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

11/3/2005 8:37:18 AM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> It seems tuning_files is too full. I deleted some of my old
> stuff but still couldn't upload this (1.9 MB). Maybe someone
> who has _lots_ of files stored there could delete some old stuff.

Don't delete old stuff!

I solved the problem the simple way -- created a new Yahoo
group!

/tuning_files2

I've already set up a bunch of folders in the "Files" section
of that group, for the people who generally upload the most
files.

*Please*, folks ... if you don't already have a folder and
want to upload files, create a folder for your own stuff!
Help to keep things organized ... it's bad enough that now
we need to have 3 separate Yahoo groups for our files!

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <gwsmith@svpal.org>

11/3/2005 9:25:18 AM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> I don't understand how any of you guys can objectively
> say that any of these are "better" or "worse". ?
>
> To my ears, they all exhibit different beat patterns,
> which are simply different rhythms, none especially
> better or worse than the others.

Better or worse isn't the relevant question. I'd phrase it as more
organized vs less organized. The synch beating example beats
coherently, which someone may or may not like the sound of. But I
think it can be helpful in making the ear think it *ought* to sound
like this. It has some of the characteristics of JI, without being JI.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/3/2005 9:49:48 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav

Of course, for circulating 12-note scales we've been discussing brats
of 2 and 4, not -1. We can get a brat of 2 from, for example,

2560:3205:3838 = 512:641:767.6

and of 4 from, for example,

512:641:766

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 11:00:19 AM

> > If the simple-ratio beating theory were right, we'd
> > expect this triad to sound very bad indeed,
> > since in some sense phi is the least rational
> > number.
>
> It seems to me that is a ridiculous conclusion.

Why?

> Certainly, I can't recall anyone claiming such a thing.

The claim that simple-ratio beat rates sound better has
been made by Bob Wendell, among several others. Including
you in the blackbeat15.scl thread.

> Is this what all this has been in aid of? Why did you pick
> these particlar chords?

Dave picked the initial two. I used his major 3rd to find
the -phi tuning, and you found the phi tuning.

> > For Gene: all four chords as mp3s, in a zip
> > file (295KB)...
> > http://lumma.org/tuning/brats3.zip
>
> I only see three chords.

A, B, B2 and B3 (.mp3) should all be there.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 11:25:16 AM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> If you choose the right
> initial phases and relative amplitudes for the three notes you may
> actually be able to get the beats to cancel each other out, or
> nearly so.

How is this possible? Each of the beatings occurs at a different pitch
(that is, for a different pair of nearly coincident partials) so I have
no idea how this could happen here. I'd love to be educated,
particularly with a demonstration!

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 11:27:36 AM

I'm curious -- How does a chord specifically designed *not* to have
proportional beating sound delightful *because* of proportional
beating?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> They both sound delightful, because of proportional beating. A is
>more energetic, B is calmer.
>
> ----- Original Message -----
> From: Carl Lumma
> To: tuning@yahoogroups.com
> Sent: 01 Kasým 2005 Salý 21:05
> Subject: [tuning] Re: the simplest pan-proportionally beating 12-
tone temperament
>
>
> > > http://lumma.org/tuning/A.wav
> > > http://lumma.org/tuning/B.wav
> > > (1MB total)
> > >
> > > Anybody care to guess which is which? Anybody feel one
> > > sounds better than the other?
> >
> > I prefer B.
>
> Thanks for weighing in!
>
> -Carl
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 11:35:21 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> I worried earlier that the intervals in synch triads
> would beat all together, and thus make the beats
> louder. Paul E. said that in real life, phase
> differences would make this unlikely.

No you must be thinking of what I said on some other topic. Here, all
the beats occur at different pitches, so relative phase should have no
effect on the individual or overall loudness of the beats. I think
Dave's all wet on this, but want to see his response.

🔗Ozan Yarman <ozanyarman@superonline.com>

11/3/2005 11:38:44 AM

Maybe I have misunderstood this from the beginning considering the subject, in that case I apologize Paul. Maybe Carl will be kind enough to remind me what the brats were.

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 03 Kasım 2005 Perşembe 21:27
Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

I'm curious -- How does a chord specifically designed *not* to have
proportional beating sound delightful *because* of proportional
beating?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> They both sound delightful, because of proportional beating. A is
>more energetic, B is calmer.
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 11:39:40 AM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > > Comments?
> > //
> > > I played with a parameter in Cool Edit called
> > > "Start Phase" (in degrees). It didn't have any
> > > audible effect.
>
> There should be separate start phases for the three notes. It
should
> only make a difference when the beats are in simple rational
ratios.
> In that case some phase combinations will give shallower beats as
> beats partly cancel each other.

I still think this is impossible. Please prove me wrong.

>/tuning/files/Keenan/beats.
> xls.zip

The requested file or directory is not found on the server.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 11:46:21 AM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> I wrote:
> > Hopefully, by the time you read this I will have uploaded it to:
> >
/tuning/files/Keenan/beats.
> > xls.zip
>
> Nope. It seems tuning_files is too full. I deleted some of my old
> stuff but still couldn't upload this (1.9 MB). Maybe someone who
has
> _lots_ of files stored there could delete some old stuff.
>
> But in any case, thanks to my friend Andrew Smith, I've put it up
here:
> http://asmith.id.au/~dkeenan/beats.xls.zip
>
> -- Dave Keenan

Can you explain the calculations in this spreadsheet and what you're
trying to represent with them? The graph makes no sense to me right
now as a depiction of any audible quantity; I'd love to be filled in.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 11:55:55 AM

There is no need to apologize. It's just that one chord was designed
to be proportionally beating; the other was designed not to be
proportionally beating. So I think it's clear from people's reactions
here that proportional beating is not at all necessary for, or even a
significant contributor to, the delightful sound of a slightly
detuned consonant triad.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> Maybe I have misunderstood this from the beginning considering the
subject, in that case I apologize Paul. Maybe Carl will be kind
enough to remind me what the brats were.
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 03 Kasým 2005 Perþembe 21:27
> Subject: [tuning] Re: the simplest pan-proportionally beating 12-
tone temperament
>
>
> I'm curious -- How does a chord specifically designed *not* to
have
> proportional beating sound delightful *because* of proportional
> beating?
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> >
> > They both sound delightful, because of proportional beating. A
is
> >more energetic, B is calmer.
> >
>

🔗Tom Dent <stringph@gmail.com>

11/3/2005 12:09:08 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav
> Gene, can you confirm if this is what you
> would call "synch tuned"?
>
> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi. If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.
>
> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd. It's fifth is
> farther from the Dave's than B2.wav's.
>
> Comments?
>

B3 is amazingly pleasant. Of course it is the nearest to pure, but
just hovering shy of exact purity. B next, then B2 (more like 1/5
comma meantone, still pleasant) then A last.

~~~T~~~

🔗Ozan Yarman <ozanyarman@superonline.com>

11/3/2005 12:15:59 PM

In my opinion, you are too impatient to reach such conclusions. You have not even ascertained objective impartiality in this experiment, nor evaluated all the necessary inputs. Now I see I have been careless and rash in listening to the beats due to the poor quality of my PC speakers. A is non-proportional beating, while B is proportional, and thus better compared to A in regards to harmonic, if not melodic, consonance.

I feel that prop-brats are significant indeed.

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 03 Kasım 2005 Perşembe 21:55
Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

There is no need to apologize. It's just that one chord was designed
to be proportionally beating; the other was designed not to be
proportionally beating. So I think it's clear from people's reactions
here that proportional beating is not at all necessary for, or even a
significant contributor to, the delightful sound of a slightly
detuned consonant triad.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> Maybe I have misunderstood this from the beginning considering the
subject, in that case I apologize Paul. Maybe Carl will be kind
enough to remind me what the brats were.

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 1:07:24 PM

> Nope. It seems tuning_files is too full. I deleted some of my old
> stuff but still couldn't upload this (1.9 MB). Maybe someone who
> has _lots_ of files stored there could delete some old stuff.
>
> But in any case, thanks to my friend Andrew Smith, I've put it up
> here: http://asmith.id.au/~dkeenan/beats.xls.zip
>
> -- Dave Keenan

Thanks!

These groups only get something like 20MB of storage, which is
really lame. But given that web hosting is so cheap ($2/month
for a 200MB at one provider I just checked), I can't see why
it's a problem for so many folks.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 1:18:14 PM

> > I worried earlier that the intervals in synch triads
> > would beat all together, and thus make the beats
> > louder. Paul E. said that in real life, phase
> > differences would make this unlikely.
>
> No you must be thinking of what I said on some other topic.
> Here, all the beats occur at different pitches, so relative
> phase should have no effect on the individual or overall
> loudness of the beats.

Oops, that was George Secor. Sorry! (Any comments you have
on the msg below would still be appreciated.)

> > Hiya George (and Ozan),
> >
> > Do you find you like the triads where all the intervals beat
> > at the same rate? I would think it would be better to stagger
> > the beats, so that they do not result in large amplitude
> > changes. Or does this not happen?
> >
> > -Carl
//
> even in the 5/17-comma temperament with its 1:1:1 beat
> ratio (which I previously incorporated in the best keys
> of my 19-tone well-temperament -- a tuning I currently
> have stored in my Scalatron). I expect that one would
> have to deliberately adjust the phase relationships
> between the tones in order for the beats to reinforce
> one another so as to produce an amplitude large enough
> to be judged objectionable.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 1:20:19 PM

> Maybe I have misunderstood this from the beginning considering
> the subject, in that case I apologize Paul. Maybe Carl will be
> kind enough to remind me what the brats were.

A brat is a Beat RATio, usually defined for approximate
4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
beat rate.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 1:23:10 PM

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > Right, so Dave's original triad of
> > 512:639:766 Hz.
> > is still at
> > http://lumma.org/tuning/A.wav
> > Gene, can you confirm if this is what you
> > would call "synch tuned"?
> >
> > Now we can compare with this triad
> > 512:639:765.505572809 Hz.
> > at
> > http://lumma.org/tuning/B2.wav
> > whose major and minor third beat rates are
> > related by the golden mean, phi. If the
> > simple-ratio beating theory were right, we'd
> > expect this triad to sound very bad indeed,
> > since in some sense phi is the least rational
> > number.
> >
> > Last but not least, check out
> > 512:639:768.094427191 Hz.
> > at
> > http://lumma.org/tuning/B3.wav
> > which is the other phi-beating triad based
> > on Dave's major 3rd. It's fifth is
> > farther from the Dave's than B2.wav's.
> >
> > Comments?
> >
>
> B3 is amazingly pleasant. Of course it is the nearest to pure, but
> just hovering shy of exact purity. B next, then B2 (more like 1/5
> comma meantone, still pleasant) then A last.
>
> ~~~T~~~

Thanks for listening, Tom. That's very close to what I had,
except I had B3 and B switched.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 1:25:20 PM

> In my opinion, you are too impatient to reach such conclusions.
> You have not even ascertained objective impartiality in this
> experiment, nor evaluated all the necessary inputs. Now I see
> I have been careless and rash in listening to the beats due to
> the poor quality of my PC speakers. A is non-proportional
> beating, while B is proportional, and thus better compared to
> A in regards to harmonic, if not melodic, consonance.
>
> I feel that prop-brats are significant indeed.

Actually Ozan, it's the reverse. B is the non-proportional
chord, yet everyone (including you) seems to prefer it. That's
what's surprising here, and we're still trying to figure
out what's going on.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 1:39:08 PM

> > Right, so Dave's original triad of
> > 512:639:766 Hz.
> > is still at
> > http://lumma.org/tuning/A.wav
>
> Of course, for circulating 12-note scales we've been discussing brats
> of 2 and 4, not -1. We can get a brat of 2 from, for example,
>
> 2560:3205:3838 = 512:641:767.6
>
> and of 4 from, for example,
>
> 512:641:766

Here they are:

http://lumma.org/tuning/2.wav
http://lumma.org/tuning/4.wav

The 'spikiness' of the waveform display in Cool Edit of
these chords seems to be a very good guide (as you'd expect)
of how prominent the beats are. What are some good
'spikiness' measures of graphs?

-Carl

🔗Ozan Yarman <ozanyarman@superonline.com>

11/3/2005 2:08:48 PM

Yes, I meant the brats of the chord examples you gave the links to.
----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 03 Kasım 2005 Perşembe 23:20
Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

> Maybe I have misunderstood this from the beginning considering
> the subject, in that case I apologize Paul. Maybe Carl will be
> kind enough to remind me what the brats were.

A brat is a Beat RATio, usually defined for approximate
4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
beat rate.

-Carl

🔗Magnus Jonsson <magnus@smartelectronix.com>

11/3/2005 2:21:02 PM

On Thu, 3 Nov 2005, Carl Lumma wrote:

> The 'spikiness' of the waveform display in Cool Edit of
> these chords seems to be a very good guide (as you'd expect)
> of how prominent the beats are. What are some good
> 'spikiness' measures of graphs?
>
> -Carl

I have an idea which might work:

1) Filter out all frequencies that we can't hear.
2) Square each sample, to get the momental energy output flow. If the signal has prominent beating, this energy output flow will probably be very uneven.
3) Lowpass filter this energy signal to get rid of energy flow variations are too quick for us to consider as beating.
4b) (optional) Highpass filter to get rid of variations that are too slow to be meaningful.
5) Calculate the standard deviation as a measurement of the unevenness of energy flow.

Here is the signal flow:

input -> bandpass -> square each sample -> lowpass(or bandpass) -> estimate standard variation

Note that I haven't tried this out in practice so I can't give any guarantees about it :)

- Magnus Jonsson

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 2:47:26 PM

Hate to have to say it, but you're saying (along with most other
listeners here) that you yourself prefer the non-proportionally-
beating triad B (which you erroneously say is proportionally beating)
to the proportionally-beating triad A (which you erroneously say is
non-proportionally beating). So I think your conclusion is unfounded.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> In my opinion, you are too impatient to reach such conclusions. You
>have not even ascertained objective impartiality in this experiment,
>nor evaluated all the necessary inputs. Now I see I have been
>careless and rash in listening to the beats due to the poor quality
>of my PC speakers. A is non-proportional beating, while B is
>proportional, and thus better compared to A in regards to harmonic,
>if not melodic, consonance.
>
> I feel that prop-brats are significant indeed.
>
> Cordially,
> Ozan
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 03 Kasým 2005 Perþembe 21:55
> Subject: [tuning] Re: the simplest pan-proportionally beating 12-
tone temperament
>
>
> There is no need to apologize. It's just that one chord was
designed
> to be proportionally beating; the other was designed not to be
> proportionally beating. So I think it's clear from people's
reactions
> here that proportional beating is not at all necessary for, or
even a
> significant contributor to, the delightful sound of a slightly
> detuned consonant triad.
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> >
> > Maybe I have misunderstood this from the beginning considering
the
> subject, in that case I apologize Paul. Maybe Carl will be kind
> enough to remind me what the brats were.
>

🔗Ozan Yarman <ozanyarman@superonline.com>

11/3/2005 2:57:40 PM

In that case I retract my statement. Obviously I am confused as to what I am talking about. Please disregard my words.

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 04 Kasım 2005 Cuma 0:47
Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Hate to have to say it, but you're saying (along with most other
listeners here) that you yourself prefer the non-proportionally-
beating triad B (which you erroneously say is proportionally beating)
to the proportionally-beating triad A (which you erroneously say is
non-proportionally beating). So I think your conclusion is unfounded.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> In my opinion, you are too impatient to reach such conclusions. You
>have not even ascertained objective impartiality in this experiment,
>nor evaluated all the necessary inputs. Now I see I have been
>careless and rash in listening to the beats due to the poor quality
>of my PC speakers. A is non-proportional beating, while B is
>proportional, and thus better compared to A in regards to harmonic,
>if not melodic, consonance.
>
> I feel that prop-brats are significant indeed.
>
> Cordially,
> Ozan
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 3:00:12 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > I worried earlier that the intervals in synch triads
> > > would beat all together, and thus make the beats
> > > louder. Paul E. said that in real life, phase
> > > differences would make this unlikely.
> >
> > No you must be thinking of what I said on some other topic.
> > Here, all the beats occur at different pitches, so relative
> > phase should have no effect on the individual or overall
> > loudness of the beats.
>
> Oops, that was George Secor. Sorry! (Any comments you have
> on the msg below would still be appreciated.)
>
> > > Hiya George (and Ozan),
> > >
> > > Do you find you like the triads where all the intervals beat
> > > at the same rate? I would think it would be better to stagger
> > > the beats, so that they do not result in large amplitude
> > > changes. Or does this not happen?
> > >
> > > -Carl
> //
> > even in the 5/17-comma temperament with its 1:1:1 beat
> > ratio (which I previously incorporated in the best keys
> > of my 19-tone well-temperament -- a tuning I currently
> > have stored in my Scalatron). I expect that one would
> > have to deliberately adjust the phase relationships
> > between the tones in order for the beats to reinforce
> > one another so as to produce an amplitude large enough
> > to be judged objectionable.
>
> -Carl

This statement (of George's) makes sense -- of course you can get the
beatings to be in phase with one another so that they all hit maximum
amplitude in synch. But they'll each sound equally loud regardless of
whether they're in phase or not. There's no way to get the beats to
cancel one another out, as Dave Keenan seemingly suggested. And there
will be plenty of other partials in the sound anyway so the overall
amplitude of everything will still remain reasonably constant.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 3:02:01 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > In my opinion, you are too impatient to reach such conclusions.
> > You have not even ascertained objective impartiality in this
> > experiment, nor evaluated all the necessary inputs. Now I see
> > I have been careless and rash in listening to the beats due to
> > the poor quality of my PC speakers. A is non-proportional
> > beating, while B is proportional, and thus better compared to
> > A in regards to harmonic, if not melodic, consonance.
> >
> > I feel that prop-brats are significant indeed.
>
> Actually Ozan, it's the reverse. B is the non-proportional
> chord, yet everyone (including you) seems to prefer it. That's
> what's surprising here, and we're still trying to figure
> out what's going on.
>
> -Carl

Maybe the beats in A should have been staggered . . . (they still
wouldn't cancel each other out, but perhaps the effect would be more
natural-sounding.)

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 3:09:11 PM

> > > In my opinion, you are too impatient to reach such conclusions.
> > > You have not even ascertained objective impartiality in this
> > > experiment, nor evaluated all the necessary inputs. Now I see
> > > I have been careless and rash in listening to the beats due to
> > > the poor quality of my PC speakers. A is non-proportional
> > > beating, while B is proportional, and thus better compared to
> > > A in regards to harmonic, if not melodic, consonance.
> > >
> > > I feel that prop-brats are significant indeed.
> >
> > Actually Ozan, it's the reverse. B is the non-proportional
> > chord, yet everyone (including you) seems to prefer it. That's
> > what's surprising here, and we're still trying to figure
> > out what's going on.
> >
> > -Carl
>
> Maybe the beats in A should have been staggered . . . (they
> still wouldn't cancel each other out, but perhaps the effect
> would be more natural-sounding.)

You mean the phases staggered? That's what I can't seem to
do in Cool Edit. Dave's spreadsheet looks like it might
produce sound, but I didn't hear any when playing with it.
The waveform graph doesn't seem to change too much when
I play with the phases there, though.

-Carl

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 3:09:47 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I don't understand how any of you guys can objectively
> > say that any of these are "better" or "worse". ?
> >
> > To my ears, they all exhibit different beat patterns,
> > which are simply different rhythms, none especially
> > better or worse than the others.
>
> Better or worse isn't the relevant question. I'd phrase it as more
> organized vs less organized. The synch beating example beats
> coherently, which someone may or may not like the sound of. But I
> think it can be helpful in making the ear think it *ought* to sound
> like this. It has some of the characteristics of JI, without being
JI.
>

Monz,

I don't think anyone's claiming "objectivity" for this. Carl asked
us to say which we preferred, and that's what we've been doing. I
happen to agree with the majority so far in finding that if your aim
is to make your temperament sound more Just, then synchronised beat
ratios may _not_ necessarily be the way to go. Or if it is, then it
may need careful attention to the relative phases of the notes, to
minimise the beat depth (this idea hasn't been checked by experiment
yet).

If you don't have the capability for actual phase locked synchrony
(or maybe even if you do), then it may be better to have quite the
opposite of beat synchrony, namely beats whose ratios are noble
numbers, e.g. of the form (a + b*phi)/(c + d*phi) where a,b,c,d are
small whole numbers. Or maybe it just doesn't matter either way.

I said earlier that it doesn't seem easy to predict, but I've just
realised (duh!) that, as Carl suggested, it is the most prominent
beat that sticks out the most (is that a tautology?).

A single variable that correlates well with the preferences of the
majority here is simply the _error_in_the_fifth_. It's the beat rate
of the fifth that dominates, since it involves lower harmonics (2
and 3), which have greater amplitude.

The logical conclusion of this for meantones is, sad to say, equal
temperament.

-- Dave Keenan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 3:34:20 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > In my opinion, you are too impatient to reach such
conclusions.
> > > > You have not even ascertained objective impartiality in this
> > > > experiment, nor evaluated all the necessary inputs. Now I see
> > > > I have been careless and rash in listening to the beats due to
> > > > the poor quality of my PC speakers. A is non-proportional
> > > > beating, while B is proportional, and thus better compared to
> > > > A in regards to harmonic, if not melodic, consonance.
> > > >
> > > > I feel that prop-brats are significant indeed.
> > >
> > > Actually Ozan, it's the reverse. B is the non-proportional
> > > chord, yet everyone (including you) seems to prefer it. That's
> > > what's surprising here, and we're still trying to figure
> > > out what's going on.
> > >
> > > -Carl
> >
> > Maybe the beats in A should have been staggered . . . (they
> > still wouldn't cancel each other out, but perhaps the effect
> > would be more natural-sounding.)
>
> You mean the phases staggered?

Yes.

> That's what I can't seem to
> do in Cool Edit.

If you can stagger the onset times, it's likely the phases will end
up staggered too.

> Dave's spreadsheet looks like it might
> produce sound, but I didn't hear any when playing with it.
> The waveform graph doesn't seem to change too much when
> I play with the phases there, though.

It does for me, though I'm skeptical that this waveform graph is
actually showing the waveform of anything meaningful.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/3/2005 3:35:47 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> A single variable that correlates well with the preferences of the
> majority here is simply the _error_in_the_fifth_. It's the beat rate
> of the fifth that dominates, since it involves lower harmonics (2
> and 3), which have greater amplitude.
>
> The logical conclusion of this for meantones is, sad to say, equal
> temperament.

How do you arrive at that conclusion?

🔗Gene Ward Smith <gwsmith@svpal.org>

11/3/2005 4:17:33 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > If the simple-ratio beating theory were right, we'd
> > > expect this triad to sound very bad indeed,
> > > since in some sense phi is the least rational
> > > number.
> >
> > It seems to me that is a ridiculous conclusion.
>
> Why?
>
> > Certainly, I can't recall anyone claiming such a thing.
>
> The claim that simple-ratio beat rates sound better has
> been made by Bob Wendell, among several others.

Wendell did not say that non-simple ratios sound very bad indeed.
You've extrapolated wildly from his claims.

Including
> you in the blackbeat15.scl thread.

No, all I said in that thread is that I thought the blackbeat version
was considerably better than the 15edo version. That suggested to me
that synch beating might have proven useful, but of course it wasn't
the only thing which differed.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/3/2005 4:27:39 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Actually Ozan, it's the reverse. B is the non-proportional
> chord, yet everyone (including you) seems to prefer it.

I neither prefer it nor fail to prefer it. I note that the synch
beating sounds quite different than the non-synch beating, producing a
different musical effect.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/3/2005 4:39:00 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> The 'spikiness' of the waveform display in Cool Edit of
> these chords seems to be a very good guide (as you'd expect)
> of how prominent the beats are. What are some good
> 'spikiness' measures of graphs?

It seems to me the question is more one of regularity than spikiness.
The differences are striking, and can also be viewed in Audacity,
Sound Forge, and WavePad. Audacity is free, and WavePad is shareware,
so it should be possible for most people to view these, though it
would be nice to learn if one of these programs will produce a visual
file (jpg, etc.)

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 4:48:13 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > If you choose the right
> > initial phases and relative amplitudes for the three notes you may
> > actually be able to get the beats to cancel each other out, or
> > nearly so.
>
> How is this possible? Each of the beatings occurs at a different
pitch
> (that is, for a different pair of nearly coincident partials) so I
have
> no idea how this could happen here. I'd love to be educated,
> particularly with a demonstration!

There are at least two perceptual aspects to beats
(a) cyclic variation of loudness with time
(b) cyclic variation of timbre with time

I'm saying we could eliminate the first, but not the second.

Here's a very contrived example.

Use a timbre that has equal amplitudes for the fundamental and all
harmonics up to the sixth, then nothing after that. And one needs to
be able to set the start-phase of each harmonic independently.

Consider the sync-beating 512:639:766 Hz chord. This will have the
following beats between the following four pairs of partials.

4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
8 Hz between 6th harmonic of root and 4th harmonic of fifth
4 Hz between 5th harmonic of root and 4th harmonic of major third
4 Hz between 6th harmonic of major third and 5th harmonic of fifth

By carefully arranging the phases of the partials in the original
timbre (sorry I'm too lazy to work out the details) it should be
possible to arrange for the three 4 Hz beats to be 120 degrees out of
phase and thereby cancel each other and leave only the 8 Hz beat.

This isn't terribly satisfactory since with equal amplitude partials
that 8 Hz beat will still be quite obvious, so the next step is to
look at a more normal timbre where the higher partials have lower
amplitude.

In this case it should be possible to make it so that the beats of the
major and minor third intervals are in phase with each other and add
to produce a beat which is the same amplitude as the 4 Hz beat in the
fifth interval but 180 degrees out of phase with it, and thereby
cancelling it. We still have the 8 Hz beat, but this time at lower
amplitude.

If one cheated and knocked out the 6th harmonic of the root and the
4th harmonic of the fifth completely, then one could have no beating
at all in the triad (in the sense of loudness variation) even though
every dyad heard alone has obvious beating. Of course we still expect
to hear a sort of beating of the triad due to the cyclic timbre
variation.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 4:58:02 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus" wrote
> This statement (of George's) makes sense -- of course you can get
the
> beatings to be in phase with one another so that they all hit
maximum
> amplitude in synch. But they'll each sound equally loud regardless
of
> whether they're in phase or not. There's no way to get the beats
to
> cancel one another out, as Dave Keenan seemingly suggested. And
there
> will be plenty of other partials in the sound anyway so the
overall
> amplitude of everything will still remain reasonably constant.

In my terms, you seem to be saying that even with ordinary
uncontrived beating, the cyclic loundess variation is relatively
unimportant and the cyclic timbre variation is mostly what we hear.
That makes sense, and I probably am all wet. But it wouldn't hurt to
do the experiment.

So now I suppose I have to come up with exact specifications for
Carl to synthesise.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 5:03:48 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Maybe I have misunderstood this from the beginning considering
> > the subject, in that case I apologize Paul. Maybe Carl will be
> > kind enough to remind me what the brats were.
>
> A brat is a Beat RATio, usually defined for approximate
> 4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
> beat rate.

Why use this obfuscatory jargon in the first place? How hard can it be
to type "beat ratio" and to be clear about exactly which intervals
you're considering the beat ratio of? [Rhetorical questions]

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 5:14:05 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> You mean the phases staggered? That's what I can't seem to
> do in Cool Edit. Dave's spreadsheet looks like it might
> produce sound, but I didn't hear any when playing with it.

No it doesn't produce sound. It only looks at the loudness
variation, which is indeed small, as Paul suggested. So maybe he's
right, that most of what we hear as beats is actually timbre
variation.

> The waveform graph doesn't seem to change too much when
> I play with the phases there, though.

It only does so with the chord you've called A (actually it's closer
to a concert-pitch C) ;-), namely 512:639:766, the synced equal
beats version. But you wont get complete cancellation of loudness
variation without contriving the relative amplitudes and phases of
the partials in the original timbre.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 5:20:39 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Actually Ozan, it's the reverse. B is the non-proportional
> > chord, yet everyone (including you) seems to prefer it.
>
> I neither prefer it nor fail to prefer it. I note that the synch
> beating sounds quite different than the non-synch beating,
producing a
> different musical effect.
>

Yes. Which is how I originally got into this discussion. By saying I
thought that synced beats had a special sound, and referring to it
as "second order JI". But I now think that if you're trying to
approximate first order JI with a temperament, then it might be
better to avoid "second order JI". Or it might simply not matter
much either way and might just show that you should try to minimise
the beat rate of the loudest harmonics.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 5:31:30 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > http://asmith.id.au/~dkeenan/beats.xls.zip
> >
> > -- Dave Keenan
>
> Can you explain the calculations in this spreadsheet and what you're
> trying to represent with them? The graph makes no sense to me right
> now as a depiction of any audible quantity; I'd love to be filled in.

It's intended to show an approximation to the envelope of the sound,
i.e. the variation of loudness with time. But as you say, that may not
be anything perceptually significant for the phenomenon of beating of
complex waveforms.

You talk about the beats happening at different "pitches". That's not
quite right, since pitch is a perceptual quality that may depend on
more than one partial. If you'd said "different frequencies" then
you'd be right, but most people I know don't hear the individual sine-
wave partials which mathematically-speaking make up a sound (unless
perhaps when one is suddenly added or taken away). They instead hear a
quality I'm calling timbre. Sorry to be so pedantic.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/3/2005 5:48:47 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > A single variable that correlates well with the preferences of the
> > majority here is simply the _error_in_the_fifth_. It's the beat
rate
> > of the fifth that dominates, since it involves lower harmonics (2
> > and 3), which have greater amplitude.
> >
> > The logical conclusion of this for meantones is, sad to say, equal
> > temperament.
>
> How do you arrive at that conclusion?
>

Simple. It's the meantone with the slowest beating fifths.

I don't consider any fifth-generated linear temperament with a fifth
wider than 700 cents to be a meantone. We are then into the
Pythagorean region.

-- Dave Keenan

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 10:07:28 PM

> > > Maybe the beats in A should have been staggered . . . (they
> > > still wouldn't cancel each other out, but perhaps the effect
> > > would be more natural-sounding.)
> >
> > You mean the phases staggered?
>
> Yes.
>
> > That's what I can't seem to
> > do in Cool Edit.
>
> If you can stagger the onset times, it's likely the phases will end
> up staggered too.

No can do. Well actually, can do, with 3 tracks and mix
down to mono....

> > Dave's spreadsheet looks like it might
> > produce sound, but I didn't hear any when playing with it.
> > The waveform graph doesn't seem to change too much when
> > I play with the phases there, though.

I was unable to get the period of the biggest humps to change
much by changing the phase. Can you give me a scenario to
look at?

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 10:17:49 PM

> > The 'spikiness' of the waveform display in Cool Edit of
> > these chords seems to be a very good guide (as you'd expect)
> > of how prominent the beats are. What are some good
> > 'spikiness' measures of graphs?
>
> It seems to me the question is more one of regularity than
> spikiness.

Not sure what the difference is...

> The differences are striking, and can also be viewed in
> Audacity, Sound Forge, and WavePad. Audacity is free, and
> WavePad is shareware, so it should be possible for most
> people to view these, though it would be nice to learn if
> one of these programs will produce a visual file (jpg, etc.)

There is such a thing as a screen shot...

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 10:27:39 PM

> > > Maybe I have misunderstood this from the beginning considering
> > > the subject, in that case I apologize Paul. Maybe Carl will be
> > > kind enough to remind me what the brats were.
> >
> > A brat is a Beat RATio, usually defined for approximate
> > 4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
> > beat rate.
>
> Why use this obfuscatory jargon in the first place? How hard can
> it be to type "beat ratio" and to be clear about exactly which
> intervals you're considering the beat ratio of? [Rhetorical
> questions]

It wasn't my idea (though I didn't disapprove).

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 10:35:31 PM

> > You mean the phases staggered? That's what I can't seem to
> > do in Cool Edit. Dave's spreadsheet looks like it might
> > produce sound, but I didn't hear any when playing with it.
>
> No it doesn't produce sound. It only looks at the loudness
> variation, which is indeed small, as Paul suggested. So maybe
> he's right, that most of what we hear as beats is actually
> timbre variation.

I didn't entirely follow your method for canceling beats...
did you address Paul's point that two 4 Hz. beats 180deg.
out of phase still won't cancel unless their 'carriers'
were at the same frequency? Or is that what you mean by
timbre variation? (Wait, maybe I'm getting it.) In any
case, what I hear as prominent beats is very clearly seen
in Cool Edit's (or any other wave editor's, as Gene points
out) waveform display as variations in the total amplitude.
Maybe that's because I haven't synthesized the right
timbers and start phases yet. But for a general-purpose
tuning theory, this is too specialized. Might be of
interest to folks like Bill Sethares, though.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/3/2005 10:38:21 PM

> Yes. Which is how I originally got into this discussion. By saying I
> thought that synced beats had a special sound, and referring to it
> as "second order JI". But I now think that if you're trying to
> approximate first order JI with a temperament, then it might be
> better to avoid "second order JI". Or it might simply not matter
> much either way and might just show that you should try to minimise
> the beat rate of the loudest harmonics.

I think you're right, Dave. Getting rid of error is enough of a
problem. And as you point out, it seems to be a good rule for
minimizing beat rates... and with weighted error, this will tend
to favor the 'loudest harmonics', wouldn't it?

-Carl

🔗monz <monz@tonalsoft.com>

11/4/2005 12:33:35 AM

Hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I don't think anyone's claiming "objectivity" for this.
> Carl asked us to say which we preferred, and that's what
> we've been doing.

OK, then my bad ... guess i wasn't paying enough attention
to the earlier posts in this thread. I thought i saw folks
saying "B2 is the best" etc.

> I happen to agree with the majority so far in finding
> that if your aim is to make your temperament sound more
> Just, then synchronised beat ratios may _not_ necessarily
> be the way to go. Or if it is, then it may need careful
> attention to the relative phases of the notes, to minimise
> the beat depth (this idea hasn't been checked by experiment
> yet).

I too have not done significant experiments in audio-testing
various aspects of phase, but all accounts that i've ever
read about digital synthesis agree that phase differences
are audibly insignificant.

> A single variable that correlates well with the preferences
> of the majority here is simply the _error_in_the_fifth_.
> It's the beat rate of the fifth that dominates, since it
> involves lower harmonics (2 and 3), which have greater
> amplitude.

The lower harmonics generally do have greater amplitude in
the timbres of acoustic instruments. But in synthesized
timbres that is not necessarily the case ... i suppose
it must be the case in the timbres Carl used.

> The logical conclusion of this for meantones is,
> sad to say, equal temperament.

By "equal temperament", do you mean 12-edo, or just the
EDO meantones in general?

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 5:20:23 AM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> I too have not done significant experiments in audio-testing
> various aspects of phase, but all accounts that i've ever
> read about digital synthesis agree that phase differences
> are audibly insignificant.

In general yes. But the question is, could they become significant in
the case of synced beats.

> > The logical conclusion of this for meantones is,
> > sad to say, equal temperament.
>
>
> By "equal temperament", do you mean 12-edo, or just the
> EDO meantones in general?

Oops! Sorry. Yes, I meant 12 equal.

-- Dave Keenan

🔗Tom Dent <stringph@gmail.com>

11/4/2005 9:25:33 AM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > How is this possible? Each of the beatings occurs at a different
> pitch
> > (that is, for a different pair of nearly coincident partials) so I
> have
> > no idea how this could happen here. I'd love to be educated,
> > particularly with a demonstration!
>
> There are at least two perceptual aspects to beats
> (a) cyclic variation of loudness with time
> (b) cyclic variation of timbre with time
>
> I'm saying we could eliminate the first, but not the second.
>
> Here's a very contrived example.
>
> Use a timbre that has equal amplitudes for the fundamental and all
> harmonics up to the sixth, then nothing after that. And one needs to
> be able to set the start-phase of each harmonic independently.
>
> Consider the sync-beating 512:639:766 Hz chord. This will have the
> following beats between the following four pairs of partials.
>
> 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> 4 Hz between 5th harmonic of root and 4th harmonic of major third
> 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
>
> By carefully arranging the phases of the partials in the original
> timbre (sorry I'm too lazy to work out the details) it should be
> possible to arrange for the three 4 Hz beats to be 120 degrees out of
> phase and thereby cancel each other and leave only the 8 Hz beat.
>
> This isn't terribly satisfactory since with equal amplitude partials
> that 8 Hz beat will still be quite obvious, so the next step is to
> look at a more normal timbre where the higher partials have lower
> amplitude.
>
> In this case it should be possible to make it so that the beats of the
> major and minor third intervals are in phase with each other and add
> to produce a beat which is the same amplitude as the 4 Hz beat in the
> fifth interval but 180 degrees out of phase with it, and thereby
> cancelling it. We still have the 8 Hz beat, but this time at lower
> amplitude.
>
> If one cheated and knocked out the 6th harmonic of the root and the
> 4th harmonic of the fifth completely, then one could have no beating
> at all in the triad (in the sense of loudness variation) even though
> every dyad heard alone has obvious beating. Of course we still expect
> to hear a sort of beating of the triad due to the cyclic timbre
> variation.
>
> -- Dave Keenan

Well, I still think the beating in timbre would be very much
noticeable. In my experience beating only produces a noticeable
oscillation of loudness at the unison and the octave.

> 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> 4 Hz between 5th harmonic of root and 4th harmonic of major third
> 4 Hz between 6th harmonic of major third and 5th harmonic of fifth

The three different 4Hz beats occur at 3 different overtone pitches
and there is no way they can cancel each other out. To put it another
way, in a close position chord *every* beat is a variation in timbre,
since every beat affects the upper partials, not the fundamental.
Beats could in principle only cancel each other out if they were all
at the same pitch (as in a violin section all playing the 'same' note).

~~~T~~~

🔗Tom Dent <stringph@gmail.com>

11/4/2005 9:37:11 AM

It rather depends on the timbral characteristics of your instrument.
If it happens to have rather weak 3rd, 6th etc. partials then the
beating of the 5ths is not so prominent and you can get good results
by making them narrower.

It might well be that harpsichord makers deliberately made the 3rd
harmonic weaker so that meantone temperaments would be more successful.
~~~T~~~

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> >
> > > A single variable that correlates well with the preferences of the
> > > majority here is simply the _error_in_the_fifth_. It's the beat
> rate
> > > of the fifth that dominates, since it involves lower harmonics (2
> > > and 3), which have greater amplitude.
> > >
> > > The logical conclusion of this for meantones is, sad to say, equal
> > > temperament.
> >
> > How do you arrive at that conclusion?
> >
>
> Simple. It's the meantone with the slowest beating fifths.
>
> I don't consider any fifth-generated linear temperament with a fifth
> wider than 700 cents to be a meantone. We are then into the
> Pythagorean region.
>
> -- Dave Keenan
>

🔗Brad Lehman <bpl@umich.edu>

11/4/2005 10:21:21 AM

> In my experience beating only produces a noticeable
> oscillation of loudness at the unison and the octave.

Gotta install more 1/5 or 1/6 comma temperaments on your harpsichord,
then, and play a bunch of open major 10ths. The pulsating effect is
quite noticeable and lovely. The playing of 12ths also brings it out.

Brad Lehman

🔗Brad Lehman <bpl@umich.edu>

11/4/2005 11:20:27 AM

> It rather depends on the timbral characteristics of your instrument.
> If it happens to have rather weak 3rd, 6th etc. partials then the
> beating of the 5ths is not so prominent and you can get good results
> by making them narrower.
>
> It might well be that harpsichord makers deliberately made the 3rd
> harmonic weaker so that meantone temperaments would be more
successful.

Wait. Why is a carefully-regulated impure 5th, and its beating, some
bad thing that would need to be hidden away (deliberately or
otherwise) to make the sound "more successful"?

In 1/4 comma meantone on harpsichords, as some might argue, it's the
noticeably impure 5ths that give the overall sound some liveliness...
counteracting the deadness and stasis of the pure major 3rds.
Complete purity (stasis) in triads is not necessarily a top *musical*
goal for the way music behaves, as forward motion is important too.
The farther away we get from 1/4 comma, the less "need" there is to
liven up the sound with non-harmonic grit in the ornamentation,
because the major 3rds themselves are contributing some motion.

The regular 1/5 and 1/6 comma temperaments have some proportionality
of impurities in the 3rds vs 5ths. But, the thing that makes them
interesting to listen to (as compared with 1/4 comma) is *not* so much
that any beats synch up sometimes; rather, it's more prominently the
sense of life in those major 3rds themselves, refusing to sit
absolutely still. Don't all of these features contribute somewhat to
making the overall sound "successful" by various criteria and
expectations? The sound gets more complex, and therefore more
interesting, when both the 5ths and major 3rds have some motion to
them...whether we're trying to line any of it up, or not. The ear is
drawn to the property that *something* is happening there in the sound
of sustained intervals.

Another important factor is the distance away from the instrument that
one is listening. What might seem "successful" in 5ths and 3rds up
close, might be just the opposite at a distance; and vice versa.
Given that the instrument and performance manner should be calculated
to sound best where the king or queen is sitting.......this being a
musical rudiment of playing the hall correctly......

Brad Lehman

🔗Tom Dent <stringph@gmail.com>

11/4/2005 12:11:59 PM

Sigh... Did you actually listen to the samples A, B, B1, B2? I was
thinking mainly about the preference that almost everyone expressed,
which turned out to be that the slower the beating of the 5th (though
not going as far as absolute purity), the more pleasant the chord.

Perhaps, though, this was a function of overtone strengths: that the
3rd harmonic was rather stronger than the 5th in the example - perhaps
relatively stronger than in most acoustic instruments.

My idea was that some harpsichord makers might have intended the 5th
harmonic stronger than the 3rd. NOT that they might have wanted to
eliminate the beating of 5ths altogether (which would make the
instrument untunable!). Taking the opinion 'X should be bigger than Y'
to a supposedly 'logical' conclusion 'X should be enormous and Y
should be nonexistent' is a straw man. In 1/4 comma meantone, the
beating 5ths are always going to be noticeable to some extent; the
question is how to stop them *dominating* the aural picture (as they
would if the 3rd harmonic was stronger).

Actually, my spinet has a rather strong 3rd harmonic, which makes it
easy to tune, but some tempered fifths rather *obtrusive* to listen
to. That's why I tend to prefer more nearly (12-)equal tunings on it.

In 1/4 comma meantone, the fifths beat really pretty quickly and in
many cases in the 'intermittence' range which psychoacoustically
causes annoyance. I would want this sort of rapid beating to be almost
submerged below other strong harmonics if the major chord is to be
perceived as a consonance.

Besides, isn't it a strong consensus in the historical tuning industry
that the size of the fifths in a circular temperament is of little
consequence except insofar as they determine the size of thirds? How
could this be true except if the beating of fifths was rather unobtrusive?

I'll let others deal with the 'deadness' of pure intervals... and the
position of the king and queen and how the strengths of harmonics can
reverse themselves in travelling through the air (??). AFAIK most
harpsichord music was written for private enjoyment.

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > It rather depends on the timbral characteristics of your instrument.
> > If it happens to have rather weak 3rd, 6th etc. partials then the
> > beating of the 5ths is not so prominent and you can get good results
> > by making them narrower.
> >
> > It might well be that harpsichord makers deliberately made the 3rd
> > harmonic weaker so that meantone temperaments would be more
> successful.
>
>
> Wait. Why is a carefully-regulated impure 5th, and its beating, some
> bad thing that would need to be hidden away (deliberately or
> otherwise) to make the sound "more successful"?
>
> In 1/4 comma meantone on harpsichords, as some might argue, it's the
> noticeably impure 5ths that give the overall sound some liveliness...
> counteracting the deadness and stasis of the pure major 3rds.
> Complete purity (stasis) in triads is not necessarily a top *musical*
> goal for the way music behaves, as forward motion is important too.
> The farther away we get from 1/4 comma, the less "need" there is to
> liven up the sound with non-harmonic grit in the ornamentation,
> because the major 3rds themselves are contributing some motion.
>
> The regular 1/5 and 1/6 comma temperaments have some proportionality
> of impurities in the 3rds vs 5ths. But, the thing that makes them
> interesting to listen to (as compared with 1/4 comma) is *not* so much
> that any beats synch up sometimes; rather, it's more prominently the
> sense of life in those major 3rds themselves, refusing to sit
> absolutely still. Don't all of these features contribute somewhat to
> making the overall sound "successful" by various criteria and
> expectations? The sound gets more complex, and therefore more
> interesting, when both the 5ths and major 3rds have some motion to
> them...whether we're trying to line any of it up, or not. The ear is
> drawn to the property that *something* is happening there in the sound
> of sustained intervals.
>
> Another important factor is the distance away from the instrument that
> one is listening. What might seem "successful" in 5ths and 3rds up
> close, might be just the opposite at a distance; and vice versa.
> Given that the instrument and performance manner should be calculated
> to sound best where the king or queen is sitting.......this being a
> musical rudiment of playing the hall correctly......
>
>
> Brad Lehman
>

🔗Tom Dent <stringph@gmail.com>

11/4/2005 12:21:16 PM

Ah, criticism by means of quote out of context. The context (clear
from Paul's previous messages) being, whether oscillation of
*loudness* or of *timbre* was more relatively important in the
perception of beating.

So, to explain once again, I hear beating at a major third and fifth
(and tenth, and twelfth, etc.) primarily as an oscillation of
*timbre*. Pulsation is a reasonable word - and ironically, it does not
imply variation of loudness.

The idea that I'm not aware of the existence of audible beats in major
thirds and tenths on a harpsichord is ludicrous.
~~~T~~~

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > In my experience beating only produces a noticeable
> > oscillation of loudness at the unison and the octave.
>
> Gotta install more 1/5 or 1/6 comma temperaments on your harpsichord,
> then, and play a bunch of open major 10ths. The pulsating effect is
> quite noticeable and lovely. The playing of 12ths also brings it out.
>
> Brad Lehman
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 1:16:32 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > If you choose the right
> > > initial phases and relative amplitudes for the three notes you
may
> > > actually be able to get the beats to cancel each other out, or
> > > nearly so.
> >
> > How is this possible? Each of the beatings occurs at a different
> pitch
> > (that is, for a different pair of nearly coincident partials) so
I
> have
> > no idea how this could happen here. I'd love to be educated,
> > particularly with a demonstration!
>
> There are at least two perceptual aspects to beats
> (a) cyclic variation of loudness with time

Ah, but do you know how to calculate the loudness of a signal
correctly?

> (b) cyclic variation of timbre with time
>
> I'm saying we could eliminate the first, but not the second.

You cannot eliminate any of the beatings, so I think I disagree.

> Here's a very contrived example.
>
> Use a timbre that has equal amplitudes for the fundamental and all
> harmonics up to the sixth, then nothing after that. And one needs
to
> be able to set the start-phase of each harmonic independently.
>
> Consider the sync-beating 512:639:766 Hz chord. This will have the
> following beats between the following four pairs of partials.
>
> 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> 4 Hz between 5th harmonic of root and 4th harmonic of major third
> 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
>
> By carefully arranging the phases of the partials in the original
> timbre (sorry I'm too lazy to work out the details) it should be
> possible to arrange for the three 4 Hz beats to be 120 degrees out
of
> phase and thereby cancel each other and leave only the 8 Hz beat.

Again, your use of the word "cancel" here strikes be as totally
incorrect. There is no cancellation because all the beats are just as
audible whether they are in phase or out of phase.

> This isn't terribly satisfactory since with equal amplitude
partials
> that 8 Hz beat will still be quite obvious, so the next step is to
> look at a more normal timbre where the higher partials have lower
> amplitude.
>
> In this case it should be possible to make it so that the beats of
the
> major and minor third intervals are in phase with each other and
add
> to produce a beat which is the same amplitude as the 4 Hz beat in
the
> fifth interval but 180 degrees out of phase with it, and thereby
> cancelling it.

Ditto. No cancellation would take place.

> We still have the 8 Hz beat, but this time at lower
> amplitude.
>
> If one cheated and knocked out the 6th harmonic of the root and the
> 4th harmonic of the fifth completely, then one could have no
beating
> at all in the triad (in the sense of loudness variation) even
though
> every dyad heard alone has obvious beating.

Well, I'm glad you agree at least about the latter.

(a) You haven't calculated loudness correctly.

(b) Have you listened to check if this supposed lack of loudness
variation is actually audible as such? And to see if "no beating at
all in the triad" is at all a fair characterization of the sound?
(Both should be checked by comparing different phase arrangements.)

> Of course we still expect
> to hear a sort of beating of the triad due to the cyclic timbre
> variation.

I don't call it a cyclic timbre variation, I call it beating, plain
and simple. Although "variation in total loudness" does mean
something, it's not what contributes to the sensation of beating
(rather it's variation in loudness at particular pitches that does),
and in any case you haven't calculated total loudness correctly.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 1:25:42 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > > http://asmith.id.au/~dkeenan/beats.xls.zip
> > >
> > > -- Dave Keenan
> >
> > Can you explain the calculations in this spreadsheet and what
you're
> > trying to represent with them? The graph makes no sense to me
right
> > now as a depiction of any audible quantity; I'd love to be filled
in.
>
> It's intended to show an approximation to the envelope of the
sound,
> i.e. the variation of loudness with time.

I'd like to know how you think your calculation acheives this
approximation, and what references/theories you're basing this on.

> But as you say, that may not
> be anything perceptually significant for the phenomenon of beating
of
> complex waveforms.
>
> You talk about the beats happening at different "pitches". That's
not
> quite right, since pitch is a perceptual quality that may depend on
> more than one partial.

But in order to hear the beating clearly, you focus on a particular
*pitch* so as to hear out the relevant pair of partials.

> If you'd said "different frequencies" then
> you'd be right, but most people I know don't hear the individual
sine-
> wave partials which mathematically-speaking make up a sound (unless
> perhaps when one is suddenly added or taken away). They instead
hear a
> quality I'm calling timbre. Sorry to be so pedantic.

It's OK. Piano tuners are trained to listen for beats *at* the pitch
of the partials which participate in the beating. Certainly, those
without such training would be hard pressed to hear all the beatings
individually. But I still don't think what they do hear is a
variation in *overall* loudness -- though it seems we may be able to
test that somewhat with a few more .wav files.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 1:30:19 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > A single variable that correlates well with the preferences of
the
> > > majority here is simply the _error_in_the_fifth_. It's the beat
> rate
> > > of the fifth that dominates, since it involves lower harmonics
(2
> > > and 3), which have greater amplitude.
> > >
> > > The logical conclusion of this for meantones is, sad to say,
equal
> > > temperament.
> >
> > How do you arrive at that conclusion?
> >
>
> Simple. It's the meantone with the slowest beating fifths.
>
> I don't consider any fifth-generated linear temperament with a
fifth
> wider than 700 cents to be a meantone. We are then into the
> Pythagorean region.

You mean the schismatic region? Or . . . (?)

Anyway, don't you think it would be prudent to see what "the
preferences of the majority" are when you actually include the equal
temperament triad, and perhaps a few others that differ from A, B,
B2, and B3, before making such a conclusion? It seems you're
extrapolating *way* beyond the realm explored with the current set of
triads.

🔗Brad Lehman <bpl@umich.edu>

11/4/2005 1:39:53 PM

Well Tom, I missed your previous material where you explained this
difference, and I'm sorry: I didn't intend any offence here. The
beats of 12ths and major 10ths, on harpsichords, simply sound like
variations of loudness to me--not timbre.

Brad Lehman

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
>
> Ah, criticism by means of quote out of context. The context (clear
> from Paul's previous messages) being, whether oscillation of
> *loudness* or of *timbre* was more relatively important in the
> perception of beating.
>
> So, to explain once again, I hear beating at a major third and fifth
> (and tenth, and twelfth, etc.) primarily as an oscillation of
> *timbre*. Pulsation is a reasonable word - and ironically, it does
not
> imply variation of loudness.
>
> The idea that I'm not aware of the existence of audible beats in
major
> thirds and tenths on a harpsichord is ludicrous.
> ~~~T~~~
>
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> >
> > > In my experience beating only produces a noticeable
> > > oscillation of loudness at the unison and the octave.
> >
> > Gotta install more 1/5 or 1/6 comma temperaments on your
harpsichord,
> > then, and play a bunch of open major 10ths. The pulsating effect
is
> > quite noticeable and lovely. The playing of 12ths also brings it
out.
> >
> > Brad Lehman
> >
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 1:42:37 PM

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> I too have not done significant experiments in audio-testing
> various aspects of phase, but all accounts that i've ever
> read about digital synthesis agree that phase differences
> are audibly insignificant.

That's true in general, but this is a particular scenario which is so
different from the general one that the meaning isn't really even the
same anymore. Yes, relative phase differences in the (pressure)
waveforms of different notes are audibly insignificant. But relative
phases of the far slower amplitude waveforms that we call beats? That's
a whole other question.

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 1:55:12 PM

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> It might well be that harpsichord makers deliberately made the 3rd
> harmonic weaker so that meantone temperaments would be more
successful.

That would require plucking the string at somewhere near 1/3 of its
length. Is this the case? Seems unlikely.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 2:02:30 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus" > I don't call it a
cyclic timbre variation, I call it beating, plain
> and simple. Although "variation in total loudness" does mean
> something, it's not what contributes to the sensation of beating
> (rather it's variation in loudness at particular pitches that does),
> and in any case you haven't calculated total loudness correctly.

OK. You've convinced me.

-- Dave Keenan

🔗Brad Lehman <bpl@umich.edu>

11/4/2005 2:12:34 PM

> Sigh... Did you actually listen to the samples A, B, B1, B2? I was
> thinking mainly about the preference that almost everyone expressed,
> which turned out to be that the slower the beating of the 5th
(though
> not going as far as absolute purity), the more pleasant the chord.

Of course I listened to them, yesterday, before writing this message:
/tuning/topicId_62112.html#62112

All of them have differently interesting qualities, and I might prefer
different ones on different days or in different musical contexts.
Triads in isolation really don't give me anything to go on, in
formulating any firm preference about pleasantness; they're not the
music that I care about (with any tensions/resolutions to deal with),
and there's no melodic factor here at all, let alone any contrast
against different triads, or any approach from a different harmony.
Totally cold and without knowing what's going on in them, I liked "A"
and "B" both a little bit better than "B2" and "B3", as far as
listening to sustained wobbly triads goes.

And all four of them seemed more regular (i.e. quickly monotonous)
than I'd expect to hear on any acoustic instrument.

So?

What good singer would ever be caught holding a note with absolutely
no variation in it, for more than a few seconds, whether there's
vibrato in it or not? :)

Brad Lehman

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 2:12:06 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > It's intended to show an approximation to the envelope of the
> sound,
> > i.e. the variation of loudness with time.
>
> I'd like to know how you think your calculation acheives this
> approximation, and what references/theories you're basing this on.

I square the instantaneous voltage or pressure to get instantaneous
power and then I filter this with a simple moving average to try to
eliminate the variations that are too rapid to hear. It's pretty crude
but surely it's good enough to see where the peaks and troughs are.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 2:30:54 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > >
> > > > A single variable that correlates well with the preferences
of
> the
> > > > majority here is simply the _error_in_the_fifth_. It's the
beat
> > rate
> > > > of the fifth that dominates, since it involves lower
harmonics
> (2
> > > > and 3), which have greater amplitude.
> > > >
> > > > The logical conclusion of this for meantones is, sad to say,
> equal
> > > > temperament.
> > >
> > > How do you arrive at that conclusion?
> > >
> >
> > Simple. It's the meantone with the slowest beating fifths.
> >
> > I don't consider any fifth-generated linear temperament with a
> fifth
> > wider than 700 cents to be a meantone. We are then into the
> > Pythagorean region.
>
> You mean the schismatic region? Or . . . (?)

Yes, or schismic or helmholtz or any other name that its been given
over the years. I think most people would have known what I meant.

> Anyway, don't you think it would be prudent to see what "the
> preferences of the majority" are when you actually include the
equal
> temperament triad, and perhaps a few others that differ from A, B,
> B2, and B3, before making such a conclusion? It seems you're
> extrapolating *way* beyond the realm explored with the current set
of
> triads.

Sure. But it's unfortunate that it can hardly be a blind test now.

-- Dave Keenan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 2:29:16 PM

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:

> In 1/4 comma meantone, the fifths beat really pretty quickly and in
> many cases in the 'intermittence' range which psychoacoustically
> causes annoyance.

Really? I haven't found that to be the case with 1/4-comma meantone,
but only with high-register fifths in temperaments that do more
damage to the fifths. Can you elaborate on your experience (as well
as on the psychoacoustics if you will)?

> Besides, isn't it a strong consensus in the historical tuning
industry
> that the size of the fifths in a circular temperament is of little
> consequence except insofar as they determine the size of thirds?

(a) The thirds in such tunings are formed from chains of three or
four fifths, so if you change each of a set of adjacent fifths by a
certain amount, you're changing the relevant thirds by three to four
times that amount. Conversely, changing a third by a given amount
only means changing each of the comprising fifths by 1/3 to 1/4 that
amount. So one has to keep the thirds much more strongly and
carefully in mind as a goal in such tunings; the individual fifths
have relatively little leeway, and thus variations in their size
relatively little impact, just by the nature of the construction.

(b) A 12-tone circular temperament has other constraints on it, such
as closure of the circle of fifths.

So it's unfair to extrapolate consensuses for 12-tone circulating
temperaments to tunings in general.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 2:33:38 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > It might well be that harpsichord makers deliberately made the 3rd
> > harmonic weaker so that meantone temperaments would be more
> successful.
>
> That would require plucking the string at somewhere near 1/3 of its
> length. Is this the case? Seems unlikely.

Why unlikely??

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 2:38:38 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > > It's intended to show an approximation to the envelope of the
> > sound,
> > > i.e. the variation of loudness with time.
> >
> > I'd like to know how you think your calculation acheives this
> > approximation, and what references/theories you're basing this on.
>
> I square the instantaneous voltage or pressure to get instantaneous
> power and then I filter this with a simple moving average to try to
> eliminate the variations that are too rapid to hear.

How rapid?

> It's pretty crude
> but surely it's good enough to see where the peaks and troughs are.

Doing the correct "convolution" or "filtering by moving average" is one
aspect of getting this calculation right; another aspect is knowing how
loudnesses *add* when one adds frequency components either (this
matters!) within the same critical band or in different critical bands
(unlike amplitudes, loudnesses do not add in a linear way!) You may be
able to find the relevant psychoacoustical formulas on the web . . .

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 2:41:42 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > > It's intended to show an approximation to the envelope of the
> > sound,
> > > i.e. the variation of loudness with time.
> >
> > I'd like to know how you think your calculation acheives this
> > approximation, and what references/theories you're basing this
on.
>
> I square the instantaneous voltage or pressure to get
instantaneous
> power and then I filter this with a simple moving average to try
to
> eliminate the variations that are too rapid to hear. It's pretty
crude
> but surely it's good enough to see where the peaks and troughs are.

I guess you're gonna say I should have taken the log or otherwise
converted to decibels in which case I would have seen how utterly
insignificant the overall loudness variation actually is. Yeah,
you're right.

-- Dave Keenan
>
> -- Dave Keenan
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 2:14:06 PM

Thanks Tom, you pretty much said what I said!

:)

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> >
> > > How is this possible? Each of the beatings occurs at a
different
> > pitch
> > > (that is, for a different pair of nearly coincident partials)
so I
> > have
> > > no idea how this could happen here. I'd love to be educated,
> > > particularly with a demonstration!
> >
> > There are at least two perceptual aspects to beats
> > (a) cyclic variation of loudness with time
> > (b) cyclic variation of timbre with time
> >
> > I'm saying we could eliminate the first, but not the second.
> >
> > Here's a very contrived example.
> >
> > Use a timbre that has equal amplitudes for the fundamental and
all
> > harmonics up to the sixth, then nothing after that. And one needs
to
> > be able to set the start-phase of each harmonic independently.
> >
> > Consider the sync-beating 512:639:766 Hz chord. This will have
the
> > following beats between the following four pairs of partials.
> >
> > 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> > 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> > 4 Hz between 5th harmonic of root and 4th harmonic of major third
> > 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
> >
> > By carefully arranging the phases of the partials in the original
> > timbre (sorry I'm too lazy to work out the details) it should be
> > possible to arrange for the three 4 Hz beats to be 120 degrees
out of
> > phase and thereby cancel each other and leave only the 8 Hz beat.
> >
> > This isn't terribly satisfactory since with equal amplitude
partials
> > that 8 Hz beat will still be quite obvious, so the next step is
to
> > look at a more normal timbre where the higher partials have lower
> > amplitude.
> >
> > In this case it should be possible to make it so that the beats
of the
> > major and minor third intervals are in phase with each other and
add
> > to produce a beat which is the same amplitude as the 4 Hz beat in
the
> > fifth interval but 180 degrees out of phase with it, and thereby
> > cancelling it. We still have the 8 Hz beat, but this time at
lower
> > amplitude.
> >
> > If one cheated and knocked out the 6th harmonic of the root and
the
> > 4th harmonic of the fifth completely, then one could have no
beating
> > at all in the triad (in the sense of loudness variation) even
though
> > every dyad heard alone has obvious beating. Of course we still
expect
> > to hear a sort of beating of the triad due to the cyclic timbre
> > variation.
> >
> > -- Dave Keenan
>
>
> Well, I still think the beating in timbre would be very much
> noticeable. In my experience beating only produces a noticeable
> oscillation of loudness at the unison and the octave.
>
> > 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> > 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> > 4 Hz between 5th harmonic of root and 4th harmonic of major third
> > 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
>
> The three different 4Hz beats occur at 3 different overtone pitches
> and there is no way they can cancel each other out. To put it
another
> way, in a close position chord *every* beat is a variation in
timbre,
> since every beat affects the upper partials, not the fundamental.
> Beats could in principle only cancel each other out if they were all
> at the same pitch (as in a violin section all playing the 'same'
note).
>
> ~~~T~~~
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/4/2005 2:45:43 PM

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > >
> > > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > > <wallyesterpaulrus@y...> wrote:
> > > >
> > > > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > >
> > > > > A single variable that correlates well with the preferences
> of
> > the
> > > > > majority here is simply the _error_in_the_fifth_. It's the
> beat
> > > rate
> > > > > of the fifth that dominates, since it involves lower
> harmonics
> > (2
> > > > > and 3), which have greater amplitude.
> > > > >
> > > > > The logical conclusion of this for meantones is, sad to
say,
> > equal
> > > > > temperament.
> > > >
> > > > How do you arrive at that conclusion?
> > > >
> > >
> > > Simple. It's the meantone with the slowest beating fifths.
> > >
> > > I don't consider any fifth-generated linear temperament with a
> > fifth
> > > wider than 700 cents to be a meantone. We are then into the
> > > Pythagorean region.
> >
> > You mean the schismatic region? Or . . . (?)
>
> Yes, or schismic or helmholtz or any other name that its been given
> over the years. I think most people would have known what I meant.

Pythagorean to many, if not most, people here would just mean pure
fifths and triads with major thirds of 408 cents and minor thirds of
294 cents. If you're talking about using diminished fourths instead
of major thirds, etc., I would think you should say so explicitly. I
don't think someone without many years of history talking to you
would be likely to correctly discern your meaning here, since you
didn't make any mention of triads, thirds, diminished fourths, etc.
(Sorry to be so pedantic.)

> > Anyway, don't you think it would be prudent to see what "the
> > preferences of the majority" are when you actually include the
> equal
> > temperament triad, and perhaps a few others that differ from A,
B,
> > B2, and B3, before making such a conclusion? It seems you're
> > extrapolating *way* beyond the realm explored with the current
set
> of
> > triads.
>
> Sure. But it's unfortunate that it can hardly be a blind test now.

Now? I thought all the triads (A, B, B2, B3) were explicitly
specified here on this list *before* any of the corresponding
listening tests. So nothing would be different now. :)

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/4/2005 2:52:22 PM

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> Well, I still think the beating in timbre would be very much
> noticeable. In my experience beating only produces a noticeable
> oscillation of loudness at the unison and the octave.

Thanks Tom. You and Paul have convinced me of that now.

-- Dave Keenan

🔗monz <monz@tonalsoft.com>

11/4/2005 3:02:33 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I too have not done significant experiments in audio-testing
> > various aspects of phase, but all accounts that i've ever
> > read about digital synthesis agree that phase differences
> > are audibly insignificant.
>
> That's true in general, but this is a particular scenario
> which is so different from the general one that the meaning
> isn't really even the same anymore. Yes, relative phase
> differences in the (pressure) waveforms of different notes
> are audibly insignificant. But relative phases of the far
> slower amplitude waveforms that we call beats? That's
> a whole other question.

Ah, OK, thanks ... i misunderstood. It's clear now.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Petr Parízek <p.parizek@chello.cz>

11/5/2005 7:53:24 AM

There is one possible case in which the beats could be cancelled out. But I
can't imagine this happening on an acoustic instrument. It would mean all of
these conditions to be met at the same time:

1. The played chord is a regular part of the subharmonic series (slightly
detuned, of course).

2. One of the intervals is rational.

3. The "almost common" overtone appears in the same intensity in all the
sounding tones.

4. The phase of each tone is carefully chosen in such a way that makes the
required harmonic overtones vanish and leaves only the inharmonic ones.

Petr

🔗Petr Parízek <p.parizek@chello.cz>

11/5/2005 8:53:26 AM

> 1. The played chord is a regular part of the subharmonic series (slightly
> detuned, of course).
>
> 2. One of the intervals is rational.
>
> 3. The "almost common" overtone appears in the same intensity in all the
> sounding tones.
>
> 4. The phase of each tone is carefully chosen in such a way that makes the
> required harmonic overtones vanish and leaves only the inharmonic ones.

I was inexact when I said "rational". Of course, if you wish, all of the
intervals may be rational but only those which fall exactly into the part of
the subharmonic series will work. For example, suppose we have a minor chord
of E4-G4-B4 where the frequency of E4 is 320Hz (i.e. as in the 18. century
or so). If it's a regular part of the subharmonic series, the frequencies
are 320:384:480. If you shift the lowest tone 1Hz higher, then two very
similar sets of overtones start beating. The lowest similar overtone in the
fifth is B5 (i.e. 960:963) and in the case of the minor third it's B6
(1920:1926). Essentially, you can't get rid of the beats in B5 but you can
get rid of the beats in B6 by removing 1920Hz which is the common overtone
of the major third. This is done by adjusting the phase of either G4 or B4
in such a way that the harmonic B6 vanishes and only the inharmonic B6
remains. This may explain another reason for promoting quarter-comma
(eventually third-comma) meantone.

Petr

🔗Carl Lumma <clumma@yahoo.com>

11/5/2005 10:00:29 AM

> Yes, or schismic or helmholtz or any other name that its been
> given over the years. I think most people would have known what
> I meant.

I did.

> Sure. But it's unfortunate that it can hardly be a blind test now.

A new blind test could always be set up, no?

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/5/2005 10:07:58 AM

> I guess you're gonna say I should have taken the log or otherwise
> converted to decibels in which case I would have seen how utterly
> insignificant the overall loudness variation actually is. Yeah,
> you're right.

Don't be intimidated, Dave. Mean-squared power is the standard
evaluation of the overall loudness of an audio signal. It's
completely obvious while looking at any of these wav files in
a waveform editor that the percieved overall loudness variations
match the instantaneous height on the display -- whatever
smoothing these editors do for a 6-second file filling half
the screen works nearly perfectly.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/5/2005 10:09:47 AM

> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > Well, I still think the beating in timbre would be very much
> > noticeable.

Agree.

> > In my experience beating only produces a noticeable
> > oscillation of loudness at the unison and the octave.

Huh??

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/5/2005 10:13:34 AM

> > Yes, relative phase differences in the (pressure)
> > waveforms of different notes are audibly insignificant.
> > But relative phases of the far slower amplitude
> > waveforms that we call beats? That's a whole other
> > question.
>
> Ah, OK, thanks ... i misunderstood. It's clear now.

However the answer in this case also seems to be that
phases don't matter all that much.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

11/6/2005 2:54:48 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> > > A brat is a Beat RATio, usually defined for approximate
> > > 4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
> > > beat rate.
> >
> > Why use this obfuscatory jargon in the first place? How hard can
> > it be to type "beat ratio" and to be clear about exactly which
> > intervals you're considering the beat ratio of? [Rhetorical
> > questions]
>
> It wasn't my idea (though I didn't disapprove).

I've said it before, and I'll say it again: it isn't obfuscatory, but
is a precisely defined term introduced exactly because "beat ratio" is
the *general* term. If f is the fifth, and t is the third, then the
brat means *precisely* this ratio:

brat = (6t - 5f)/(4t - 5)

It does not mean some other beat ratio. It does not mean
(5f-6t)/(4t-5). It means exactly *this* ratio. This allows it to be
given, and convey a precise meaning. Otherwise confusion can and will
ensue. Admittedly, it might have been better to base everything on

(4t - 5)/(2f - 3)

and also to state things in terms of the approximate "3", a, and the
approximate "5", b, and so as

(b - 5)/(a - 3)

but I was attempting to follow Wendell.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/6/2005 4:11:46 PM

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> In 1/4 comma meantone on harpsichords, as some might argue, it's the
> noticeably impure 5ths that give the overall sound some liveliness...
> counteracting the deadness and stasis of the pure major 3rds.
> Complete purity (stasis) in triads is not necessarily a top *musical*
> goal for the way music behaves, as forward motion is important too.

I don't think this is a linear relationship; I think it does not take
much detuning of the pure major thirds of 1/4 comma meantone to
produce a different sound. Anyway, I like the sound of 31-et, for my part.

🔗Gene Ward Smith <gwsmith@svpal.org>

11/6/2005 4:14:21 PM

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
>
> Sigh... Did you actually listen to the samples A, B, B1, B2? I was
> thinking mainly about the preference that almost everyone expressed,
> which turned out to be that the slower the beating of the 5th (though
> not going as far as absolute purity), the more pleasant the chord.

Dave made this point also. Given that the tuning of the major third
was fixed, I think it is pretty well meaningless.

🔗Carl Lumma <clumma@yahoo.com>

11/6/2005 7:10:59 PM

> > Sigh... Did you actually listen to the samples A, B, B1, B2?
> > I was thinking mainly about the preference that almost
> > everyone expressed, which turned out to be that the slower
> > the beating of the 5th (though not going as far as absolute
> > purity), the more pleasant the chord.
>
> Dave made this point also. Given that the tuning of the major
> third was fixed, I think it is pretty well meaningless.

The major 3rd was not the same in all chords, but this is a
worthwhile point.

-Carl

🔗Tom Dent <stringph@gmail.com>

11/7/2005 7:57:57 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > > Well, I still think the beating in timbre would be very much
> > > noticeable.
>
> Agree.
>
> > > In my experience beating only produces a noticeable
> > > oscillation of loudness at the unison and the octave.
>
> Huh??
>
> -Carl

Without knowing what 'Huh??' means, I can only restate in more
pedantically explicit form as follows:

Beats produced by a particular out-of-tune interval have the twin
effects of oscillating loudness and/or timbre.

In my opinion, beats in a mistuned unison (or equison, to be yet more
pedantic) or an octave partake substantially of both effects. But
beats in fifths and thirds sound more like oscillations of timbre than
of loudness.

Of course, this question depends on relative strengths of the
fundamental and overtones. In the limit where the fundamental is weak
and the overtones are strong, beating fifths, thirds etc. *will* have
a quite appreciable effect on overall perceived loudness. But in most
musical situations it is the other way round.

Incidentally, what happens with the .wav generation program if you
simply tell it to stagger the notes of the chord by some fractions of
a second?

~~~T~~~

🔗Kraig Grady <kraiggrady@anaphoria.com>

11/7/2005 11:08:15 AM

Hi Gene!
Since there are others triads that could be the basis besides the 5 limit. is it possible to apply this method to other triads?
For instance in Meta Slendro the basic triad is the 6-7-8. it seems this method chord be applied to any proportional triad?
>Message: 4 > Date: Sun, 06 Nov 2005 22:54:48 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
>Subject: Re: the simplest pan-proportionally beating 12-tone temperament
>
>--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> >
>>>>A brat is a Beat RATio, usually defined for approximate
>>>>4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
>>>>beat rate.
>>>> >>>>
>>>Why use this obfuscatory jargon in the first place? How hard can
>>>it be to type "beat ratio" and to be clear about exactly which
>>>intervals you're considering the beat ratio of? [Rhetorical
>>>questions]
>>> >>>
>>It wasn't my idea (though I didn't disapprove).
>> >>
>
>I've said it before, and I'll say it again: it isn't obfuscatory, but
>is a precisely defined term introduced exactly because "beat ratio" is
>the *general* term. If f is the fifth, and t is the third, then the
>brat means *precisely* this ratio:
>
>brat = (6t - 5f)/(4t - 5)
>
>It does not mean some other beat ratio. It does not mean
>(5f-6t)/(4t-5). It means exactly *this* ratio. This allows it to be
>given, and convey a precise meaning. Otherwise confusion can and will
>ensue. Admittedly, it might have been better to base everything on
>
>(4t - 5)/(2f - 3)
>
>and also to state things in terms of the approximate "3", a, and the
>approximate "5", b, and so as
>
>(b - 5)/(a - 3)
>
>but I was attempting to follow Wendell.
>
>
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Gene Ward Smith <gwsmith@svpal.org>

11/7/2005 11:54:14 AM

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> Hi Gene!
> Since there are others triads that could be the basis besides the 5
> limit. is it possible to apply this method to other triads?

It certainly is! It is also possible to apply the idea to *tetrads*
and beyond; I've posted some stuff on tuning-math on how you can take
a single-comma temperament and get synchronized beating of tetrads;
chosing the right single-comma temperament can give near-tunings of
rank two ("linear") temperaments also.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/8/2005 10:24:19 AM

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@c...> wrote:
>
> > 1. The played chord is a regular part of the subharmonic series
(slightly
> > detuned, of course).
> >
> > 2. One of the intervals is rational.
> >
> > 3. The "almost common" overtone appears in the same intensity in
all the
> > sounding tones.
> >
> > 4. The phase of each tone is carefully chosen in such a way that
makes the
> > required harmonic overtones vanish and leaves only the inharmonic
ones.
>
> I was inexact when I said "rational". Of course, if you wish, all
of the
> intervals may be rational but only those which fall exactly into
the part of
> the subharmonic series will work. For example, suppose we have a
minor chord
> of E4-G4-B4 where the frequency of E4 is 320Hz (i.e. as in the 18.
century
> or so). If it's a regular part of the subharmonic series, the
frequencies
> are 320:384:480. If you shift the lowest tone 1Hz higher, then two
very
> similar sets of overtones start beating. The lowest similar
overtone in the
> fifth is B5 (i.e. 960:963) and in the case of the minor third it's
B6
> (1920:1926). Essentially, you can't get rid of the beats in B5 but
you can
> get rid of the beats in B6 by removing 1920Hz which is the common
overtone
> of the major third. This is done by adjusting the phase of either
G4 or B4
> in such a way that the harmonic B6 vanishes and only the inharmonic
B6
> remains.

Oh, I think I get what you're saying now. You're considering
harmonics of the lowest tone "inharmonic" since you've shifted the
lower tone?

> This may explain another reason for promoting quarter-comma
> (eventually third-comma) meantone.

On acoustic instruments, you'd have little control over the phase,
and get constructive interference as often as destructive
interference. So I don't think this effect was relevant in the era
when meantone was being promoted or the era in which it dominated in
Western music.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/8/2005 10:31:24 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > I guess you're gonna say I should have taken the log or otherwise
> > converted to decibels in which case I would have seen how utterly
> > insignificant the overall loudness variation actually is. Yeah,
> > you're right.
>
> Don't be intimidated, Dave. Mean-squared power is the standard
> evaluation of the overall loudness of an audio signal.

In psychoacoustics, far from it, particularly when some components
fall into the same critical band and others don't. Also, Dave needed
to use some time-smoothing, which you don't mention at all.

> It's
> completely obvious while looking at any of these wav files in
> a waveform editor that the percieved overall loudness variations
> match the instantaneous height on the display

Are you talking about the case where the beatings were in phase with
one another? If so, that's no surprise. How about when the phases are
arranged to keep the overall loudness (by your or Dave's definition)
constant?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/8/2005 10:33:01 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Yes, relative phase differences in the (pressure)
> > > waveforms of different notes are audibly insignificant.
> > > But relative phases of the far slower amplitude
> > > waveforms that we call beats? That's a whole other
> > > question.
> >
> > Ah, OK, thanks ... i misunderstood. It's clear now.
>
> However the answer in this case also seems to be that
> phases don't matter all that much.

I don't recall anyone creating synched-beating sound files to compare
with different phases in each, so I don't know on what basis you're
claiming this "answer".

🔗Carl Lumma <clumma@yahoo.com>

11/8/2005 1:14:54 PM

> > Don't be intimidated, Dave. Mean-squared power is the standard
> > evaluation of the overall loudness of an audio signal.
>
> In psychoacoustics, far from it, particularly when some
> components fall into the same critical band and others don't.
> Also, Dave needed to use some time-smoothing, which you
> don't mention at all.

Years ago I read a Terhardt paper on this stuff, but I don't
think it's necessary here. In musical applications, ms power
is good enough. I did mention smoothing farther down, but
you snipped it.

> > It's completely obvious while looking at any of these wav
> > files in a waveform editor that the percieved overall
> > loudness variations match the instantaneous height on the
> > display
>
> Are you talking about the case where the beatings were in
> phase with one another? If so, that's no surprise. How about
> when the phases are arranged to keep the overall loudness (by
> your or Dave's definition) constant?

That's a very special case that's not likely to occur in
the real world.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/8/2005 1:28:56 PM

> > > > Yes, relative phase differences in the (pressure)
> > > > waveforms of different notes are audibly insignificant.
> > > > But relative phases of the far slower amplitude
> > > > waveforms that we call beats? That's a whole other
> > > > question.
> > >
> > > Ah, OK, thanks ... i misunderstood. It's clear now.
> >
> > However the answer in this case also seems to be that
> > phases don't matter all that much.
>
> I don't recall anyone creating synched-beating sound files to
> compare with different phases in each, so I don't know on what
> basis you're claiming this "answer".

Dave's spreadsheet. But here's one randomly-phased version
of A.wav...

http://lumma.org/tuning/Aphase.wav

It sounds the same and looks the same as A.wav.

-Carl

🔗oyarman@ozanyarman.com

11/8/2005 1:55:31 PM

Aha, this random phased version sounds as consonant as B, assuming that it
is still proportional beating.

Cordially,
Ozan

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 08 Kas�m 2005 Sal� 23:28
Subject: [tuning] brats test! (was Re: the simplest pan-proportionally
beating...)

> >
> > I don't recall anyone creating synched-beating sound files to
> > compare with different phases in each, so I don't know on what
> > basis you're claiming this "answer".
>
> Dave's spreadsheet. But here's one randomly-phased version
> of A.wav...
>
> http://lumma.org/tuning/Aphase.wav
>
> It sounds the same and looks the same as A.wav.
>
> -Carl
>
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/10/2005 12:58:17 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Don't be intimidated, Dave. Mean-squared power is the standard
> > > evaluation of the overall loudness of an audio signal.
> >
> > In psychoacoustics, far from it, particularly when some
> > components fall into the same critical band and others don't.
> > Also, Dave needed to use some time-smoothing, which you
> > don't mention at all.
>
> Years ago I read a Terhardt paper on this stuff, but I don't
> think it's necessary here. In musical applications, ms power
> is good enough.

Carl, have you ever heard of phons?

> I did mention smoothing farther down, but
> you snipped it.

Sorry. Please restore it.

> > > It's completely obvious while looking at any of these wav
> > > files in a waveform editor that the percieved overall
> > > loudness variations match the instantaneous height on the
> > > display
> >
> > Are you talking about the case where the beatings were in
> > phase with one another? If so, that's no surprise. How about
> > when the phases are arranged to keep the overall loudness (by
> > your or Dave's definition) constant?
>
> That's a very special case that's not likely to occur in
> the real world.

But it's exactly the case that Dave was trying to draw attention to
with all his work, and so far we've neither listened to it nor
attempted to calculate its loudness (not RMS power convoluted with
some not-too-long, not-too-short averaging window, which is what Dave
did) profile. All sorts of phase relationships *do* occur in the real
world, so we shouldn't be shy about investigating particular ones.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/10/2005 1:00:20 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > Yes, relative phase differences in the (pressure)
> > > > > waveforms of different notes are audibly insignificant.
> > > > > But relative phases of the far slower amplitude
> > > > > waveforms that we call beats? That's a whole other
> > > > > question.
> > > >
> > > > Ah, OK, thanks ... i misunderstood. It's clear now.
> > >
> > > However the answer in this case also seems to be that
> > > phases don't matter all that much.
> >
> > I don't recall anyone creating synched-beating sound files to
> > compare with different phases in each, so I don't know on what
> > basis you're claiming this "answer".
>
> Dave's spreadsheet. But here's one randomly-phased version
> of A.wav...
>
> http://lumma.org/tuning/Aphase.wav
>
> It sounds the same and looks the same as A.wav.

I found somne phases that changed the shape of Dave's curve quite a
bit, but maybe that was just Excel changing the scale of the y-
axis . . . maybe Dave would like to make a specific suggestion here,
and then would could get everyone to listen to the resulting .wav and
make comparisons?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/10/2005 1:18:12 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Don't be intimidated, Dave. Mean-squared power is the standard
> > > evaluation of the overall loudness of an audio signal.
> >
> > In psychoacoustics, far from it, particularly when some
> > components fall into the same critical band and others don't.
> > Also, Dave needed to use some time-smoothing, which you
> > don't mention at all.
>
> Years ago I read a Terhardt paper on this stuff, but I don't
> think it's necessary here. In musical applications, ms power
> is good enough.

How do you do it without the time-smoothing Dave needed to use?

🔗Carl Lumma <clumma@yahoo.com>

11/11/2005 1:16:28 PM

> > Years ago I read a Terhardt paper on this stuff, but I don't
> > think it's necessary here. In musical applications, ms power
> > is good enough.
>
> How do you do it without the time-smoothing Dave needed to use?

For applications like replaygain, you just use the whole
file. For applications like beating in mildly tempered
chords, whatever smoothing Cool Edit is using is obviously
working very well.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/11/2005 2:53:38 PM

> I found somne phases that changed the shape of Dave's curve quite
> a bit, but maybe that was just Excel changing the scale of the
> y-axis . . .

I was able to change the height, trend, and dynamic range of the
curve, but not its period much.

> maybe Dave would like to make a specific suggestion
> here, and then would could get everyone to listen to the
> resulting .wav and make comparisons?

I think I may have just figured out how to get precise phase
differences in Cool Edit... but maybe you could do it easier...

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/11/2005 2:59:29 PM

> > Years ago I read a Terhardt paper on this stuff, but I don't
> > think it's necessary here. In musical applications, ms power
> > is good enough.
>
> Carl, have you ever heard of phons?

Yes, but let's not get distracted.

> > I did mention smoothing farther down, but
> > you snipped it.
>
> Sorry. Please restore it.

No time...

> > > Are you talking about the case where the beatings were in
> > > phase with one another? If so, that's no surprise. How about
> > > when the phases are arranged to keep the overall loudness
> > > (by your or Dave's definition) constant?
> >
> > That's a very special case that's not likely to occur in
> > the real world.
>
> But it's exactly the case that Dave was trying to draw attention
> to with all his work,

...which you yourself said couldn't cause the beats to cancel.

> and so far we've neither listened to it nor attempted to
> calculate its loudness (not RMS power convoluted with
> some not-too-long, not-too-short averaging window, which
> is what Dave did) profile.

I have no trouble believing the overall amplitude can be
flattened, but like you I think the "timbre change" would
be significant. But even if it's not...

> All sorts of phase relationships *do* occur in
> the real world, so we shouldn't be shy about investigating
> particular ones.

...such a particular condition would almost never happen
without special-purpose hardware. It'd be interesting
to hear it, and I'd be happy to try and synthesize whatever
test chords are proposed, but I'm not inclined to bend over
backward for any of this. I'd be much more interested in
how to calculate the heard beat rate of a chord....

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/14/2005 2:46:44 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > think it's necessary here. In musical applications, ms power
> > > is good enough.
> >
> > How do you do it without the time-smoothing Dave needed to use?
>
> For applications like replaygain, you just use the whole
> file.

Well that wouldn't tell you anything useful, as a file with wildly
varying amplitude and one with constant amplitude could ned up with
exactly the same number, right?

> For applications like beating in mildly tempered
> chords, whatever smoothing Cool Edit is using is obviously
> working very well.

You said ms power, but in reality you don't know what it is --
"whatever smoothing Cool Edit is doing" -- that you say is "good
enough"? Or are we misunderstanding one another?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/14/2005 2:49:28 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > I found somne phases that changed the shape of Dave's curve quite
> > a bit, but maybe that was just Excel changing the scale of the
> > y-axis . . .
>
> I was able to change the height, trend, and dynamic range of the
> curve, but not its period much.

Of course not -- each of the beat rates remain the same regardless of
phase, and they're all equal rates!

> > maybe Dave would like to make a specific suggestion
> > here, and then would could get everyone to listen to the
> > resulting .wav and make comparisons?
>
> I think I may have just figured out how to get precise phase
> differences in Cool Edit... but maybe you could do it easier...

Dave?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/14/2005 2:52:47 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > think it's necessary here. In musical applications, ms power
> > > is good enough.
> >
> > Carl, have you ever heard of phons?
>
> Yes, but let's not get distracted.

OK . . .

> > > I did mention smoothing farther down, but
> > > you snipped it.
> >
> > Sorry. Please restore it.
>
> No time...

OK again . . .

> > > > Are you talking about the case where the beatings were in
> > > > phase with one another? If so, that's no surprise. How about
> > > > when the phases are arranged to keep the overall loudness
> > > > (by your or Dave's definition) constant?
> > >
> > > That's a very special case that's not likely to occur in
> > > the real world.
> >
> > But it's exactly the case that Dave was trying to draw attention
> > to with all his work,
>
> ...which you yourself said couldn't cause the beats to cancel.

And he agreed. But the idea that total loudness could remain constant
remains intact.

> > and so far we've neither listened to it nor attempted to
> > calculate its loudness (not RMS power convoluted with
> > some not-too-long, not-too-short averaging window, which
> > is what Dave did) profile.
>
> I have no trouble believing the overall amplitude can be
> flattened, but like you I think the "timbre change" would
> be significant. But even if it's not...
>
> > All sorts of phase relationships *do* occur in
> > the real world, so we shouldn't be shy about investigating
> > particular ones.
>
> ...such a particular condition would almost never happen
> without special-purpose hardware. It'd be interesting
> to hear it, and I'd be happy to try and synthesize whatever
> test chords are proposed, but I'm not inclined to bend over
> backward for any of this. I'd be much more interested in
> how to calculate the heard beat rate of a chord....

You never hear more than one beat rate in a chord? Try some tetrads
with two fifths . . .

🔗Carl Lumma <clumma@yahoo.com>

11/14/2005 6:21:24 PM

> > > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > > think it's necessary here. In musical applications, ms power
> > > > is good enough.
> > >
> > > How do you do it without the time-smoothing Dave needed to use?
> >
> > For applications like replaygain, you just use the whole
> > file.
>
> Well that wouldn't tell you anything useful, as a file with wildly
> varying amplitude and one with constant amplitude could ned up with
> exactly the same number, right?

I've applied this kind of gain to 20GB of music of all kinds,
and I can tell you it works fantastically well, when the amount
of gain is limited to prevent clipping.

> > For applications like beating in mildly tempered
> > chords, whatever smoothing Cool Edit is using is obviously
> > working very well.
>
> You said ms power, but in reality you don't know what it is --
> "whatever smoothing Cool Edit is doing" -- that you say is "good
> enough"? Or are we misunderstanding one another?

Dave said he used ms power. Cool Edit plots the values of the
samples on a linear scale between zero and the max value
possible at the given bit depth, then connects the dots using
some bezier-looking thing. That's if you can see every sample.
When time smoothing is needed, it divides the number of columns
of x-axis pixels by the number of samples to display, finds the
maximum abs sample value in each resulting bin, then colors
pixels between + and - that value on the column.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/14/2005 6:27:00 PM

> > > But it's exactly the case that Dave was trying to draw
> > > attention to with all his work,
> >
> > ...which you yourself said couldn't cause the beats to cancel.
>
> And he agreed. But the idea that total loudness could remain
> constant remains intact.

Sure. It's a neat idea. And something like a Hammond organ
has all the wheels running whether the keys are down are not,
so these phases might be set in the factory, as Dave pointed
out.

> > > All sorts of phase relationships *do* occur in
> > > the real world, so we shouldn't be shy about investigating
> > > particular ones.
> >
> > ...such a particular condition would almost never happen
> > without special-purpose hardware. It'd be interesting
> > to hear it, and I'd be happy to try and synthesize whatever
> > test chords are proposed, but I'm not inclined to bend over
> > backward for any of this. I'd be much more interested in
> > how to calculate the heard beat rate of a chord....
>
> You never hear more than one beat rate in a chord?

No no, I didn't say that. In fact beat rates on my piano
are nothing like those in these synthesized examples.
So I wouldn't want to rule out brats just yet. But this
is the third time I've synthesized comparisons (using
different methods) and they've never held up to Robert's
claims. But piano acoustics are a strange bunch....

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/15/2005 10:42:58 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > > > think it's necessary here. In musical applications, ms
power
> > > > > is good enough.
> > > >
> > > > How do you do it without the time-smoothing Dave needed to
use?
> > >
> > > For applications like replaygain, you just use the whole
> > > file.
> >
> > Well that wouldn't tell you anything useful, as a file with
wildly
> > varying amplitude and one with constant amplitude could ned up
with
> > exactly the same number, right?
>
> I've applied this kind of gain

This kind of gain?

> to 20GB of music of all kinds,
> and I can tell you it works fantastically well, when the amount
> of gain is limited to prevent clipping.

I can't tell if you've completely changed the subject, or what. What
does this have to do with calculating loudness as a function of time?

> > > For applications like beating in mildly tempered
> > > chords, whatever smoothing Cool Edit is using is obviously
> > > working very well.
> >
> > You said ms power, but in reality you don't know what it is --
> > "whatever smoothing Cool Edit is doing" -- that you say is "good
> > enough"? Or are we misunderstanding one another?
>
> Dave said he used ms power.

Read his posts again. He had to use time-averaging (or smoothing)
over some (fairly arbitrarily chosen) time window.

> Cool Edit plots the values of the
> samples on a linear scale between zero and the max value
> possible at the given bit depth, then connects the dots using
> some bezier-looking thing. That's if you can see every sample.
> When time smoothing is needed, it divides the number of columns
> of x-axis pixels by the number of samples to display, finds the
> maximum abs sample value in each resulting bin, then colors
> pixels between + and - that value on the column.

Go on. So far none of this sounds like an ms power calculation of any
kind.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/15/2005 10:44:59 AM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> > > ...such a particular condition would almost never happen
> > > without special-purpose hardware. It'd be interesting
> > > to hear it, and I'd be happy to try and synthesize whatever
> > > test chords are proposed, but I'm not inclined to bend over
> > > backward for any of this. I'd be much more interested in
> > > how to calculate the heard beat rate of a chord....
> >
> > You never hear more than one beat rate in a chord?
>
> No no, I didn't say that.

Why do you think calculating beat rates in a chord presents any
special problems, then?

> In fact beat rates on my piano
> are nothing like those in these synthesized examples.

How did you tune it?

> So I wouldn't want to rule out brats just yet. But this
> is the third time I've synthesized comparisons (using
> different methods) and they've never held up to Robert's
> claims. But piano acoustics are a strange bunch....

🔗Carl Lumma <clumma@yahoo.com>

11/15/2005 3:19:11 PM

> > > > > > In musical applications, ms power
> > > > > > is good enough.
> > > > >
> > > > > How do you do it without the time-smoothing?
> > > >
> > > > For applications like replaygain, you just use the
> > > > whole file.
> > >
> > > Well that wouldn't tell you anything useful,
> >
> > I've applied this kind of gain
>
> This kind of gain?

Replaygain. It's a constant amount of amplification
that brings the ms power of the file to a predetermined
level.

> > Dave said he used ms power.
>
> Read his posts again.

"I square the instantaneous voltage or pressure to get
instantaneous power and then I filter this with a simple moving
average"

Sounds like ms to me (ms voltage, to be more accurate).

> He had to use time-averaging (or smoothing) over
> some (fairly arbitrarily chosen) time window.

So?

> > Cool Edit plots the values of the
> > samples on a linear scale between zero and the max value
> > possible at the given bit depth, then connects the dots using
> > some bezier-looking thing. That's if you can see every sample.
> > When time smoothing is needed, it divides the number of columns
> > of x-axis pixels by the number of samples to display, finds the
> > maximum abs sample value in each resulting bin, then colors
> > pixels between + and - that value on the column.
>
> Go on.

Nothing more to tell.

> So far none of this sounds like an ms power calculation of
> any kind.

It isn't.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/15/2005 3:22:58 PM

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > ...such a particular condition would almost never happen
> > > > without special-purpose hardware. It'd be interesting
> > > > to hear it, and I'd be happy to try and synthesize whatever
> > > > test chords are proposed, but I'm not inclined to bend over
> > > > backward for any of this. I'd be much more interested in
> > > > how to calculate the heard beat rate of a chord....
> > >
> > > You never hear more than one beat rate in a chord?
> >
> > No no, I didn't say that.
>
> Why do you think calculating beat rates in a chord presents any
> special problems, then?

I'm completely at a loss for what you're asking here.

> > In fact beat rates on my piano
> > are nothing like those in these synthesized examples.
>
> How did you tune it?

It's far out of tune at the moment. But I mean, beating
chords on a piano don't seem to behave like synthesized
ones.

-Carl

🔗Dave Keenan <d.keenan@bigpond.net.au>

11/15/2005 4:12:47 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > I found somne phases that changed the shape of Dave's curve quite
> > a bit, but maybe that was just Excel changing the scale of the
> > y-axis . . .

No. I'm pretty sure I fixed the Y axis.

> > maybe Dave would like to make a specific suggestion
> > here, and then would could get everyone to listen to the
> > resulting .wav and make comparisons?
>
> I think I may have just figured out how to get precise phase
> differences in Cool Edit... but maybe you could do it easier...

I think it is so unlikely to produce significant audible differences
that I'm not willing to spend the time, or ask others to spend
theirs.

I was very interested in what Petr Parizek pointed out in
/tuning/topicId_59689.html#62226
and
/tuning/topicId_59689.html#62227
although I don't think it has anything to do with the historical
popularity of quarter-comma or third-comma meantone.

i.e. with a utonal chord such as a minor triad the beating
occurs "in one place". i.e. around the frequency of the guidetone
(and its multiples). But I soon convinced myself that one can't
cancel all beats in a mistuned chord there either.

-- Dave Keenan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/16/2005 1:06:25 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > > > In musical applications, ms power
> > > > > > > is good enough.
> > > > > >
> > > > > > How do you do it without the time-smoothing?
> > > > >
> > > > > For applications like replaygain, you just use the
> > > > > whole file.
> > > >
> > > > Well that wouldn't tell you anything useful,
> > >
> > > I've applied this kind of gain
> >
> > This kind of gain?
>
> Replaygain. It's a constant amount of amplification
> that brings the ms power of the file to a predetermined
> level.

I can't see what this has to do with this discussion.

> > > Dave said he used ms power.
> >
> > Read his posts again.
>
> "I square the instantaneous voltage or pressure to get
> instantaneous power and then I filter this with a simple moving
> average"
>
> Sounds like ms to me (ms voltage, to be more accurate).

My point was the "moving average" Dave refers to.

> > He had to use time-averaging (or smoothing) over
> > some (fairly arbitrarily chosen) time window.
>
> So?

I thought we were talking about how to calculate loudness as a
function of time.

> > > Cool Edit plots the values of the
> > > samples on a linear scale between zero and the max value
> > > possible at the given bit depth, then connects the dots using
> > > some bezier-looking thing. That's if you can see every sample.
> > > When time smoothing is needed, it divides the number of columns
> > > of x-axis pixels by the number of samples to display, finds the
> > > maximum abs sample value in each resulting bin, then colors
> > > pixels between + and - that value on the column.
> >
> > Go on.
>
> Nothing more to tell.
>
> > So far none of this sounds like an ms power calculation of
> > any kind.
>
> It isn't.

I guess I lost your train of thought, then.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/16/2005 1:09:37 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > > > > ...such a particular condition would almost never happen
> > > > > without special-purpose hardware. It'd be interesting
> > > > > to hear it, and I'd be happy to try and synthesize whatever
> > > > > test chords are proposed, but I'm not inclined to bend over
> > > > > backward for any of this. I'd be much more interested in
> > > > > how to calculate the heard beat rate of a chord....
> > > >
> > > > You never hear more than one beat rate in a chord?
> > >
> > > No no, I didn't say that.
> >
> > Why do you think calculating beat rates in a chord presents any
> > special problems, then?
>
> I'm completely at a loss for what you're asking here.

Carl, *you* were the one who wrote, "I'd be much more interested in
how to calculate the heard beat rate of a chord...." (but snipped it
above).

> > > In fact beat rates on my piano
> > > are nothing like those in these synthesized examples.
> >
> > How did you tune it?
>
> It's far out of tune at the moment. But I mean, beating
> chords on a piano don't seem to behave like synthesized
> ones.

Are you talking about high-quality synthesized piano emulations, or
some other kind of synth sound?

🔗Carl Lumma <clumma@yahoo.com>

11/16/2005 1:51:20 PM

> > > > > > I'd be much more interested in
> > > > > > how to calculate the heard beat rate of a chord....
> > > > >
> > > > > You never hear more than one beat rate in a chord?
> > > >
> > > > No no, I didn't say that.
> > >
> > > Why do you think calculating beat rates in a chord presents
> > > any special problems, then?
> >
> > I'm completely at a loss for what you're asking here.
>
> Carl, *you* were the one who wrote, "I'd be much more interested
> in how to calculate the heard beat rate of a chord...." (but
> snipped it above).

I didn't snip it, it's right there. I didn't say it represented
a special problem, but Dave said he didn't know how to do it,
and I don't know how to do it. But I have a few ideas.

> > > > In fact beat rates on my piano
> > > > are nothing like those in these synthesized examples.
> > >
> > > How did you tune it?
> >
> > It's far out of tune at the moment. But I mean, beating
> > chords on a piano don't seem to behave like synthesized
> > ones.
>
> Are you talking about high-quality synthesized piano
> emulations, or some other kind of synth sound?

I'm talking about the kind of synth sounds I've used to
test brats.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/16/2005 7:38:58 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > > > I'd be much more interested in
> > > > > > > how to calculate the heard beat rate of a chord....
> > > > > >
> > > > > > You never hear more than one beat rate in a chord?
> > > > >
> > > > > No no, I didn't say that.
> > > >
> > > > Why do you think calculating beat rates in a chord presents
> > > > any special problems, then?
> > >
> > > I'm completely at a loss for what you're asking here.
> >
> > Carl, *you* were the one who wrote, "I'd be much more interested
> > in how to calculate the heard beat rate of a chord...." (but
> > snipped it above).
>
> I didn't snip it, it's right there. I didn't say it represented
> a special problem, but Dave said he didn't know how to do it,
> and I don't know how to do it. But I have a few ideas.

If you hear more than one beat rate in a chord, it's not only not a
special problem, it's not a problem at all. You just calculate the
beat rates for every interval.

> > > > > In fact beat rates on my piano
> > > > > are nothing like those in these synthesized examples.
> > > >
> > > > How did you tune it?
> > >
> > > It's far out of tune at the moment. But I mean, beating
> > > chords on a piano don't seem to behave like synthesized
> > > ones.
> >
> > Are you talking about high-quality synthesized piano
> > emulations, or some other kind of synth sound?
>
> I'm talking about the kind of synth sounds I've used to
> test brats.

Are you keeping that secret or something?

🔗Carl Lumma <clumma@yahoo.com>

11/17/2005 10:17:35 AM

> If you hear more than one beat rate in a chord, it's not only not a
> special problem, it's not a problem at all. You just calculate the
> beat rates for every interval.

In the chords I've been listening to, there is one primary
beat.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

11/17/2005 10:17:00 AM

> > I'm talking about the kind of synth sounds I've used to
> > test brats.
>
> Are you keeping that secret or something?

I've been posting files...

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

11/17/2005 12:20:07 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > If you hear more than one beat rate in a chord, it's not only not a
> > special problem, it's not a problem at all. You just calculate the
> > beat rates for every interval.
>
> In the chords I've been listening to, there is one primary
> beat.

Probably just the loudest of the interval-beats, no?

🔗George D. Secor <gdsecor@yahoo.com>

11/17/2005 2:45:22 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
[to Gene Ward Smith:]
> ... I still don't think either one of us has answered Aaron's
> original question:
> /tuning/topicId_59689.html#61731
> which I repeat here:
>
> << what is the simplest possible 12-note temperament where all 24
> major and minor triads have rationally proportional beating?
> here 'simplest' means that the brats (beat ratios for the un-
> initiated) are the lowest numbers in the numerator and denominator
> that they can be..... >>
>
> I also assume the following additional requirements:
>
> That it be a *circulating* temperament, and
> That it should, at the very least, be reasonably useful from a
> musical standpoint -- or should I go beyond that by insisting that
it
> should sound good (just so we don't overlook the most important
point
> of this exercise)?
>
> --George

I've come to the conclusion that it's almost impossible to construct
a *musically useful* 12-note temperament where *all 24* major and
minor triads have rationally proportional beating that's reasonably
simple, even if you allow close-to-rational brats (as seems to be
necessary for the minor triads with tempered fifths).

Here are the two best temperaments I've been able to come up with so
far:

1) A well-temperament containing a chain of five ~1/5-(Didymus-)comma
fifths:

! secor_WT08.scl
!
George Secor's 12-tone well-temperament, proportional beating
(attempt #8)
12
!
90.22500
195.30322
294.13500
390.60644
498.04500
588.05203
697.65161
792.18000
892.95483
996.09000
1088.25804
2/1

Here are the brats and total absolute error for each triad:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.5000 34.41
Bb... ------- ------- 1.5000 25.80
F.... ------- ------- 1.5000 17.19
C.... 1.6667 5.0000 3.0000 17.19
G.... 1.6667 5.0000 3.0000 17.19
D.... 2.5000 6.2500 2.5000 21.48
A.... 4.2621 8.8931 2.0866 30.52
E.... 5.9435 11.4153 1.9206 39.13
B.... 15.1830 25.2745 1.6647 43.45
F#... 167.150 248.224 1.4850 43.45
C#... ------- ------- 1.5000 43.01
G#... ------- ------- 1.5000 43.01

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.0000 43.45
Bb... ------- ------- 1.0000 43.01
F.... ------- ------- 1.0000 43.01
C.... 7.9456 9.9456 1.2517 43.01
G.... 5.9653 7.9653 1.3353 34.41
D.... 3.9802 5.9802 1.5025 25.80
A.... 1.9901 3.9901 2.0050 17.19
E.... 1.9901 3.9901 2.0050 17.19
B.... 5.9409 7.9409 1.3367 17.19
F#... 100.222 98.2225 1.9800 21.91
C#... ------- ------- 1.0000 30.52
G#... ------- ------- 1.0000 39.13

I count 22 proportionally beating triads (or 24 if you're willing to
consider brats of 1.92 and 2.09 a close enough approximation to 2).

2) A temperament (extra)ordinaire containing a chain of six ~1/5-
(Didymus-)comma fifths:

! secor_TEO8.scl
!
George Secor's 12-tone temperament (extra)ordinaire, proportional
beating (attempt #8)
12
!
87.84337
195.30322
294.39757
390.60644
500.19803
585.90965
697.65161
789.76651
892.95483
998.24303
1088.25804
2/1

And here are the brats and total absolute error for each triad in
this one:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... 15.0000 20.0000 1.3333 33.88
Bb... ------- ------- 1.5000 21.49
F.... 5.0000 10.0000 2.0000 17.19
C.... 1.6667 5.0000 3.0000 17.19
G.... 1.6667 5.0000 3.0000 17.19
D.... 1.6667 5.0000 3.0000 17.19
A.... 3.3333 7.5000 2.2500 25.76
E.... 5.0000 10.0000 2.0000 34.30
B.... 7.7321 14.0982 1.8233 48.26
F#... 2052.42 3081.13 1.5012 52.08
C#... 1732.84 2061.75 1.5018 52.15
G#... 14.9892 19.9838 1.3332 47.84

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... 27.3363 25.3363 0.9268 52.04
Bb... ------- ------- 1.0000 52.08
F.... 22.0531 24.0531 1.0907 52.15
C.... 7.8249 9.8249 1.2556 42.49
G.... 4.9728 6.9728 1.4022 30.10
D.... 2.9851 4.9851 1.6700 21.49
A.... 1.9901 3.9901 2.0050 17.19
E.... 1.9901 3.9901 2.0050 17.19
B.... 1.9901 3.9901 2.0050 17.19
F#... 803.568 805.568 1.0025 17.19
C#... 803.590 805.590 1.0025 25.76
G#... 14.7441 12.7441 0.8644 39.65

For this one I can claim no more than 21 proportionally beating
triads, counting a brat of 0.8644 as 6/7 (or 23, if you're willing to
consider brats of 0.93 and 1.09 as approximately 1).

I should emphasize that my first priority was to follow sound
principles in designing a circulating temperament, with proportional
beating being a secondary consideration.

I was wondering if Gene could *rationalize* each of these, as he did
with my previous "latest" temperament (extra)ordinaire:
/tuning/topicId_59689.html#62013
for which I should express my belated thanks! For this one I can
claim proportional beating in 11 (or possibly 12) major triads, but
only 6 minor triads. (I should add that the major-triad brats are
simpler than in either of the newer temperaments given above.)

BTW, Gene, I now consider the ratios you gave as the official
description of the final version of the original temperament (extra)
ordinaire about which I enthused back in August:
/tuning/topicId_59689.html#59999
with the only significant difference between the original and final
versions being a brat change from 7/3 to 2 for the A major triad.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

11/17/2005 5:18:36 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> I was wondering if Gene could *rationalize* each of these, as he did
> with my previous "latest" temperament (extra)ordinaire:

Here they are:

! secorwt08.scl
George Secor well-temperament, rationalized version
12
!
256/243
124661/111375
32/27
502429/400950
4/3
1024/729
499946/334125
128/81
671389/400950
16/9
11392/6075
2

! secorte08.scl
George Secor extraordinare temperament, rationalized version
12
!
5075/4824
75/67
28591/24120
2015/1608
805/603
5075/3618
401/268
5075/3216
1010/603
3220/1809
15/8
2

🔗Carl Lumma <clumma@yahoo.com>

11/17/2005 11:27:47 PM

> > In the chords I've been listening to, there is one primary
> > beat.
>
> Probably just the loudest of the interval-beats, no?

I think so.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

11/18/2005 11:01:24 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
>
> > I was wondering if Gene could *rationalize* each of these, as he
did
> > with my previous "latest" temperament (extra)ordinaire:
>
> Here they are:
> ...

Thanks, Gene -- that was quick!

These rational versions are quite interesting and (to me) a bit
surprising -- I'll reply in more detail later (when I have more time),
on tuning-math.

--George

🔗George D. Secor <gdsecor@yahoo.com>

1/3/2006 9:03:32 AM

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
>
> it seems that all this exploration into equal-beating well
temperaments has
> given me a great question:
>
> what is the simplest possible 12-note temperament where all 24
major and minor
> triads have rationally proportional beating? here 'simplest' means
that the
> brats (beat ratios for the un-initiated) are the lowest numbers in
the
> numerator and denominator that they can be.....
>
> anyone? this is ripe for george or gene to explore.
>
> -aaron.

Aaron, it turned out to be something of a challenge to find a tuning
with reasonably simple brats for the 12 major triads alone. One
example that was brought to my attention is Paul Bailey's well-
temperament, for which Gene gave ratios:
/tuning-math/message/13670

On the first day of this new year, while using Gene's "Christmas
present for George" (on tuning-math), I stumbled across the following
solution (a high-contrast well-temperament) with major-triad brats
(starting on C) of:
4, 4, 3, 2, 1.5, 16/9, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5
and minor-triad brats (starting on C) of :
1.125, 1.5, 15/7, 4.2, 1, 1.5, 1, 1, 1, 1, 1, 1

For the actual ratios (in .scl format) see:
/tuning-math/message/13733

There's also another (2nd-best, lower-contrast) well-temperament at
the above link that may be more useful.

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

4/26/2006 1:39:34 PM

This is in reply to
/makemicromusic/topicId_13080.html#13088
which is quoted here:

--- In MakeMicroMusic@yahoogroups.com, "Aaron Krister Johnson"
<aaron@...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@> wrote:
>
> > Is there something microtonal about it?
>
> in truth, i believe i had the korg X5DR set to neidhardt 1. so in a
> *very mild* sense, yes.
>
> failing that, you could say that each chromatic step is close to
18/17
> apart!
>
> -aaron

/makemicromusic/topicId_13080.html#13080

Aaron, this is really very nice. I especially liked variation 3.

But should I infer from your using Neidhard 1 that you haven't
settled on any particular modern (low-contrast) well-temperament to
replace 12-ET?

At one point you were experimenting with a 1/7-comma circulating
temperament, observing that it has "some very sweet builtin beat
ratios":
/tuning/topicId_59689.html#60068

In checking this out, I found that 1/7-comma major brats are
approximately 2, and the minor brats ~4/3, which is very sweet,
indeed!

About a month later I came up with a proportional-beating 1/7-comma
well-temperament:
/tuning/topicId_59689.html#60252
with a correction at:
/tuning/topicId_59689.html#60264
but it appears that you didn't stick around this list long enough to
see it.

A couple months after that Gene Ward Smith transformed it into a
rational tuning (tuning-math #13615), which I transposed to arrive at
the following modern well-temperament:

! Secor1_7WT.scl
!
George Secor's 1/7-comma well-temperament (Gene Ward Smith rational
version)
12
!
6264/5929
33232/29645
105592/88935
5322/4235
39597/29645
8352/5929
31706/21175
9396/5929
9952/5929
52796/29645
55782/29645
2/1

All of the major brats (in the 3rd column of numbers) except one (on
B) are simple ratios:

Major -----Beat Ratios------ Total absolute
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.5000 30.69
Bb... ------- ------- 1.5000 24.52
F.... 5.0000 10.0000 2.0000 24.56
C.... 5.0000 10.0000 2.0000 24.59
G.... 5.0000 10.0000 2.0000 24.63
D.... 5.0000 10.0000 2.0000 24.41
A.... 6.6667 12.5000 1.8750 30.54
E.... 8.3333 15.0000 1.8000 36.68
B.... 8.7211 15.5817 1.7867 39.31
F#... ------- ------- 1.5000 39.31
C#... ------- ------- 1.5000 39.31
G#... 15.0000 25.0000 1.6667 36.86

Most of the minor brats are very close approximations of simple
ratios:

Minor -----Beat Ratios------ Total absolute
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.0000 39.31
Bb... ------- ------- 1.0000 39.31
F.... 10.7109 12.7109 1.1867 39.31
C.... 9.9048 11.9048 1.2019 36.86
G.... 7.9089 9.9089 1.2529 30.69
D.... 5.9943 7.9943 1.3336 24.52
A.... 5.9924 7.9924 1.3338 24.56
E.... 5.9899 7.9899 1.3339 24.59
B.... 5.7516 7.7516 1.3477 24.63
F#... ------- ------- 1.0000 24.41
C#... ------- ------- 1.0000 30.54
G#... 17.7124 19.7124 1.1129 36.68

As pretty as the numbers are, it's the sound the counts. Please try
it and let me know what you think.

Best,

--George

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

4/28/2006 5:37:37 PM

With all due respect to your efforts, could you not do a little better dear
George? I had the impression that the previous temperament extra-ordinaire
had it all.

One more thing if I may... could you send me all your related 12-tone Scala
files in a zip folder? I got them all messed up here in my hardrive.

Also, it might be a good idea to build a public scale archive for general
use, much like in wikipedia?

Cordially,
Oz.

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 26 Nisan 2006 �ar�amba 23:39
Subject: [tuning] 1/7-comma proportional-beating well-temperament

> This is in reply to
> /makemicromusic/topicId_13080.html#13088
> which is quoted here:
>
> --- In MakeMicroMusic@yahoogroups.com, "Aaron Krister Johnson"
> <aaron@...> wrote:
> >
> > --- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >
> > > Is there something microtonal about it?
> >
> > in truth, i believe i had the korg X5DR set to neidhardt 1. so in a
> > *very mild* sense, yes.
> >
> > failing that, you could say that each chromatic step is close to
> 18/17
> > apart!
> >
> > -aaron
>
> /makemicromusic/topicId_13080.html#13080
>
> Aaron, this is really very nice. I especially liked variation 3.
>
> But should I infer from your using Neidhard 1 that you haven't
> settled on any particular modern (low-contrast) well-temperament to
> replace 12-ET?
>
> At one point you were experimenting with a 1/7-comma circulating
> temperament, observing that it has "some very sweet builtin beat
> ratios":
> /tuning/topicId_59689.html#60068
>
> In checking this out, I found that 1/7-comma major brats are
> approximately 2, and the minor brats ~4/3, which is very sweet,
> indeed!
>
> About a month later I came up with a proportional-beating 1/7-comma
> well-temperament:
> /tuning/topicId_59689.html#60252
> with a correction at:
> /tuning/topicId_59689.html#60264
> but it appears that you didn't stick around this list long enough to
> see it.
>
> A couple months after that Gene Ward Smith transformed it into a
> rational tuning (tuning-math #13615), which I transposed to arrive at
> the following modern well-temperament:
>
> ! Secor1_7WT.scl
> !
> George Secor's 1/7-comma well-temperament (Gene Ward Smith rational
> version)
> 12
> !
> 6264/5929
> 33232/29645
> 105592/88935
> 5322/4235
> 39597/29645
> 8352/5929
> 31706/21175
> 9396/5929
> 9952/5929
> 52796/29645
> 55782/29645
> 2/1
>
> All of the major brats (in the 3rd column of numbers) except one (on
> B) are simple ratios:
>
> Major -----Beat Ratios------ Total absolute
> Triad M3/5th. m3/5th. m3/M3 error (cents)
> ----- ------- ------- ------ -------------
> Eb... ------- ------- 1.5000 30.69
> Bb... ------- ------- 1.5000 24.52
> F.... 5.0000 10.0000 2.0000 24.56
> C.... 5.0000 10.0000 2.0000 24.59
> G.... 5.0000 10.0000 2.0000 24.63
> D.... 5.0000 10.0000 2.0000 24.41
> A.... 6.6667 12.5000 1.8750 30.54
> E.... 8.3333 15.0000 1.8000 36.68
> B.... 8.7211 15.5817 1.7867 39.31
> F#... ------- ------- 1.5000 39.31
> C#... ------- ------- 1.5000 39.31
> G#... 15.0000 25.0000 1.6667 36.86
>
> Most of the minor brats are very close approximations of simple
> ratios:
>
> Minor -----Beat Ratios------ Total absolute
> Triad M3/5th. m3/5th. m3/M3 error (cents)
> ----- ------- ------- ------ -------------
> Eb... ------- ------- 1.0000 39.31
> Bb... ------- ------- 1.0000 39.31
> F.... 10.7109 12.7109 1.1867 39.31
> C.... 9.9048 11.9048 1.2019 36.86
> G.... 7.9089 9.9089 1.2529 30.69
> D.... 5.9943 7.9943 1.3336 24.52
> A.... 5.9924 7.9924 1.3338 24.56
> E.... 5.9899 7.9899 1.3339 24.59
> B.... 5.7516 7.7516 1.3477 24.63
> F#... ------- ------- 1.0000 24.41
> C#... ------- ------- 1.0000 30.54
> G#... 17.7124 19.7124 1.1129 36.68
>
> As pretty as the numbers are, it's the sound the counts. Please try
> it and let me know what you think.
>
> Best,
>
> --George
>
>
>

🔗George D. Secor <gdsecor@yahoo.com>

5/1/2006 1:18:33 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> With all due respect to your efforts, could you not do a little
better dear
> George? I had the impression that the previous temperament extra-
ordinaire
> had it all.

Oh, yes, I quite agree that it did (and still does) have it all!

However, "all" is evidently too much for some people, as Paul Bailey
wrote me (off-list) that although he was initially impressed by it,
he wouldn't be forwarding my 1/7-comma temperament to "a few other
piano tuners" because he remembered that:

<< I've learned from other tuners [that] 18 cents is about the
biggest [major] third that's acceptable to 'academic' pianists. >>

My 1/7th-comma WT has two major thirds 19.65 cents wide. I sent the
following to Paul today for comment (with widest major 3rds +18.49
cents, but many of the brats not as simple):

! Secor1_7MCRWT.scl
!
George Secor's 1/7-comma minimum-contrast rational well-temperament
12
!
14285/13512
5049/4504
428221/360320
1415/1126
752/563
25407/18016
843/563
1785/1126
945/563
80249/45040
8475/4504
2/1

> One more thing if I may... could you send me all your related 12-
tone Scala
> files in a zip folder? I got them all messed up here in my hardrive.

It's no problem to list them all here (in ascending order of key
contrast). They all have proportional-beating triads, both major and
minor. If a temperament is too tame (or too wild) for your purposes,
then just try one farther down (or up) the list. You'll notice that
Gene Ward Smith is credited with determining the ratios for some of
these:

! Secor1_7WT.scl
!
George Secor's 1/7-comma well-temperament (ratios supplied by G. W.
Smith)

12
!
6264/5929
33232/29645
105592/88935
5322/4235
39597/29645
8352/5929
31706/21175
9396/5929
9952/5929
52796/29645
55782/29645
2/1

! Secor2_11WT.scl
!
George Secor's rational 2/11-comma well-temperament
12
!
560/531
2643/2360
70/59
74/59
315/236
2240/1593
883/590
280/177
890/531
105/59
443/236
2/1

! Secor1_5WT.scl
!
George Secor's 1/5-comma well-temperament (ratios supplied by G. W.
Smith)
12
!
256/243
75/67
32/27
2015/1608
4/3
9101/6480
401/268
128/81
1010/603
16/9
15/8
2/1

! Secor5_23WT.scl
!
George Secor's rational 5/23-comma proportional-beating well-
temperament
12
!
256/243
966664/863865
32/27
120176/95985
4/3
1024/729
143588/95985
128/81
160624/95985
16/9
4096/2187
2/1

! Secor1_5TX.scl
!
George Secor's 1/5-comma temperament extraordinaire (ratios supplied
by G. W. Smith)
12
!
5075/4824
75/67
28591/24120
2015/1608
805/603
5075/3618
401/268
5075/3216
1010/603
3220/1809
15/8
2/1

! Secor5_23TX.scl
!
George Secor's rational 5/23-comma temperament extraordinaire
12
!
390/371
3321/2968
4397/3710
929/742
2476/1855
8325/5936
555/371
9365/5936
621/371
9904/5565
5559/2968
2/1

! Secor1_4TX.scl
!
George Secor's rational 1/4-comma temperament extraordinaire
12
!
4873/4644
481/430
61837/52245
484/387
7747/5805
8132/5805
193/129
1219/774
1942/1161
30988/17415
3621/1935
2/1

The latest version of the original temperament (extra)ordinaire that
you like so well is the 5/23-comma tuning (Secor5_23TX.scl).

In addition to the above, I also have two well-temperaments in which
all 24 major and minor triads have relatively simple brats; see:

/tuning-math/message/13733

> Also, it might be a good idea to build a public scale archive for
general
> use, much like in wikipedia?

What's wrong with the Scala archive -- if Manuel will add these?

Best,

--George

🔗Kraig Grady <kraiggrady@anaphoria.com>

5/2/2006 12:19:53 AM

Hello George!
I hate to bring up a tangential thought to your post so early, but it was something that came up during the Johnston Symposium both from Monzo and myself.
This was the idea of notating ratios in terms of factors instead of the resulting high number we sometimes use.
The advantage is that, at least, i have to figure it out this way anyway when running across a ratio, i hadn't used before or very frequently.
This way one would also know the implied basic function of the ratio.
admittedly we don't always use intervals in this way. Thought you might have some thoughts on this as a possibility.

I have BTW quite liked many of these scales you have come up along these lines and look forward to trying this one out in the future when i have a little less on my table.
It seems thirds this large might be useful to Margo Schulter

> Message: 8 > Date: Mon, 01 May 2006 20:18:33 -0000
> From: "George D. Secor" <gdsecor@yahoo.com>
> Subject: 1/7-comma proportional-beating WT - and all the others as well!
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >> With all due respect to your efforts, could you not do a little >> > better dear
> >> George? I had the impression that the previous temperament extra-
>> > ordinaire
> >> had it all.
>> >
> Oh, yes, I quite agree that it did (and still does) have it all!
>
> However, "all" is evidently too much for some people, as Paul Bailey > wrote me (off-list) that although he was initially impressed by it, > he wouldn't be forwarding my 1/7-comma temperament to "a few other > piano tuners" because he remembered that:
>
> << I've learned from other tuners [that] 18 cents is about the > biggest [major] third that's acceptable to 'academic' pianists. >>
>
> My 1/7th-comma WT has two major thirds 19.65 cents wide. I sent the > following to Paul today for comment (with widest major 3rds +18.49 > cents, but many of the brats not as simple):
>
> ! Secor1_7MCRWT.scl
> !
> George Secor's 1/7-comma minimum-contrast rational well-temperament
> 12
> !
> 14285/13512
> 5049/4504
> 428221/360320
> 1415/1126
> 752/563
> 25407/18016
> 843/563
> 1785/1126
> 945/563
> 80249/45040
> 8475/4504
> 2/1
>
> >> One more thing if I may... could you send me all your related 12-
>> > tone Scala
> >> files in a zip folder? I got them all messed up here in my hardrive.
>> >
> It's no problem to list them all here (in ascending order of key > contrast). They all have proportional-beating triads, both major and > minor. If a temperament is too tame (or too wild) for your purposes, > then just try one farther down (or up) the list. You'll notice that > Gene Ward Smith is credited with determining the ratios for some of > these:
>
>
> ! Secor1_7WT.scl
> !
> George Secor's 1/7-comma well-temperament (ratios supplied by G. W. > Smith)
>
> 12
> !
> 6264/5929
> 33232/29645
> 105592/88935
> 5322/4235
> 39597/29645
> 8352/5929
> 31706/21175
> 9396/5929
> 9952/5929
> 52796/29645
> 55782/29645
> 2/1
>
>
> ! Secor2_11WT.scl
> !
> George Secor's rational 2/11-comma well-temperament
> 12
> !
> 560/531
> 2643/2360
> 70/59
> 74/59
> 315/236
> 2240/1593
> 883/590
> 280/177
> 890/531
> 105/59
> 443/236
> 2/1
>
>
> ! Secor1_5WT.scl
> !
> George Secor's 1/5-comma well-temperament (ratios supplied by G. W. > Smith)
> 12
> !
> 256/243
> 75/67
> 32/27
> 2015/1608
> 4/3
> 9101/6480
> 401/268
> 128/81
> 1010/603
> 16/9
> 15/8
> 2/1
>
>
> ! Secor5_23WT.scl
> !
> George Secor's rational 5/23-comma proportional-beating well-
> temperament
> 12
> !
> 256/243
> 966664/863865
> 32/27
> 120176/95985
> 4/3
> 1024/729
> 143588/95985
> 128/81
> 160624/95985
> 16/9
> 4096/2187
> 2/1
>
>
> ! Secor1_5TX.scl
> !
> George Secor's 1/5-comma temperament extraordinaire (ratios supplied > by G. W. Smith)
> 12
> !
> 5075/4824
> 75/67
> 28591/24120
> 2015/1608
> 805/603
> 5075/3618
> 401/268
> 5075/3216
> 1010/603
> 3220/1809
> 15/8
> 2/1
>
>
> ! Secor5_23TX.scl
> !
> George Secor's rational 5/23-comma temperament extraordinaire
> 12
> !
> 390/371
> 3321/2968
> 4397/3710
> 929/742
> 2476/1855
> 8325/5936
> 555/371
> 9365/5936
> 621/371
> 9904/5565
> 5559/2968
> 2/1
>
>
> ! Secor1_4TX.scl
> !
> George Secor's rational 1/4-comma temperament extraordinaire
> 12
> !
> 4873/4644
> 481/430
> 61837/52245
> 484/387
> 7747/5805
> 8132/5805
> 193/129
> 1219/774
> 1942/1161
> 30988/17415
> 3621/1935
> 2/1
>
>
> The latest version of the original temperament (extra)ordinaire that > you like so well is the 5/23-comma tuning (Secor5_23TX.scl).
>
> In addition to the above, I also have two well-temperaments in which > all 24 major and minor triads have relatively simple brats; see:
>
> /tuning-math/message/13733
>
> >> Also, it might be a good idea to build a public scale archive for >> > general
> >> use, much like in wikipedia?
>> >
> What's wrong with the Scala archive -- if Manuel will add these?
>
> Best,
>
> --George
>
>
>
>
>
>
> __
> > ------------------------------------------------------------------------
>
>
>
>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗George D. Secor <gdsecor@yahoo.com>

5/2/2006 11:02:47 AM

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Hello George!
> I hate to bring up a tangential thought to your post so early, but
it
> was something that came up during the Johnston Symposium both from
Monzo
> and myself.
> This was the idea of notating ratios in terms of factors instead
of the
> resulting high number we sometimes use.
> The advantage is that, at least, i have to figure it out this way
> anyway when running across a ratio, i hadn't used before or very
frequently.
> This way one would also know the implied basic function of the
ratio.
> admittedly we don't always use intervals in this way. Thought you
might
> have some thoughts on this as a possibility.

Hi Kraig,

I am in complete agreement with you on this point, so much so, in
fact, that for the past few years I have made it a general practice
in my tuning spreadsheets to express the numbers of ratios as the
calculated products of their factors whenever it's meaningful to do
so.

> I have BTW quite liked many of these scales you have come up along
> these lines and look forward to trying this one out in the future
when i
> have a little less on my table.

OOPS! Please disregard this latest one!!! Although the paint's not
dry yet, I've already come to the conclusion that I can do better
(and expect that I'll eventually replace my 1/7-comma and 2/11-comma
well-temperaments with something else).

BTW, getting back to factoring ratios, I hope you won't try to factor
the ones with the largest numbers in the following one:

> > ! Secor1_5WT.scl
> > !> > George Secor's 1/5-comma well-temperament (ratios supplied
by G. W.
> > Smith)
> > 12
> > !
> > 256/243
> > 75/67
> > 32/27
> > 2015/1608
> > 4/3
> > 9101/6480
> > 401/268
> > 128/81
> > 1010/603
> > 16/9
> > 15/8
> > 2/1

Most of the numbers are so small that you're apt to think it's JI.
However, if you calculate the number of cents in each tone, you'll
find that all of the fifths in a chain from C to B are 4.3 cents
(rounded to 1 decimal place), so essentially this really is a *1/5-
comma temperament*.

--George

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

5/3/2006 6:59:14 PM

Dear George,

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > With all due respect to your efforts, could you not do a little
> better dear
> > George? I had the impression that the previous temperament extra-
> ordinaire
> > had it all.
>
> Oh, yes, I quite agree that it did (and still does) have it all!
>

Glad to hear it!

>
> However, "all" is evidently too much for some people, as Paul Bailey
> wrote me (off-list) that although he was initially impressed by it,
> he wouldn't be forwarding my 1/7-comma temperament to "a few other
> piano tuners" because he remembered that:
>
> << I've learned from other tuners [that] 18 cents is about the
> biggest [major] third that's acceptable to 'academic' pianists. >>
>
> My 1/7th-comma WT has two major thirds 19.65 cents wide. I sent the
> following to Paul today for comment (with widest major 3rds +18.49
> cents, but many of the brats not as simple):
>

Nonsense. They just don't wish to exploit the possibilities.

> ! Secor1_7MCRWT.scl
> !
> George Secor's 1/7-comma minimum-contrast rational well-temperament
> 12
> !
> 14285/13512
> 5049/4504
> 428221/360320
> 1415/1126
> 752/563
> 25407/18016
> 843/563
> 1785/1126
> 945/563
> 80249/45040
> 8475/4504
> 2/1
>

Urm. I did not like this one too much I'm afraid.

> > One more thing if I may... could you send me all your related 12-
> tone Scala
> > files in a zip folder? I got them all messed up here in my hardrive.
>
> It's no problem to list them all here (in ascending order of key
> contrast). They all have proportional-beating triads, both major and
> minor. If a temperament is too tame (or too wild) for your purposes,
> then just try one farther down (or up) the list. You'll notice that
> Gene Ward Smith is credited with determining the ratios for some of
> these:
>

They are all swell, but the copy-paste procedure is a tad tiresome. I would
be truly glad if you could send not only the SCL files, but also the excel
documents detailing the BRATs for further analysis.

>
> ! Secor1_7WT.scl
> !
> George Secor's 1/7-comma well-temperament (ratios supplied by G. W.
> Smith)
>
> 12
> !
> 6264/5929
> 33232/29645
> 105592/88935
> 5322/4235
> 39597/29645
> 8352/5929
> 31706/21175
> 9396/5929
> 9952/5929
> 52796/29645
> 55782/29645
> 2/1
>

This one is wild and brilliant.

>
> ! Secor2_11WT.scl
> !
> George Secor's rational 2/11-comma well-temperament
> 12
> !
> 560/531
> 2643/2360
> 70/59
> 74/59
> 315/236
> 2240/1593
> 883/590
> 280/177
> 890/531
> 105/59
> 443/236
> 2/1
>

More tamed, but still sparkly.

>
> ! Secor1_5WT.scl
> !
> George Secor's 1/5-comma well-temperament (ratios supplied by G. W.
> Smith)
> 12
> !
> 256/243
> 75/67
> 32/27
> 2015/1608
> 4/3
> 9101/6480
> 401/268
> 128/81
> 1010/603
> 16/9
> 15/8
> 2/1
>

Quite balanced, save a few beating glitches.

>
> ! Secor5_23WT.scl
> !
> George Secor's rational 5/23-comma proportional-beating well-
> temperament
> 12
> !
> 256/243
> 966664/863865
> 32/27
> 120176/95985
> 4/3
> 1024/729
> 143588/95985
> 128/81
> 160624/95985
> 16/9
> 4096/2187
> 2/1
>

Not quite to my liking.

>
> ! Secor1_5TX.scl
> !
> George Secor's 1/5-comma temperament extraordinaire (ratios supplied
> by G. W. Smith)
> 12
> !
> 5075/4824
> 75/67
> 28591/24120
> 2015/1608
> 805/603
> 5075/3618
> 401/268
> 5075/3216
> 1010/603
> 3220/1809
> 15/8
> 2/1

Intriguing.

>
>
> ! Secor5_23TX.scl
> !
> George Secor's rational 5/23-comma temperament extraordinaire
> 12
> !
> 390/371
> 3321/2968
> 4397/3710
> 929/742
> 2476/1855
> 8325/5936
> 555/371
> 9365/5936
> 621/371
> 9904/5565
> 5559/2968
> 2/1
>

This one is simply terrific, though lacking the brilliance of the previous
listed, which can be compensated by sheer volume of sound I believe.

I made a hasty recording using this temperament on FTS:

http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3

My hands were too crude for the plastic keyboard of Yamaha P-200. My
apologies beforehand for the reviling pianism.

>
> ! Secor1_4TX.scl
> !
> George Secor's rational 1/4-comma temperament extraordinaire
> 12
> !
> 4873/4644
> 481/430
> 61837/52245
> 484/387
> 7747/5805
> 8132/5805
> 193/129
> 1219/774
> 1942/1161
> 30988/17415
> 3621/1935
> 2/1
>
>

This one is extra rugged.

> The latest version of the original temperament (extra)ordinaire that
> you like so well is the 5/23-comma tuning (Secor5_23TX.scl).
>
> In addition to the above, I also have two well-temperaments in which
> all 24 major and minor triads have relatively simple brats; see:
>
> /tuning-math/message/13733
>

Are they not extraordinary?

> > Also, it might be a good idea to build a public scale archive for
> general
> > use, much like in wikipedia?
>
> What's wrong with the Scala archive -- if Manuel will add these?
>

The issue is, this really should not be a job cast over the shoulders of one
person, however hard-working. There are simply too many scales to track and
catalogue. Minute changes and corrections may be forgotten or omitted
altogether due to confusion. Scale archiving could be made a public effort
within the tuning list. In this way, we could have a Wikipedia-like database
and supply extra information for each scale. I for one would like to
describe my 79 MOS 159-tET in detail in several pages. CVs are also
important! It should be possible to view scales not only by the name of
their creator, but date of creation, cardinality, equalness of intervals,
etc...

> Best,
>
> --George
>
>
>

Cordially,
Oz.

🔗Carl Lumma <clumma@yahoo.com>

5/3/2006 8:09:12 PM

> > ! Secor5_23TX.scl
> > !
> > George Secor's rational 5/23-comma temperament extraordinaire
> > 12
> > !
> > 390/371
> > 3321/2968
> > 4397/3710
> > 929/742
> > 2476/1855
> > 8325/5936
> > 555/371
> > 9365/5936
> > 621/371
> > 9904/5565
> > 5559/2968
> > 2/1
>
> This one is simply terrific, though lacking the brilliance of
> the previous listed, which can be compensated by sheer volume
> of sound I believe.
>
> I made a hasty recording using this temperament on FTS:
>
> http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
>
> My hands were too crude for the plastic keyboard of Yamaha P-200.
> My apologies beforehand for the reviling pianism.

Not at all. This is quite nice. Though for people like me,
and I think we're in the minority around here, sounds like
those at 1:02, though bareable and sometimes even flavorful,
are in the end not worth the improved consonance of the 'near'
keys relative to 12-equal.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

5/4/2006 2:10:45 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> ...
> [GS:]
> > My 1/7th-comma WT has two major thirds 19.65 cents wide. I sent
the
> > following to Paul today for comment (with widest major 3rds +18.49
> > cents, but many of the brats not as simple):
> >
> > ! Secor1_7MCRWT.scl
> > !
> > George Secor's 1/7-comma minimum-contrast rational well-
temperament
> > ...
>
> Urm. I did not like this one too much I'm afraid.

I don't either -- I said I was scrapping it, because I'm working on
something that's much better, which I believe will be acceptable
to 'academic' pianists. (I expect even Carl will like it. :-)

> > > One more thing if I may... could you send me all your related
12-tone Scala
> > > files in a zip folder? I got them all messed up here in my
hardrive.
> >
> > It's no problem to list them all here (in ascending order of key
> > contrast). They all have proportional-beating triads, both major
and
> > minor. If a temperament is too tame (or too wild) for your
purposes,
> > then just try one farther down (or up) the list. You'll notice
that
> > Gene Ward Smith is credited with determining the ratios for some
of
> > these:
>
> They are all swell, but the copy-paste procedure is a tad tiresome.
I would
> be truly glad if you could send not only the SCL files, but also
the excel
> documents detailing the BRATs for further analysis.

Okay, I'll have to send those directly to you (soon, when I have a
little more time), since they would take up too much space in the
files section.

> > ! Secor1_7WT.scl
> > !
> > George Secor's 1/7-comma well-temperament (ratios supplied by G.
W.
> > Smith)
> > ...
>
> This one is wild and brilliant.

Hmmm, that's interesting, because I think I can improve on this one
as well.

> >
> > ! Secor2_11WT.scl
> > !
> > George Secor's rational 2/11-comma well-temperament
> > 12
> > !
> > 560/531
> > 2643/2360
> > 70/59
> > 74/59
> > 315/236
> > 2240/1593
> > 883/590
> > 280/177
> > 890/531
> > 105/59
> > 443/236
> > 2/1
>
> More tamed, ...

I don't quite understand why you would think so, because it has a
greater key contrast the the 1/7-comma temperament.

> >
> > ! Secor1_5WT.scl
> > !
> > George Secor's 1/5-comma well-temperament (ratios supplied by G.
W.
> > Smith)
> > ...
>
> Quite balanced, save a few beating glitches.

Okay.

> >
> > ! Secor5_23WT.scl
> > !
> > George Secor's rational 5/23-comma proportional-beating well-
> > temperament
> > ...
>
> Not quite to my liking.

I guess you like the consonance spread out to more keys, as in the
1/5-comma WT that precedes it. The 5/23-comma WT one has 7 just
fifths, hence is relatively easy to tune. Also, it contains a lot of
simple brats, hence is one of my favorites.

> >
> > ! Secor1_5TX.scl
> > !
> > George Secor's 1/5-comma temperament extraordinaire (ratios
supplied
> > by G. W. Smith)
> > ...
>
> Intriguing.

It has a lot of exact brats (both major & minor), and most of the
others approximate relatively simple numbers.

> >
> > ! Secor5_23TX.scl
> > !
> > George Secor's rational 5/23-comma temperament extraordinaire
> > ...
>
> This one is simply terrific, though lacking the brilliance of the
previous
> listed, which can be compensated by sheer volume of sound I believe.

As I indicated, this is the latest version of the original
temperament (extra)ordinaire that you liked so well.

> I made a hasty recording using this temperament on FTS:
>
> http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
>
> My hands were too crude for the plastic keyboard of Yamaha P-200. My
> apologies beforehand for the reviling pianism.

Sorry, I won't be able to listen to this until later.

> >
> > ! Secor1_4TX.scl
> > !
> > George Secor's rational 1/4-comma temperament extraordinaire
> > ...
>
> This one is extra rugged.

You betcha! Its maximum contrast is not for the faint of heart. Did
you notice how consonant the A major triad is?

> ...
> > > Also, it might be a good idea to build a public scale archive
for general
> > > use, much like in wikipedia?
> >
> > What's wrong with the Scala archive -- if Manuel will add these?
>
> The issue is, this really should not be a job cast over the
shoulders of one
> person, however hard-working. There are simply too many scales to
track and
> catalogue. Minute changes and corrections may be forgotten or
omitted
> altogether due to confusion. Scale archiving could be made a public
effort
> within the tuning list. In this way, we could have a Wikipedia-like
database
> and supply extra information for each scale. I for one would like to
> describe my 79 MOS 159-tET in detail in several pages. CVs are also
> important! It should be possible to view scales not only by the
name of
> their creator, but date of creation, cardinality, equalness of
intervals,
> etc...

Are you volunteering to chair a committee? ;-)

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

5/4/2006 2:32:26 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> [GS:]
> > > ! Secor5_23TX.scl
> > > !
> > > George Secor's rational 5/23-comma temperament extraordinaire
> > > ...
> > [Oz:]
> > I made a hasty recording using this temperament on FTS:
> >
> > http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
> >
> > My hands were too crude for the plastic keyboard of Yamaha P-200.
> > My apologies beforehand for the reviling pianism.
>
> Not at all. This is quite nice. Though for people like me,
> and I think we're in the minority around here, sounds like
> those at 1:02, though bareable and sometimes even flavorful,

I'm answering this without having heard the recording. Your comment
seems to indicate that I accomplished my objectives with this
temperament (extra)ordinaire, which were:

1) to have no wolf fifths (in this I was successful, since all fifths
are tempered by less than 4.7c), and
2) to have no triads (or thirds) that are tempered so much as to be
judged 'unbearable' (open to debate, since it's a matter of opinion
whether a major 3rd tempered 27c wide is bearable or not).

I'd be interested in your reaction to the "extra-rugged" 1/4-comma
temperament, if Oz would be so kind as to make a recording of that
one.

> are in the end not worth the improved consonance of the 'near'
> keys relative to 12-equal.

It depends on what you're using the temperament for. It was intended
principally for playing older music with no more than 3 sharps or 2
flats in the key signature (i.e., the meantone keys), while allowing
brief modulations into other keys without encountering any wolves.

--George

🔗Carl Lumma <clumma@yahoo.com>

5/5/2006 6:18:36 PM

> > Urm. I did not like this one too much I'm afraid.
>
> I don't either -- I said I was scrapping it, because I'm working on
> something that's much better, which I believe will be acceptable
> to 'academic' pianists. (I expect even Carl will like it. :-)

I'll watch this space!

-C.

🔗Carl Lumma <clumma@yahoo.com>

5/5/2006 6:24:56 PM

> > Not at all. This is quite nice. Though for people like me,
> > and I think we're in the minority around here, sounds like
> > those at 1:02, though bareable and sometimes even flavorful,
> > are in the end not worth the improved consonance of the 'near'
> > keys relative to 12-equal.
>
> It depends on what you're using the temperament for. It was
> intended principally for playing older music with no more than
> 3 sharps or 2 flats in the key signature (i.e., the meantone
> keys), while allowing brief modulations into other keys without
> encountering any wolves.

Of course. Meantone is great for Byrd, etc. But for Bach
and later, 27-wide 3rds just don't work for me *on the piano*.
I can stand them, but I don't prefer them.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

5/8/2006 1:19:25 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > Urm. I did not like this one too much I'm afraid.
> >
> > I don't either -- I said I was scrapping it, because I'm working
on
> > something that's much better, which I believe will be acceptable
> > to 'academic' pianists. (I expect even Carl will like it. :-)
>
> I'll watch this space!
>
> -C.

Okay, Carl, here's my all-purpose 12-ET replacement!

Offlist, Paul Bailey directed my attention to the Broadwood Victorian
well-temperament (which was devised by none other than Alexander J.
Ellis), with which I'm highly impressed, to say the least! I
concluded that the only way that it might be improved was to make
some of the triads proportional-beating, which I did this past
weekend:

! Secor-VRWT.scl
!
George Secor's Victorian rational well-temperament (based on Ellis #2)
12
!
545/516
6071/5418
707447/595980
755/602
30118/22575
30535/21672
451/301
34315/21672
1514/903
2755/1548
20365/10836
2/1

Here are the brats and total absolute error for each triad:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... 35.0000 55.0000 1.5714 35.40
Bb... 25.0000 40.0000 1.6000 27.14
F.... 15.0000 25.0000 1.6667 20.64
C.... 5.0000 10.0000 2.0000 15.32
G.... 2.0000 5.5000 2.7500 21.89
D.... 4.0000 8.5000 2.1250 29.04
A.... 5.0000 10.0000 2.0000 36.50
E.... 17.0000 28.0000 1.6471 37.88
B.... 43.0553 67.0829 1.5581 37.83
F#... 35.6667 56.0000 1.5701 37.91
C#... 30.0780 47.6170 1.5831 38.23
G#... 41.2901 64.4352 1.5605 37.62

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3 error (cents)
----- ------- ------- ------ -------------
Eb... 44.5383 46.5383 1.0449 37.83
Bb... 42.3038 44.3038 1.0473 37.91
F.... 34.7415 36.7415 1.0576 38.23
C.... 17.5177 19.5177 1.1142 37.62
G.... 5.0786 7.0786 1.3938 35.40
D.... 4.3200 6.3200 1.4630 27.14
A.... 2.5000 4.5000 1.8000 20.64
E.... 7.0000 9.0000 1.2857 15.32
B.... 28.8000 30.8000 1.0694 21.89
F#... 32.0000 34.0000 1.0625 29.04
C#... 34.0000 36.0000 1.0588 36.50
G#... 49.3547 51.3547 1.0405 37.88

I judged that the most important thing was to have whole numbers for
as many M3:5th ratios in the major triads as possible (beginning with
the 4 or 5 most consonant, considering that the rest may amount to
not much more than "eye candy"), and in the process I managed to get
some fairly simple brat numbers in the minor triads.

The brats aren't as simple as in the 1/7-comma WT that I posted
previously (as Secor1_7WT.scl), but the most and least consonant
triads (and arguably the harmonic balance) are all better,
respectively, in this latest one.

Comments, anyone?

--George

🔗Carl Lumma <clumma@yahoo.com>

5/8/2006 1:44:54 PM

> Okay, Carl, here's my all-purpose 12-ET replacement!
>
> Offlist, Paul Bailey directed my attention to the Broadwood
> Victorian well-temperament (which was devised by none other
> than Alexander J. Ellis), with which I'm highly impressed,
> to say the least!

Do you have Jorgensen? Looks like something that ought to
be on Wikisource. (It's over $400 on Amazon.)

Could you post the Broadwood tuning for comparison?
A Jorgensen TOC I found on the web shows two of
them; "usual" and "best".

> I concluded that the only way that it might be improved was
> to make some of the triads proportional-beating, which I did
> this past weekend:
>
> ! Secor-VRWT.scl
> !
> George Secor's Victorian rational well-temperament
> 12
> !
> 545/516
> 6071/5418
> 707447/595980
> 755/602
> 30118/22575
> 30535/21672
> 451/301
> 34315/21672
> 1514/903
> 2755/1548
> 20365/10836
> 2/1
>
//
> The brats aren't as simple as in the 1/7-comma WT that I posted
> previously (as Secor1_7WT.scl), but the most and least consonant
> triads (and arguably the harmonic balance) are all better,
> respectively, in this latest one.
>
> Comments, anyone?

With a worst major 3rd of 405 cents, this is something I'll
want to play. Having just moved, however, I don't have my
keyboard set up at the moment...

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

5/8/2006 2:42:20 PM

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Could you post the Broadwood tuning for comparison?
> A Jorgensen TOC I found on the web shows two of
> them; "usual" and "best".

This is the "usual", which I like better than the "best". ;-)

! Broadwood.scl
!
Broadwood's Usual (Ellis #2) Victorian Well-temperament
12
!
95.00000
197.00000
297.00000
392.00000
499.00000
594.00000
700.00000
796.00000
894.00000
998.00000
1093.00000
2/1

--George

🔗George D. Secor <gdsecor@yahoo.com>

5/9/2006 11:16:38 AM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> ...
> > > One more thing if I may... could you send me all your related
12-tone Scala
> > > files in a zip folder? I got them all messed up here in my
hardrive.
> >
> > It's no problem to list them all here (in ascending order of key
> > contrast). They all have proportional-beating triads, both major
and
> > minor. If a temperament is too tame (or too wild) for your
purposes,
> > then just try one farther down (or up) the list. You'll notice
that
> > Gene Ward Smith is credited with determining the ratios for some
of
> > these:
>
> They are all swell, but the copy-paste procedure is a tad tiresome.
I would
> be truly glad if you could send not only the SCL files, but also
the excel
> documents detailing the BRATs for further analysis.

I'll be sending these to you shortly. Anyone else wanting to analyze
the brats in a rational temperament can download the file in the
following link and input the numbers themselves from any of the
rational-temperament .scl file listings that Gene and I have posted
(into the cyan-colored cells):

/tuning-math/files/secor/RT-wksht.xls

(You need to be a member of the tuning-math group to access the file.)

This is the actual worksheet that I've been using to construct
rational temperaments. The beat ratios are in columns O thru Q (page
downward to see the minor triads); the formal "brat" numbers that
Gene & I have been using are in column Q. The beat rates of the
individual intervals in each triad are in columns S thru U, and the
frequencies of each tone are in columns W thru Y. If you wish, you
can change the reference frequency (of A) in cell K2, but that will
not affect the numbers in any columns to the right of S.

Should you want to see the brats for a temperament where the pitches
are given in cents, you can input those numbers directly into cells
E8-10,12-19. (This will overwrite the formulas in those cells, but
they can be restored easily by copying cell E11.)

--George

🔗Aaron Krister Johnson <aaron@akjmusic.com>

5/9/2006 6:24:15 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> /tuning/topicId_59689.html#66183

George,

I haven't been checking out the main 'tuning list' lately, but I was
happy to come across you above mentioned post when I did a casual
glance the other day....

We both share an attraction for the beat-ratios in 1/7-comma...and I'm
intrigued by your rational variation that was inspired by my post a
while back on 1/7-comma a while back...thanks for post this, and I'll
check it out when I have time to be away from cleaning my daughter's
diapers! ;)

Best,
Aaron

🔗George D. Secor <gdsecor@yahoo.com>

5/10/2006 10:17:53 AM

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> > /tuning/topicId_59689.html#66183
>
> George,
>
> I haven't been checking out the main 'tuning list' lately, but I was
> happy to come across you above mentioned post when I did a casual
> glance the other day....
>
> We both share an attraction for the beat-ratios in 1/7-comma...and
I'm
> intrigued by your rational variation that was inspired by my post a
> while back on 1/7-comma a while back...thanks for post this, and
I'll
> check it out when I have time to be away from cleaning my daughter's
> diapers! ;)
>
> Best,
> Aaron

Aaron, thanks for taking time out from some highly important
unschedulable tasks to reply. This brings back memories of times
past, when I was in the very same situation. Once my daughter was
born, I found not much time over the next 20 years to spend on
microtonality (with no regrets :-).

Should you find a few spare moments ;-), I'd be interested in your
opinion regarding the following. I wrote (excerpt from msg. #62222):

<< Paul Bailey wrote me (off-list) that although he was initially
impressed by it, he wouldn't be forwarding my 1/7-comma temperament
to "a few other piano tuners" because he remembered that, "I've
learned from other tuners [that] 18 cents is about the biggest
[major] third that's acceptable to 'academic' pianists." >>

Since my 1/7-comma WT has a max M3 error of +19.66c, it was back to
the drawing board. After an insufficiently successful attempt to
modify it, I came up with a rational temperament based on Ellis #2,
a/k/a Broadwood's "usual" WT, with max M3 error of +18.20c:
/tuning/topicId_59689.html#66329

At the moment, I think I prefer this to the 1/7-comma RWT, but I'd
like to get a few more opinions.

BTW, do you remember your challenge
/tuning/topicId_59689.html#61731
to find a temperament with the simplest proportional-beating major
*and* minor triads? If you revisit that thread, you'll find a bunch
of replies with a few nice tries, none of which met all of the
requirements. I also inferred that the solution "should, at the very
least, be reasonably useful from a musical standpoint [and] ...
should sound good" (msg. #62031).

At the beginning of January, when I was messing around with the
numbers in some spreadsheets containing equations that Gene had given
me (as a "Christmas present for George"), I unexpectedly stumbled
upon an answer (in the form of a very-high-contrast well-temperament):

/tuning/topicId_59689.html#63293

Best,

--George

🔗Aaron Krister Johnson <aaron@akjmusic.com>

5/11/2006 1:01:17 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> << Paul Bailey wrote me (off-list) that although he was initially
> impressed by it, he wouldn't be forwarding my 1/7-comma temperament
> to "a few other piano tuners" because he remembered that, "I've
> learned from other tuners [that] 18 cents is about the biggest
> [major] third that's acceptable to 'academic' pianists." >>

I can definately see how when M3rds get too wide, it might sound a bit
too subversive for most classical music. BTW, I assume you mean 18
cents wide of a 5/4?

> Since my 1/7-comma WT has a max M3 error of +19.66c, it was back to
> the drawing board. After an insufficiently successful attempt to
> modify it, I came up with a rational temperament based on Ellis #2,
> a/k/a Broadwood's "usual" WT, with max M3 error of +18.20c:
> /tuning/topicId_59689.html#66329
>
> At the moment, I think I prefer this to the 1/7-comma RWT, but I'd
> like to get a few more opinions.

I'm excited to check it out, even more so about what you write below!:

> BTW, do you remember your challenge
> /tuning/topicId_59689.html#61731
> to find a temperament with the simplest proportional-beating major
> *and* minor triads? If you revisit that thread, you'll find a bunch
> of replies with a few nice tries, none of which met all of the
> requirements. I also inferred that the solution "should, at the very
> least, be reasonably useful from a musical standpoint [and] ...
> should sound good" (msg. #62031).
>
> At the beginning of January, when I was messing around with the
> numbers in some spreadsheets containing equations that Gene had given
> me (as a "Christmas present for George"), I unexpectedly stumbled
> upon an answer (in the form of a very-high-contrast well-temperament):
>
> /tuning/topicId_59689.html#63293

Bravo! this looks incredible....from a pure-math point of view, is it
possible to rigorously prove that this or anything else is the optimal
solution?

I'll check out the *sound* when I get a chance later (diapers come first)

Best,
Aaron.

🔗George D. Secor <gdsecor@yahoo.com>

5/11/2006 2:39:26 PM

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > << Paul Bailey wrote me (off-list) that although he was initially
> > impressed by it, he wouldn't be forwarding my 1/7-comma
temperament
> > to "a few other piano tuners" because he remembered that, "I've
> > learned from other tuners [that] 18 cents is about the biggest
> > [major] third that's acceptable to 'academic' pianists." >>
>
> I can definately see how when M3rds get too wide, it might sound a
bit
> too subversive for most classical music. BTW, I assume you mean 18
> cents wide of a 5/4?

Yep.

> > Since my 1/7-comma WT has a max M3 error of +19.66c, it was back
to
> > the drawing board. ...
>
> I'm excited to check it out, even more so about what you write
below!:
>
> > BTW, do you remember your challenge
> > /tuning/topicId_59689.html#61731
> > to find a temperament with the simplest proportional-beating
major
> > *and* minor triads? ...
> >
> > At the beginning of January ... I unexpectedly stumbled
> > upon an answer (in the form of a very-high-contrast well-
temperament):
> >
> > /tuning/topicId_59689.html#63293
>
> Bravo! this looks incredible....from a pure-math point of view, is
it
> possible to rigorously prove that this or anything else is the
optimal
> solution?

That's something Gene would be much better equipped to answer. There
would need to be some sort of constraint on how the pitches are
related to one another. For example, if all 12 tones had the ratio
1/1, then there would be 24 extremely simple brats -- but this isn't
what we had in mind. Possibly it would need to be specified that the
tones should be in a circle of fifths, with limits on how much the
fifths could be tempered.

> I'll check out the *sound* when I get a chance later (diapers come
first)

Have fun! ;-) -- and congratulations! :-D

--George

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

5/11/2006 9:24:44 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> > Bravo! this looks incredible....from a pure-math point of view, is
> it
> > possible to rigorously prove that this or anything else is the
> optimal
> > solution?
>
> That's something Gene would be much better equipped to answer.

The most important requirement is a precise definition of "optimal".

🔗Aaron Krister Johnson <aaron@akjmusic.com>

5/12/2006 11:15:40 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > > Bravo! this looks incredible....from a pure-math point of view, is
> > it
> > > possible to rigorously prove that this or anything else is the
> > optimal
> > > solution?
> >
> > That's something Gene would be much better equipped to answer.
>
> The most important requirement is a precise definition of "optimal".
>

we migght say 'minimal' instead of 'optimal'--perhaps the least sum of
numerators and denominators of the brats would serve?

🔗Aaron Krister Johnson <aaron@akjmusic.com>

5/12/2006 11:51:19 AM

George,

In examining the 'WTPB-24a.scl' and 'WTPB-24b.scl' files, I see they
both contain chains of 6 pure 3/2s--putting it in the Valotti-Young
category...

question--would it be possible to have this low level of brats without
so many consecutive 3/2s? Or does it depend on them?

I notice that Wendell's Natural Synchronous doesn't use more than 3
3/2s in a row, but then again, it doesn't have the low brats for minor
triads that yours do.

Best,
Aaron.

🔗George D. Secor <gdsecor@yahoo.com>

5/12/2006 12:04:49 PM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > > Bravo! this looks incredible....from a pure-math point of view,
is it
> > > possible to rigorously prove that this or anything else is the
optimal
> > > solution?
> >
> > That's something Gene would be much better equipped to answer.
>
> The most important requirement is a precise definition of "optimal".

For starters, look at Aaron's wording:

<< what is the simplest possible 12-note temperament where all 24
major and minor triads have rationally proportional beating?
here 'simplest' means that the brats (beat ratios for the un-
initiated) are the lowest numbers in the numerator and denominator
that they can be..... >>

This leaves it pretty much up to your discretion as to what the order
of simplicity for brats should be, but I think you already have some
idea what this ought to be, since you've already put major brats of
3/2 and 4 at the top of the list, followed by 2 and 3. Other simple
numbers are 7/3 (giving 3 for M3/5th and 7 for m3/5th) and 11/4
(giving 2 for M3/5th and 11/2 for m3/5th). I think it's important
that all 3 ratios -- M3/5th, m3/5th, and m3/M3 -- be relatively
simple: the more integers, the better.

The simplest minor brats can be determined similarly (with 1, 2, and
3/2 topping the list).

My additional requirements are, at face value, a bit more subjective:

<< the solution should, at the very least, be reasonably useful from
a musical standpoint [and] ... should sound good >>

For example, we could specify that no fifth be tempered by more than
1/2-comma. Also, in a well-constructed temperament the brats
shouldn't zig-zag wildly as one goes around the circle of fifths.

If this isn't specific enough for you, then perhaps you could send me
the simultaneous equations containing 11 independent major-triad brat
variables that you've been using to determine the ratios for the 11
pitches (relative to 1/1) that will produce those brats. If I put
those into a spreadsheet, then by trial and error I might find some
more 24-simple-brat solutions, or at least, in the process, gravitate
toward some sort of algorithm that you could use in a program to
generate possible solutions.

We may want to continue this on tuning-math.

--George

🔗George D. Secor <gdsecor@yahoo.com>

5/12/2006 1:32:08 PM

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> George,
>
> In examining the 'WTPB-24a.scl' and 'WTPB-24b.scl' files, I see they
> both contain chains of 6 pure 3/2s--putting it in the Valotti-Young
> category...

In general, it's a good practice in a well-temperament not to have
the sizes of the fifths jumping back and forth as you go around the
circle. Instead, the narrowest ones should be between two natural
notes, and the widest ones will usually be pure. This will result in
a smooth progression of mood or color as one goes around the circle.

24a is slightly less desirable than 24b in this respect -- but it
does have better brat numbers (and higher contrast).

> question--would it be possible to have this low level of brats
without
> so many consecutive 3/2s? Or does it depend on them?

Any triad with a pure 5th, either major or minor, will automatically
have an exact brat (of 3/2 or 1, respectively), so having a lot of
pure 5ths in a temperament is an easy way to get a lot of exact
brats. It has nothing to do with whether or not the 3/2's are
consecutive.

> I notice that Wendell's Natural Synchronous doesn't use more than 3
> 3/2s in a row, but then again, it doesn't have the low brats for
minor
> triads that yours do.

It's only because Robert didn't attempt to get low minor brats. I
don't think it has anything to do with the zig-zag fifth sizes.

Having the fifth sizes jump back and forth also tends to lower the
key contrast. It's rather wasteful IMO to have fifths tempered by
more than 4 or 5 cents and not have any highly consonant triads to
show for it. (Compare this with my 2/11 or 1/5-comma rational WT's,
which have all of the fifths tempered by less than 4.0 and 4.32
cents, respectively.)

Pssst! If there's any one trick for getting 24 simple brats, it's
this: make the numerators and denominators in your pitch ratios
fairly small, keeping as many as you can under 4 digits. (Egad! My
secret is out! I hope Gene doesn't read this. ;-)

--George

🔗Carl Lumma <clumma@yahoo.com>

5/12/2006 2:47:55 PM

> In general, it's a good practice in a well-temperament not to have
> the sizes of the fifths jumping back and forth as you go around the
> circle. Instead, the narrowest ones should be between two natural
> notes, and the widest ones will usually be pure. This will result
> in a smooth progression of mood or color as one goes around the
> circle.

Which might not be important to those who like to modulate by
thirds.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

5/12/2006 3:44:25 PM

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> This leaves it pretty much up to your discretion as to what the order
> of simplicity for brats should be, but I think you already have some
> idea what this ought to be, since you've already put major brats of
> 3/2 and 4 at the top of the list, followed by 2 and 3.

That's for brats in the vicinity of the 12-et brat. Simpler brats are
-1, 0, and infinity, of course.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

5/19/2006 11:10:09 AM

Dear George,

SNIP

> >
> > ! Secor1_4TX.scl
> > !
> > George Secor's rational 1/4-comma temperament extraordinaire
> > ...
>
> This one is extra rugged.

You betcha! Its maximum contrast is not for the faint of heart. Did
you notice how consonant the A major triad is?

~~~~~ Indeed, it is very similar to my 79MOS159-tET tempered A major triad:

1. Copy the scala file at www.ozanyarman.com\anonymous to the Scala folder.
2. Edit file and point to the actual scala folder on your hardrive.
3. Download and run the 79MOS159tET.cmd file in SCALA.

Also check out the incomplete Excel file at the same address and tell me what you think!

> ...
> > > Also, it might be a good idea to build a public scale archive
for general
> > > use, much like in wikipedia?
> >
> > What's wrong with the Scala archive -- if Manuel will add these?
>
> The issue is, this really should not be a job cast over the
shoulders of one
> person, however hard-working. There are simply too many scales to
track and
> catalogue. Minute changes and corrections may be forgotten or
omitted
> altogether due to confusion. Scale archiving could be made a public
effort
> within the tuning list. In this way, we could have a Wikipedia-like
database
> and supply extra information for each scale. I for one would like to
> describe my 79 MOS 159-tET in detail in several pages. CVs are also
> important! It should be possible to view scales not only by the
name of
> their creator, but date of creation, cardinality, equalness of
intervals,
> etc...

Are you volunteering to chair a committee? ;-)

~~~~~~~ Ack! no! I'm not the kind of guy competent for that stuff.

Best,

--George

~~~~~~

Cordially,
Oz.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

5/19/2006 10:30:03 AM

George,

SNIP

>
> I'd be interested in your reaction to the "extra-rugged" 1/4-comma
> temperament, if Oz would be so kind as to make a recording of that
> one.
>

Unfortunately, I will be busy these few weeks, so the recording shall have
to wait a while.

Cordially,
Oz.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

5/19/2006 11:08:28 AM

I'm pleased that you like it. Sorry for the late reply. I have an important
presentation to prepare in 9 days on my extended and improved 79-tone Qanun.
Returned from Izmir, will tell more later.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 04 Mayıs 2006 Perşembe 6:09
Subject: [tuning] Re: 1/7-comma proportional-beating WT - and all the others
as well!

> > > ! Secor5_23TX.scl
> > > !
> > > George Secor's rational 5/23-comma temperament extraordinaire
> > > 12
> > > !
> > > 390/371
> > > 3321/2968
> > > 4397/3710
> > > 929/742
> > > 2476/1855
> > > 8325/5936
> > > 555/371
> > > 9365/5936
> > > 621/371
> > > 9904/5565
> > > 5559/2968
> > > 2/1
> >
> > This one is simply terrific, though lacking the brilliance of
> > the previous listed, which can be compensated by sheer volume
> > of sound I believe.
> >
> > I made a hasty recording using this temperament on FTS:
> >
> > http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
> >
> > My hands were too crude for the plastic keyboard of Yamaha P-200.
> > My apologies beforehand for the reviling pianism.
>
> Not at all. This is quite nice. Though for people like me,
> and I think we're in the minority around here, sounds like
> those at 1:02, though bareable and sometimes even flavorful,
> are in the end not worth the improved consonance of the 'near'
> keys relative to 12-equal.
>
> -Carl
>

🔗George D. Secor <gdsecor@yahoo.com>

5/19/2006 1:06:21 PM

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Dear George,
>
> SNIP
>
> > >
> > > ! Secor1_4TX.scl
> > > !
> > > George Secor's rational 1/4-comma temperament extraordinaire
> > > ...
> >
> > This one is extra rugged.
>
> You betcha! Its maximum contrast is not for the faint of heart.
Did
> you notice how consonant the A major triad is?
>
> ~~~~~ Indeed, it is very similar to my 79MOS159-tET tempered A
major triad:
>
> 1. Copy the scala file at www.ozanyarman.com\anonymous to the Scala
folder.
> 2. Edit file and point to the actual scala folder on your hardrive.
> 3. Download and run the 79MOS159tET.cmd file in SCALA.
>
> Also check out the incomplete Excel file at the same address and
tell me what you think!

Okay, I've downloaded these, but I'm taking a break (away from the
Internet) for the next week or so and won't be able to respond until
after next week.

Best,

--George