Lehman's 'Bach' scale

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

--- 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

<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

--- 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

> > 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

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

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.

--- 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

--- 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

> > > 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

--- 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

> 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

--- 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

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

--- 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

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

> > 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

--- 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.

--- 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.

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.

> > 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

> 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

--- 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

--- 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

--- 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

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

--- 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

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
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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.

--- 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?

--- 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.

--- 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

--- 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

--- 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

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

> > > 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

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

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):
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------------------------------------------------------------------------------
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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

--- 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

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

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

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> ! 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

--- 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

--- 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?

> > 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

--- 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.

--- 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 . . .

> > > > 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

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

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

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

--- 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.

--- 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?

--- 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.

--- 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?

--- 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?

--- 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.

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.

--- 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.

--- 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.

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

--- 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.

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.

--- 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.

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.

--- 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.

--- 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.

> > > 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

> 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

--- 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.

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

>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.

>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.

--- 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?

--- 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.

--- 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.

--- 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).

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).

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).

--- 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.)

--- 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.

--- 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

> 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

--- 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

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:

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

--- 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

--- 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

> > 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

--- 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.

--- 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

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

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

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

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

Period = identity interval.

-Carl

--- 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.

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

> > 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

--- 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.

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

--- 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.

--- 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.

--- 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

--- 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

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

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.

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

--- 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

--- 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.

--- 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.

--- 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.

--- 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

--- 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.

--- 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.

--- 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.

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?

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?

--- 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.

--- 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

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

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
>
>

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?

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

--- 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

--- 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.

--- 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

--- 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?

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?

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

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

> 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

--- 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?

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.

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

--- 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

--- 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

> 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

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?

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

--- 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

> 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

--- 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.

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?

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

--- 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

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.

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.

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.

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.

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

--
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--- 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?

--- 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

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

--
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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

--- 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.

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

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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

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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

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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

--- 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

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.

--- 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.

> > 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

--- 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.

--- 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

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

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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

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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

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

--- 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.

--- 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.

--- 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

--- 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.

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

--- 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

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

--- 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"...

> 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

--- 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

--- 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.

--- 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.

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

--- 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

--- 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?

--- 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!

--- 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.

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"...

--- 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.

> > > 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

> 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

--- 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

--- 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

--- 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

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