Yahoo Tuning Groups Ultimate Backup tuning Grady, Monzo, Comma, Temperament

If you got the previous versions of this message, delete them -- this
one is corrected.

Here are two essentially identical graphs that show the 5-prime-limit
intervals, focusing on the small ones, and omitting entirely those
whose ratios require about 9 or more digits and those which are
powers of smaller 5-prime-limit intervals. In other words, we're
looking at the 'commas' of 5-limit music.

/tuning-math/files/Paul/com5monz.gif

/tuning-math/files/Paul/com5rat.gif

The cyan lines with green numbers show the number of cents that each
interval, when played in JI, measures. The horizontal position of an
interval on this graph shows its "taxicab distance in the Tenney
lattice" -- or just as well, the number of digits in the interval's
ratio. The vertical position of an interval in this graph shows the
typical error you would expect in the prime intervals as a result of
tempering the interval out. Herman has explored many such
temperaments in his music and on his webpages.

The first graph shows the intervals in lattice-vector notation --
that is, each interval has a frequency ratio of 2^e2 * 3^e3 * 5^e5
and is represented as [e2 e3 e5] on the graph. Intervals so notated
have been named (by Gene, I believe) monzos in honor of our friend
Joe Monzo who likes such notation, and in this guise the word monzo
makes frequent appearances on the tuning-math list (even if Joe Monzo
doesn't :) ).

The second graph shows the interval ratios as fractions. This seems
to be a favored notation of Kraig Grady, who prefers it over the
monzo notation. Therefore it would be equally logical to call these
gradys -- with due respect to the fact that Kraig doesn't tend to
operate in such a 'limiting' scenario.

Both should be of particular interest to Herman Miller.

The further to the right an interval is on these graphs, the more
complex the temperament is. Clearly the temperaments of the most
interest will be the least complex ones for a given level of error of
the prime intervals, or those which give the lowest error for a given
level of complexity. On the graphs, at least if you consider it
unreasonable to temper out intervals as large or simple as the major
whole tone and minor whole tone, there are 13 commas which look most
useful to temper out in order to derive easily manageable two-
dimensional pitch systems (such as the familiar meantone) from the
three-dimensional 5-limit JI lattice. Note that I don't assume octave-
equivalence in any of this so far.

Here is a table showing some properties of these 13 commas &
temperaments -- the 'period' column does assume octave-equivalence,
and the 'generator' column assumes 'period-equivalence' -- ordered
from 'coarse' to 'fine':

lowest complexity, highest error -- these imply extremely simple
scales and require specially designed electronic timbres to
sound 'consonant':
Grady.....Monzo...Temperament...Notes/Octave..Period....Generator
27/25....[0 3 2]...."beep"........4,5,9.......1 oct....~268 cent
16/15....[4 -1 -1].."father"......3,5,8.......1 oct....~442 cent
25/24....[-3 -1 2].."dicot"......3,5,7,10.....1 oct....~351 cent

low complexity, high error -- these imply scales of ordinary
complexity and typically require specially designed electronic
timbres to sound 'just' with the possible exception of meantone:
Grady.....Monzo...Temperament...Notes/Octave..Period...Generator
135/128..[-7 3 1].."pelogic"....7,9,16,23.....1 oct....~523 cent
256/243..[8 5 0].."Blackwood"...5,10,15,25....1/5 oct...~85 cent
648/625..[3 4 -4]..diminished..4,8,12,16,28...1/4 oct...~94 cent
250/243..[1 -5 3]..porcupine..7,8,15,22,29....1 oct....~163 cent
128/125..[7 0 3]...augmented.3,9,12,15,18,27..1/3 oct...~91 cent
81/80....[-4 4 1]..meantone..5,7,12,19,26,31..1 oct....~504 cent

medium complexity, medium error -- these imply slightly more complex
scales; a reasonably 'just' sound is generally a bit easier to come
by:
Grady.....Monzo....Temperament..Notes/Octave......Period..Generator
3125/3072.[-10 -1 5]."magic".3,16,19,22,25,53,41..1 oct...~380 cent
2048/2025.[11 -4 -2].diaschismic.10,12,22,34,46...1/2 oct..~105 cent

high complexity, low error -- still more complex scales; sound is
quite close to 'just' with any harmonic timbre:
Grady.......Monzo..Temperament........Notes/Octave....Period.Generator
15625/15552.[-6 -5 6].kleismic/Hanson.15,19,23,34,53..1 oct.~317 cent

highest complexity, lowest error -- most complex scales; sound is
generally indistinguishable from 'just' with any harmonic timbre:
Grady.......Monzo.....Temperament..Notes/Octave...Period.Generator
32805/32768.[-15 8 1].schismic*..12,29,41,53,65...1 oct...~498.3 cent

*also known as Helmholtz or Groven, and used on Eduardo Sabat-
Garibaldi's 53-note-per-octave guitars ("Dinarras")

🔗Herman Miller <hmiller@IO.COM>

On Tue, 30 Dec 2003 13:32:02 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>The cyan lines with green numbers show the number of cents that each
>interval, when played in JI, measures. The horizontal position of an
>interval on this graph shows its "taxicab distance in the Tenney
>lattice" -- or just as well, the number of digits in the interval's
>ratio. The vertical position of an interval in this graph shows the
>typical error you would expect in the prime intervals as a result of
>tempering the interval out. Herman has explored many such
>temperaments in his music and on his webpages.

I don't know about "many", but I've played around with a few of these on
the keyboard, even if I haven't written much music around them. I suppose
you could count the Warped Canon page, where I began to realize how scales
could be classified by their commas, but Johann Pachelbel wrote the music,
and all I did was warp it. But one of my New Year's resolutions is to get
more involved in writing music, as part of an effort to enrich the
development of my fictional alien cultures, so I'm planning to spend more
time investigating these temperaments. It'd be interesting to see a graph
like this for 7-limit commas.

>The first graph shows the intervals in lattice-vector notation --
>that is, each interval has a frequency ratio of 2^e2 * 3^e3 * 5^e5
>and is represented as [e2 e3 e5] on the graph. Intervals so notated
>have been named (by Gene, I believe) monzos in honor of our friend
>Joe Monzo who likes such notation, and in this guise the word monzo
>makes frequent appearances on the tuning-math list (even if Joe Monzo
>doesn't :) ).

What's interesting about this graph is how the "good" temperaments seem to
stand out so much from the crowd of uninteresting ones -- especially
meantone and schismic, but also orwell, kleismic, diaschismic, and others.
Würschmidt temperament also stands out, but it's one that although it looks
good mathematically, I haven't found it to be as useful as it might seem to
be. So there are other factors involved, but at first glance, something as
simple as the size of a comma in cents seems to be a pretty good guide to
whether it might be of interest. Certainly, most of the "good" commas are
under 50 cents, and it's only when you get to the left hand side of the
graph that you start coming across "interesting" temperaments with larger
commas like diminished [3 4 -4] (62.6c), Blackwood [8 -5 0] (90.2c), and
pelogic [-7 3 1] (92.1c).

Another comma of interest, although it's just above the 100c curve, is [-11
7 0] 2187;2048 (113.7c). Easley Blackwood's 21-ET etude takes advantage of
this comma, modulating around the scale a whole step at a time until it
arrives in the original key after 7 steps.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

/tuning/topicId_50628.html#50907

>
> What's interesting about this graph is how the "good" temperaments
seem to
> stand out so much from the crowd of uninteresting ones -- especially
> meantone and schismic, but also orwell, kleismic, diaschismic, and
others.
> Würschmidt temperament also stands out, but it's one that although
it looks
> good mathematically, I haven't found it to be as useful as it might
seem to
> be. So there are other factors involved, but at first glance,
something as
> simple as the size of a comma in cents seems to be a pretty good
guide to
> whether it might be of interest. Certainly, most of the "good"
commas are
> under 50 cents, and it's only when you get to the left hand side of
the
> graph that you start coming across "interesting" temperaments with
larger
> commas like diminished [3 4 -4] (62.6c), Blackwood [8 -5 0]
(90.2c), and
> pelogic [-7 3 1] (92.1c).
>
> Another comma of interest, although it's just above the 100c curve,
is [-11
> 7 0] 2187;2048 (113.7c). Easley Blackwood's 21-ET etude takes
advantage of
> this comma, modulating around the scale a whole step at a time
until it
> arrives in the original key after 7 steps.
>

***Just as a point of information: you're considering "good"
temperaments the ones that more closely approach our traditional
diatonic? Or am I off the track here... Thanks!

J. Pehrson

🔗Paul Erlich <paul@stretch-music.com>

Herman, thanks for looking at this. I was afraid no one would.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On Tue, 30 Dec 2003 13:32:02 -0000, "Paul Erlich" <paul@s...>
> wrote:
>
> >The cyan lines with green numbers show the number of cents that
each
> >interval, when played in JI, measures. The horizontal position of
an
> >interval on this graph shows its "taxicab distance in the Tenney
> >lattice" -- or just as well, the number of digits in the
interval's
> >ratio. The vertical position of an interval in this graph shows
the
> >typical error you would expect in the prime intervals as a result
of
> >tempering the interval out. Herman has explored many such
> >temperaments in his music and on his webpages.
>
> I don't know about "many", but I've played around with a few of
these on
> the keyboard, even if I haven't written much music around them. I
suppose
> you could count the Warped Canon page, where I began to realize how
scales
> could be classified by their commas, but Johann Pachelbel wrote the
music,
> and all I did was warp it. But one of my New Year's resolutions is
to get
> more involved in writing music, as part of an effort to enrich the
> development of my fictional alien cultures,

I CAN'T WAIT!!

> so I'm planning to spend more
> time investigating these temperaments. It'd be interesting to see a
graph
> like this for 7-limit commas.

Pretty soon, I'll get to it -- and those will of course
represent 'planar temperaments', not 'linear temperaments'.

Meanwhile, I'm working on quantifying the exact form these
temperaments should take to fulfill the presentation on the graph
(particularly the vertical axis which measures the severity of the
errors that the tempering imparts), and that may truly be 'best' in
some rather universal sense that doesn't assume any dichotomy between
consonances and dissonances. See the tuning-math list. If I'm right,
it seems that meantone should be realized with octaves of 1201.6985
cents, fifths of 697.5644 cents, hence fourths of 504.1341 cents,
major sixths of 890.9947 cents, major thirds of 386.8606 cents, minor
thirds of 310.7038 cents, minor sixths of 814.8379 cents, as well as
major ninths of 1395.1288 cents, etc . . . I'll call this "Top
Meantone", where "Top" stands for "Tempered Octaves, Please" or more
technically, "Tenney-OPtimal" . . . For another 'octave&fifth-
generated' example, I think "Top Pelogic" will have octaves of
1206.5482 cents, fifths of 685.0280 cents, hence fourths of 521.5203
cents, etc . . . More will appear on tuning-math . . . I'm hoping you
and others will create originals and 'covers' in these tunings for my
listening pleasure ;) !

> >The first graph shows the intervals in lattice-vector notation --
> >that is, each interval has a frequency ratio of 2^e2 * 3^e3 * 5^e5
> >and is represented as [e2 e3 e5] on the graph. Intervals so
notated
> >have been named (by Gene, I believe) monzos in honor of our friend
> >Joe Monzo who likes such notation, and in this guise the word
monzo
> >makes frequent appearances on the tuning-math list (even if Joe
Monzo
> >doesn't :) ).
>
> What's interesting about this graph is how the "good" temperaments
seem to
> stand out so much from the crowd of uninteresting ones -- especially
> meantone and schismic, but also orwell, kleismic, diaschismic, and
others.
> Würschmidt temperament also stands out, but it's one that although
it looks
> good mathematically, I haven't found it to be as useful as it might
seem to
> be.

Less useful that 5-limit orwell?

> So there are other factors involved, but at first glance, something
as
> simple as the size of a comma in cents seems to be a pretty good
guide to
> whether it might be of interest.

How about the size of the font I used? Is that a better guide? The
font size was a monotonic function of log(n-d)/log(n*d), where n/d is
the ratio of the comma.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> /tuning/topicId_50628.html#50907
>
> >
> > What's interesting about this graph is how the "good"
temperaments
> seem to
> > stand out so much from the crowd of uninteresting ones --
especially
> > meantone and schismic, but also orwell, kleismic, diaschismic,
and
> others.

> ***Just as a point of information: you're considering "good"
> temperaments the ones that more closely approach our traditional
> diatonic?

No, not at all. "Good" ones, ones with low "badness" in some parlance
you may have seen before, are those which:

*SIMPLIFY the space of possible pitches considerably, relative to JI;
*APPROXIMATE the JI intervals somewhat well;

and some trade-off between these two desiderata is implied.

Being further to the left on horizontal axis implies "goodness" on
the first criterion, while being lower down on the vertical axis
implies "goodness" on the second.

As you can see, on the left side of the graph, good old-fashioned
meantone pretty much dominates the picture, but that's not because it
corresponds to the traditional diatonic, rather it may be that the
diatonic is so traditional just *because* meantone dominates the left
side of the graph. Meanwhile, the lower part of the graph is
dominated by schismic, something that has actually popped up in our
culture from time to time, such as the early 1400s, and again
sporadically as with Groven's organ and Sabat-Garibaldi's guitar . . .

🔗Herman Miller <hmiller@IO.COM>

On Sat, 03 Jan 2004 19:32:18 -0000, "Joseph Pehrson" <jpehrson@rcn.com>
wrote:

>***Just as a point of information: you're considering "good"
>temperaments the ones that more closely approach our traditional
>diatonic? Or am I off the track here... Thanks!

I put "good" in quotation marks precisely because I'm using it
subjectively, but it doesn't really have anything to do with the
traditional diatonic scale. Pelogic is about as far from diatonic as you
can get, but it's one of what I'd call the more "interesting" temperaments.
Würschmidt has good consonances, but not good enough to justify the
complexity of the temperament as far as I can tell from my (admittedly
limited) exploration and experimentation. The one tuning that really stands
out on the chart that I haven't played with is Amity, which never really
sounded like it would be all that useful, but should be better than Orwell
according to the chart (its predicted error and Tenney Complexity are both
lower). I've had some "interesting" results with Orwell, although that was
a long time ago, and it takes some time to feel your way around an
unfamiliar tuning.

Generally, the kind of temperament that I'm looking for should have good
consonances (and dissonances, but those tend to come naturally whether you
intend them or not), and should have useful enharmonic equivalents
(otherwise you could just use an arbitrary high-numbered ET, or start with
JI and randomly detune notes until you get the desired degree of beating).
It should also have a useful melodic subset within a fairly small number
(around 7-12 or so) of iterations of the generator (which is a problem with
temperaments that have generators around the size of a major third).

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On Sat, 03 Jan 2004 19:32:18 -0000, "Joseph Pehrson" <jpehrson@r...>
> wrote:
>
> >***Just as a point of information: you're considering "good"
> >temperaments the ones that more closely approach our traditional
> >diatonic? Or am I off the track here... Thanks!
>
> I put "good" in quotation marks precisely because I'm using it
> subjectively, but it doesn't really have anything to do with the
> traditional diatonic scale. Pelogic is about as far from diatonic
as you
> can get, but it's one of what I'd call the more "interesting"
temperaments.

I see things somewhat differently. Pelogic is quite similar to
diatonic to me, because:

*Assuming octave-repetition, it's generated by 'fifths' (the
approximate 3:2s) and 'octaves' (the approximate 2:1s);
*It has MOSs at 5 and 7 notes;
*The approximate 5-limit triads occur exactly where they do in the
corresponding diatonic scales, only major and minor are switched.

> Würschmidt has good consonances, but not good enough to justify the
> complexity of the temperament as far as I can tell from my
(admittedly
> limited) exploration and experimentation. The one tuning that
really stands
> out on the chart that I haven't played with is Amity, which never
really
> sounded like it would be all that useful, but should be better than
Orwell
> according to the chart (its predicted error and Tenney Complexity
are both
> lower).

Better than Orwell in 5-limit, but probably not in higher limits.

What did you think of the list of temperaments that I typed in the
original message in this thread?

Another musical mad scientist,
Paul

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> I put "good" in quotation marks precisely because I'm using it
> subjectively, but it doesn't really have anything to do with the
> traditional diatonic scale. Pelogic is about as far from diatonic
as you
> can get, but it's one of what I'd call the more "interesting"
temperaments.
> Würschmidt has good consonances, but not good enough to justify the
> complexity of the temperament as far as I can tell from my
(admittedly
> limited) exploration and experimentation. The one tuning that
really stands
> out on the chart that I haven't played with is Amity, which never
really
> sounded like it would be all that useful, but should be better than
Orwell
> according to the chart (its predicted error and Tenney Complexity
are both
> lower). I've had some "interesting" results with Orwell, although
that was
> a long time ago, and it takes some time to feel your way around an
> unfamiliar tuning.

Once again, note this is a 5-limit discussion. Orwell, which has
generators of an approximate 7/6, hardly comes into its own in the 5-
limit. The same, of course, is true of Miracle.

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> The one tuning that really stands
> out on the chart that I haven't played with is Amity, which never
really
> sounded like it would be all that useful, but should be better than
Orwell
> according to the chart (its predicted error and Tenney Complexity
are both
> lower). I've had some "interesting" results with Orwell, although
that was
> a long time ago, and it takes some time to feel your way around an
> unfamiliar tuning.

Herman, I need to clarify something.

[-21 3 7] = 2109375/2097152 is a comma (actually, it's called
the 'semicomma', due to Fokker).

'Orwell', on the other hand, is Gene's term for any 'linear
temperament' with a generator of about 19/84 octave and a period of 1
octave. Gene usually thinks of it in terms of higher limits than 5.

If you temper the semicomma out from the 5-limit lattice, you indeed
get a linear temperament that fits the 'Orwell' description. However,
if you temper it out from the 7-limit lattice, you get a 'planar
temperament'; temper it out from the 11-limit lattice, a 'spacial
temperament'. In these cases, the information on my charts remains
valid.

However, in order to get 7-limit Orwell, which is a 'linear
temperament', you need to temper out an additional comma from the 7-
limit lattice. Alternatively, you could start over and use the two
commas 225/224 and 1728/1715. Since these are a lot simpler than
2109375/2097152, it's a pretty good guess that you're dealing with a
simpler temperament here. To get 11-limit Orwell, you need to temper
out three commas -- you could use two of the above and one more, or
you could use 99/98, 121/120, and 176/175. Note that these are all
very simple, and all superparticular (thus being the smallest size
possible for their complexity)!

I foresaw this confusion which is why, on the big list of commas
and 'linear temperaments' on Monz's ET page, you'll see that the
entry for orwell actually says "orwell (5-limit)".

For a different sort of example, note the appearance of [-19 12 0] =
531441/524288 on my charts. This is the famous Pythagorean comma.
Tempering it out is usually associated with 12-equal -- in fact, in
my Top scheme, it leads to an equal division of a 1200.61705-cent
octave into 12 equal parts, or 12ED2.00071297. But as is well known,
the 5-limit errors of 12-equal are much larger than the 5-limit
errors of meantone, and yet on the graph, the Pythagorean comma is
lower, meaning it has lower errors, than the syntonic comma. Why?
Because tempering out the Pythagorean comma from the *5-limit*
lattice does not lead to 12-equal, instead it leads to an infinite
number of parallel 12-equal systems at a ratio of 5:1 apart from one
another (the Pythagorean comma doesn't touch the 5-axis, as you can
see from the third entry in its monzo being 0). This is a 'linear
temperament' with a period of about 100 cents and a generator of
about 15 cents; it was the subject of my second or third post ever to
the tuning list in 1996; and we now call it 'aristoxenean'. A true 5-
limit use of 12-equal involves tempering out *two* commas, not one,
and so can't be evaluated from my charts. My charts *will* read
correctly whether you are considering the effect of tempering out the
Pythagorean comma in the 3-limit, 5-limit, or any higher limit -- as
long as you aren't employing a hidden assumption about other commas
being tempered out.

🔗Herman Miller <hmiller@IO.COM>

On Sat, 03 Jan 2004 22:58:06 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>I see things somewhat differently. Pelogic is quite similar to
>diatonic to me, because:
>
>*Assuming octave-repetition, it's generated by 'fifths' (the
>approximate 3:2s) and 'octaves' (the approximate 2:1s);
>*It has MOSs at 5 and 7 notes;
>*The approximate 5-limit triads occur exactly where they do in the
>corresponding diatonic scales, only major and minor are switched.

Well, switching major and minor (and not just the triads, but everywhere)
seems to be a pretty radical difference to me. But there are certainly
similarities.

>What did you think of the list of temperaments that I typed in the
>original message in this thread?

It seems to be a good list of 5-limit temperaments; the only other one I
can think of that might be worth adding to the list is the [-11 7 0]
(2187;2048) that I mentioned (used in Blackwood's 21-ET etude, which takes
advantage of the circle of fifths closing after 7 steps). "Father" [4 -1
-1] seems to be what I've been calling "anti-pentatonic", and works well
with 13-ET (as well as anything can be said to work with 13-ET). You just
have to be careful with your choice of timbres, as you pointed out:

file:///C:/web/midi/canon13-ap.mid

"Dicot" [-3 -1 2] is the "neutral third" scale in Graham Breed's list
(http://x31eq.com/7plus3.htm), which has good fifths, and so you
don't need any special timbres if you like the flavor of the neutral
thirds; my 24-ET Warped Canon uses this tuning.

file:///C:/web/midi/canon24.mid

I'm not familiar with "beep".

🔗Herman Miller <hmiller@IO.COM>

On Sat, 03 Jan 2004 21:54:55 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>> so I'm planning to spend more
>> time investigating these temperaments. It'd be interesting to see a
>graph
>> like this for 7-limit commas.
>
>Pretty soon, I'll get to it -- and those will of course
>represent 'planar temperaments', not 'linear temperaments'.

Naturally. In any case, the reason I mention this is that Zireen music uses
lots of 7-limit harmony; one of the basic scales is something like this:

1/1 15/14 5/4 9/7 3/2 (7/4 = 12/7)

where the 49/48 is tempered out. Mizarian music will tend to use porcupine
temperament, extended to 11-limit (with the use of the [6 -2 0 -1] 64;63
and [-7 -1 1 1 1] 385;384 commas), but I'd like to explore alternatives.

>Meanwhile, I'm working on quantifying the exact form these
>temperaments should take to fulfill the presentation on the graph
>(particularly the vertical axis which measures the severity of the
>errors that the tempering imparts), and that may truly be 'best' in
>some rather universal sense that doesn't assume any dichotomy between
>consonances and dissonances. See the tuning-math list. If I'm right,
>it seems that meantone should be realized with octaves of 1201.6985
>cents, fifths of 697.5644 cents, hence fourths of 504.1341 cents,
>major sixths of 890.9947 cents, major thirds of 386.8606 cents, minor
>thirds of 310.7038 cents, minor sixths of 814.8379 cents, as well as
>major ninths of 1395.1288 cents, etc . . .

Sounds good to me.

>> Würschmidt temperament also stands out, but it's one that although
>it looks
>> good mathematically, I haven't found it to be as useful as it might
>seem to
>> be.
>
>Less useful that 5-limit orwell?

Orwell has a useful 9-note MOS for melodic purposes, and it seems to work
well with extended tertian harmonies from what I recall (although I haven't
spent much time with either one).

>> So there are other factors involved, but at first glance, something
>as
>> simple as the size of a comma in cents seems to be a pretty good
>guide to
>> whether it might be of interest.
>
>How about the size of the font I used? Is that a better guide? The
>font size was a monotonic function of log(n-d)/log(n*d), where n/d is
>the ratio of the comma.

It looks pretty good until you get to the far left hand side of the chart,
where commas like [-1 1 0] and [2 -1 0], and [-2 0 1] show up as huge
(would anyone actually temper these out?) It also makes temperaments like
pelogic [-7 3 1] and diminished [3 4 -4] appear less interesting than I
think they are, while exaggerating the importance of others such as father
[4 -1 -1].

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> "Father" [4 -1
> -1] seems to be what I've been calling "anti-pentatonic", and works
well
> with 13-ET (as well as anything can be said to work with 13-ET).

It would probably work even better in 8-equal. 11-equal, 13-equal,
and 14-equal can all support it to some degree, but you have to be a
bit careful because these latter three are inconsistent.

> You just
> have to be careful with your choice of timbres, as you pointed out:
>
> file:///C:/web/midi/canon13-ap.mid

I'm not sure if that's the link you intended :)

Anyway, the Top tuning for "father" temperament, which should work
best with the least "warped" timbres, has the following values for
the simplest intervals:

2:1 -- 1191.18 cents
3:1 -- 1915.94 cents
4:1 -- 2382.35 cents
5:1 -- 2806.80 cents
3:2 -- 707.12 cents
6:1 -- 3107.12 cents

Hope you will be encouraged to use this, Top meantone, Top pelogic,
and Top everything for my listening enjoyment ;)

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> > "Father" [4 -1
> > -1] seems to be what I've been calling "anti-pentatonic", and
works
> well
> > with 13-ET (as well as anything can be said to work with 13-ET).
>
> It would probably work even better in 8-equal. 11-equal, 13-equal,
> and 14-equal can all support it to some degree, but you have to be
a
> bit careful because these latter three are inconsistent.
>
> > You just
> > have to be careful with your choice of timbres, as you pointed
out:
> >
> > file:///C:/web/midi/canon13-ap.mid
>
> I'm not sure if that's the link you intended :)
>
> Anyway, the Top tuning for "father" temperament, which should work
> best with the least "warped" timbres, has the following values for
> the simplest intervals:
>
> 2:1 -- 1191.18 cents
> 3:1 -- 1915.94 cents
> 4:1 -- 2382.35 cents
> 5:1 -- 2806.80 cents
> 3:2 -- 707.12 cents
> 6:1 -- 3107.12 cents

Whoops! That's wrong -- way too close to JI. Here are the correct
values:

2:1 -- 1185.87 cents
3:1 -- 1924.35 cents
4:1 -- 2371.74 cents
5:1 -- 2819.12 cents
3:2 -- 738.48 cents
6:1 -- 3110.22 cents

> Hope you will be encouraged to use this, Top meantone, Top pelogic,
> and Top everything for my listening enjoyment ;)

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <paul@stretch-music.com>

🔗Herman Miller <hmiller@IO.COM>

On Mon, 05 Jan 2004 22:10:15 -0000, "wallyesterpaulrus"
<paul@stretch-music.com> wrote:

>--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
>> "Father" [4 -1
>> -1] seems to be what I've been calling "anti-pentatonic", and works
>well
>> with 13-ET (as well as anything can be said to work with 13-ET).
>
>It would probably work even better in 8-equal. 11-equal, 13-equal,
>and 14-equal can all support it to some degree, but you have to be a
>bit careful because these latter three are inconsistent.

It ís pretty nice in 8-equal. I used to have an MP3 of this on the Tuning
Punks site, before MP3.com shut down:
http://www.io.com/~hmiller/midi/canon8-alt.mid

>> You just
>> have to be careful with your choice of timbres, as you pointed out:
>>
>> file:///C:/web/midi/canon13-ap.mid
>
>I'm not sure if that's the link you intended :)

Try this one. :-)
http://www.io.com/~hmiller/midi/canon13-ap.mid

>Anyway, the Top tuning for "father" temperament, which should work
>best with the least "warped" timbres, has the following values for
>the simplest intervals:
>
>2:1 -- 1191.18 cents
>3:1 -- 1915.94 cents
>4:1 -- 2382.35 cents
>5:1 -- 2806.80 cents
>3:2 -- 707.12 cents
>6:1 -- 3107.12 cents

What's the period and generator for this? 1191.18 and somewhere around 442?

>Hope you will be encouraged to use this, Top meantone, Top pelogic,
>and Top everything for my listening enjoyment ;)

🔗Herman Miller <hmiller@IO.COM>

🔗wallyesterpaulrus <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_50628.html#50943

>
> No, not at all. "Good" ones, ones with low "badness" in some
parlance
> you may have seen before, are those which:
>
> *SIMPLIFY the space of possible pitches considerably, relative to
JI;
> *APPROXIMATE the JI intervals somewhat well;
>
> and some trade-off between these two desiderata is implied.
>

***Got it. I remember this specific term from before...

> Being further to the left on horizontal axis implies "goodness" on
> the first criterion, while being lower down on the vertical axis
> implies "goodness" on the second.
>
> As you can see, on the left side of the graph, good old-fashioned
> meantone pretty much dominates the picture, but that's not because
it
> corresponds to the traditional diatonic, rather it may be that the
> diatonic is so traditional just *because* meantone dominates the
left
> side of the graph. Meanwhile, the lower part of the graph is
> dominated by schismic, something that has actually popped up in our
> culture from time to time, such as the early 1400s, and again
> sporadically as with Groven's organ and Sabat-Garibaldi's
guitar . . .

***This is very interesting, but I can't find the original
temperament graph at all now. This is the one associated with ETs,
yes??

Monzo's dictionary seems inaccessible at the moment. Anybody else
having a problem.

I try by typing in "Tonalsoft" to get it that way, but I think it may
be momentarily down... :(

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

/tuning/topicId_50628.html#50944

> On Sat, 03 Jan 2004 19:32:18 -0000, "Joseph Pehrson" <jpehrson@r...>
> wrote:
>
> >***Just as a point of information: you're considering "good"
> >temperaments the ones that more closely approach our traditional
> >diatonic? Or am I off the track here... Thanks!
>
> I put "good" in quotation marks precisely because I'm using it
> subjectively, but it doesn't really have anything to do with the
> traditional diatonic scale. Pelogic is about as far from diatonic
as you
> can get, but it's one of what I'd call the more "interesting"
temperaments.
> Würschmidt has good consonances, but not good enough to justify the
> complexity of the temperament as far as I can tell from my
(admittedly
> limited) exploration and experimentation. The one tuning that
really stands
> out on the chart that I haven't played with is Amity, which never
really
> sounded like it would be all that useful, but should be better than
Orwell
> according to the chart (its predicted error and Tenney Complexity
are both
> lower). I've had some "interesting" results with Orwell, although
that was
> a long time ago, and it takes some time to feel your way around an
> unfamiliar tuning.
>
> Generally, the kind of temperament that I'm looking for should have
good
> consonances (and dissonances, but those tend to come naturally
whether you
> intend them or not), and should have useful enharmonic equivalents
> (otherwise you could just use an arbitrary high-numbered ET, or
start with
> JI and randomly detune notes until you get the desired degree of
beating).
> It should also have a useful melodic subset within a fairly small
number
> (around 7-12 or so) of iterations of the generator (which is a
problem with
> temperaments that have generators around the size of a major third).
>

***Well, just "off the cuff" this seems like a much more interesting
approach than the one that Easily Blackwood favors. He always seems
to be hunting (both musically and mathematically) with temperaments
that resemble the traditional diatonic, correct??

Thanks!

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> > Being further to the left on horizontal axis implies "goodness"
on
> > the first criterion, while being lower down on the vertical axis
> > implies "goodness" on the second.
> >
> > As you can see, on the left side of the graph, good old-fashioned
> > meantone pretty much dominates the picture, but that's not
because
> it
> > corresponds to the traditional diatonic, rather it may be that
the
> > diatonic is so traditional just *because* meantone dominates the
> left
> > side of the graph. Meanwhile, the lower part of the graph is
> > dominated by schismic, something that has actually popped up in
our
> > culture from time to time, such as the early 1400s, and again
> > sporadically as with Groven's organ and Sabat-Garibaldi's
> guitar . . .
>
> ***This is very interesting, but I can't find the original
> temperament graph at all now. This is the one associated with ETs,
> yes??

No, it's a pair of synonymous graphs of commas:

/tuning-math/files/Paul/com5rat.gif
/tuning-math/files/Paul/com5monz.gif

If you start with the 5-limit JI lattice, and temper out one of these
commas, you'll have a linear temperament. Tempering out 81/80 gives
you meantone. Tempering out 32805/32768 gives you schismic. Etc.

> Monzo's dictionary seems inaccessible at the moment.

These graphs aren't there (yet) anyway.

> Anybody else
> having a problem.

Yes, but the individual pages still seem accessible at sonic-arts.

> I try by typing in "Tonalsoft" to get it that way, but I think it
may
> be momentarily down... :(

Yep.

🔗wallyesterpaulrus <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@attglobal.net>

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> What's interesting about this graph is how the "good" temperaments
seem to
> stand out so much from the crowd of uninteresting ones

Well, that's kind of how I arranged things. You'll note that the
vertical (error) axis was logarithmic. If I make it linear, and zoom
in a bit on the lower left, here's how the picture looks:

/tuning/files/Erlich/herman1.gif

Now things aren't more crowded at higher errors than at lower errors.
But they're still more crowded at higher complexity than lower
complexity. All this would change if I measured complexity with n*d
instead of log(n*d). Then the *left* (low complexity) side of the
graph becomes extremely crowded (since I'm not plotting powers of
commas in these graphs) . . .

Anyway, the graph chart shows that if you're interested in tempering
out 2187/2048, you might also be interested in tempering out . . .
well never mind, you can see it from the chart, at least if your
eyesight is better than mine . . .

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_50628.html#51112

> >***Well, just "off the cuff" this seems like a much more
interesting
> >approach than the one that Easily Blackwood favors. He always
seems
> >to be hunting (both musically and mathematically) with
temperaments
> >that resemble the traditional diatonic, correct??
>
> I think he has been fairly criticized for this, but let's not
> overdo it. The much-derided "diatonic" scale in 15 is actually
> a subset of the much-praised Blackwood decatonic scale. He's
> used analogs of augmented and dimininished scales in various
> temperaments (the aforementioned decatonic scale is an analog
> of the diminished scale in a sense). Anyway, Blackwood's music
> rocks, to me. Sure, he missed a lot of interesting stuff, but
> he was a *pioneer* in the field.
>
> -Carl

***Hi Carl,

Oh no... I wasn't deprecating the end result at all (!), only
discussing the *approach...* Blackwoods _Etudes_ is one of my most
highly prized scores on my shelf...

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Herman Miller <hmiller@IO.COM>

On Tue, 06 Jan 2004 03:39:28 -0000, "wallyesterpaulrus"
<paul@stretch-music.com> wrote:

>> http://www.io.com/~hmiller/midi/canon-top-father.mid
>
>Hmm . . . I can't say that this is a good choice of timbres -- I
>think the ones you used for 8-equal or 13-equal would be better . . .

That was just a quick retuning with the original timbres; I played around
with the timbres and uploaded a new version (plus a "top pelogic" and "top
meantone" version) to the Warped Canon page. See the section at the very
end of the page.

http://www.io.com/~hmiller/music/warped-canon.html

🔗Carl Lumma <ekin@lumma.org>

🔗Herman Miller <hmiller@IO.COM>

On Tue, 06 Jan 2004 04:28:01 -0000, "Joseph Pehrson" <jpehrson@rcn.com>
wrote:

>***Well, just "off the cuff" this seems like a much more interesting
>approach than the one that Easily Blackwood favors. He always seems
>to be hunting (both musically and mathematically) with temperaments
>that resemble the traditional diatonic, correct??

He's also explored the pelog-like properties of 23-ET, not to mention what
he calls the "sub-minor" mode of 13-ET, what we'd now call the 8-note MOS
of "Father" temperament (which he notates as Ab Bb Cbb Dbb Ebb E F# G#). I
never noticed this resemblance until just now, but it makes sense; it's one
of the more reasonable things you can do with 13-ET. Considering that his
etudes were written back in 1979-1980 (almost a quarter century ago!), this
seems pretty sophisticated to me. But he does tend to miss out on some of
the more interesting and exotic features of the various ET's, with his
emphasis on the more traditional harmonies.

🔗Herman Miller <hmiller@IO.COM>

On Wed, 07 Jan 2004 00:19:47 -0000, "wallyesterpaulrus"
<paul@stretch-music.com> wrote:

>/tuning/files/Erlich/herman1.gif
>
>Now things aren't more crowded at higher errors than at lower errors.
>But they're still more crowded at higher complexity than lower
>complexity. All this would change if I measured complexity with n*d
>instead of log(n*d). Then the *left* (low complexity) side of the
>graph becomes extremely crowded (since I'm not plotting powers of
>commas in these graphs) . . .

I find it easier to remember the temperaments from the names than the
ratios.

http://www.io.com/~hmiller/gif/labeled.gif

>Anyway, the graph chart shows that if you're interested in tempering
>out 2187/2048, you might also be interested in tempering out . . .
>well never mind, you can see it from the chart, at least if your
>eyesight is better than mine . . .

That's 3125/2916. Potentially of interest in relation to 18-ET, but not
much else. Actually, since not much else works with 18-ET, that might be of
some usefulness; you could modulate down a fifth and five minor thirds and
get back to where you started. It's also available as an option in 25-ET
(where it could be analyzed as a combination of the blackwood and magic
commas), and it could be used as a basis for 7-nominal notation of 18-ET or
25-ET.

🔗Herman Miller <hmiller@IO.COM>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> > http://www.io.com/~hmiller/music/warped-canon.html
>
> Awesome!! I love the TOP-pelogic version!

It is kind of k00l. I'll need to think more about pelogic.

And the TOP-meantone
> sounds like the best one yet [though I have different hardware
> when I determined 69-tET was my fav, and I'm only comparing to
> that now.

Clearly, we need a Wilson meantone added to this page.

In general my hardware these days isn't as good as
> it was then. Herman, how does the MIDI retuning tech compare
> these days to when this page debuted?].

These days Scala will tune using MTS, hooray! Of course, your midi
player will ignore the tuning message, so we are far better off with
pitch bends.

I want a midi player which supports MTS. :(

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On Wed, 07 Jan 2004 00:19:47 -0000, "wallyesterpaulrus"
> <paul@s...> wrote:
>
> >/tuning/files/Erlich/herman1.gif
> >
> >Now things aren't more crowded at higher errors than at lower
errors.
> >But they're still more crowded at higher complexity than lower
> >complexity. All this would change if I measured complexity with
n*d
> >instead of log(n*d). Then the *left* (low complexity) side of the
> >graph becomes extremely crowded (since I'm not plotting powers of
> >commas in these graphs) . . .
>
> I find it easier to remember the temperaments from the names than
the
> ratios.
>
> http://www.io.com/~hmiller/gif/labeled.gif
>
> >Anyway, the graph chart shows that if you're interested in
tempering
> >out 2187/2048, you might also be interested in tempering out . . .
> >well never mind, you can see it from the chart, at least if your
> >eyesight is better than mine . . .
>
> That's 3125/2916.

Actually, I was referring to *all* the commas along a band
connecting "beep" with "semisixths". But I think most of those, as
well as 2187/2048, may have to be left out of whatever paper gets
written, just because we'll have to stop somewhere and there's a nice
big "gulf" before you get to 2187/2048, on both this graph and the
original, logarithmic-scaled one.

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On Tue, 06 Jan 2004 03:39:28 -0000, "wallyesterpaulrus"
> <paul@s...> wrote:
>
> >> http://www.io.com/~hmiller/midi/canon-top-father.mid
> >
> >Hmm . . . I can't say that this is a good choice of timbres -- I
> >think the ones you used for 8-equal or 13-equal would be
better . . .
>
> That was just a quick retuning with the original timbres; I played
around
> with the timbres and uploaded a new version (plus a "top pelogic"
and "top
> meantone" version) to the Warped Canon page. See the section at the
very
> end of the page.
>
> http://www.io.com/~hmiller/music/warped-canon.html

Herman, that's PHAT! I might go insane if I listen to too many
versions of this piece in a row, but it's fantastic that there's a
place on the web where someone can hear what all these tunings sound
like -- if only for a piece that wasn't especially intended for them.

Apparently 9-equal did get some play in the construction of authentic
gamelans in Indonesia for a time. See Daniel Wolf's posts on SpecMus.
So your comment about more than two step sizes might be toned down a
touch.

Now, the other 10 "Tops":

dicot:
prime 2 becomes 1207.66 cents -- this is the period
prime 3 becomes 1914.09 cents
prime 5 becomes 2768.53 cents
generator is 353.22 cents

beep:
prime 2 remains just at 1200 cents -- this is the period
prime 3 becomes 1879.49 cents
prime 5 becomes 2819.23 cents
generator is 260.26 cents

augmented:
prime 2 becomes 1197.06 cents
prime 3 remains just at 1901.96 cents
prime 5 becomes 2793.14 cents
period is 399.02 cents
generator is 93.15 cents

porcupine:
prime 2 becomes 1196.91 cents -- this is the period
prime 3 becomes 1906.86 cents
prime 5 becomes 2779.13 cents
generator is 162.32 cents

blackwood:
prime 2 becomes 1194.33 cents
prime 3 becomes 1910.93 cents
prime 5 remains just at 2786.31 cents
period is 238.87 cents
generator is 80.09 cents

diminished:
prime 2 becomes 1196.64 cents
prime 3 becomes 1896.63 cents
prime 5 becomes 2794.11 cents
period is 299.16 cents
generator is 101.67 cents

diaschismic:
prime 2 becomes 1199.11 cents
prime 3 becomes 1903.36 cents
prime 5 becomes 2788.38 cents
period is 599.56 cents
generator is 104.70 cents

magic:
prime 2 becomes 1201.28 cents -- this is the period
prime 3 becomes 1903.98 cents
prime 5 becomes 2783.35 cents
generator is 380.80 cents

kleismic:
prime 2 becomes 1200.29 cents -- this is the period
prime 3 becomes 1902.42 cents
prime 5 becomes 2785.64 cents
generator is 317.07 cents

schismic:
prime 2 becomes 1200.07 cents -- this is the period
prime 3 becomes 1901.85 cents
prime 5 becomes 2786.16 cents
generator is 498.28 cents

I think those are correct . . .

Happy tuning!

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_50628.html#51096
>
> > > ***This is very interesting, but I can't find the original
> > > temperament graph at all now. This is the one associated with
> ETs,
> > > yes??
> >
> > No, it's a pair of synonymous graphs of commas:
> >
> > /tuning-math/files/Paul/com5rat.gif
> > /tuning-math/files/Paul/com5monz.gif
> >
> > If you start with the 5-limit JI lattice, and temper out one of
> these
> > commas, you'll have a linear temperament. Tempering out 81/80
gives
> > you meantone. Tempering out 32805/32768 gives you schismic. Etc.
> >
>
> ***Oh... I meant to comment on these when I first saw them...
>These
> are particularly beautiful graphs...

Thanks. They might be useful centerpieces for, say, Monz's "linear
temperament" dictionary entry, which is pretty impoverished at the
moment . . . So now that you've found them, do you care to return to
where we were in the discussion? (I'm always eager to "teach" . . .)

-P

🔗Stephen Szpak <stephen_szpak@hotmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> ***Whew! Thanks, Monz. I realize I'm coming to *depend* on
these
> >> pages... :)
> >
> >I keep thinking we need a nonprofit corporation with longterm
goals
> >of preserving important web resources.
>
>Who has the resources to start such a thing?

Anyone here know a lawyer? You need a few people willing to be a
corporate officers as well. You don't need to meet in person; the
Internet would be fine. California or New York would seem the best
states to start one in, or wherever there is a lawyer willing to do a
little free work. If there was a corporation, it could be a tax
writeoff situation or something you could stick in your will with
enough money to keep your web page up for the next 100 years,
assuming html lasts that long.
--- End forwarded message ---

Since we're planning until the year 2104 now I have a suggestion. What about a tutorial?
For all the people that want to get into microtonality but don't know there are 1200 cents
to an octave, I personally feel the Tonalsoft encyclopedia is too complicated. Also, new
people coming to Yahoo tuning over the years will keep asking the same basic questions,
which of course have to be answered by Paul Erlich and others.

Stephen Szpak
stephen_szpak@hotmail.com

_________________________________________________________________
Have fun customizing MSN Messenger � learn how here! http://www.msnmessenger-download.com/tracking/reach_customize

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> What
> about a tutorial?
> For all the people that want to get into microtonality but don't
know
> there are 1200 cents
> to an octave, I personally feel the Tonalsoft encyclopedia is too
> complicated. Also, new
> people coming to Yahoo tuning over the years will keep asking the
same
> basic questions,
> which of course have to be answered by Paul Erlich and others.

The idea of a Tuning List FAQ (which actually exists but in a very
limited form) or tutorial has been thrown around here, but
encountered some strenuous objections from prominent list members,
followed up by personal phone calls . . . In the meantime, there are
some good "intro" articles like

http://www.cix.co.uk/~gbreed/start.htm

or maybe a "historical" intro article is best like

http://home.swipnet.se/~w-37192/eng/handbook/Tuning/history.html

http://home.earthlink.net/~kgann/histune.html

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Stephen,

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> Since we're planning until the year 2104 now I
> have a suggestion. What about a tutorial?
> For all the people that want to get into microtonality
> but don't know there are 1200 cents to an octave,
> I personally feel the Tonalsoft encyclopedia is too
> complicated. Also, new people coming to Yahoo tuning
> over the years will keep asking the same basic questions,
> which of course have to be answered by Paul Erlich and
> others.

eventually, a tutorial on the basics of tuning will be
part of the Tonalsoft Encyclopaedia.

... right now i'm busy just getting release 1.0 of the
software done, and transforming the webpages i've already
written in the Tonalsoft format.

within the next year i'd say that i'll have a tutorial.

-monz

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
>
> > Now, the other 10 "Tops":
> >
> > dicot:
> > prime 2 becomes 1207.66 cents -- this is the period
> > prime 3 becomes 1914.09 cents
> > prime 5 becomes 2768.53 cents
> > generator is 353.22 cents
> >
> > beep:
> > prime 2 remains just at 1200 cents -- this is the period
> > prime 3 becomes 1879.49 cents
> > prime 5 becomes 2819.23 cents
> > generator is 260.26 cents
> >
> > <etc. -- snip>
>
>
>
> it looks to me like i should have these on a page
> somewhere in the Encyclopaedia of Tuning. please advise.
>
> (i've been extremely busy with other stuff lately and
> have no idea what "Top" refers to.)
>
>
> -monz

These are *almost* the same as the linear temperaments with the same
names currently listed in the big table on your ET page -- the most
obvious difference, though, is that most of them have tempered
octaves.

"Top" means "Tempered Octaves, Please" or "Tenney-OPtimal". See
tuning-math for more details.

Perhaps your linear temperament page should be spruced up. Add this
pair of complementary graphs:

/tuning-math/files/Paul/com5rat.gif
/tuning-math/files/Paul/com5monz.gif

(which should also link to, and be linked to from, your small 5-limit
intervals page)

and then I'll make a table for you, kinda like the one on your ET
page, but with the slightly different octaves and other intervals
that "Top" gives these temperaments.

How does that sound?

🔗Herman Miller <hmiller@IO.COM>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_50628.html#51193

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > On Tue, 06 Jan 2004 03:39:28 -0000, "wallyesterpaulrus"
> > <paul@s...> wrote:
> >
> > >> http://www.io.com/~hmiller/midi/canon-top-father.mid
> > >
> > >Hmm . . . I can't say that this is a good choice of timbres -- I
> > >think the ones you used for 8-equal or 13-equal would be
> better . . .
> >
> > That was just a quick retuning with the original timbres; I
played
> around
> > with the timbres and uploaded a new version (plus a "top pelogic"
> and "top
> > meantone" version) to the Warped Canon page. See the section at
the
> very
> > end of the page.
> >
> > http://www.io.com/~hmiller/music/warped-canon.html
>
> Herman, that's PHAT! I might go insane if I listen to too many
> versions of this piece in a row, but it's fantastic that there's a
> place on the web where someone can hear what all these tunings
sound
> like -- if only for a piece that wasn't especially intended for
them.
>

***Yes, I agree this is a very valuable, rare and interesting
experience... and there's not much available in this vein, on the Web
or otherwise...

J. Pehrson

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_50628.html#51194

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > /tuning/topicId_50628.html#51096
> >
> > > > ***This is very interesting, but I can't find the original
> > > > temperament graph at all now. This is the one associated
with
> > ETs,
> > > > yes??
> > >
> > > No, it's a pair of synonymous graphs of commas:
> > >
> > > /tuning-math/files/Paul/com5rat.gif
> > > /tuning-
math/files/Paul/com5monz.gif
> > >
> > > If you start with the 5-limit JI lattice, and temper out one of
> > these
> > > commas, you'll have a linear temperament. Tempering out 81/80
> gives
> > > you meantone. Tempering out 32805/32768 gives you schismic. Etc.
> > >
> >
> > ***Oh... I meant to comment on these when I first saw them...
> >These
> > are particularly beautiful graphs...
>
> Thanks. They might be useful centerpieces for, say, Monz's "linear
> temperament" dictionary entry, which is pretty impoverished at the
> moment . . . So now that you've found them, do you care to return
to
> where we were in the discussion? (I'm always eager to "teach" . . .)
>
> -P

***Yes, that's why I'm glad you're back. I need more "personal
training..."...

So, what was I supposed to be thinking about again?? :)

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_50628.html#51194
>
> > --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> wrote:
> > > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> > wrote:
> > >
> > > /tuning/topicId_50628.html#51096
> > >
> > > > > ***This is very interesting, but I can't find the original
> > > > > temperament graph at all now. This is the one associated
> with
> > > ETs,
> > > > > yes??
> > > >
> > > > No, it's a pair of synonymous graphs of commas:
> > > >
> > > > /tuning-
math/files/Paul/com5rat.gif
> > > > /tuning-
> math/files/Paul/com5monz.gif
> > > >
> > > > If you start with the 5-limit JI lattice, and temper out one
of
> > > these
> > > > commas, you'll have a linear temperament. Tempering out 81/80
> > gives
> > > > you meantone. Tempering out 32805/32768 gives you schismic.
Etc.
> > > >
> > >
> > > ***Oh... I meant to comment on these when I first saw them...
> > >These
> > > are particularly beautiful graphs...
> >
> > Thanks. They might be useful centerpieces for, say,
Monz's "linear
> > temperament" dictionary entry, which is pretty impoverished at
the
> > moment . . . So now that you've found them, do you care to return
> to
> > where we were in the discussion? (I'm always eager
to "teach" . . .)
> >
> > -P
>
>
> ***Yes, that's why I'm glad you're back. I need more "personal
> training..."...
>
> So, what was I supposed to be thinking about again?? :)

I don't know . . . you could "snap back" to
/tuning/topicId_50628.html#51094
and see if you have any questions . . .

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_50628.html#51236

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > /tuning/topicId_50628.html#51194
> >
> > > --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> > wrote:
> > > > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
<paul@s...>
> > > wrote:
> > > >
> > > > /tuning/topicId_50628.html#51096
> > > >
> > > > > > ***This is very interesting, but I can't find the
original
> > > > > > temperament graph at all now. This is the one associated
> > with
> > > > ETs,
> > > > > > yes??
> > > > >
> > > > > No, it's a pair of synonymous graphs of commas:
> > > > >
> > > > > /tuning-
> math/files/Paul/com5rat.gif
> > > > > /tuning-
> > math/files/Paul/com5monz.gif
> > > > >
> > > > > If you start with the 5-limit JI lattice, and temper out
one
> of
> > > > these
> > > > > commas, you'll have a linear temperament. Tempering out
81/80
> > > gives
> > > > > you meantone. Tempering out 32805/32768 gives you schismic.
> Etc.
> > > > >
> > > >
> > > > ***Oh... I meant to comment on these when I first saw
them...
> > > >These
> > > > are particularly beautiful graphs...
> > >
> > > Thanks. They might be useful centerpieces for, say,
> Monz's "linear
> > > temperament" dictionary entry, which is pretty impoverished at
> the
> > > moment . . . So now that you've found them, do you care to
return
> > to
> > > where we were in the discussion? (I'm always eager
> to "teach" . . .)
> > >
> > > -P
> >
> >
> > ***Yes, that's why I'm glad you're back. I need more "personal
> > training..."...
> >
> > So, what was I supposed to be thinking about again?? :)
>
> I don't know . . . you could "snap back" to
> /tuning/topicId_50628.html#51094
> and see if you have any questions . . .

***Well, for starters, I'm not understanding these beautiful graphs.
I take it the black area is the "big infinity" of ratios, but what
exactly is being shown in these? It's fascinating, in any case...

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> ***Well, for starters, I'm not understanding these beautiful
graphs.
> I take it the black area is the "big infinity" of ratios, but what
> exactly is being shown in these? It's fascinating, in any case...

The position of a comma along horizontal direction shows the
complexity of the temperament. Firstly, it shows how far in the
lattice you have to go to traverse the comma. If you temper this
comma out, it then shows how far you have to go to get from a given
pitch to its duplicate, say through a "comma pump" progression. It
shows the complexity of the temperament: if you take a big giant
chunk of the just lattice, and apply this tempering to it, you'll
only have a small percentage of the number of pitches you started
with (since so many of the pitches have become the same as one
another). The horizontal distance (from the left edge) is
proportional to this percentage.

The position of a comma along the vertical direction is a (weighted)
measure of the error from JI that the intervals have when you temper
out the commas in a particular way (uniformly per unit length along
the rungs in the lattice) that minimizes this measure for the given
temperament. Note that the vertical axis is *logarithmic* on the
original pair of graphs, so that each big step along that axis
corresponds to *10 times* the error as the next lower big step. No
commas smaller, or resulting in less error, than the schisma are to
be found in this complexity range (0-60).

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_50628.html#51243

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > ***Well, for starters, I'm not understanding these beautiful
> graphs.
> > I take it the black area is the "big infinity" of ratios, but
what
> > exactly is being shown in these? It's fascinating, in any case...
>
> The position of a comma along horizontal direction shows the
> complexity of the temperament. Firstly, it shows how far in the
> lattice you have to go to traverse the comma. If you temper this
> comma out, it then shows how far you have to go to get from a given
> pitch to its duplicate, say through a "comma pump" progression. It
> shows the complexity of the temperament: if you take a big giant
> chunk of the just lattice, and apply this tempering to it, you'll
> only have a small percentage of the number of pitches you started
> with (since so many of the pitches have become the same as one
> another). The horizontal distance (from the left edge) is
> proportional to this percentage.
>
> The position of a comma along the vertical direction is a
(weighted)
> measure of the error from JI that the intervals have when you
temper
> out the commas in a particular way (uniformly per unit length along
> the rungs in the lattice) that minimizes this measure for the given
> temperament. Note that the vertical axis is *logarithmic* on the
> original pair of graphs, so that each big step along that axis
> corresponds to *10 times* the error as the next lower big step. No
> commas smaller, or resulting in less error, than the schisma are to
> be found in this complexity range (0-60).

***Well, I'm definitely seeing commas here... :) and they look about
the right cents values 81/80 is around 20 cents... 25/24 is close to
100 cents, and 5/4 is close to 700 cents, etc., etc.

So the ones toward the top of the chart must be *very large* commas,
yes? Is that why there are so many of them?? Because they don't
*tighten* things up the way the smaller commas do?? Am I totally off
the track here?? :)

Thanks!

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>

> ***Well, I'm definitely seeing commas here... :) and they look
about
> the right cents values 81/80 is around 20 cents... 25/24 is close
to
> 100 cents, and 5/4 is close to 700 cents, etc., etc.
>
> So the ones toward the top of the chart must be *very large*
commas,
> yes?

Yeah, and it's kind of silly to even call them commas.

> Is that why there are so many of them?? Because they don't
> *tighten* things up the way the smaller commas do??
> Am I totally off
> the track here?? :)

Since I'm using a *logarithmic* error scale (for the vertical axis),
the ratios thin out as you go lower down -- in fact the zero-error
line is infinitely far away (though you won't find any more commas if
you just move towards it, in this complexity range (0-60)). On this
(slightly zoomed-in) variant,

/tuning/files/Erlich/herman1.gif

the (vertical) error scale is linear. Twice as high means twice as
much error, etc. And here the ratios *don't* thin out toward the
bottom. All the small ones that took up the lower half of the log-
scaled graph occupy a small band near the zero-error line here.

🔗monz <monz@attglobal.net>

hi paul,

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

> These are *almost* the same as the linear temperaments
> with the same names currently listed in the big table
> on your ET page -- the most obvious difference, though,
> is that most of them have tempered octaves.
>
> "Top" means "Tempered Octaves, Please" or "Tenney-OPtimal".
> See tuning-math for more details.

aha! very interesting!

just a few months ago, Chris and i were using the Tonalsoft
software to come up with tunings with tempered 8ves ...
mostly just playing around with them, to see what we could
come up with. i am very interested in these!!!

we were discussing ways to implement Partch's
"Three Observations" in our software, and were suddenly
inspired to make use of his idea that the higher odd
or prime a factor is, the more precisely it needs
to be tuned -- and of course the converse of this is
that the lower the prime/odd, the more intonational
leeway a listener is willing to accept.

so we came up with some tunings which tempered the 8ve
(2:1 ratio) most of all, the 3:1 less than that, the
5:1 less than that, and the 7:1 least.

i guess the best place for these "Tops" would be on
my "non-just non-equal" page:

http://tonalsoft.com/enc/njne.htm

they certainly fit the bill, having neither 8ves nor
just intervals -- and yet they approximate JI harmony
very nicely.

this is a point that McLaren emphasized to me a lot:
tuning with are neither just nor 8ve-equivalent, but
which sound lovely.

> Perhaps your linear temperament page should be spruced up.
> Add this pair of complementary graphs:
>
> /tuning-math/files/Paul/com5rat.gif
> /tuning-math/files/Paul/com5monz.gif
>
> (which should also link to, and be linked to from, your
> small 5-limit intervals page)

i'm really under the gun to get some other work done
right now. maybe in a couple of days i can spare the
few minutes it will take to do that.

if you want to alter my webpages's HTML source-code
yourself to include the graphics, please feel free.
then you can just email it to me and i'll upload it,
along with the pretty pictures. :)

> and then I'll make a table for you, kinda like the one
> on your ET page, but with the slightly different octaves
> and other intervals that "Top" gives these temperaments.
>
> How does that sound?

thanks so much for the offer. of course i'll take it.

... but no guarantees on when any Encyclopaedia updates
get made.

-monz

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > These are *almost* the same as the linear temperaments
> > with the same names currently listed in the big table
> > on your ET page -- the most obvious difference, though,
> > is that most of them have tempered octaves.
> >
> > "Top" means "Tempered Octaves, Please" or "Tenney-OPtimal".
> > See tuning-math for more details.
>
>
>
> aha! very interesting!
>
> just a few months ago, Chris and i were using the Tonalsoft
> software to come up with tunings with tempered 8ves ...
> mostly just playing around with them, to see what we could
> come up with. i am very interested in these!!!

If you use a lattice that includes prime 2, and rungs for each prime
p with length log(p) (the log can be to any base you like) -- in
other words the Tenney lattice -- then the "Top" tempering is the
most natural one to apply, it tempers out the rungs comprising the
comma a uniform amount per unit length.

> we were discussing ways to implement Partch's
> "Three Observations" in our software, and were suddenly
> inspired to make use of his idea that the higher odd
> or prime a factor is, the more precisely it needs
> to be tuned -- and of course the converse of this is
> that the lower the prime/odd, the more intonational
> leeway a listener is willing to accept.
>
> so we came up with some tunings which tempered the 8ve
> (2:1 ratio) most of all, the 3:1 less than that, the
> 5:1 less than that, and the 7:1 least.

I'm doing the opposite -- take a look at the harmonic entropy graphs
to see why. The simplest ratios have the steepest increase in
discordance as a function of mistuning.

> i guess the best place for these "Tops" would be on
> my "non-just non-equal" page:
>
> http://tonalsoft.com/enc/njne.htm

Not necessarily, because most "non-just non-equal" tunings are not
temperaments, but here we're talking specifically about temperaments,
and if you start with the 5-limit lattice these are "linear
temperaments", which is where I suggested they go.

> they certainly fit the bill, having neither 8ves nor
> just intervals -- and yet they approximate JI harmony
> very nicely.

Well sure, you can put them there too. The Top meantone, pelogic, and
father temperaments were described in previous posts. Of course, they
have "8ves", just not just ones.

> this is a point that McLaren emphasized to me a lot:
> tuning with are neither just nor 8ve-equivalent, but
> which sound lovely.

How do you define an '8ve-equivalent' tuning?

I did assume octave-repetition to derive the period and generator of
each.

🔗monz <monz@attglobal.net>

hi paul,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > we were discussing ways to implement Partch's
> > "Three Observations" in our software, and were suddenly
> > inspired to make use of his idea that the higher odd
> > or prime a factor is, the more precisely it needs
> > to be tuned -- and of course the converse of this is
> > that the lower the prime/odd, the more intonational
> > leeway a listener is willing to accept.
> >
> > so we came up with some tunings which tempered the 8ve
> > (2:1 ratio) most of all, the 3:1 less than that, the
> > 5:1 less than that, and the 7:1 least.
>
> I'm doing the opposite -- take a look at the harmonic entropy
> graphs to see why. The simplest ratios have the steepest
> increase in discordance as a function of mistuning.

hmm ... i remember when i visited you, you spoke about
harmonic entropy as a "validation of Partch's Observations".

now it appears that harmonic entropy gives precisely
the opposing view. am i off track here?

> > i guess the best place for these "Tops" would be on
> > my "non-just non-equal" page:
> >
> > http://tonalsoft.com/enc/njne.htm
>
> Not necessarily, because most "non-just non-equal" tunings
> are not temperaments,

right. gotcha.

> but here we're talking specifically about temperaments,
> and if you start with the 5-limit lattice these are "linear
> temperaments", which is where I suggested they go.
>
> > they certainly fit the bill, having neither 8ves nor
> > just intervals -- and yet they approximate JI harmony
> > very nicely.
>
> Well sure, you can put them there too. The Top meantone,
> pelogic, and father temperaments were described in previous
> posts. Of course, they have "8ves", just not just ones.

well, i haven't spoken about it with McLaren for a long
time, but i believe that his point was that the 2:1 ratio
was not *exactly* present, as it is in most "traditional"
Euro-centric tunings.

he made a big deal about the fact that some research
showed that the interval most often perceived as an 8ve
as actually around 1215 cents. 2^(1215/1200) = 2.017403968.

> > this is a point that McLaren emphasized to me a lot:
> > tuning with are neither just nor 8ve-equivalent, but
> > which sound lovely.
>
> How do you define an '8ve-equivalent' tuning?
>
> I did assume octave-repetition to derive the period
> and generator of each.

yes, i see. i was too hasty in writing that. scratch it.

-monz

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > we were discussing ways to implement Partch's
> > > "Three Observations" in our software, and were suddenly
> > > inspired to make use of his idea that the higher odd
> > > or prime a factor is, the more precisely it needs
> > > to be tuned -- and of course the converse of this is
> > > that the lower the prime/odd, the more intonational
> > > leeway a listener is willing to accept.
> > >
> > > so we came up with some tunings which tempered the 8ve
> > > (2:1 ratio) most of all, the 3:1 less than that, the
> > > 5:1 less than that, and the 7:1 least.
> >
> > I'm doing the opposite -- take a look at the harmonic entropy
> > graphs to see why. The simplest ratios have the steepest
> > increase in discordance as a function of mistuning.
>
>
>
> hmm ... i remember when i visited you, you spoke about
> harmonic entropy as a "validation of Partch's Observations".

Van Eck's model, which underlies harmonic entropy, does validate
Observation One in terms of the "clustering of satellites".

> now it appears that harmonic entropy gives precisely
> the opposing view. am i off track here?

I think we need to be more careful about what Partch actually said.
What is the wording again?

Note that Partch reacts thinks it rather absurd to consider 3:2
maintaining its identity under tempering of as much as 7 cents (when
he speaks about 19-equal), yet admits the (albeit poor) ability of 12-
equal to imply odd limits of 5, 7, and even 9.

> > > i guess the best place for these "Tops" would be on
> > > my "non-just non-equal" page:
> > >
> > > http://tonalsoft.com/enc/njne.htm
> >
> > Not necessarily, because most "non-just non-equal" tunings
> > are not temperaments,
>
>
> right. gotcha.
>
>
> > but here we're talking specifically about temperaments,
> > and if you start with the 5-limit lattice these are "linear
> > temperaments", which is where I suggested they go.
> >
> > > they certainly fit the bill, having neither 8ves nor
> > > just intervals -- and yet they approximate JI harmony
> > > very nicely.
> >
> > Well sure, you can put them there too. The Top meantone,
> > pelogic, and father temperaments were described in previous
> > posts. Of course, they have "8ves", just not just ones.
>
>
>
> well, i haven't spoken about it with McLaren for a long
> time, but i believe that his point was that the 2:1 ratio
> was not *exactly* present, as it is in most "traditional"
> Euro-centric tunings.

Well, not on the tuning of the piano, which is normally
stretched . . .

> he made a big deal about the fact that some research
> showed that the interval most often perceived as an 8ve
> as actually around 1215 cents. 2^(1215/1200) = 2.017403968.

For melodic sine waves, this may be true, though it's highly
dependent on register. For precisely harmonic timbres, which include
the human voice, bowed strings, winds and brass, and all synthesized
timbres based on a repeating waveform, it's fairly easy to tune 1200
cent octaves by ear, particularly harmonically (rather than
melodically). But these "Top" temperaments allow this to be traded
off against the other simple-integer ratios implied by the precise
harmonics, though the octave gets tempered less (reflecting, for one
thing, the fact that it involves the strongest overtone pair 1:2 as
well as many others like 2:4, 3:6, etc.) . . .

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗czhang23@aol.com

In a message dated 2004:01:09 01:41:01 AM, joe p writes:

>So Dave Keenan feels that "temperament" implies relationship to JI,
>and this has been stated several times before on this list in
>reference to the terms ET and EDO for scales.

Some say "temperament" is harmonically stuck in certain types of
Euro-ruts...
(hmmm... does that mean we might be playin' with Euro-mutts?)

::quickly shakes head to shake Dr. Seuss off from possessin' his body::

>However, Carl Lumma feels the opposite: that temperaments are
>animals (or machines?) all in themselves which have taken of on their
>own to make their own sonic worlds.

Yepyep, AFAIK, mecha-organisms for making music (sorta like Le
Corbusier's architecture: machines for living in). I like this, personally.

---|-----|--------|-------------|---------------------|
Hanuman Zhang, musical mad scientist
"Space is a practiced place." -- Michel de Certeau
"Space is the Place for the Human Race." -- William S. Burroughs

"... simple, chaotic, anarchic and menacing.... This is what people of today
have lost and need most - the ability to experience permanent bodily and
mental ecstasy, to be a receiving station for messages howling by on the ether from
other worlds and nonhuman entities, those peculiar short-wave messages which
come in static-free in the secret pleasure center in the brain." - Slava Ranko
(Donald L. Philippi)

The German word for "noise" _Geräusch_ is derived from _rauschen_ "the
sound of the wind," related to _Rausch_ "ecstasy, intoxication" hinting at some
of the possible aesthetic, bodily effects of noise in music. In Japanese
Romaji: _uchu_ = "universe"... _uchoten_ = "ecstasty," "rapture"..._uchujin_ =
[space] alien!

"When you're trying to do something you should feel absolutely alone, like a
spark in the blackness of the universe."-Xenakis

"For twenty-five centuries, Western knowledge has tried to look upon the
world. It has failed to understand that the world is not for the beholding. It
is for the hearing. It is not legible, but audible. ... Music is a herald,
for change is inscribed in noise faster than it transforms society. ...
Listening to music is listening to all noise, realizing that its appropriation and
control is a reflection of power, that is essentially political." - Jacques
Attali, _Noise: The Political Economy of Music_

"The sky and its stars make music in you." - Dendera, Egypt wall
inscription

"Sound as an isolated object of reproduction, call it our collective memory
bank... Any sound can be you." - DJ Spooky that Subliminal Kid (a.k.a. Paul D.
Miller)

"Overhead, without any fuss, the stars were going out."
--Arthur C. Clarke, _The Nine Billion Names of God_

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
>
> /tuning/topicId_50628.html#51262
>
> So Dave Keenan feels that "temperament" implies relationship to JI,
> and this has been stated several times before on this list in
> reference to the terms ET and EDO for scales.
>
> However, Carl Lumma feels the opposite: that temperaments are
> animals (or machines?) all in themselves which have taken of on their
> own to make their own sonic worlds.

Poetic though this is, it doesn't seem to say anything about how
Carl's "temperament" category differs from mine.

I think history is with me, and I think Carl should be happy with the
term "regular tuning" with sub-categories "linear tuning", "planar
tuning" etc., if there is no intention to approximate just intervals.
Or if I've misunderstood, then I still think we can find other terms
for what Carl means, rather than redefine "temperament".

In any case, what we were talking about were "5-limit" whatevers and
if you say n-"limit" you are automatically talking about something
that either _is_ JI or attempts to approximate it.

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_50628.html#51262
>
> So Dave Keenan feels that "temperament" implies relationship to JI,
> and this has been stated several times before on this list in
> reference to the terms ET and EDO for scales.

Actually, at one point Dave Keenan was using a more inclusive
definition of "temperament" than anyone else on the tuning-math list,
because the rest of us were all talking about tuning systems derived
directly from JI, while Dave was considering some systems
like "double meantone" where only *half* the notes derive from JI, or
at best, you have two "incompatible" JI systems stuck together.

> However, Carl Lumma feels the opposite: that temperaments are
> animals (or machines?) all in themselves which have taken of on
their
> own to make their own sonic worlds.

I'll have to leave it to Carl to clarify his position.

> So, which is it? Or is it an undefined area? I guess,
historically,
> the reference was with regard to JI...

Yes, and as Dave suggested, there's no reason not to use the
term "tuning" for anything else, be it a "linear tuning", a "random
tuning", a "found tuning" or whatnot.

🔗Carl Lumma <ekin@lumma.org>

>> /tuning/topicId_50628.html#51262
>>
>> So Dave Keenan feels that "temperament" implies relationship to JI,
>> and this has been stated several times before on this list in
>> reference to the terms ET and EDO for scales.
>>
>> However, Carl Lumma feels the opposite: that temperaments are
>> animals (or machines?) all in themselves which have taken of on their
>> own to make their own sonic worlds.
>
>Poetic though this is, it doesn't seem to say anything about how
>Carl's "temperament" category differs from mine.
>
>I think history is with me, and I think Carl should be happy with the
>term "regular tuning" with sub-categories "linear tuning", "planar
>tuning" etc., if there is no intention to approximate just intervals.
>Or if I've misunderstood, then I still think we can find other terms
>for what Carl means, rather than redefine "temperament".

Temperament does mean approximating JI, but it doesn't need to do
so within the absolute max error that happens to suit you. The ear
is vigilant -- I've heard enough 11-tET music to convince me of
that.

And what if, as I suggest in my last post, we temper only melodic
intervals but not harmonic ones? Any comma can be seen as vanishing
in pitch space, interval space, or both.

-Carl

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> /tuning/topicId_50628.html#51262
> Temperament does mean approximating JI,

I'm glad we've agreed on that.

> but it doesn't need to do
> so within the absolute max error that happens to suit you.

I just have this idea that most folk on this list find the idea of a
"JI-approximation with 50 cent errors" to be an oxymoron, at least for
dyads. This would mean 12-tET would provide an approxmation of any JI
interval you want. Do you hear ratios of 11 and 13 in 12-tET?

> The ear
> is vigilant -- I've heard enough 11-tET music to convince me of
> that.

There seems to be a myth (which I'm not accusing you of succumbing to,
but thought I'd mention) that there are no good JI approximations in
11-tET, which of course isn't true since it has all the even-stepped
approximations of 22-ET including its minor thirds (approximate 5:6s).
But of course it has no good approximation of a complete 5-limit
otonality. Do you claim to hear these in it?

I've asked Paul to find a linear tuning that Pachelbel's Canon can be
warped into that he does _not_ consider to be an approximation of
5-limit JI. So far he doesn't think it can be done.

Obviously if absolutely anything can be considered an "approximation
of JI" then it becomes a fairly useless category, not deserving of a
special name like "temperament".

> And what if, as I suggest in my last post, we temper only melodic
> intervals but not harmonic ones? Any comma can be seen as vanishing
> in pitch space, interval space, or both.

I'm not sure what you mean here, or how it is relevant to the
consideration of how much error a linear tuning can have and still be
considered an approximation of 5-limit JI.

🔗Carl Lumma <ekin@lumma.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Carl Lumma <ekin@lumma.org>

>> >Do you hear ratios of 11 and 13 in 12-tET?
>>
>> Depending on the context, yes. I'm sure that there are at least
>> a few here who would agree.
>
>In dyads? In triads? ...

Usually it requires at least tetrads (even in JI bare extended-limit
dyads sometimes don't work), statically. But once a tonality/context
has been established even bare dyads will work.

>> >> And what if, as I suggest in my last post, we temper only melodic
>> >> intervals but not harmonic ones? Any comma can be seen as vanishing
>> >> in pitch space, interval space, or both.
>> >
>> >I'm not sure what you mean here, or how it is relevant to the
>> >consideration of how much error a linear tuning can have and still
>> >be considered an approximation of 5-limit JI.
>>
>> JI includes both pitch space and interval space. There's nothing
>> about "dicot" that says where the error has to go. If I put it
>> all in pitch space, am I tempering?
>
>I still don't understand. I thought we were talking about fixed
>tunings, not adaptive ones.

A temperament is an abstract object, which can be used to define
both fixed or "adaptive" tunings.

-Carl

🔗monz <monz@attglobal.net>

hi Carl and Dave,

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > Do you hear ratios of 11 and 13 in 12-tET?
>
> Depending on the context, yes. I'm sure that there are
> at least a few here who would agree.

that was certainly Schoenberg's opinion.
i've gotten into arguments with others here about that.

in particular, about two years ago when i was trying
to come up with an 11-limit TM-reduced lattice that
explained Schoenberg's concepts, paul erlich said "well,
if that's the case, then 12-ET can represent *anything*",
and in fact, that *was* Schoenberg's view.

just for the record, i certainly agree with Dave that
"temperament" should only be used to describe a tuning
which is intended to imply JI.

if the tuning is used without reference to JI, as in
most 12-tone serialism, then "12edo" or "12ed2" or
"12-tone equal tuning" or some such is a better name.

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

/tuning/topicId_50628.html#51339

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> >
> > /tuning/topicId_50628.html#51262
> >
> > So Dave Keenan feels that "temperament" implies relationship to
JI,
> > and this has been stated several times before on this list in
> > reference to the terms ET and EDO for scales.
> >
> > However, Carl Lumma feels the opposite: that temperaments are
> > animals (or machines?) all in themselves which have taken of on
their
> > own to make their own sonic worlds.
>
> Poetic though this is, it doesn't seem to say anything about how
> Carl's "temperament" category differs from mine.
>

***"Art," obviously... that is, beginning with an "f"... :)

> I think history is with me, and I think Carl should be happy with
the
> term "regular tuning" with sub-categories "linear tuning", "planar
> tuning" etc., if there is no intention to approximate just
intervals.
> Or if I've misunderstood, then I still think we can find other terms
> for what Carl means, rather than redefine "temperament".
>
> In any case, what we were talking about were "5-limit" whatevers and
> if you say n-"limit" you are automatically talking about something
> that either _is_ JI or attempts to approximate it.

***Sure *sounds* logical to me... !

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_50628.html#51353

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > /tuning/topicId_50628.html#51262
> >
> > So Dave Keenan feels that "temperament" implies relationship to
JI,
> > and this has been stated several times before on this list in
> > reference to the terms ET and EDO for scales.
>
> Actually, at one point Dave Keenan was using a more inclusive
> definition of "temperament" than anyone else on the tuning-math
list,
> because the rest of us were all talking about tuning systems
derived
> directly from JI, while Dave was considering some systems
> like "double meantone" where only *half* the notes derive from JI,
or
> at best, you have two "incompatible" JI systems stuck together.
>
> > However, Carl Lumma feels the opposite: that temperaments are
> > animals (or machines?) all in themselves which have taken of on
> their
> > own to make their own sonic worlds.
>
> I'll have to leave it to Carl to clarify his position.
>
> > So, which is it? Or is it an undefined area? I guess,
> historically,
> > the reference was with regard to JI...
>
> Yes, and as Dave suggested, there's no reason not to use the
> term "tuning" for anything else, be it a "linear tuning", a "random
> tuning", a "found tuning" or whatnot.

***That seems to make a lot of sense to somebody who has absolutely
no authority on this subject... (moi)... :)

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_50628.html#51358

> >> /tuning/topicId_50628.html#51262
> >>
> >> So Dave Keenan feels that "temperament" implies relationship to
JI,
> >> and this has been stated several times before on this list in
> >> reference to the terms ET and EDO for scales.
> >>
> >> However, Carl Lumma feels the opposite: that temperaments are
> >> animals (or machines?) all in themselves which have taken of on
their
> >> own to make their own sonic worlds.
> >
> >Poetic though this is, it doesn't seem to say anything about how
> >Carl's "temperament" category differs from mine.
> >
> >I think history is with me, and I think Carl should be happy with
the
> >term "regular tuning" with sub-categories "linear tuning", "planar
> >tuning" etc., if there is no intention to approximate just
intervals.
> >Or if I've misunderstood, then I still think we can find other
terms
> >for what Carl means, rather than redefine "temperament".
>
> Temperament does mean approximating JI, but it doesn't need to do
> so within the absolute max error that happens to suit you. The ear
> is vigilant -- I've heard enough 11-tET music to convince me of
> that.
>
> And what if, as I suggest in my last post, we temper only melodic
> intervals but not harmonic ones? Any comma can be seen as vanishing
> in pitch space, interval space, or both.
>
> -Carl

***But, isn't the point that the term creates *confusion,* even
though it's used in an "innocent" way??

🔗Carl Lumma <ekin@lumma.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

on 1/9/04 7:40 PM, monz <monz@attglobal.net> wrote:

> just for the record, i certainly agree with Dave that
> "temperament" should only be used to describe a tuning
> which is intended to imply JI.

Maybe this is a little obscure...

but the implication of JI might sometimes be contextual (e.g. via harmonic
function), and such context is not a property of the tuning itself, although
I suppose you could say the tuning allows for the potential of such context.
Still I'd rather say "approximate JI" rather than "imply JI". In fact I can
imagine wanting a different name for something which fails to approximate JI
very well but can still imply the familiar functions. However, I think the
historical use of "temperament" is much less fussy than this. So I am
guessing you would mean to say that "temperament" would be used to describe
a tuning that is able to be used to imply JI with the possible aid of
contextual cues. Maybe this goes without saying, but I detected a possible
point of confusion in the sense that a tuning implies JI. The question is
whether the tuning as a static object is meant to do this, or only as it is
used.

Also consider that one JI tuning may be an approximation of another, either
by accident or by design. For example Carl just demonstrated to me that the
following chord which occurs rooted on C# in various basic 5-limit 12-tone
JI scales:

16/15 : 4/3 : 8/5 : 15/8
or
128 : 160 : 192 : 225

is a darn good approximation to

128 : 160 : 192 : 224
or
4:5:6:7

But would that make the original scale a temperament, if we imagine it was
actually designed to include the 6:7 approximation?

So if I were defining temperament it would seem to require an intention to
temper, or to *deviate* from JI. I don't see any reason the result could
not be *another* JI, as long as the process was one of tempering. However,
correct me if I'm wrong, but I have the impression that when it comes right
down do it, what is tempered is intervals, not tunings, and a given tuning
could contain some tempered and some untempered intervals, or a mixure of
tempered intervals from two different other scales. Thus tempering would be
involved, and the result is therefore a temperament. I guess more to the
point is that a given temperament might not have a unique JI scale that it
is a temperament *of*.

And so in the end I'm a little confused. Perhaps "tempering" involves
intervals, but a "temperament" is a tuning/scale. Yet a temperament is not
necessarily a tempering of a uniquely-determinable JI scale--it may contain
intervals which are approximations of intervals which could not have
co-occured in a single JI scale having the same number of scale degrees as
the temperament.

-Kurt

>
> if the tuning is used without reference to JI, as in
> most 12-tone serialism, then "12edo" or "12ed2" or
> "12-tone equal tuning" or some such is a better name.
>
>
>
> -monz

🔗monz <monz@attglobal.net>

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Also consider that one JI tuning may be an approximation
> of another, either by accident or by design. For example
> Carl just demonstrated to me that the following chord which
> occurs rooted on C# in various basic 5-limit 12-tone
> JI scales:
>
> 16/15 : 4/3 : 8/5 : 15/8
> or
> 128 : 160 : 192 : 225
>
> is a darn good approximation to
>
> 128 : 160 : 192 : 224
> or
> 4:5:6:7
>
> But would that make the original scale a temperament,
> if we imagine it was actually designed to include the
> 6:7 approximation?

i think a broad definition of "temperament" could include
the substitution of one JI tuning for another. others
may disagree and prefer a more narrow definition.

i'm well aware of this kind of thing: in fact, it's the
entire musical point of the 4th section ("Meditation")
of my 1999 piece _A Noiseless Patient Spider_.

http://tonalsoft.com/monzo/spider/spider.htm

i quote from the webpage (the accidentals use my HEWM notation):

>> The 'Meditation', where the poet's thoughts turn inward,
>> is tuned to the symmetrical 5-limit system centered on 'A'
>> portrayed on the prime-factor lattice diagram - one kind
>> of ultimate rational understanding of musical harmony -
>> with a very soft drone on a low 'A'. The entire section
>> uses only three main instrumental parts which each hold
>> a pitch for six very slow beats as they move around the
>> lattice.
>>
>> These three parts overlap and explore various 5-limit
>> triad subsets of the total lattice of 13 pitches, and
>> about two-thirds of the way thru this section they form a
>> 'first inversion A major' triad, but the '3rd of the chord'
>> (in the bass) is not the 'correct' 5:8 ratio with the
>> pitch C#-, but rather the 16:25 Db>, giving a proportion
>> for the whole triad of 32:50:75, instead of 5:8:12, so the
>> exploration continues.
>>
>> It finally comes to an end when the triad forms the
>> outline of an 'augmented 6th' chord on a Bb+ 'root',
>> with the proportion 128:160:225, which strongly suggests
>> a 'harmonic dominant 7th' chord with the proportion
>> 4:5:7 [= 128:160:224].
>>
>> This suggestion is due to the 224:225 'bridge' function
>> of the highest pitch in the chord, the G#-, which was
>> also the highest pitch in the first chord heard in this
>> section. After this triad fades away, unresolved, a hint
>> of the drone on 'A' can be heard as the section ends.

do you recognize that pseudo-harmonic-7th chord?
it's the same one Carl showed you.

in the final section of the poem, Whitman actually uses
the word "bridge", and i repeat this musical pun at that point.

and in the "Meditation", i also made a play on the
major triad, as described above.

-monz

🔗Graham Breed <graham@microtonal.co.uk>

Kurt Bigler wrote:

> So if I were defining temperament it would seem to require an intention to
> temper, or to *deviate* from JI. I don't see any reason the result could
> not be *another* JI, as long as the process was one of tempering. However,
> correct me if I'm wrong, but I have the impression that when it comes right
> down do it, what is tempered is intervals, not tunings, and a given tuning
> could contain some tempered and some untempered intervals, or a mixure of
> tempered intervals from two different other scales. Thus tempering would be
> involved, and the result is therefore a temperament. I guess more to the
> point is that a given temperament might not have a unique JI scale that it
> is a temperament *of*.

Tunings are made up of intervals, so a tuning that contains tempered intervals is a temperament. Temperaments can certainly contain just intervals -- quarter comma meantone does, and it'd be crazy revisionism to start saying that it isn't a temperament.

> And so in the end I'm a little confused. Perhaps "tempering" involves
> intervals, but a "temperament" is a tuning/scale. Yet a temperament is not
> necessarily a tempering of a uniquely-determinable JI scale--it may contain
> intervals which are approximations of intervals which could not have
> co-occured in a single JI scale having the same number of scale degrees as
> the temperament.

A temperament will generally not be a tempering of a unique JI scale. The obvious reason for using a temperament is that you can approximate more than one JI scale with it. It can also be the case that all intervals in a single chord approximate JI, but the chord as a whole can't be played in JI. Or that some intervals in a temperament can't be identified with a single, obvious just interval. Like the whole tone in meantone -- it could be 9:8 or 10:9.

Graham

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Carl Lumma <ekin@lumma.org>

🔗wallyesterpaulrus <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> I guess more to the
> point is that a given temperament might not have a unique JI scale
that it
> is a temperament *of*.

Well, in a sense that's true, for example since according to Gene,
the Top 5-limit meantone (which is a temperament of 5-limit JI that
eliminates the 81:80) and the Top 7-limit meantone (which is a
temperament of 7-limit JI that eliminates the 81:80 and the 225:224).

> And so in the end I'm a little confused. Perhaps "tempering"
involves
> intervals, but a "temperament" is a tuning/scale. Yet a
temperament is not
> necessarily a tempering of a uniquely-determinable JI scale--it may
contain
> intervals which are approximations of intervals which could not have
> co-occured in a single JI scale having the same number of scale
degrees as
> the temperament.

Kurt, that's *always* the case with temperament.

🔗wallyesterpaulrus <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

🔗wallyesterpaulrus <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

>> >> >> we temper only melodic
>> >> >> intervals but not harmonic ones?
>> >> >
>> >> >Adaptive JI for the relevant scale, right?
>> >>
>> >> Yep.
>> >>
>> >> >> Any comma can be seen as vanishing
>> >> >> in pitch space, interval space, or both.
>> >> >
>> >> >????????
>> >>
>> >> One can also imagine an adaptive tuning in which
>> >> vertical interval are tempered (say, to get magic
>> >> chords) while melodic intervals are kept pure.
>> >
>> >Well, taking your word for it that one can imagine this, what
>> >does this have to do with "pitch space" vs. "interval space"?
>>
>> The comma vanishes in interval space but not pitch space.
>>
>> -Carl
>
>No comprendo. Can you construct, for me, an "interval space" and
>a "pitch space" for, say, adaptive JI (since I understand that),
>and show where the comma vanishes and where it doesn't?

Sure. To the unweighted Tenney lattice add a time dimension.
For a given prime limit, mark the points with ratios written
in / slash notation and the rungs with ratios written in : colon
notation. In plain temperament, these ratios must be replaced
with irrational numbers at T_0, and they remain unchanged at
T_x for any x. In "adaptive JI" the slash ratios must be replaced
at T_(>0) but colon ratios at T_x (but not those between T_(x-1)
and T_x) remain. In the system I mention above, at least one
colon ratio between T_(x-1) and T_x can remain.

-Carl