least squares best fit between Pythagorean and equal-tempered

First let me introduce myself: I am, by profession, a professor of
mathematics, and I am also an amateur musician (I play trumpet in a
community concert band).

I had an idea a few years ago which I have developed and am currently
drafting as an article for an online mathematics journal, the Journal
of Online Mathematics and its Applications (joma.org), published by
the Mathematics Association of America (maa.org). I would like to
find a place for the idea to get some attention in the music world,
and in particular among those interested in different tuning systems.

So let me tell you the idea, and how I have developed it: I know
enough about the relationships between mathematics and music (more
from the mathematics side) to know about the differences between a
modern equal-tempered scale and the ancient Pythagorean scale. It
occurred to me that mathematical techniques may apply in finding an
improved equal-tempered scale, provided I was willing to adjust the
perfect 2:1 octave ratio. This does have some precedent, for example
in the practice of "octave stretching" when tuning a concert piano.
So I introduced a parameter into the definition of the equal-tempered
scale, to represent the extent of a possible stretch, then applied a
least-squares best fit technique to optimize the stretch so that the
notes of the diatonic major scale in the equal-tempered definition
better fit the Pythagorean diatonic major scale. I have tried several
options, and I have shown my results to some musician friends of mine
who agree that there is something to what I have found. I have come
to the conclusion that a weighted least-squares fit, giving extra
weight to the Pythagorean fourth, fifth, and octave, and resulting in
a stretch of about 2.27 cents (applied to every octave), gives the
best sounding equal-tempered scale, by comparison with the Pythagorean
scale. This isn't much of a stretch, but for applications requiring
high-precesion tuning, it can make a usable difference. My article
will include a Java applet demonstrating the different stretching
options along with a Pythagorean scale for comparison.

I e-mailed Joe Monzo already with this idea, and he suggested I post
to this group. He also directed me to look at Paul Erlich's
Tenney-optimal tunings -- online references to this were scarce, but
what I could find (best was in Herman Miller's "Warped Canon" page)
indicated that the closest to what I have done is the "Tenney-optimal
meantone" scale, which includes a stretch of 1.70 cents, equal-tempered.

I would be interested in any comments or observations about this idea.

Regards,
Prof. Tom Leathrum
MCIS Dept.
Jacksonville State Univ.

Distinguished professor,

By stretching the octave as much as 2.27 cents, you have created a 12-tone scale with the following values:

0: 1/1 C Dbb unison, perfect prime
1: 100.189 cents C# Db
2: 200.378 cents D Ebb
3: 300.568 cents D# Eb
4: 400.757 cents E Fb
5: 500.946 cents F Gbb
6: 601.135 cents F# Gb
7: 701.324 cents G Abb
8: 801.513 cents G# Ab
9: 901.702 cents A Bbb
10: 1001.892 cents A# Bb
11: 1102.081 cents B Cb
12: 1202.270 cents C Dbb

With that operation, you have made the fifths almost just with a negligable error of 0.631 cents. However, the major and minor thirds are not much affected, seeing as the results are less than a cent.

Besides, according to my understanding, a 12-tone temperament of this nature not only requires consistency of transposition at every significant degree, but should also aim to approximate JI intervals as best as one can achieve.

In my opinion, a 390-394 cent major third is a much better interval as compared to 401 cents.

Cordially,
Ozan Yarman*

*Istanbul Technical Univesity
Turkish Music State Conservatory
Musicology and Music Theory Deparment
Doctorate Course
----- Original Message -----
From: leathrum
To: tuning-math@yahoogroups.com
Sent: 02 Ekim 2005 Pazar 3:28
Subject: [tuning-math] least squares best fit between Pythagorean and equal-tempered

First let me introduce myself: I am, by profession, a professor of
mathematics, and I am also an amateur musician (I play trumpet in a
community concert band).

I had an idea a few years ago which I have developed and am currently
drafting as an article for an online mathematics journal, the Journal
of Online Mathematics and its Applications (joma.org), published by
the Mathematics Association of America (maa.org). I would like to
find a place for the idea to get some attention in the music world,
and in particular among those interested in different tuning systems.

So let me tell you the idea, and how I have developed it: I know
enough about the relationships between mathematics and music (more
from the mathematics side) to know about the differences between a
modern equal-tempered scale and the ancient Pythagorean scale. It
occurred to me that mathematical techniques may apply in finding an
improved equal-tempered scale, provided I was willing to adjust the
perfect 2:1 octave ratio. This does have some precedent, for example
in the practice of "octave stretching" when tuning a concert piano.
So I introduced a parameter into the definition of the equal-tempered
scale, to represent the extent of a possible stretch, then applied a
least-squares best fit technique to optimize the stretch so that the
notes of the diatonic major scale in the equal-tempered definition
better fit the Pythagorean diatonic major scale. I have tried several
options, and I have shown my results to some musician friends of mine
who agree that there is something to what I have found. I have come
to the conclusion that a weighted least-squares fit, giving extra
weight to the Pythagorean fourth, fifth, and octave, and resulting in
a stretch of about 2.27 cents (applied to every octave), gives the
best sounding equal-tempered scale, by comparison with the Pythagorean
scale. This isn't much of a stretch, but for applications requiring
high-precesion tuning, it can make a usable difference. My article
will include a Java applet demonstrating the different stretching
options along with a Pythagorean scale for comparison.

I e-mailed Joe Monzo already with this idea, and he suggested I post
to this group. He also directed me to look at Paul Erlich's
Tenney-optimal tunings -- online references to this were scarce, but
what I could find (best was in Herman Miller's "Warped Canon" page)
indicated that the closest to what I have done is the "Tenney-optimal
meantone" scale, which includes a stretch of 1.70 cents, equal-tempered.

I would be interested in any comments or observations about this idea.

Regards,
Prof. Tom Leathrum
MCIS Dept.
Jacksonville State Univ.

--- In tuning-math@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Besides, according to my understanding, a 12-tone temperament of
> this nature not only requires consistency of transposition at every
> significant degree, but should also aim to approximate JI intervals
> as best as one can achieve.

Yes, but there is a balancing act to perform when adjusting an
equal-tempered scale to better fit JI: using the least squares
technique, it amounts to asking which JI intervals should be given
greater weight in the computation. I decided to use all JI ratios for
the diatonic major scale, with double weight on the fourth and fifth
and quadruple weight on the octave, to get my 2.27 cents stretch.

I have looked at lots of other options, including giving all JI
diatonic major scale intervals equal weight -- that gave a stretch of
4.87 cents. For comparison, getting the fifth to exactly match the JI
perfect fifth gives a stretch of 3.35 cents (exactly 1/7 of a
Pythagorean comma). Including only the JI fourth, fifth, and octave,
with no other intervals in the computation, but giving these three
equal value, gave only a negligible 0.22 cent stretch.

> In my opinion, a 390-394 cent major third is a much better interval
> as compared to 401 cents.

I did also consider the effect of including a 5:4 major third in the
computation. With the 5:4 major third, 4:3 fourth, 3:2 fifth, and 2:1
octave, all given equal weight, I got an octave shrink of 2.61 cents.
That may be more to your liking, but I think it is unlikely to catch
on with western ears unaccustomed to the 5:4 major third ratio. As a
trumpet player, I saw the computation as confirmation that the
trumpet's written fourth-space E, played open, really is flat -- again
by western standards.

Regards,
Prof. Tom Leathrum
MCIS Dept.
Jacksonville State Univ.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:

> First let me introduce myself: I am, by profession, a professor of
> mathematics, and I am also an amateur musician (I play trumpet in a
> community concert band).

Well hello, and I hope you will stick around. What sort of mathematics
would you regard as your specialties? (Sort of the same nosy question
the AMS asks, I guess.)

> It
> occurred to me that mathematical techniques may apply in finding an
> improved equal-tempered scale, provided I was willing to adjust the
> perfect 2:1 octave ratio.

This has been the subject of a lot of discussion on this group,
particularly in connection with the Riemann Zeta function and the TOP
tunings you mention below. How the latter works you can glean from
this web page:

http://66.98.148.43/~xenharmo/top.htm

The connection with the Riemann Zeta function is something I
discovered in the seventies and have given a couple of talks on, but
it remains unpublished.

I have come
> to the conclusion that a weighted least-squares fit, giving extra
> weight to the Pythagorean fourth, fifth, and octave, and resulting in
> a stretch of about 2.27 cents (applied to every octave), gives the
> best sounding equal-tempered scale, by comparison with the Pythagorean
> scale.

This is the reverse of the Zeta tuning. For that you would take the
local maximum of |Zeta(0.5 + it)| near t = 12*(2*pi/log(2)) and divide
the result through by 2*pi/log(2), which gives 12.023 notes to the
octave, or 2.32 cents *flatter* than pure octaves. I wonder if you
don't have this reversed, as a flattened octave is what I'd generally
expect, since the major thirds are sharp, as is the 7/4 if you are
counting that.

The 5-limit TOP tuning is also flat, being 2.326 cents flat, very
close to the Zeta tuning. The 7-limit TOP tuing, though I think not
relevant to what you are doing, is even flatter. Here by "limit" I
mean the highest odd prime appearing in the factorization of the
ratios you are considering.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:

> I did also consider the effect of including a 5:4 major third in the
> computation. With the 5:4 major third, 4:3 fourth, 3:2 fifth, and 2:1
> octave, all given equal weight, I got an octave shrink of 2.61 cents.

Now you're talking! Including thirds is absolutely crucial to
common-practice music. It makes little sense to ignore thirds and
sixths if you are going to do this sort of thing.

> That may be more to your liking, but I think it is unlikely to catch
> on with western ears unaccustomed to the 5:4 major third ratio.

My experience tuning to meantone is that people unused to the better
thirds it gives generally like them. The major thirds of 12-edo are
really very sharp, and the whole tradition of Western common practice
music suggests that something closer to a pure 5/4 is in fact preferable.

As a
> trumpet player, I saw the computation as confirmation that the
> trumpet's written fourth-space E, played open, really is flat -- again
> by western standards.

"Western standards" only more recently. Older musicians often thought
the thirds of equal temperament were intolerably sharp.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
>I have come
> to the conclusion that a weighted least-squares fit, giving extra
> weight to the Pythagorean fourth, fifth, and octave, and resulting in
> a stretch of about 2.27 cents (applied to every octave), gives the
> best sounding equal-tempered scale, by comparison with the Pythagorean
> scale.

I didn't read carefully what you wrote the first time through, and
missed that you were working with the 3-limit, and with the fourth,
fifth and octave in particular. The 3-limit TOP tuning for 12 *is*
stretched, but by a much smaller amount that you give it. You give a
pretty helfty stretch, and I don't see how it is possible to get it
from a weighted least squares; the octave is now 2.27 cents sharp,
sharper than the fifth is flat in 12-edo, which seems fairly
pointless, especially since experience seems to show this amount of
octave mistuning is much more disturbing to many people than the same
amount of mistuning of a fifth. The fifth is down to 0.63 cents flat,
but this doesn't provide much compensation for the octave. The fourth,
at 2.9 cents sharp, isn't improved either.

Dear professor, I agree with Gene Ward Smith's last comments, even though I know that the Western ear for the past century and a half inclines more favorably towards Pythagorean intonation than Just.

Cordially,
Ozan Yarman

----- Original Message -----
From: leathrum
To: tuning-math@yahoogroups.com
Sent: 02 Ekim 2005 Pazar 4:21
Subject: [tuning-math] Re: least squares best fit between Pythagorean and equal-tempered

--- In tuning-math@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Besides, according to my understanding, a 12-tone temperament of
> this nature not only requires consistency of transposition at every
> significant degree, but should also aim to approximate JI intervals
> as best as one can achieve.

Yes, but there is a balancing act to perform when adjusting an
equal-tempered scale to better fit JI: using the least squares
technique, it amounts to asking which JI intervals should be given
greater weight in the computation. I decided to use all JI ratios for
the diatonic major scale, with double weight on the fourth and fifth
and quadruple weight on the octave, to get my 2.27 cents stretch.

I have looked at lots of other options, including giving all JI
diatonic major scale intervals equal weight -- that gave a stretch of
4.87 cents. For comparison, getting the fifth to exactly match the JI
perfect fifth gives a stretch of 3.35 cents (exactly 1/7 of a
Pythagorean comma). Including only the JI fourth, fifth, and octave,
with no other intervals in the computation, but giving these three
equal value, gave only a negligible 0.22 cent stretch.

> In my opinion, a 390-394 cent major third is a much better interval
> as compared to 401 cents.

I did also consider the effect of including a 5:4 major third in the
computation. With the 5:4 major third, 4:3 fourth, 3:2 fifth, and 2:1
octave, all given equal weight, I got an octave shrink of 2.61 cents.
That may be more to your liking, but I think it is unlikely to catch
on with western ears unaccustomed to the 5:4 major third ratio. As a
trumpet player, I saw the computation as confirmation that the
trumpet's written fourth-space E, played open, really is flat -- again
by western standards.

Regards,
Prof. Tom Leathrum
MCIS Dept.
Jacksonville State Univ.

My library of useful tempering functions isn't set up for the 3-limit, because it's just too darn simple. So I may have this wrong, but I get a 0.6 cent octave stretch for 12-equal doing a least squares optimization of the fit to primes 2 and 3, with Tenney weighting. This is an approximation to a measure with all Pythagorean intervals weighted according to their complexity. Not a perfect approximation, but acceptable considering the assumptions behind any of this having a musical relevance.

In the 5-limit, I get a prime weighted, least squares, optimum shrinkage of 1.6 cents. That's in the same ball park as the equally arbitrary 2.3 cent TOP shrinkage.

leathrum wrote:
> Yes, but there is a balancing act to perform when adjusting an
> equal-tempered scale to better fit JI: using the least squares
> technique, it amounts to asking which JI intervals should be given
> greater weight in the computation. I decided to use all JI ratios for
> the diatonic major scale, with double weight on the fourth and fifth
> and quadruple weight on the octave, to get my 2.27 cents stretch.

So you started with a good idea of what kind of stretch you wanted, and fudged the criteria until you got the right result?

This all depends on what diatonic scale you look at. Dorian should give negligible stretching because of its symmetry. Any improvement in Peter is stolen from Paul. With minimax, you'll get exactly zero stretch. With least squares, you'll get a small result because larger intervals are affected more than small ones. To me, this is a pure artifact of the method of calculation.

So you're looking at the major scale. It's built from 5 fifths, and only 1 fourth from the tonic. The result will therefore favour fifths over fourths, and so give a stretched octave. Adding extra weight to the fourth, fifth and octave will give a more symmetrical result, and so dampen down the stretch. Choose the weighting carefully, and you can get exactly the stretch you want.

All this assuming that you're calculating errors of intervals relative to the tonic. If you consider all intervals within the scale, you'll hopefully get negligible results again.

If you really want a diatonic scale in Pythagorean intonation, I suggest you tune a diatonic scale to Pythagorean intervals.

> I have looked at lots of other options, including giving all JI
> diatonic major scale intervals equal weight -- that gave a stretch of
> 4.87 cents. For comparison, getting the fifth to exactly match the JI
> perfect fifth gives a stretch of 3.35 cents (exactly 1/7 of a
> Pythagorean comma). Including only the JI fourth, fifth, and octave,
> with no other intervals in the computation, but giving these three
> equal value, gave only a negligible 0.22 cent stretch.

Like any symmetrical, unweighted measure, it gives an arbitrary result. But yes, 0.22 cents is correct.

> I did also consider the effect of including a 5:4 major third in the
> computation. With the 5:4 major third, 4:3 fourth, 3:2 fifth, and 2:1
> octave, all given equal weight, I got an octave shrink of 2.61 cents.
> That may be more to your liking, but I think it is unlikely to catch
> on with western ears unaccustomed to the 5:4 major third ratio. As a
> trumpet player, I saw the computation as confirmation that the
> trumpet's written fourth-space E, played open, really is flat -- again
> by western standards.

The 4:3 and 3:2 will roughly cancel out, so this calculation will be dominated by the 5:4. It's lousy as an optimization for 5-limit harmony. If you include 6:5, 8:5, and 5:3, I expect you'll get negligible deviation again.

Graham

Let me try to answer as many of the replies as I can in one note:

From genewardsmith:
> Well hello, and I hope you will stick around. What sort of
> mathematics would you regard as your specialties? (Sort of the
> same nosy question the AMS asks, I guess.)

My grad school specialty was logic and set theory (advisor was
Jim Baumgartner at Dartmouth), but in recent years most of my
work has had to do with writing applets and online materials
for visualizing various topics in calculus and precalculus.
Oh, and of course teaching.

My work writing applets actually led me into this: for precalculus,
I wrote an applet demonstrating transformations of periodic functions
with sound, so looking at the effects of the coefficients of
y = A sin(B x + C) + D -- A is amplitude, B is frequency, C is
phase shift (inaudible), and D is vertical shift (inaudible).
I followed this applet up with one showing the effects of combining
two tones (especially how small differences in frequency introduce
beats, and how whole number multiples of frequencies sound
harmonic, and how phase shift does make a difference when there are
two tones around) and one doing the first several terms of a Fourier
series (approximating some simple waveforms).

From genewardsmith:
> This has been the subject of a lot of discussion on this group,

Good -- I am definitely getting the impression that this is the right
place to bring these ideas.

From genewardsmith:
> This is the reverse of the Zeta tuning. For that you would take the
[...]
> which gives 12.023 notes to the
> octave, or 2.32 cents *flatter* than pure octaves.
[...]
> The 5-limit TOP tuning is also flat, being 2.326 cents flat, very
> close to the Zeta tuning.

I have looked at a lot of possibilities, but I have come to the
computations using the least squares optimization technique, not
using series or algebraic approaches. The idea I had was simply
to introduce a stretch parameter into the equal-tempered scale,
then look at using least squares to optimize.

From genewardsmith:
> Now you're talking! Including thirds is absolutely crucial to
> common-practice music. It makes little sense to ignore thirds and
> sixths if you are going to do this sort of thing.
[...]
> My experience tuning to meantone is that people unused to the better
> thirds it gives generally like them. The major thirds of 12-edo are
> really very sharp, and the whole tradition of Western common practice
> music suggests that something closer to a pure 5/4 is in fact
preferable.

There are certainly arguments to be made here, which is why I did
investigate the 5:4 major third ratio as well.

From genewardsmith:
> I didn't read carefully what you wrote the first time through, and
> missed that you were working with the 3-limit, and with the fourth,
> fifth and octave in particular. The 3-limit TOP tuning for 12 *is*
> stretched, but by a much smaller amount that you give it. You give a
> pretty helfty stretch, and I don't see how it is possible to get it
> from a weighted least squares; the octave is now 2.27 cents sharp,
> sharper than the fifth is flat in 12-edo, which seems fairly
> pointless, especially since experience seems to show this amount of
> octave mistuning is much more disturbing to many people than the same
> amount of mistuning of a fifth. The fifth is down to 0.63 cents flat,
> but this doesn't provide much compensation for the octave. The fourth,
> at 2.9 cents sharp, isn't improved either.

I arrived at the 2.27 cent stretch number with the least squares fit
using *all* of the diatonic major scale Pythagorean ratios, but
giving double weight to fourth and fifth and quadruple to octave,
as much for historical reasons as anything -- those ratios have
been regarded as "more important" historically, even to the point
of considering the octave sacrosanct. I was just curious about
whether including some of the other ratios in a least squares fit but
allowing the octave to be sacrificed to some modest degree would
result in a scale with some advantages over the modern equal-temp.
scale with its strict 2:1 octave.

From ozanyarman:
> Dear professor, I agree with Gene Ward Smith's last comments, even
> though I know that the Western ear for the past century and a half
> inclines more favorably towards Pythagorean intonation than Just.

I have been regarding the Pythagorean ratios as the "ideal" and
looking mathematically for a way to get the equal-tempered intervals
to better fit the Pythagorean ratios.

From x31eq:
> So you started with a good idea of what kind of stretch you wanted, and
> fudged the criteria until you got the right result?

Well, yes and no -- I came into this knowing about the practice of
"octave stretching" when tuning a piano, but wondering if there
was a way to justify stretching or shrinking without relying on the
physics of a particular instrument. There is also some precedent
for stretching in the behavior of brasswind instruments, with
which I am more familiar -- the flared shape of the bell causes
a foreshortening effect for the vibrating column of air as the
frequency of the vibration goes higher at the higher harmonics,
effectively stretching the octaves again.

But I looked at lots of possibilities, and only arrived at a
conclusion about what sounded right after I built an applet to
play the scales and chords in the scales, and listened to the
results. I included lots of possibilities in the applet, and
computed several possibilities that I decided not to include
in the applet because they were either too small (0.12 cents
in one case) or too large (one computation, using an exact match
to the Pythagorean major third only, gave me a stretch of a full
Pythagorean comma, which was painfully sharp).

My demonstration applet includes two one-octave keyboards, one
with a fixed Pythagorean tuning, and the other with radio buttons
that allow for selecting any of 7 possibilities, including the
standard equal-temp. scale along with 5 stretches and the
one shrink I got from including the 5:4 major third.
For comparison, I even included a scale computed from an
exact match to the Pythagorean perfect fifth, which gave
a 3.35 cent stretch, exactly 1/7 of a Pythagorean comma.
With the applet, I listened to the results of the computations
and came to the conclusion only then that what sounded best was
the 2.27 cent stretch. I have used the same demonstration with
half a dozen or so musician friends, and they all see the same
result -- that stretch makes the equal-temp. scale sound better.

From x31eq:
> This all depends on what diatonic scale you look at. Dorian should give
> negligible stretching because of its symmetry. Any improvement in Peter
> is stolen from Paul. With minimax, you'll get exactly zero stretch.
> With least squares, you'll get a small result because larger intervals
> are affected more than small ones. To me, this is a pure artifact of
> the method of calculation.

Well, by including the entire diatonic major scale in the computation,
I hoped to eliminate the need to consider other modes. If I get a
good fit for a C major diatonic, then for example its relative minor
will be just as good a fit, at least for the Pythagorean definitions.
This is another reason why I only gave extra weight to the "perfect"
intervals, since the whole idea behind them being "perfect" is that
they don't change between major and minor.

From x31eq:
> So you're looking at the major scale. It's built from 5 fifths, and
> only 1 fourth from the tonic. The result will therefore favour fifths
> over fourths, and so give a stretched octave. Adding extra weight to
> the fourth, fifth and octave will give a more symmetrical result, and so
> dampen down the stretch. Choose the weighting carefully, and you can
> get exactly the stretch you want.

Actually the Pythagorean ratios, including the fourth, can be built
entirely from fifths and octaves only. The question is what to do
about the resulting Pythagorean comma. So yes, there is a certain
extra weight to the fifth and fourth that comes naturally from the
whole premise of using the Pythagorean ratios in the first place. And
yes, it is possible to get any result I want by tweaking the weights,
which is why I considered so many different ways of tweaking them, in
order to get something that actually sounded better. The math gave
the computations, but I wasn't willing to judge the results until I
heard them, and let other people with more practiced ears hear them, too.

Regards,
Prof. Tom Leathrum
MCIS Dept.
Jacksonville State Univ.

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

> My library of useful tempering functions isn't set up for the 3-limit,
> because it's just too darn simple. So I may have this wrong, but I get
> a 0.6 cent octave stretch for 12-equal doing a least squares
> optimization of the fit to primes 2 and 3, with Tenney weighting.

3-limit TOP tuning is 0.617 cents, so this agrees. I haven't
calculated any least-squares values since I don't know the weighting.

> leathrum wrote:
> > Yes, but there is a balancing act to perform when adjusting an
> > equal-tempered scale to better fit JI: using the least squares
> > technique, it amounts to asking which JI intervals should be given
> > greater weight in the computation. I decided to use all JI ratios for
> > the diatonic major scale, with double weight on the fourth and fifth
> > and quadruple weight on the octave, to get my 2.27 cents stretch.

It seems to me that thus conflating the Pythagorean and diatonic
scales is a bad idea if the objective is tuning common-practice music.
One won't get an octave stretch if, as I think is most reasonable,
meantone is used to define the diatonic scale. In that case, we might
get 5-limit TOP meantone tuning, which has octaves sharp by 1.7 cents,
and fifths flat by 4.4 cents. Now, however, it has circulation
problems; a stretched well-temperament might be an interesting try.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:

> My grad school specialty was logic and set theory (advisor was
> Jim Baumgartner at Dartmouth)...

Ken Ribet at Berkeley here. Have you ever run into the phase "musical
set theory", by the way?

but in recent years most of my
> work has had to do with writing applets and online materials
> for visualizing various topics in calculus and precalculus.
> Oh, and of course teaching.

Java is a wonderful invention. If you ever get interested in the Zeta
function and tuning, there is (or was) an on-line applet which
calculated the Z function and graphed it.

> I arrived at the 2.27 cent stretch number with the least squares fit
> using *all* of the diatonic major scale Pythagorean ratios, but
> giving double weight to fourth and fifth and quadruple to octave,
> as much for historical reasons as anything -- those ratios have
> been regarded as "more important" historically, even to the point
> of considering the octave sacrosanct.

I wouldn't call seven notes obtained from a chain of pure fifths the
diatonic scale, but rather Pythagorean. The fourth and fifth are
important 3-limit intervals, but so is the twelvth. In common practice
music historically, claiming they are more important than thirds and
sixths is a little dubious. The fourth in particular was sometimes
regarded as a dissonance, and the second inversion of a triad, with
the fifth at the bottom, was not regarded as a stable chord like the
first inversion or (especially) the root position. In two-part
harmony, the thirds and the sixths are generally preferred.

Bottom line, it's really not possible to correctly analyze
common-practice harmony in pure 3-limit terms.

I was just curious about
> whether including some of the other ratios in a least squares fit but
> allowing the octave to be sacrificed to some modest degree would
> result in a scale with some advantages over the modern equal-temp.
> scale with its strict 2:1 octave.

It's certainly arguable, but I think it makes more sense to flatten
the octave a smidgen for common-practice music.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Ken Ribet at Berkeley here.

Hi! We may have met several years ago at a conference.... Speaking
of which, are you going to be in San Antonio in January? Do you know
of any math/tuning related stuff going on there?

> Have you ever run into the phase "musical
> set theory", by the way?

Heard of it, yes, but source references seem to be hard to find, so
I'm afraid my knowledge of it goes no deeper than the Wikipedia essay
on it.

> Java is a wonderful invention.

Tell me about it! My work with Java and JOMA (see joma.org) pretty
much got me tenure.

> If you ever get interested in the Zeta
> function and tuning, there is (or was) an on-line applet which
> calculated the Z function and graphed it.

I did have a thought related to Zeta that I have been meaning to
pursue, but it is really off-field for this discussion -- let me see
if I can give a Reader's Digest version (those who aren't interested
can tune out until the next paragraph): Hugh Woodin has pretty much
concluded his work using inner models to characerize the first-order
theory of the hereditary model H_{\omega_1}, which includes the theory
of the natural numbers, and he has concluded that Projective
Determinacy is the right axiom to use (in addition to ZFC) in that
context; so does PD have any bearing on RH? It really boils down to
asking whether there is a first-order way to state RH -- if yes, then
Woodin's work would imply that PD should at least say something. But
the usual ways of stating RH are second-order. (Woodin has more
recently moved on to H_{\omega_2} and its bearing on CH, with still
more interesing ideas....)

> I wouldn't call seven notes obtained from a chain of pure fifths the
> diatonic scale, but rather Pythagorean. The fourth and fifth are
> important 3-limit intervals, but so is the twelvth. In common practice
> music historically, claiming they are more important than thirds and
> sixths is a little dubious. The fourth in particular was sometimes
> regarded as a dissonance, and the second inversion of a triad, with
> the fifth at the bottom, was not regarded as a stable chord like the
> first inversion or (especially) the root position. In two-part
> harmony, the thirds and the sixths are generally preferred.
>
> Bottom line, it's really not possible to correctly analyze
> common-practice harmony in pure 3-limit terms.

There seems to be some confusion of terminology here -- by the
references I have seen, "diatonic" just refers to a scale constructed
from 5 whole steps and 2 half steps, with the half steps distributed
as evenly as possible. That leaves open the question of how to build
the whole steps and half steps. The Pythagorean system is one way to
do that, but so is modern equal-temp.

I am still picking up on some of the other terminology, like 3-limits
and 5-limits, so please bear with me for a while.

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:

> Hi! We may have met several years ago at a conference.... Speaking
> of which, are you going to be in San Antonio in January? Do you know
> of any math/tuning related stuff going on there?

No, but I'd like to hear of any tuning-math stuff discussed there.

> I did have a thought related to Zeta that I have been meaning to
> pursue, but it is really off-field for this discussion -- let me see
> if I can give a Reader's Digest version (those who aren't interested
> can tune out until the next paragraph): Hugh Woodin has pretty much
> concluded his work using inner models to characerize the first-order
> theory of the hereditary model H_{\omega_1}, which includes the theory
> of the natural numbers, and he has concluded that Projective
> Determinacy is the right axiom to use (in addition to ZFC) in that
> context; so does PD have any bearing on RH? It really boils down to
> asking whether there is a first-order way to state RH -- if yes, then
> Woodin's work would imply that PD should at least say something.

So are you suggesting there may be a first-order statement of RH in
which it is true but provable only under PD? It seems clear, by the
way, that RH is indeed stateable in first-order terms since there are
equivalents to it which are in elementary terms. Examples of this are
the equivalence of RH to a statements about the rate of growth of
arithmetic functions such as the sum of the Mobius function (the
Mertens function) or the divisor function, or statements about the
degree of evenness of the Farey sequence.

It would be interesting for a logician to sort out how strong the
axioms need to be to prove some well-known theorems of number theory.
The proof of Fermat's Last Theorem is probably far more high-powered
than it needs to be in terms of axiomatic strength; I would guess it
might really be provable in Peano Arithmetic, or without choice.
Trying to prove as much arithmetic algebraic geometry as one can in ZF
without C sounds like a fine thesis topic for someone.

> There seems to be some confusion of terminology here -- by the
> references I have seen, "diatonic" just refers to a scale constructed
> from 5 whole steps and 2 half steps, with the half steps distributed
> as evenly as possible.

That's how Easley Blackwood defines in in The Structure of
Recongizable Diatonic Tunings. However, "diatonic scale" in practice
normally means the 7-note scale with the familiar functional harmonic
relationships and triadic harmony, and that entails meantone of some sort.

Here's Wikipedia's take on it:

http://en.wikipedia.org/wiki/Diatonic_scale

> I am still picking up on some of the other terminology, like 3-limits
> and 5-limits, so please bear with me for a while.

It's a very useful terminology due to Harry Partch. The p-limit
rationals are the positive rationals which are factorizable by primes
less than or equal to the prime p, and usually one considers that in
terms of the rank phi(p) free group structure of these under
multiplication, with the primes as generators.

leathrum wrote:

> Well, yes and no -- I came into this knowing about the practice of
> "octave stretching" when tuning a piano, but wondering if there
> was a way to justify stretching or shrinking without relying on the
> physics of a particular instrument. There is also some precedent
> for stretching in the behavior of brasswind instruments, with
> which I am more familiar -- the flared shape of the bell causes
> a foreshortening effect for the vibrating column of air as the
> frequency of the vibration goes higher at the higher harmonics,
> effectively stretching the octaves again.

A preference for stretched octaves does have a psychoacoustical backing. See this site for starters:

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/scalestretch.html

I'm not sure how this percentage is calculated, but it must be roughly the percentage of the octave in cents. So a 3% stretch would be 36 cents per octave. That's more substantial than anything you consider. The explanation, IIRC, is to do with the pitch of a harmonic note disagreeing slightly with that of a pure sine wave with the same frequency as the fundamental. In turn, that has something to do with the pitch of a sine wave having a small variation with intensity.

> But I looked at lots of possibilities, and only arrived at a
> conclusion about what sounded right after I built an applet to
> play the scales and chords in the scales, and listened to the
> results. I included lots of possibilities in the applet, and
> computed several possibilities that I decided not to include
> in the applet because they were either too small (0.12 cents
> in one case) or too large (one computation, using an exact match
> to the Pythagorean major third only, gave me a stretch of a full
> Pythagorean comma, which was painfully sharp).

What timbres does your applet use? According to the literature, that's critical in determining the octave stretch. The brighter the timbre, the greater the stretch.

> My demonstration applet includes two one-octave keyboards, one
> with a fixed Pythagorean tuning, and the other with radio buttons
> that allow for selecting any of 7 possibilities, including the
> standard equal-temp. scale along with 5 stretches and the
> one shrink I got from including the 5:4 major third.
> For comparison, I even included a scale computed from an
> exact match to the Pythagorean perfect fifth, which gave
> a 3.35 cent stretch, exactly 1/7 of a Pythagorean comma.
> With the applet, I listened to the results of the computations
> and came to the conclusion only then that what sounded best was
> the 2.27 cent stretch. I have used the same demonstration with
> half a dozen or so musician friends, and they all see the same
> result -- that stretch makes the equal-temp. scale sound better.

Your conclusion may be correct, as a small octave stretch is a compromise between the greater tendency for melodic stretching and the smaller stretches that harmonic factors usually lead to.

> Well, by including the entire diatonic major scale in the computation,
> I hoped to eliminate the need to consider other modes. If I get a
> good fit for a C major diatonic, then for example its relative minor
> will be just as good a fit, at least for the Pythagorean definitions.
> This is another reason why I only gave extra weight to the "perfect"
> intervals, since the whole idea behind them being "perfect" is that
> they don't change between major and minor.

If you're doing what I think you're doing, that won't work. Try the scale of the Phrygian mode and see if you get the same result. If you do then, well, I'm surprised the stretch comes out so high.

> Actually the Pythagorean ratios, including the fourth, can be built
> entirely from fifths and octaves only. The question is what to do
> about the resulting Pythagorean comma. So yes, there is a certain
> extra weight to the fifth and fourth that comes naturally from the
> whole premise of using the Pythagorean ratios in the first place. And
> yes, it is possible to get any result I want by tweaking the weights,
> which is why I considered so many different ways of tweaking them, in
> order to get something that actually sounded better. The math gave
> the computations, but I wasn't willing to judge the results until I
> heard them, and let other people with more practiced ears hear them, too.

The question is where to put the Pythagorean comma, relative to the tonic. Fifths and fourths should roughly cancel each other out in their tendencies. If you measure a major diatonic scale relative to the tonic, it amounts to weighting the fifths higher than fourths.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> >
> So are you suggesting there may be a first-order statement of RH in
> which it is true but provable only under PD? It seems clear, by the
> way, that RH is indeed stateable in first-order terms since there are
> equivalents to it which are in elementary terms. Examples of this are
> the equivalence of RH to a statements about the rate of growth of
> arithmetic functions such as the sum of the Mobius function (the
> Mertens function) or the divisor function, or statements about the
> degree of evenness of the Farey sequence.

This isn't really the place for this discussion, but let me just clear
up my question a bit. In order for PD to really apply, I would need a
statement of RH that could at least be translated into something about
the winning set of an infinite game. If that set can be shown to be
projective (really a sort of first-order-ness question) then PD
applies, the game has a winning strategy, and that can say something
about RH. What exactly it would say, I don't know. Just an idea.

Even more fascinating would be if such a translation of RH could be
shown to be equivalent to some form of PD, which is known to be
independent of ZFC.... But I suspect that RH is more specific than that.

> It would be interesting for a logician to sort out how strong the
> axioms need to be to prove some well-known theorems of number theory.
> The proof of Fermat's Last Theorem is probably far more high-powered
> than it needs to be in terms of axiomatic strength; I would guess it
> might really be provable in Peano Arithmetic, or without choice.
> Trying to prove as much arithmetic algebraic geometry as one can in ZF
> without C sounds like a fine thesis topic for someone.

Oh yes, there is lots of wonderful "reverse math" going on these days
-- the big name in that field is Ted Slaman. He and his group work
mostly on fragments of Peano arithmetic, in particular what initial
fragment of induction is necessary for a particular proof. So you get
into some recursive math that way, hence Ted's involvement. As for
algebraic geometry, given that there are forms of the Nullstellensatz
equivalent to Zorn's Lemma, it seems that there wouldn't be a whole
lot left without Choice, but yes, it would be a great research topic.
I even remember seeing a series of talks when I was a grad student
about how, if you take away the Power Set Axiom, the equivalence
between Choice, Well-Ordering Axiom, and Zorn's Lemma breaks down.
Fun stuff.

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> A preference for stretched octaves does have a psychoacoustical
backing.
> See this site for starters:
>
>
http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/scalestretch.html
>
> I'm not sure how this percentage is calculated, but it must be roughly
> the percentage of the octave in cents. So a 3% stretch would be 36
> cents per octave. That's more substantial than anything you consider.
> The explanation, IIRC, is to do with the pitch of a harmonic note
> disagreeing slightly with that of a pure sine wave with the same
> frequency as the fundamental. In turn, that has something to do with
> the pitch of a sine wave having a small variation with intensity.

Thanks for the link -- that is yet another approach. The precedent I
saw in piano tuning is best represented near the end of the document
at this link:

http://www.precisionstrobe.com/apps/pianotemp/temper.html

It's on a site from a company that makes high-precesion electronic
strobe tuners, and it is talking about how the tuner can be set up to
take "octave stretching" into account, so it looks pretty closely at
the practice, even graphing out common stretches. It's justification
is in terms of the physics of a vibrating string, though, not
phychoacoustics. It seems (according to the article) that the higher
harmonics of the fat strings at the bottom and the short strings at
the top are a bit off, so the stretch tries to fix that.

> What timbres does your applet use? According to the literature, that's
> critical in determining the octave stretch. The brighter the timbre,
> the greater the stretch.

I'm a math person -- I'm using pure sine tones in the applet.
Anything else being picked up is an artifact of the computer's
speakers, but I have tried this on several different computers with
several different qualities of speakers, with the same results, so I
don't think that is having an effect.

> If you're doing what I think you're doing, that won't work. Try the
> scale of the Phrygian mode and see if you get the same result. If you
> do then, well, I'm surprised the stretch comes out so high.

I can try that, but it will take some time to set up the right
computations. I'll let you know.

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:

> Even more fascinating would be if such a translation of RH could be
> shown to be equivalent to some form of PD, which is known to be
> independent of ZFC.... But I suspect that RH is more specific than
that.

It's stateable in Peano arthmetic, and I suspect there's a proof out
there which does not require any extra set theoretical assumptions.
But then, that's what most algebraists would have thought about
Whitehead's Conjecture, and that turned out to be wrong.

> As for
> algebraic geometry, given that there are forms of the Nullstellensatz
> equivalent to Zorn's Lemma, it seems that there wouldn't be a whole
> lot left without Choice, but yes, it would be a great research topic.

I thought the existence of a maximal ideal was strictly weaker than
choice. For sure, you can get a weak Nullstellensatz with even less
than that.

Gene Ward Smith wrote:

> 3-limit TOP tuning is 0.617 cents, so this agrees. I haven't
> calculated any least-squares values since I don't know the weighting.

Couldn't you guess what weighting makes sense, and then act surprised if you guessed wrong?

Oh, all right then. Weight by the size of the prime interval such that the result is dimensionless. Probably the same way you do TOP. It's fairly obvious.

The twelfth in 12-equal is 19x/12, where x is the stretched octave. The error is 19x/12-log2(3) octaves. The weighted error is 19x/12log2(3)-1. The weighted error in the octave is x-1.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> If you're doing what I think you're doing, that won't work. Try the
> scale of the Phrygian mode and see if you get the same result. If you
> do then, well, I'm surprised the stretch comes out so high.

You're right -- Phrygian mode (E to E on the white keys) computing
using the entire scale in the least squares, but unweighted, gave me a
shrink of 3.39 cents, and when I weighted (double on fourth and fifth,
quadruple on octave), the shrink went down to 1.40 cents. Hmmm....
This is kinda cool, now all I have to do is find a way to include this
in the article draft. Thanks for the pointer. I'll need to double
check the minor key mode, too. Any other things you think I should
check out?

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > If you're doing what I think you're doing, that won't work. Try the
> > scale of the Phrygian mode and see if you get the same result. If
you
> > do then, well, I'm surprised the stretch comes out so high.
>
> You're right -- Phrygian mode (E to E on the white keys) computing
> using the entire scale in the least squares, but unweighted, gave me a
> shrink of 3.39 cents, and when I weighted (double on fourth and fifth,
> quadruple on octave), the shrink went down to 1.40 cents.

I recalculated for all of the modes (except the hyper- ones), and this
is what I got -- first number is unweighted, second is weighted with
double weight on 4:3 fourth and 3:2 fifth and quadruple on 2:1 octave
(so these weights don't make sense in Lydian, with its different
fourth, or Locrian, with different fifth):

Ionian 4.8691 2.2703
Dorian 0.7568 0.3871
Phrygian -3.3689 -1.3986
Lydian 6.9571 ***
Mixolydian 0.7568 0.3871
Aeolian -2.7958 -1.1572
Locrian -6.2137 ***

(Positive is stretch, negative is shrink.) I'm not sure yet what to
make of these numbers, especially the correspondence between Dorian
and Mixolydian (I double-checked these because I didn't believe them,
and they are right). The extreme number for Locrian makes sense given
the peculiar nature of that mode, but I would have thought Lydian
would have been more reasonable. The funny thing is that the numbers
for Ionian actually seem large, by comparison with the other numbers.

Regards,
Tom Leathrum

Hi Tom and Graham and all,

>> A preference for stretched octaves does have a psychoacoustical
>backing.
>> See this site for starters:
> http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/scalestretch.html
>>
>> I'm not sure how this percentage is calculated, but it must be roughly
>> the percentage of the octave in cents. So a 3% stretch would be 36
>> cents per octave. That's more substantial than anything you consider.
>> The explanation, IIRC, is to do with the pitch of a harmonic note
>> disagreeing slightly with that of a pure sine wave with the same
>> frequency as the fundamental. In turn, that has something to do with
>> the pitch of a sine wave having a small variation with intensity.
>
>Thanks for the link -- that is yet another approach. The precedent I
>saw in piano tuning is best represented near the end of the document
>at this link:
>
>http://www.precisionstrobe.com/apps/pianotemp/temper.html
>
>It's on a site from a company that makes high-precesion electronic
>strobe tuners, and it is talking about how the tuner can be set up to
>take "octave stretching" into account, so it looks pretty closely at
>the practice, even graphing out common stretches. It's justification
>is in terms of the physics of a vibrating string, though, not
>phychoacoustics.

FWIW, I've never found Terhardt's psychoacoustic stretch very
convincing, except for pure tones. Not that I've rigorously tested
it. When tuning an electronic keyboard, though, I never found any
stretch desirable, and he seems to be reaching for evidence in
remarks made to upright bass players and by clarinet makers. Also
the way he starts the article with a description of piano stretch,
even though he admits that has a different cause. If there is
something universal going on with complex tones, I suspect its
effects are mostly above 1-2KHz.

>> What timbres does your applet use? According to the literature,
>> that's critical in determining the octave stretch. The brighter
>> the timbre, the greater the stretch.
>
>I'm a math person -- I'm using pure sine tones in the applet.
>Anything else being picked up is an artifact of the computer's
>speakers, but I have tried this on several different computers with
>several different qualities of speakers, with the same results, so
>I don't think that is having an effect.

But all computer speakers (and pretty much all speakers, not to
mention tube amps) will likely introduce harmonic distortion of
one degree or another. Even at very low volumes, I haven't been
able to get good sine tones out of my computer sound system.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> FWIW, I've never found Terhardt's psychoacoustic stretch very
> convincing, except for pure tones. Not that I've rigorously tested
> it. When tuning an electronic keyboard, though, I never found any
> stretch desirable, and he seems to be reaching for evidence in
> remarks made to upright bass players and by clarinet makers. Also
> the way he starts the article with a description of piano stretch,
> even though he admits that has a different cause. If there is
> something universal going on with complex tones, I suspect its
> effects are mostly above 1-2KHz.

Yes, there is clearly an inconsistency here -- if the stretch has
psychoacoustic reasons, electronic keyboards would need it, too, but
you rarely if ever see electronic keyboards with stretched octaves.
If it is physical, from the piano's vibrating strings, then the
electronic keyboards would have no need for stretching. Either
electronic keyboard designers are missing something or the
psychoacoustic reasoning doesn't wash.

> >> What timbres does your applet use? According to the literature,
> >> that's critical in determining the octave stretch. The brighter
> >> the timbre, the greater the stretch.
> >
> >I'm a math person -- I'm using pure sine tones in the applet.
> >Anything else being picked up is an artifact of the computer's
> >speakers, but I have tried this on several different computers with
> >several different qualities of speakers, with the same results, so
> >I don't think that is having an effect.
>
> But all computer speakers (and pretty much all speakers, not to
> mention tube amps) will likely introduce harmonic distortion of
> one degree or another. Even at very low volumes, I haven't been
> able to get good sine tones out of my computer sound system.

Yes, point taken. That's why I tried it on several different
computers with different kinds of speakers. All speakers would
introduce something, but different kinds of speakers would introduce
different harmonic profiles, so if the effect persists over different
speakers, there must be something common, independent of the different
speakers.

I think ultimately, though, when the article I am drafting finally
gets into good form, the position I am going to take isn't going to be
leaning toward a particular stretch, but rather simply suggesting that
the 2:1 octave not be considered sacrosanct, even for electronic
keyboards, because some degree of stretching or shrinking can help the
equal-tempered scale sound better. But what stretch or shrink to use
might depend on the application. For that reason, it will probably
only affect electronic keyboard design, and even then it will be
somewhat esoteric -- not everyone is going to be interested in a
keyboard that can do this.

One thing I am working on right now, based on other parts of this
discussion, is looking at adding to the demonstration applet an option
for the shrink I computed for Aeolian mode. That was a result I was
not expecting, but it is already making for some interesting changes
in the draft.

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
I post
> to this group. He also directed me to look at Paul Erlich's
> Tenney-optimal tunings -- online references to this were scarce, but
> what I could find (best was in Herman Miller's "Warped Canon" page)
> indicated that the closest to what I have done is the "Tenney-optimal
> meantone" scale, which includes a stretch of 1.70 cents, equal-
>tempered.

Meantone is not equal-tempered, in case you don't know that.

> I would be interested in any comments or observations about this idea.
>
> Regards,
> Prof. Tom Leathrum
> MCIS Dept.
> Jacksonville State Univ.

If you will provide me with your full address, I'll snail-mail you a
copy of my paper that discusses Tenney-optimal tunings.

In there, I find that the best stretch to apply to 12-equal when only
considering 3-limit (i.e., Pythagorean) ratios is 0.62 cents per octave.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:

> > analyze
> > common-practice harmony in pure 3-limit terms.
>
> There seems to be some confusion of terminology here -- by the
> references I have seen, "diatonic" just refers to a scale
constructed
> from 5 whole steps and 2 half steps, with the half steps distributed
> as evenly as possible.

That's correct.

> That leaves open the question of how to build
> the whole steps and half steps. The Pythagorean system is one way
to
> do that,

And was used for it from about 800 to 1450 A.D.

> but so is modern equal-temp.

In between the medieval and modern, there was a long meantone period.

Meanwhile, I'm intrigued to learn that Gene Ward Smith is in fact Ken
Ribet. Or is the latter just using the former's e-mail address??

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

> > There seems to be some confusion of terminology here -- by the
> > references I have seen, "diatonic" just refers to a scale
constructed
> > from 5 whole steps and 2 half steps, with the half steps
distributed
> > as evenly as possible.
>
> That's how Easley Blackwood defines in in The Structure of
> Recongizable Diatonic Tunings. However, "diatonic scale" in practice
> normally means the 7-note scale with the familiar functional
harmonic
> relationships and triadic harmony, and that entails meantone of
some sort.

I strongly disagree. The diatonic scale was used in the West long
before functional or triadic harmony, and ancient Greece before that
(isn't that where the name 'diatonic' comes from), not to mention in
other parts of the world throughout history. There's no question that
the 7-note Pythagorean chain is a diatonic scale.

> Here's Wikipedia's take on it:
>
> http://en.wikipedia.org/wiki/Diatonic_scale

I'll have to look at this later, it's not coming up on my computer.

> > I am still picking up on some of the other terminology, like 3-
limits
> > and 5-limits, so please bear with me for a while.
>
> It's a very useful terminology due to Harry Partch.

Harry Partch's definition was different and included a 9-limit, so be
careful.

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

> What timbres does your applet use? According to the literature,
that's
> critical in determining the octave stretch. The brighter the timbre,
> the greater the stretch.

Quite the contrary -- if you have harmonic partials, then the brighter
the timbre, the less the stretch. With sine waves, the darkest timbre,
you want the most stretch.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > If you're doing what I think you're doing, that won't work. Try
the
> > scale of the Phrygian mode and see if you get the same result. If
you
> > do then, well, I'm surprised the stretch comes out so high.
>
> You're right -- Phrygian mode (E to E on the white keys) computing
> using the entire scale in the least squares, but unweighted, gave me a
> shrink of 3.39 cents, and when I weighted (double on fourth and fifth,
> quadruple on octave), the shrink went down to 1.40 cents. Hmmm....
> This is kinda cool, now all I have to do is find a way to include this
> in the article draft. Thanks for the pointer. I'll need to double
> check the minor key mode, too. Any other things you think I should
> check out?
>
> Regards,
> Tom Leathrum

I think you may want to reconsider your methods -- to me the fact that
different modes lead to different results is evidence that you're doing
something "wrong", IMHO. I believe that you really want to optimize are
the *intervals*, not the pitches, in the scale, and the intervals are
exactly the same no matter what mode you choose.

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > FWIW, I've never found Terhardt's psychoacoustic stretch very
> > convincing, except for pure tones. Not that I've rigorously
tested
> > it. When tuning an electronic keyboard, though, I never found any
> > stretch desirable, and he seems to be reaching for evidence in
> > remarks made to upright bass players and by clarinet makers. Also
> > the way he starts the article with a description of piano stretch,
> > even though he admits that has a different cause. If there is
> > something universal going on with complex tones, I suspect its
> > effects are mostly above 1-2KHz.
>
> Yes, there is clearly an inconsistency here -- if the stretch has
> psychoacoustic reasons, electronic keyboards would need it, too, but
> you rarely if ever see electronic keyboards with stretched octaves.
> If it is physical, from the piano's vibrating strings, then the
> electronic keyboards would have no need for stretching. Either
> electronic keyboard designers are missing something or the
> psychoacoustic reasoning doesn't wash.

Excuse me -- what are you saying? I don't see how you come to this
conclusion.

> > >> What timbres does your applet use? According to the
literature,
> > >> that's critical in determining the octave stretch. The
brighter
> > >> the timbre, the greater the stretch.
> > >
> > >I'm a math person -- I'm using pure sine tones in the applet.
> > >Anything else being picked up is an artifact of the computer's
> > >speakers, but I have tried this on several different computers
with
> > >several different qualities of speakers, with the same results,
so
> > >I don't think that is having an effect.
> >
> > But all computer speakers (and pretty much all speakers, not to
> > mention tube amps) will likely introduce harmonic distortion of
> > one degree or another. Even at very low volumes, I haven't been
> > able to get good sine tones out of my computer sound system.
>
> Yes, point taken. That's why I tried it on several different
> computers with different kinds of speakers. All speakers would
> introduce something, but different kinds of speakers would introduce
> different harmonic profiles, so if the effect persists over
different
> speakers, there must be something common, independent of the
different
> speakers.

You're getting something close to sine waves in any case, and the
psychoacoustical evidence for "octave stretch" for sine waves or the
like is quite solid.

leathrum wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>FWIW, I've never found Terhardt's psychoacoustic stretch very
>>convincing, except for pure tones. Not that I've rigorously tested
>>it. When tuning an electronic keyboard, though, I never found any
>>stretch desirable, and he seems to be reaching for evidence in
>>remarks made to upright bass players and by clarinet makers. Also
>>the way he starts the article with a description of piano stretch,
>>even though he admits that has a different cause. If there is
>>something universal going on with complex tones, I suspect its
>>effects are mostly above 1-2KHz.

I thought Terhardt said that pure tones lead to octave shrinkage (not on the page I gave before). He certainly has experimental backing. And he says, of somebody else's experiments, "Although these instruments produce truly harmonic complex tones, a stretch of the tone scale was found that resembles that of the piano - with the only exception that even the middle octaves were not unstretched." That doesn't look like below 1kHz.

> Yes, there is clearly an inconsistency here -- if the stretch has
> psychoacoustic reasons, electronic keyboards would need it, too, but
> you rarely if ever see electronic keyboards with stretched octaves. > If it is physical, from the piano's vibrating strings, then the
> electronic keyboards would have no need for stretching. Either
> electronic keyboard designers are missing something or the
> psychoacoustic reasoning doesn't wash.

If the psychoacoustic reasoning doesn't wash, then neither do your experiments. You can't really get around psychoacoustics as long as you deal with people listening to things. As you have a size of stretch that's difficult to justify by approximations to JI, and there is psychoacoustical evidence for an even greater stretch, well, check the literature.

Electronic keyboard designers are probably missing something, because keyboards with piano sounds usually don't have a knob to simulate the stretching in a real piano. I know that Sound Fonts don't have a direct way to specify this. If they want to sound like a piano (and I'm sure they do) they need to use realistic piano sounds, which have the same inharmonicity as a physical piano's vibrating strings. Maybe they fudge this by setting the sample pitch wrong. But it'd also be nice if algorithmic synths could play in tune with a piano, and they can't. So being a little bit wrong can't matter much.

One psychoacoustical phenomenon of preferred scale stretching doesn't mean that all scales should be stretched. There's also psychoacoustic backing for consonance models where the most consonant octave matches the octaves in the timbre. In most cases that means 1200 cent octaves. In many cases, exact harmonic timbres are convenient for technical reasons. Even a 1200 cent octave has technical advantages. It's really easy on a digital computer to multiply and divide by two, and use lookup tables for pitches within the octave. So 1200 cent octaves might still be optimal in the general case despite a tendency for stretched octaves in melody. Poor octaves in harmony are more noticeable than slight deviations from your personal interval template (which differs from everybody else's anyway).

The amount of stretch mentioned is smaller than I thought before, BTW. The percentages in that page simply mean there's a 50 cent stretch over several octaves. Divide that through, and you get a stretch of several cents per octave. Read the other pages, and I'm sure you'll find that the amount of stretch depends on both the listener and the timbre, with some listeners preferring shrunk octaves. Terhardt makes it clear that he isn't suggesting an alternative octave standard, but saying that no fixed tuning can be universally optimal.

As 1200 cents is within the acceptable range, you may as well use that as the standard. Having a standard means that different instruments that are tuned to be the same in one register will still be the same in others.

There's still a fine tuning knob so that you can adjust the tuning of instruments that only play in a restricted range. A lot of the psychoscoustic stretching is simply about high pitched instruments being slightly higher than you would have otherwise thought. As the lead parts are usually relatively high pitched, making them a bit sharp also helps them to stand out.

Graham

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

> I thought Terhardt said that pure tones lead to octave shrinkage

Nope; octave stretch.

> He certainly has experimental backing.

Huh? All the experimental backing I've seen, including in Terhardt's
papers, supports octave stretch for sine waves.

In other words, when you play a pure 2:1 melodically with sine waves
(particularly in the low or high registers), the upper tone sounds
like a lower pitch class. The 2:1 needs to be stretched in order for
the tones to sound like the same pitch class.

> And he
> says, of somebody else's experiments, "Although these instruments
> produce truly harmonic complex tones, a stretch of the tone scale
was
> found that resembles that of the piano - with the only exception
that
> even the middle octaves were not unstretched." That doesn't look
like
> below 1kHz.

I don't know what you're referring to, but if Terhardt was somehow
unaware of the inharmonicity of the piano, there are much better
sources on this, such as a couple of pages in Hall's textbook.

> > Yes, there is clearly an inconsistency here -- if the stretch has
> > psychoacoustic reasons, electronic keyboards would need it, too,
but
> > you rarely if ever see electronic keyboards with stretched
octaves.
> > If it is physical, from the piano's vibrating strings, then the
> > electronic keyboards would have no need for stretching. Either
> > electronic keyboard designers are missing something or the
> > psychoacoustic reasoning doesn't wash.
>
> If the psychoacoustic reasoning doesn't wash, then neither do your
> experiments. You can't really get around psychoacoustics as long
as you
> deal with people listening to things. As you have a size of
stretch
> that's difficult to justify by approximations to JI, and there is
> psychoacoustical evidence for an even greater stretch, well, check
the
> literature.

Now I don't know what *either* of you are saying.

> One psychoacoustical phenomenon of preferred scale stretching
doesn't
> mean that all scales should be stretched.

Right -- especially if you're using timbres radically different from
those in the given experiment. Use timbres with a strong series of
partials resembling a *compressed* harmonic series, and surely your
experiment will find a preference *against* stretching.

>> What timbres does your applet use? According to the literature,
>> that's critical in determining the octave stretch. The brighter
>> the timbre, the greater the stretch.
>
>Quite the contrary -- if you have harmonic partials, then the brighter
>the timbre, the less the stretch. With sine waves, the darkest timbre,
>you want the most stretch.

That's been my experience.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Meanwhile, I'm intrigued to learn that Gene Ward Smith is in fact Ken
> Ribet. Or is the latter just using the former's e-mail address??

This is news to me. Ken was my thesis advisor. He's probably most
famous for proving the "epilon conjecture", to the effect that
Taniyama-Shimura entails Fermat's Last Theorem. When Ken proved that,
Wiles set out to prove Taniyama-Shimura, which is actually a much more
important theorem.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> I strongly disagree. The diatonic scale was used in the West long
> before functional or triadic harmony, and ancient Greece before that
> (isn't that where the name 'diatonic' comes from), not to mention in
> other parts of the world throughout history. There's no question that
> the 7-note Pythagorean chain is a diatonic scale.

You were the guy who chewed me out for calling the Zarlino/Ptolemy
scale "JI diatonic" once.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Meanwhile, I'm intrigued to learn that Gene Ward Smith is in fact
Ken
> > Ribet. Or is the latter just using the former's e-mail address??
>
> This is news to me. Ken was my thesis advisor.

Oh, I got seriously confused when you wrote:

/tuning-math/message/12761

"Ken Ribet at Berkeley here."

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > I strongly disagree. The diatonic scale was used in the West long
> > before functional or triadic harmony, and ancient Greece before
that
> > (isn't that where the name 'diatonic' comes from), not to mention
in
> > other parts of the world throughout history. There's no question
that
> > the 7-note Pythagorean chain is a diatonic scale.
>
> You were the guy who chewed me out for calling the Zarlino/Ptolemy
> scale "JI diatonic" once.

Umm . . . what was the full context of that, Gene? Are you sure
you're remembering quite correctly and completely? I recall something
about Western common practice being referenced in that conversation,
which is quite a bit narrower than diatonicity in general.

My online papers "Gentle Introduction to Periodicity Blocks" and "The
Forms Of Tonality" both refer to 5-limit JI diatonic scales, as
does "Tuning, Tonality, and Twenty-Two Tone Temperament" in a
footnote.

And, not that it's necessarily a defining point, but the Pythagorean
diatonic scale doesn't have any wolf fifths.

At 10:21 PM 10/6/2005, you wrote:
>--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
>wrote:
>>
>> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
>wrote:
>>
>> > Meanwhile, I'm intrigued to learn that Gene Ward Smith is in fact
>Ken
>> > Ribet. Or is the latter just using the former's e-mail address??
>>
>> This is news to me. Ken was my thesis advisor.
>
>Oh, I got seriously confused when you wrote:
>
>/tuning-math/message/12761
>
>"Ken Ribet at Berkeley here."

:)

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> I think you may want to reconsider your methods -- to me the fact that
> different modes lead to different results is evidence that you're doing
> something "wrong", IMHO.

Yes, that's why I was surprised by the results when I checked the
different modes -- I was expecting the least-squares technique to be
mode-independent, but it wasn't.

> I believe that you really want to optimize are
> the *intervals*, not the pitches, in the scale, and the intervals are
> exactly the same no matter what mode you choose.

I *am* optimizing intervals, not pitches. I base the least-squares
computation on logarithms of interval ratios, not on frequencies.
Here is a sample function for computing the stretch factor by
least-squares:

f(x)=(ln(4/3)-5*ln(2*x)/12)^2+(ln(3/2)-7*ln(2*x)/12)^2+(ln(x))^2

The critical number for this function gives a minimum value for f(x),
and then x can be translated into cents. This function just includes
the Pythagorean fourth (4/3), fifth (3/2), and octave (2:1) terms (the
octave term simplifies and the unison 1:1 term is zero).

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > I think you may want to reconsider your methods -- to me the fact
that
> > different modes lead to different results is evidence that you're
doing
> > something "wrong", IMHO.
>
> Yes, that's why I was surprised by the results when I checked the
> different modes -- I was expecting the least-squares technique to be
> mode-independent, but it wasn't.
>
> > I believe that you really want to optimize are
> > the *intervals*, not the pitches, in the scale, and the intervals
are
> > exactly the same no matter what mode you choose.
>
> I *am* optimizing intervals, not pitches.

It can't be the case that you're optimizing the intervals in the
scale, since all the modes of the diatonic scale have the same
intervals.

> I base the least-squares
> computation on logarithms of interval ratios, not on frequencies.
> Here is a sample function for computing the stretch factor by
> least-squares:
>
> f(x)=(ln(4/3)-5*ln(2*x)/12)^2+(ln(3/2)-7*ln(2*x)/12)^2+(ln(x))^2

I don't understand why you have 2*x instead of x for the first two
terms above, and it also looks like you're missing a "ln(2)-" in the
last term.

> The critical number for this function gives a minimum value for f
(x),
> and then x can be translated into cents. This function just
includes
> the Pythagorean fourth (4/3), fifth (3/2), and octave (2:1) terms
(the
> octave term simplifies and the unison 1:1 term is zero).

I see only three intervals here; an eight-note diatonic scale
(spanning one octave) would have 7*8/2 = 28 intervals, and a fifteen-
note diatonic scale (spanning two octaves) would have 15*14/2 = 105
intervals.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...>
wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > I think you may want to reconsider your methods -- to me the
fact
> that
> > > different modes lead to different results is evidence that
you're
> doing
> > > something "wrong", IMHO.
> >
> > Yes, that's why I was surprised by the results when I checked the
> > different modes -- I was expecting the least-squares technique to
be
> > mode-independent, but it wasn't.
> >
> > > I believe that you really want to optimize are
> > > the *intervals*, not the pitches, in the scale, and the
intervals
> are
> > > exactly the same no matter what mode you choose.
> >
> > I *am* optimizing intervals, not pitches.
>
> It can't be the case that you're optimizing the intervals in the
> scale, since all the modes of the diatonic scale have the same
> intervals.
>
> > I base the least-squares
> > computation on logarithms of interval ratios, not on frequencies.
> > Here is a sample function for computing the stretch factor by
> > least-squares:
> >
> > f(x)=(ln(4/3)-5*ln(2*x)/12)^2+(ln(3/2)-7*ln(2*x)/12)^2+(ln(x))^2
>
> I don't understand why you have 2*x instead of x for the first two
> terms above, and it also looks like you're missing a "ln(2)-" in
the
> last term.
>
> > The critical number for this function gives a minimum value for f
> (x),
> > and then x can be translated into cents. This function just
> includes
> > the Pythagorean fourth (4/3), fifth (3/2), and octave (2:1) terms
> (the
> > octave term simplifies and the unison 1:1 term is zero).
>
> I see only three intervals here; an eight-note diatonic scale
> (spanning one octave) would have 7*8/2 = 28 intervals, and a
fifteen-
> note diatonic scale (spanning two octaves) would have 15*14/2 = 105
> intervals.

Now that I wrote that, I see that different modes could indeed give
different optimal tunings, if you are only including the notes from
one tonic to a higher-octave tonic in each case, the tonic in each
case being defined by the mode in question.

When the octave is meant to remain just, though, one only has to deal
with a finite number of pitch-classes, and optimizing on these should
indeed yield the same tuning regardless of mode.

In my own investigations into tempered-octave tunings (which I'd love
to mail to you if you give me your full address), I in effect
consider *all* intervals, no matter how wide -- so again mode doesn't
matter.

Hence, I incorrectly assumed mode wouldn't matter for you.

I'm still unsure of the exact formulae you are using. Here's an
example of how *I* did a least-squares optimization for the diatonic
scale given fixed pure octaves -- from someone who did the same thing
as me over a century earlier:

http://sonic-arts.org/monzo/woolhouse/essay.htm

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> It can't be the case that you're optimizing the intervals in the
> scale, since all the modes of the diatonic scale have the same
> intervals.

I am optimizing based on intervals over tonic -- i.e. using the
following Pythagorean ratios for Ionian and Aeolian modes:

Ionian: 1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1
Aeolian: 1:1 9:8 32:27 4:3 3:2 128:81 16:9 2:1

> > I base the least-squares
> > computation on logarithms of interval ratios, not on frequencies.
> > Here is a sample function for computing the stretch factor by
> > least-squares:
> >
> > f(x)=(ln(4/3)-5*ln(2*x)/12)^2+(ln(3/2)-7*ln(2*x)/12)^2+(ln(x))^2
>
> I don't understand why you have 2*x instead of x for the first two
> terms above, and it also looks like you're missing a "ln(2)-" in the
> last term.

The independent variable x is the factor by which an octave is to be
stretched or shrunk -- in any case, x is near 1, >1 if stretched, <1
if shrunk, and I can determine the exact stretch or shrink in cents by
first finding x then solving x=2^(q/1200) for q. If I just used ln(x)
instead of ln(2*x), then the left side of this equation would be the
interval ratio between octave and x, i.e. x/2. Easier to just use
ln(2*x) in the function in the first place. In the last term, written
out in all its glory it would be (ln(2)-12*ln(2*x)/12)^2, but this
simplifies via properties of logarithms to (ln(x))^2.

> > The critical number for this function gives a minimum value for f
> (x),
> > and then x can be translated into cents. This function just
> includes
> > the Pythagorean fourth (4/3), fifth (3/2), and octave (2:1) terms
> (the
> > octave term simplifies and the unison 1:1 term is zero).
>
> I see only three intervals here; an eight-note diatonic scale
> (spanning one octave) would have 7*8/2 = 28 intervals, and a fifteen-
> note diatonic scale (spanning two octaves) would have 15*14/2 = 105
> intervals.

This was just an example including 4:3 fourth, 3:2 fifth, and 2:1
octave. This gave a rather small stretch, around 0.27 cents. More
complete computation would include all of the Pythagorean ratios for a
mode. Using all of the ratios for Ionian mode, unweighted, gave more
than 4 cents stretch. For comparison, getting the ET scale to match
exactly the 3:2 fifth gave a stretch of exactly 1/7 of a Pythagorean
comma, about 3.3 cents.

> Now that I wrote that, I see that different modes could indeed give
> different optimal tunings, if you are only including the notes from
> one tonic to a higher-octave tonic in each case, the tonic in each
> case being defined by the mode in question.

Yes, that is how I set up the optimization -- notes within one octave,
from tonic to tonic, based on interval ratios over tonic, no absolute
frequencies or pitches.

> When the octave is meant to remain just, though, one only has to
> deal with a finite number of pitch-classes, and optimizing on these
> should indeed yield the same tuning regardless of mode.

But the point of my investigation was to see if least-squares could
justify an octave which was *not* a strict 2:1 JI octave.

> In my own investigations into tempered-octave tunings (which I'd
> love to mail to you if you give me your full address), I in effect
> consider *all* intervals, no matter how wide -- so again mode
> doesn't matter.

Paul Erlich offered to ground-mail me some stuff, too, but I don't
want to post my ground-mail address on the group. As soon as I track
down e-mail addresses for both of you, I will directly e-mail you my
ground-mail address.

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > It can't be the case that you're optimizing the intervals in the
> > scale, since all the modes of the diatonic scale have the same
> > intervals.
>
> I am optimizing based on intervals over tonic

Aha -- then you're neglecting a whole lot of musically important
intervals! Most of them don't involve the tonic at all.

> -- i.e. using the
> following Pythagorean ratios for Ionian and Aeolian modes:
>
> Ionian: 1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1
> Aeolian: 1:1 9:8 32:27 4:3 3:2 128:81 16:9 2:1

Whether anyone can detect justness or deviation in a ratio as complex
as 32:27, let alone 128:81, is another question -- perhaps we should
return to that later.

> > > I base the least-squares
> > > computation on logarithms of interval ratios, not on
frequencies.
> > > Here is a sample function for computing the stretch factor by
> > > least-squares:
> > >
> > > f(x)=(ln(4/3)-5*ln(2*x)/12)^2+(ln(3/2)-7*ln(2*x)/12)^2+(ln(x))^2
> >
> > I don't understand why you have 2*x instead of x for the first
two
> > terms above, and it also looks like you're missing a "ln(2)-" in
the
> > last term.
>
> The independent variable x is the factor by which an octave is to be
> stretched or shrunk

Aha!

> > > The critical number for this function gives a minimum value for
f
> > (x),
> > > and then x can be translated into cents. This function just
> > includes
> > > the Pythagorean fourth (4/3), fifth (3/2), and octave (2:1)
terms
> > (the
> > > octave term simplifies and the unison 1:1 term is zero).
> >
> > I see only three intervals here; an eight-note diatonic scale
> > (spanning one octave) would have 7*8/2 = 28 intervals, and a
fifteen-
> > note diatonic scale (spanning two octaves) would have 15*14/2 =
105
> > intervals.
>
> This was just an example including 4:3 fourth, 3:2 fifth, and 2:1
> octave. This gave a rather small stretch, around 0.27 cents. More
> complete computation would include all of the Pythagorean ratios
for a
> mode. Using all of the ratios for Ionian mode, unweighted, gave
more
> than 4 cents stretch. For comparison, getting the ET scale to match
> exactly the 3:2 fifth gave a stretch of exactly 1/7 of a Pythagorean
> comma, about 3.3 cents.
>
> > Now that I wrote that, I see that different modes could indeed
give
> > different optimal tunings, if you are only including the notes
from
> > one tonic to a higher-octave tonic in each case, the tonic in each
> > case being defined by the mode in question.
>
> Yes, that is how I set up the optimization -- notes within one
octave,
> from tonic to tonic, based on interval ratios over tonic, no
absolute
> frequencies or pitches.

But I still think you should include the other intervals too -- those
that don't involve the tonic -- too, unless you're dealing with a
form of Eastern, drone-based music.

BTW, pitch-ratios are the same as ratios with respect to the tonic
(1/1), so when I said "pitches", I meant the same thing as intervals
from the tonic. IMO, almost all Western music, from whatever era,
would be better assessed using *all* the intervals, not just those
formed against the tonic.

> > When the octave is meant to remain just, though, one only has to
> > deal with a finite number of pitch-classes, and optimizing on
these
> > should indeed yield the same tuning regardless of mode.
>
> But the point of my investigation was to see if least-squares could
> justify an octave which was *not* a strict 2:1 JI octave.

Of course. I was just trying to give you some background.

> > In my own investigations into tempered-octave tunings (which I'd
> > love to mail to you if you give me your full address), I in effect
> > consider *all* intervals, no matter how wide -- so again mode
> > doesn't matter.
>
> Paul Erlich

That's me.

> offered to ground-mail me some stuff, too, but I don't
> want to post my ground-mail address on the group.

Your first post had it except for the last line. I could probably
figure out the rest if I really needed to.

> As soon as I track
> down e-mail addresses for both of you, I will directly e-mail you my
> ground-mail address.

Both of you? I'm not sure why my posts would appear to be coming from
different people (?). But you can just e-mail me by clicking on my
name / truncated e-mail address right on the Yahoo page where you're
probably reading this.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> Both of you? I'm not sure why my posts would appear to be coming from
> different people (?). But you can just e-mail me by clicking on my
> name / truncated e-mail address right on the Yahoo page where you're
> probably reading this.

I'm sorry -- for some reason I thought that last note came from Gene
Smith. My mistake. As far as the snail-mail address, it would be
better to send to my home address instead of my work address, just so
that I don't have to wait around for campus mail to finally get it to
me. As far as clicking on things in the message board, when I click
on your name I get your profile, which does not include your e-mail
address, and when I click on the e-mail icon by your name on the
message list, I am told that I am posting a message on the board, not
sending an e-mail. The "truncated" form that shows up in the
"so-and-so wrote:" lines of messages isn't enough for me to send an
e-mail. I'll tell you what: My e-mail address is leathrum(at)jsu.edu
-- e-mail me something, and I can get the return e-mail address off
that, then I can e-mail you my ground address.

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> In my own investigations into tempered-octave tunings (which I'd love
> to mail to you if you give me your full address), I in effect
> consider *all* intervals, no matter how wide -- so again mode doesn't
> matter.

I did some more number-crunching today -- if I also include some of
the other Pythagorean intervals (256/243 half-step, 32/27 minor third,
128/81 aug. 5th, 16/9 aug. 6th), but equal weight on all intervals, I
get about 1.45 cents stretch. Using super-octave intervals (e.g. a
9/4 ninth) would be effectively the same as assigning greater weight
to a sub-octave interval (9/8 whole step for the example). There are
a couple of problematic intervals, too, like a different half-step
(one Pythagorean comma short of the 256/243 half-step) and two options
for the aug. 4th / dim. 5th interval ratio (differing by a comma).
For this computation, I didn't include any of these. I tried coming
up with some weights for the ratios based on how many times the
intervals appear as intervals over tonic in different modes, to try to
reduce the mode-dependence that way, and I got very small stretches
(0.1 cents if I don't include Lydian and Locrian, which have those
strange aug. 4th and
dim. 5th intervals, and 0.2 cents if I do).

Regards,
Tom Leathrum

--- In tuning-math@yahoogroups.com, "leathrum" <leathrum@j...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > Both of you? I'm not sure why my posts would appear to be coming
from
> > different people (?). But you can just e-mail me by clicking on
my
> > name / truncated e-mail address right on the Yahoo page where
you're
> > probably reading this.
>
> I'm sorry -- for some reason I thought that last note came from Gene
> Smith. My mistake. As far as the snail-mail address, it would be
> better to send to my home address instead of my work address, just
so
> that I don't have to wait around for campus mail to finally get it
to
> me. As far as clicking on things in the message board, when I click
> on your name I get your profile, which does not include your e-mail
> address, and when I click on the e-mail icon by your name on the
> message list, I am told that I am posting a message on the board,
not
> sending an e-mail.

Don't worry -- you aren't. Here, I'll e-mail you using this method
and then we should be all set to being our off-list correspondence.