source file: m1545.txt
Date: Sun, 4 Oct 1998 22:29:07 -0700 (PDT)

Subject: Re: Paul Erlich's "entropy" and 16:9

From: "M. Schulter" <mschulter@value.net>

Hello, there.

Paul Erlich's post on "harmonic entropy, neutral zones" leads me to
ask a curious question and offer a few odd observations on Pythagorean
and "neo-Pythagorean" tunings.

The whole "neutral zone" idea, extended from thirds to the whole
spectrum of the intervals, is a really creative one, and I'd like to
express my appreciation for the effort that went into both the
conceptualization and implementation of this study.

Please let me emphasize that my expression of surprise at one point
below (the question of 16:9) isn't meant in any way as a criticism of
the method (sometimes an electronic medium can make such nuances
ambiguous), only an expression of pleasurable puzzlement and
anticipation of an interesting reply in one direction or another,
maybe an unsuspected one.

> The local minima and maxima were as follows (maxima denoted with *):

> N=80:

> 264 (7/6=267)
> *285
> 316 (6/5=316)

Curiously 17-tet, with m3 at about 282.4 cents, comes quite close to
the maximum here. A 24-note Xeno-Gothic tuning, with two 12-note
Pythagorean keyboards a Pythagorean comma (about 23.46 cents) apart,
offers a "minimal 3rd" of about 270.67 cents, only what I call a
"septimal schisma" of about 3.80 cents from a just 7:6. The usual
Pythagorean m3 at 32:27, or about 294.13 cents, is actually not too
far from the "maximum."

> 387 (5/4=386)
> *423
> 437 (9/7=435)

Interestingly, a 17-tet M3 at about 423.5 cents at almost exactly at
the point of "maximum." A Xeno-Gothic "maximal third" of about 431.28
cents is again only 3.8 cents or so from a just 9:7. This time the
usual Pythagorean M3 of 81:64 or about 407.82 cents is a bit further
from the "maximum" point.

> 885 (5/3=884)
> *924

This time, the 17-tet M6 at around 917.6 cents is a bit further from
the "maximum," but still quite close. The Xeno-Gothic "maximal sixth"
is at about 929.33 cents, again about 3.80 cents from a just 12:7 at
around 933.12 -- I notice that 12:7 isn't on the chart, although 7:6
is, of course.. The regular Pythagorean M6 at 27:16 (about 905.87
cents) is about halfway between your minimum around 5:3 and your
maximum around 924 cents.

> 970 (7/4=969)
> *999
> 1021 (9/5=1018)

My immediate medievalist reaction: shouldn't 16:9 be listed here as a
basic ratio at 996 cents? Somehow my Pythagorean predilections find it
curious that this interval should be treated mainly as a point of
maximum ambiguity between the 5-prime and 7-prime standards, rather
than a 3-prime standard in its own right. Also, this interval seems
very basic to me since it is derived from two pure 4:3 fourths.
However, these are only, of course, my biases at first blush <grin>.

Anyway, calming down, I note that the 17-tet m7 of about 988.2 cents
is in this case actually _further_ from the "maximum" point than the
regular Pythagorean 16:9 at about 996.09 cents. As usual, the
Xeno-Gothic "minimal 7th" at about 972.63 cents is only 3.80 cents or
so from a just 7:4.

[with N=40]

> 968 (7/4=969)
> *996
> 1021 (9/5=1018)

Here, interestingly, a just 16:9 is placed at just about exactly the
maximum point of entropy.

I'm curious if this result for 16:9 might say anything about the
Pythagorean m7 (which often gets used prominently in 13th-14th century
music, and which I tend to agree with some medieval theorists has a
certain degree of "compatibility" or even "concord"), or for that
matter about the 12-tone equal temperament (12-tet) m7 at 1000 cents?

Also, from the viewpoint of the analytical method involved here, I
wonder if this result for 16:9 would be an expected or unexpected one.

Anyway, the "neutral zone" concept is really intriguing, and thanks
for such an interesting survey.

Most appreciatively,

Margo Schulter
mschulter@value.net