Re: Schismas, tuning systems, and tastes (Graham)

In a recent Tuning Digest, graham@microtonal.co.uk writes:

> Maybe this is a trivial point, but that's never stopped me before.
> The septimal schisma of 3.8 cents is a fairly arbitrary interval.
> Simpler ratios are 225/224 and 5120/5103, which are between 5 and 8
> cents.

Hello, there, Graham and everyone; I hope that in what follows I may
show some moderation in discussing a very interesting point: the way
that a feature of a given tuning can seem "arbitrary" from one
viewpoint and "natural" from another.

In tuning systems, as in computer languages and operating systems, one
person may describe as "fairly arbitrary" what another finds "natural
and indeed elegant." As with music itself, so with tuning systems and
theories, this is a matter of taste as well of "logic" or of
"objective performance," for example in approximating a given interval
ratio.

Also, sometimes "arbitrary" can be a synonym for "less familiar";
being unfamiliar with a ratio such as 225:224, I may find it
"arbitrary," and someone else may react in the same way to what is for
me a "familiar and elegant" Pythagorean septimal schisma.

Maybe the matter of the 5120:5103, and my reaction to it, may give
some clue as to how these personal tastes and associations (musical as
well as mathematical) can give color to one's theoretical
response. When I first saw these numbers, I was a bit unsure what they
meant; but when I ran them through GNU Emacs Calc, and found that they
defined an interval of about 5.7578 cents, I recognized that
measurement in cents as something like the difference between a
familiar (to me) Pythagorean diminished fifth (1024:729) and a 7:5.
When I divided 1024:729 by 7:5 (a handy Emacs Calc feature for tuning
enthusiasts), I indeed got 5120:5103.

Here my first musical reaction from a medievalist view is that the
diminished fifth, although it certainly occurs in Gothic or modern
Xeno-Gothic music, may be less important for directed verticality
because it is the one unstable interval which cannot resolve to a
stable 3-limit interval by stepwise contrary motion. Seconds, thirds,
sixths, and sevenths are the unstable intervals which play a leading
role in directed cadential progressions -- and all of these intervals
have septimal schisma versions only 3.80 cents or so from 7-based
values.

Further, in finding this septimal schisma of 3.80 cents "natural"
rather than "arbitrary," I should emphasize that I approach the
problem mainly from a perspective of Xeno-Gothic tuning. The purpose
of this "neo-medieval" Pythagorean or 3-limit just intonation (JI)
tuning is not to approximate 5-limit or 7-limit JI, but to provide the
usual Pythagorean intervals plus some interesting variants a
Pythagorean comma wider or narrower.

While it may be poetic to speak of "xenharmonic bridges" as both Joe
Monzo and I do, maybe the term "commatic variants" would be more
descriptive of what these extra Xeno-Gothic intervals are intended to
do. If the septimal schisma were actually 5 cents or even 10 cents
rather than 3.8 cents, this wouldn't bother me; I like the idea of an
extended Pythagorean tuning with commatic variants, whether or not
these variants happen to match very closely intervals produced in some
other kind of JI system.

Here there may be a difference of taste among people in our Tuning
List community on the mathematical as well as musical level.
Personally I love "large-integer JI," and the septimal schisma of
Xeno-Gothic might be an example par excellence. I find beauty in
having a simple linear chain of fifths providing such a variety of
intervals, such a subtle range of ratios. Here we might apply a phrase
which is also the title of one of my favorite childhood books: _The
Lore of Large Numbers_.

For other people, who are oriented to "small-integer JI,"
approximations involving "simpler" ratios may seem more elegant even
if they are also less accurate.

For example, the septimal schisma of 33554432:33480783 (the
difference, for example, between a Pythagorean minor seventh a comma
smaller than 16:9, 8388608:4782969, and a 7:4) is about 3.80 cents. In
comparison, as already noted, a 5120:5103 is about 5.76 cents; and a
225:224 is about 7.71 cents.

Thus the septimal schisma is indeed a _closer_ approximation than
either of these "simpler" ratios. As you observe, it is, however, not
so close an approximation as the Pythagorean 5-limit schisma of
32805:32768 (~1.95 cents).

Comparing these approximations raises another knotty question:
"how close is close" in approaching simple integer ratios in different
odd-limits? While this may not be a big issue in Xeno-Gothic with its
"commatic variants" seen as alternate flavors of the usual unstable
Pythagorean intervals, with septimal schisma intervals inviting
"superefficient" resolutions to 3-limit stability, it may be an
important issue if someone is using Pythagorean to approximate 7-limit
JI.

For example, if we consider a 3:2 fifth, then the Pythagorean 5-limit
schisma of ~1.95 cents is almost identical to the tempering of the
fifth in 12-tone equal temperament (12-tet). The septimal schisma of
~3.80 cents is comparable (on the largish side) to 1/6-comma meantone;
a 5120:5103 of ~5.76 cents to something between 1/4-comma and
2/7-comma; and a 225:224 at 7.71 cents to 1/3 _Pythagorean_ comma,
about the limit (in some 18th-century well-temperaments) of historical
practice in the European compositional tradition.

However, this is for a 3:2 fifth; for a 5:4 third, I would consider
3.8 cents a quite close approximation, and might guess that for a 7:4
or 9:7, etc., it would be quite close also. However, someone seeking
accurate 7-limit JI as opposed to commatic variants on Pythagorean
intervals might say "not close enough" -- and it has been commented
that even the tuning errors of conventional microtunable synthesizers
(on the order of 1-2 cents) may be "not good enough" for JI at its
most exacting.

There has been an interesting discussion here about the "zone of
attraction" for various intervals such as 5:4 or 9:7, and the various
schismas might be considered as part of this open dialogue.

However, at least for me, "objective" measures of "accuracy" in
approximating a given interval may not be the decisive factors in
choosing a given tuning system. I was an enthusiastic exponent of
Pythagorean tuning for medieval music and modern offshoots long before
John Chalmers informed me of its quasi-septimal approximations; and I
am also fascinated by the corresponding intervals in 17-tet, much
further removed from 7-based ratios (e.g. major third, ~423.5 cents).

For other people, 225:224 or 5120:5203 may be intervals of prime
theoretical importance, intervals which fit into some elegant system
and thus share this elegance. In such a system, the Pythagorean
septimal schisma may seem like an "odd" or less elegant approach.

Thus your response to my comments on the septimal schisma may
illustrate how we evaluate intervals based on such factors as musical
setting and theoretical taste: this theme may be a very nontrivial one
as the Tuning List provides an opportunity to discuss these diverse
musics and tuning systems.

Most appreciatively,

Margo Schulter
mschulter@value.net

Margo Schulter wrote;

> Maybe the matter of the 5120:5103, and my reaction to it, may give
> some clue as to how these personal tastes and associations (musical as
> well as mathematical) can give color to one's theoretical
> response. When I first saw these numbers, I was a bit unsure what they
> meant; but when I ran them through GNU Emacs Calc, and found that they
> defined an interval of about 5.7578 cents, I recognized that
> measurement in cents as something like the difference between a
> familiar (to me) Pythagorean diminished fifth (1024:729) and a 7:5.
> When I divided 1024:729 by 7:5 (a handy Emacs Calc feature for tuning
> enthusiasts), I indeed got 5120:5103.

Yes, that explains a lot.

I had no idea that the 3.8 cent schisma is the difference between a Pythagorean
interval and the 7:4. I don't usually look at intervals that small. I prefer
to think in terms of commas. The interval 7:4 is flat of 16:9 by the septimal
comma of 64:63. That's a 27.3 cent interval.

Septimal schismic temperament is defined by setting this comma to be equal to
both the syntonic comma of 81:80 or 21.5 cents and the Pythagorean comma 23.5
cents (I don't know the ratio, and can't be bothered to work it out).

I know that 7:4 is a comma flat of 16:9, and 8:5 is a comma sharp of 16:9. I
also know that 7:6, 32:27 and 6:5 are each separated by commas, as are 5:4,
81:64 and 9:7. Although I could work out the number of steps on the spiral of
fifths that correspond to all these intervals, the equivalent Pythagorean
intervals don't interest me at all. In fact, I had to work out 81:64 for this
post: I only know it as the Pythagorean major third, a comma above the just
major third.

So a schisma is defined as the interval which is equivalent to a unison in a
schismic temperament. The difference between two commas will a ways be a
schisma. So, if you define your commas, the schismas take care of themselves.
As they're all approximated away, they aren't of themselves at all interesting
in a schismic temperament.

Approximating to a unison is a bit like setting equal to zero. So, as 2*0=0,
that means adding or subtracting schismas will always leave you with more
schismas. Hence 225:224, 5120:5103, and the one at 3.8 cents are all separated
by the usual 5-limit comma. This is why the choice is arbitrary. The smaller
they get (or larger from 225:224) the more complex. So, as they're all
equivalent, and we need two to define the temperament, I take the two simplest
ones. But it doesn't matter which you choose: the result is the same. all
commas are equal.

It may be that a temperament that zeros the larger schismas would be more
accurately described as kleismic. But if I called it that, you can get people
would think it had something to do with Klezma.

Right, let's find out where these schismas come from. One must be the
difference between a septimal and syntonic comma.

64/63*80/81 = 5120/5103

What a suprise. The 3.8 cent schisma is a 5-limit schisma flat of this, and a
Pythagorean comma is a 5-limit schisma sharp of a syntonic comma, so the 3.8
cent schisma must be the difference between the Pythagorean and septimal commas.
I don't even need to work out the ratios! (Of course, knowing the cent values
helps). 225/224, then, isn't the difference between any named commas, but is
still dimensionally a schisma.

> Here my first musical reaction from a medievalist view is that the
> diminished fifth, although it certainly occurs in Gothic or modern
> Xeno-Gothic music, may be less important for directed verticality
> because it is the one unstable interval which cannot resolve to a
> stable 3-limit interval by stepwise contrary motion. Seconds, thirds,
> sixths, and sevenths are the unstable intervals which play a leading
> role in directed cadential progressions -- and all of these intervals
> have septimal schisma versions only 3.80 cents or so from 7-based
> values.

I go by the odd-limit, and consider all 7-limit intervals of equal importance.
The 3.8 cent schisma is the only septimal schisma with no factors of 5, which
may be relevant.. It would be more accurate to say I'm interested in the
9-limit, but I'll keep talking 7-limit for simplicity.

I find it hard to see why, with 17 notes available, you wouldn't be pulled away
from the Pythagorean intervals, and towards the 5- or 7-limit. Quite likely
the European tradition would have moved the same way, had 17 notes become
popular, but it's fruitless to speculate.

> Further, in finding this septimal schisma of 3.80 cents "natural"
> rather than "arbitrary," I should emphasize that I approach the
> problem mainly from a perspective of Xeno-Gothic tuning. The purpose
> of this "neo-medieval" Pythagorean or 3-limit just intonation (JI)
> tuning is not to approximate 5-limit or 7-limit JI, but to provide the
> usual Pythagorean intervals plus some interesting variants a
> Pythagorean comma wider or narrower.

Is that right? I thought Xeno-Gothic was 7-limit schismic with Pythagorean
tuning -- or untempered schismic temperament, so to speak.

> While it may be poetic to speak of "xenharmonic bridges" as both Joe
> Monzo and I do, maybe the term "commatic variants" would be more
> descriptive of what these extra Xeno-Gothic intervals are intended to
> do. If the septimal schisma were actually 5 cents or even 10 cents
> rather than 3.8 cents, this wouldn't bother me; I like the idea of an
> extended Pythagorean tuning with commatic variants, whether or not
> these variants happen to match very closely intervals produced in some
> other kind of JI system.

Well, if it reached 10 cents it would be getting on for half a comma, and so
ambiguous. But yes, there are plenty of interesting intervals that don't match
simple ratios. Pythagorean thirds are important in schismic tuning, and can be
used to add a bit of spice to 5-limit music, or make 7-limit music more
ambiguous, or simple to get the melodies "right". I consider the relative
simplicity of there ratios unimportant: they're beyond the threshold where the
resulting consonance will be audible.

> Here there may be a difference of taste among people in our Tuning
> List community on the mathematical as well as musical level.
> Personally I love "large-integer JI," and the septimal schisma of
> Xeno-Gothic might be an example par excellence. I find beauty in
> having a simple linear chain of fifths providing such a variety of
> intervals, such a subtle range of ratios. Here we might apply a phrase
> which is also the title of one of my favorite childhood books: _The
> Lore of Large Numbers_.
>

> For other people, who are oriented to "small-integer JI,"
> approximations involving "simpler" ratios may seem more elegant even
> if they are also less accurate.

That's the stuff! Although I still keep the linear chain of fifths, but temper
them. This has a conceptual beauty, rather than being something I'd expect to
be audible.

> However, this is for a 3:2 fifth; for a 5:4 third, I would consider
> 3.8 cents a quite close approximation, and might guess that for a 7:4
> or 9:7, etc., it would be quite close also. However, someone seeking
> accurate 7-limit JI as opposed to commatic variants on Pythagorean
> intervals might say "not close enough" -- and it has been commented
> that even the tuning errors of conventional microtunable synthesizers
> (on the order of 1-2 cents) may be "not good enough" for JI at its
> most exacting

My experience so far, after noodling on a 29-note schismic keyboard, is that
the 7-limit intervals only come into their own when the fifth is tempered very
close to the point where the 7:4 becomes just. I don't know exactly why this
is, although it does represent the minimax temperament. In the 7-limit, it's
comparable to 1/4-comma meantone, but the improved fifths do make a difference.
In the 9-limit it's that bit better.

With Pythagorean tuning, the 7-limit intervals are close enough to be 7-limit,
but I find they don't really "ring" as consonances. I don't know why this is:
I prefer meantones with theoretically poorer approximations. Something magic
happened when the 7:4 became just, and it may be have been some kind of
artifact. I will investigate more sometime.

> Thus your response to my comments on the septimal schisma may
> illustrate how we evaluate intervals based on such factors as musical
> setting and theoretical taste: this theme may be a very nontrivial one
> as the Tuning List provides an opportunity to discuss these diverse
> musics and tuning systems.

Yes, it is an interesting topic, and there does seem to be a cultural
divergence. But I'll leave it here, as I have a bus to catch.

Graham

"I think therefore I toss" -- Sartre