Yahoo Tuning Groups Ultimate Backup tuning Re: Set-consistency, flavor-consistency, and 24-tET

Hello there to Todd Wilcox and Paul Erlich, whose points lead me to
offer a few comments about musical and intonational pluralism, and
about "consistency" as a kind of transitivity property which could
apply to various ideals or classificatory schemes for intervals.

Focusing on 24-tET, since this is one of the better-known equal
temperaments and also a topic of discussion in this thread, I would
like to propose concepts of "set-consistency" and "flavor-consistency"
for musics based on just intonation (JI) sets with non-contiguous
prime factors (e.g. ratios of 2-3-7, but not 5), or on a spectrum of
intonational "flavors."

In a neo-Gothic setting, for example, here's a summary of the range of
intonational "flavors" of major thirds I use, also including neutral
thirds which divide a fifth into roughly or often precisely equal
parts:

--------------------------------------------------------------------
flavor ratio zone rough size description/example
--------------------------------------------------------------------
49-flavor 49:40 ~351 cents neutral 3rd -- 17-tET
17-flavor 21:17 ~366 cents submajor 3rd -- 46-tET
5-flavor 5:4 ~386 cents Pythagorean schisma 3rd
3-flavor 81:64 ~408 cents Pythagorean major 3rd
11-flavor 14:11 ~418 cents regular 3rd -- 29-tET or 46-tET
23-flavor 23:18 ~424 cents regular 3rd -- 17-tET
7-flavor 9:7 ~435 cents regular 3rd -- 22-tET, 3-7 JI
13-flavor 13:10 ~454 cents "diesis 3rd" -- 29-tET

What "consistency" says to me in a general sense is simply that
intervals "add up" so as to yield the best approximations for all
desired ratios in a given sonority. As a layperson, I might call it a
kind of "transitive property": if ratios a + b = c, then the best
approximations of a + b yield the best approximation of c also.[1]

Note that this concept can be applied to whatever rational or
irrational ratios we happen to be approximating for a given style or
piece of music: it can apply to a Partchian 7-limit model, or a
Pythagorean model, or a neo-Gothic 17-flavor.

Turning to 24-tET, I would like to explore the alternative approaches
of "set-consistency" and "flavor-consistency."

Please let me emphasize that what follows is not a rigorous
demonstration or proof, only a suggestion that 24-tET _may_ be
consistent for certain interesting sets or flavors when these concepts
are applied.

To keep this discussion musically grounded, I have chosen sets and
flavors of special interest for my own musicmaking, and which have
made 24-tET an intriguing alternative for me.

Needless to say, readers can and should substitute their own favorite
alternative musics and intonational styles in applying and testing
these concepts. In neo-Gothic music, it may be the unstable nature of
thirds which invites their intonation in a wide spectrum of flavors;
but a 20th-21st century outlook seeking the emancipation of intervals
at all points of the continuum could have a similar effect quite
outside Gothic or other historical norms of stability/instability.

-------------------------------------
1. Set-consistency (2-3-7) and 24-tET
-------------------------------------

Let's suppose, for example, that we want to use 24-tET for music in a
neo-Gothic style based on ratios of 3 and 7, but not 5. We can speak
of the "set-consistency" of 24-tET for ratios of 2-3-7.

Starting with basic Pythagorean ratios of 2 and 3, we might begin by
quickly confirming that the best approximations of 3:2, 4:3, and 9:8
"add up" as expected, so that a ~4:3 fourth plus a ~9:8 major second
equal a ~3:2 fifth. We can alternatively test whether two fourths add
up to the best approximation of a 16:9 minor seventh. In the following
diagrams, I use a MIDI-style octave notation with C4 as middle C:

~3:2 ~16:9
700 1000
|-----------------------| |-------------------------|
D3 G3 A3 D3 G3 C4
0 500 700 0 500 1000
|----------------|------| |-------------|-----------|
~4:3 ~9:8 ~4:3 ~4:3
500 200 500 500

As we might have guessed, since the 24-tET fifth at 700 cents is quite
close to the Pythagorean 3:2 (~701.955 cents), these 3-based intervals
behave in a consistent manner. The 200-cent major second, to use a
medieval phrase, defines "the difference of the consonances," that is
the difference between the 500-cent fourth and the 700-cent fifth. Two
500-cent fourths likewise add up to a 1000 cent minor seventh, the
best approximation of 16:9 (~996.09 cents).

Turning to 7-based intervals within our relevant set, we now consider
8:7 and 7:4; 7:6 and 12:7; and 9:7 or 14:9.

One informal way to approach the question of consistency is to look at
two typical 7-based cadential sonorities in their closest 24-tET
approximations: 12:14:18:21 (a rounded 0-267-702-969 cents in JI), and
14:18:21:24 (a rounded 0-435-702-933 cents in JI).

To provide 24-tET spellings for examples of these sonorities, I use an
asterisk (*) to show a note raised by 50 cents, and a "d" to show a
note lowered by this same amount.

Let's look at a ~12:14:18:21 and a ~14:18:21:24 built above the same
note, C3, diagramming all six intervals of these sonorities. What we
find suggests 3-7 set-consistency while revealing a fascinating
pattern of ambiguity:

~12:14:18:21 in 24-tET ~14:18:21:24 in 24-tET
(0-250-700-950 cents) (0-450-700-950 cents)

~7:4 ~12:7
950 950
|---------------------------| |----------------------------|
C3 Ebd3 G3 Bbd3 C3 E*3 G3 A*3
0 250 700 950 0 450 700 950
|--------|----------|-------| |-----------|--------|-------|
~7:6 ~9:7 ~7:6 ~9:7 ~7:6 ~8:7
250 450 250 450 250 250
|------------------| |----------------|
~3:2 ~4:3
700 500

For each sonority, we are indeed using the best 24-tET approximations
for all six intervals, with 7-based intervals quite "consistently"
and not-so-accurately varying by about 15-19 cents from pure.

At the same time, we notice that the 250-cent interval of 24-tET
serves as a consistent approximation of _either_ an 8:7 (~231 cents)
or a 7:6 (~267 cents). Likewise, 950 cents is the best and consistent
approximation for either the 12:7 (~933 cents) of the first sonority
or the 7:4 (~969 cents) of the second sonority.

For the three-voice 4:6:7 (0-702-967 cents in JI) and 14:21:24
(0-702-933 cents in JI), in fact, the two approximations are an
identical as well as consistent 0-700-950 cents:

~7:4 ~12:7
950 950
|-------------------------| |-------------------------|
C3 G3 Bbd3 C3 G3 A*3
0 700 950 0 700 950
|----------------|--------| |----------------|--------|
~3:2 ~7:6 ~3:2 ~8:7
700 250 700 250

The distinction between these two interpretations of 0-700-950 cents
could be made by their standard contractive or expansive resolutions:
the small minor seventh and upper minor third of the first example
contracting respectively to the fifth and unison; and the wide major
sixth and upper major second of the second example expanding to the
octave and upper fourth of the resolving sonority:

Bbd3 A3 A*3 Bb3
G3 A3 G3 F3
C3 D3 C3 Bb2

(m7-5 + m3-1) (M6-8 + M2-4)

Thus it turns out that 24-tET is apparently set-consistent for ratios
of 2-3-7, although not especially accurate for 7-based intervals -- an
"inaccuracy" inviting the kind of musical puns for which 12-divisible
n-tET's are famous.

This ambiguity, as compared to a JI or near-JI system for the relevant
set of ratios, might be seen as a special attraction of 24-tET.

A theorist who takes simple JI ratios as the central focus for
psychoacoustics might characterize 24-tET as "consistent/ambiguous for
ratios of 2-3-7," and point to the ambiguity of the system as a prime
artistic virtue, an artful distortion, as it were.

At the same time, if we are seeking in a given context to realize
rather than to neutralize the intonational distinction between a 12:7
and a 7:4, for example, looking at accuracy as well as set-consistency
suggests that other tunings might be more likely choices.

If looking for a 24-note tuning with the symmetry of a 12-divisible
equal temperament, we might consider 24-out-of-36-tET with two manuals
tuned 1/6-tone (~33.33 cents) apart.

This gives us superb approximations of 7-based intervals, plus a fine
17-flavor of "submajor/supraminor" thirds at ~333.33 cents (~17:14)
and 366.67 cents (~21:17). Unlike 24-tET, this system involves the
asymmetry that not all intervals are available from all steps --
possibly an extra motivation for 33-cent "diesis shifts" between the
two manuals.

----------------------------------
2. "Flavor-consistency" and 24-tET
----------------------------------

From another point of view, we can use a concept which might be termed
"flavor-consistency" to say more specifically what 24-tET does most
accurately.

As is well known, either 12-tET or its superset of 24-tET is
consistent and rather accurate for the usual Pythagorean intervals,
with fifths and fourths within 2 cents of pure; major seconds and
minor sevenths within 4 cents of 9:8 and 16:9; and the various types
of thirds and sixths within 6-8 cents of their ideal Pythagorean or
"3-flavor" sizes, albeit on the "subdued" or "diluted" side.

Additionally, however, we notice that 24-tET seems to have excellent
approximations of what I call "13-flavor" intervals: major thirds at
around 13:10 (~454 cents); wide major seconds or narrow minor thirds
at around 15:13 (~248 cents); wide major sixths or narrow minor
sevenths at around 26:15 (~952 cents); and narrow minor sixths at
around 20:13 (~746 cents).

Looking again at our four-voice sonorities suggests that 24-tET is
both "flavor-consistent" (best approximations used for every interval)
and admirably accurate (if not quite as close as 29-tET) for the
13-flavor:

~26:15 ~26:15
950 950
|---------------------------| |----------------------------|
C3 Ebd3 G3 Bbd3 C3 E*3 G3 A*3
0 250 700 950 0 450 700 950
|--------|----------|-------| |-----------|--------|-------|
~15:13 ~13:10 ~15:13 ~13:10 ~15:13 ~15:13
250 450 250 450 250 250
|------------------| |----------------|
~3:2 ~4:3
700 500

For these 13-flavor sonorities, 24-tET is not only consistent but
accurate to within 5 cents for all intervals.

Additionally, 24-tET also appears to be both consistent and accurate
for the neo-Gothic "49-flavor" of neutral thirds and sixths with
ratios of around 49:40 (~351 cents) and 80:49 (~849 cents), with its
approximations of 350 cents and 850 cents.[1] Again, the intervals
seem to "add up" as expected, for example in this 49-flavor variation
on a minor sixth sonority also featuring a ~12:11 "neutral second"
between the two highest voices (the difference between the 700-cent
fifth and 850-cent neutral sixth above the lowest part):

~80:49
850
|-------------------------|
C3 Eb*3 G3 Ab*3
0 350 700 850
|---------|--------|------|
~49:40 ~49:40 ~12:11
350 350 150
|---------------|
~4:3
500

To sum up, taking a "flavor-consistency" approach we might say that
24-tET offers a somewhat diluted 3-flavor (fifths and fourths quite
accurate, regular "semi-Pythagorean" thirds and sixths a bit on the
mild side), and fine approximations of 13-flavor and 49-flavor
sonorities. Each of these flavors seems to add up consistently, at
least at first blush.

An advantage of these offshoots of the classic Erlich "n-limit"
paradigm of consistency is that we can look at the possibilities of an
equal temperament from different musical angles. The question might be
summed up: "Consistency for what?" with an answer which may be
different from piece to piece.

-----
Notes
-----

1. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament,"
_Xenharmonikon_ 17:12-40 (Spring 1998), at 13 n. 8, provides a
definition of classic n-limit consistency. A tuning is not consistent
within a given limit "if for some odd numbers a, b, and c less than or
equal to the limit, the best approximation of b:a plus the best
approximation of c:b does not equal the best approximation of c:a."

2. For typical neo-Gothic tunings along the spectrum from Pythagorean
to 22-tET, with fifths at a pure 3:2 or wider (~701.96-709.09 cents),
49:40 (~351.34 cents) divides such a fifth into two almost precisely
equal parts, and so may be a more convenient ratio to represent a
neutral third than the simpler 11:9 (~347.41 cents). Although 24-tET
is more of a "semi-Gothic" tuning with its slightly narrow fifths at
700 cents, its neutral third at 350 cents remains slightly closer to
49:40 than to 11:9. If we were to take 11:9 as an ideal (with a
neutral sixth at 18:11), this would not seem to alter our impression
that 24-tET is both consistent and accurate to within 3 cents for
sonorities combining these neutral intervals with regular fifths or
fourths, and/or neutral seconds or sevenths at ~12:11 and ~11:6.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Set-consistency, flavor-consistency, and 24-tET

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Monz <MONZ@JUNO.COM>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>