Yahoo Tuning Groups Ultimate Backup tuning Re: Manuel-tone at ~233.536 cents -- 14:18:21:24 sum of squares

Hello, there, everyone

Please let me thank Paul Erlich for an excellent suggestion regarding
optimization of the 12:14:18:21 or 14:18:21:24 sonority: that we take
into account all six intervals of this type of four-voice sonority,
thus counting the 3:2 or 4:3 category and the 7:6 or 12:7 category
twice in seeking a least sum of squares.

When I proceeded accordingly, I got a mildly surprisingly result: the
solution doesn't change, the generator remaining at ~233.536 cents, or
in other words an 8:7 tempered by 7/25 of a 1029:1024 comma.

Before getting into the math -- inviting any "debugging" -- I might
comment briefly on terminology in some musical contexts for the
12:14:18:21 or 14:18:21:24, starting with a fine naming suggestion by
Dave Keenan which would fit many meantone-oriented or similar styles.

In many 20th-century settings, as Dave suggests, the 12:14:18:21 is
very happily described as a "subminor seventh chord," a septimal
version of the very common sonority with a minor third, fifth, and
minor seventh above the lowest note (e.g. E3-G3-B3-D4). In this
tuning, we have a rounded 0-267-702-969 cents.

Following Dave's lead, I might similarly proposse in this type of
musical setting that the 14:18:21:24 might be termed a "supermajor
added sixth chord," a variation on the usual added sixth chord with a
major third and fifth (the Classic triad) plus major sixth. As this
name suggests, the third and sixth are here "supermajor" from a
meantone or 12n-tET viewpoint, in a tuning of 0-435-702-933 cents.

In a neo-Gothic setting, these same sonorities are known as "complete
cadential quads," a term reflecting the fact that they have not only
four voices, but also four unstable intervals which in a standard
resultion all progress to stable ones by means of ideally efficient
stepwise contrary motion.

To illustrate this, let's pick a regular temperament outstanding for
these sonorities, and I hope not uncongenial to your tastes, Paul:
22-tET.

Here are standard 22-tET resolutions of the 14:18:21:24 and
12:14:18:21 approximations on F, using the natural spelling of these
sonorities which results in a neo-Gothic or "Pythagorean" scale where
each regular whole-tone is equal to 4 steps (~218.18 cents), and each
diatonic semitone to 1 step (~54.55 cents):

14:18:21:24 Contractive/Intensive

E4 ----- +55 ----- F4 D4 ----- -218 ----- C4
(218) (491) (273) (0)
D4 ------ -218 ----- C4 B3 ----- +55 ----- C4
(491,273) (491,0) (709,436) (709,709)
B3 ----- +55 ----- C4 G3 ----- -218 ----- F3
(927,709,436) (1200,709,709) (982,709,273) (709,709,0)
G3 ------ -218 ----- F3 E3 ----- +55 ----- F3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Here numbers in parentheses show vertical intervals in rounded cents,
while signed numbers show the motion of each voice in ascending or
descending rounded cents. The 9:7 major third is almost pure, and the
7:6 minor third or 12:7 major sixth rather close to pure also; the 3:2
fifth and especially the 7:4 minor seventh are more heavily tempered,
one might say a necessary compromise involved in the heroic effort
needed to produce these quads in a chain of only four fifths or
fourths.

From a neo-Gothic standpoint, all four resolutions of unstable
two-voice intervals (shown in parentheses beneath the examples) are
"closest approach" progressions, involving motion of a whole-tone in
one voice and a semitone in the other. More specifically, we here have
ascending semitones and descending whole-tones, often more conclusion
than the converse arrangement.

Given the discussions about 22-tET scales and modes, I might note that
this kind of cadence occurs naturally (without accidentals) in what
might be termed the "22-tET Pythagorean Lydian" mode on F, how here in
22-tET steps and rounded cents:

F G A B C D E F
0 4 8 12 13 17 21 22
0 218 436 655 709 927 1145 1200
4 4 4 1 4 4 1
218 218 218 55 218 218 55

This "Pythagorean-style" diatonic scale provides all the notes needed
for our cadences from a complete quad to a stable trine (2:3:4) on F
(here F3-C4-F4).

The example of 22-tET illustrates Dave Keenan's more general point
than an ideal nomenclature for one musical setting may invite
modification in a different setting.

Here _regular_ major and minor thirds, the usual versions of these
intervals, have ratios near 9:7 and 7:6, so to call them "supermajor"
or "subminor" might fit a different kind of setting such as meantone
or much 20th-century music in 12-tET, where the usual or at least
"ideal" ratios for these thirds are at or near 5:4 and 6:5.

However, Dave's terminology might nicely fit Paul's decatonic tonality
based on stable tetrads of 4:5:6:7, where thirds of around 381.81 and
327.27 cents (from a "Pythagorean" point of view, respectively
augmented seconds and diminished fourths) are treated as the usual
major and minor sizes, in contrast to the "dissonant" ~9:7.

Let us now return to the optimization problem. In my earlier post, I
counted each category of interval in 12:14:18:21 or 14:18:21:24 only
once:

7:4 or 8:7 = ~231.174 cents (pure at t = 0)
7:6 or 12:7 = ~266.871 cents (pure at t = 1/4, ~233.282 cents)
9:7 or 14:9 = ~435.084 cents (pure at t = 2/7, ~233.583 cents)
3:2 or 4:3 = ~701.955 cents (pure at t = 1/3, ~233.985 cents)

Using thie approach, I got a minimum at

150t - 42 = 0
t = 42/150 = 7/25

Counting 3:2 (or 4:3) and 7:6 (or 12:7) twice each, as suggested, thus
accounting for all six intervals of a 14:18:21:24 or 12:14:18:21
sonority, I curiously got a minimum at:

200t - 56 = 0
t = 56/200 = 7/25

If this is correct, then my original result of ~233.536 cents, and
accompanying Scala file of a 31-note tuning based on this generator,
would represent the optimization by the six-interval method also.

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: Manuel-tone at ~233.536 cents -- 14:18:21:24 sum of squares

🔗mschulter <MSCHULTER@VALUE.NET>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>