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

--------------------------------------------------------
An optimized "Manuel-tone" temperament at ~233.536 cents
Finding the least sum of squares for 14:18:21:24
--------------------------------------------------------

Hello, there, everyone.

Recently Michael Saunders and Paul Erlich have been having an
interesting dialogue on optimization, and the concurrent discussion
about the "Miracle Tuning" and its subsets provides an opportunity to
share an interesting kind of temperament problem involving a very
closely related tuning.

Please let me say that Paul Erlich has much stimulated my interest in
this kind of theory by introducing me to the meantone optimization
approach based on least sum of squares error (as he would say) or
"variation" (as I might say) in respect to pure ratios. Using this
approach, we find that a 7/26-comma tuning gives the optimization of
this kind for the ideal meantone concords of the 3:2 fifth, 5:4 major
third, and 6:5 minor third.

Also, reading and rereading _Xenharmonikon_ 17 has introduced me,
Michael Saunders, to your most readable and engaging article on
"Rusty" and fuzzy logic in optimizing a tuning. What an honor it is to
have an opportunity to exchange ideas with such an author.[1]

Interestingly, Manuel Op de Coul's Scala program uses a "Pythagorean"
category for any tuning built from a single repeated generator, and
terms this generator the "formal fifth," regardless of the size of the
interval.

Here I shall consider a variation on a tuning documented by Manuel in
the Scala archive file temp31g3.scl with a generator of ~233.985
cents, or precisely 1/3 the size of a pure fifth at 3:2 (~701.955
cents). In "fine-tuning" this scheme according to Paul's least sum of
squares ideal, we deal with some interval quite different than those
in the meantone problem, and yet come up with a temperament involving
a curiously similar fraction of a comma (or maybe more accurately a
schisma).

A Scala file for this temperament in a 31-note version is included
below.

In approaching this problem, we might first consider its
"non-meantone" nature, starting with a possible definition of
"meantone" as fulfilling these conditions, with some room for
flexibility in the "border" regions, so to speak:

(1) The tuning is built from a chain of identically
sized generators, specifically "fifths" not too far
from a ratio of 3:2;

(2) Regular major thirds are formed from chains of four
such identical fifths up (e.g. F-A from F-C-G-D-A),
or from two equal major seconds or whole-tones
(e.g. F-A from F-G-A);

(3) In a meantone tuning, these major thirds are more
specifically targeted at 5:4, and minor thirds at
6:5, with fifths tempered in the narrow direction
to achieve or approximate these ratios.

Here it may bear emphasis that a tuning may fulfill conditions (1) and
(2) but target sizes of thirds other than 5:4 or 6:5, and thus fall in
some category other than a "meantone" as usually defined.

Pythagorean tuning is one obvious example: regular major and minor
thirds at 81:64 and 32:27 are superb for much of the music of Gothic
Europe, and for "neo-Gothic" music in similar styles.

Also, various tunings I favor in my "neo-Gothic" music temper the
fifths gently in the _wide_ direction in order to attain ratios such
as 14:11 for major thirds, or 13:11 or 33:28 for minor thirds. In
22-tET, four fifths give us a major third very close to a pure 9:7,
also a favorite neo-Gothic size.

Thus meantone is only one part of the spectrum of "tunings with
regular fifths fairly close to 3:2 where major thirds are regularly
formed from chains of four such intervals."

For our optimization problem, however, we move to a tuning built on
quite a different paradigm, with identical generators at around 1/3
the size of a pure fifth (~233.985 cents), or close to the size of an
8:7 or septimal whole-tone (~231.174 cents).

--------------------------------------------------------
1. A four-interval optimization for least sum of squares
--------------------------------------------------------

In Manuel Op de Coul's tuning, apparently related to an observation by
the great Netherlands physicist and musician Adriaan Fokker famed for
the modern revival of 31-tET, the generator is equal precisely to a
third part of 3:2, so that three such generators form a pure fifth.

While Manuel is the person to decide on a name for this tuning, I here
refer to it informally as a "Manuel-tone" tuning, by playful analogy
to "meantone."

Here we approach the problem of optimizing one of the favorite
sonorities of neo-Gothic style, a septimal "quad" or tetrad with
ratios of 12:14:18:21 (a rounded 0-267-702-969 cents) or 14:18:21:24
(a rounded 0-435-702-933 cents).

In seeking a least sum of squares variation from pure ratios for this
sonority, we must consider four intervals:

1. The 3:2 fifth or 4:3 fourth;
2. The 9:7 major third;
3. The 7:6 minor third or 12:7 major sixth; and
4. The 8:7 major second or 7:4 minor seventh.

While it is impossible to get more than one of these four categories
of intervals pure in a repeated-identical-generator tuning, our
purpose is to minimize the overall sum of squares error or variation.

Here our generator is around the size of the 8:7, or of 3:2^1/3, so
that our first step is to consider the number of generators required
to produce each of our four categories of intervals, in the case of
the 9:7 subtracting an octave:

1 generator up = ~8:7
3 generators up = ~3:2
4 generators up = ~12:7
7 generators up = ~9:7

In seeking the best compromise between these four categories, our
relevant comma -- also arguably a kind of schisma[2] -- is the
1029:1024 (~8.43 cents) defining the difference between three pure 8:7
septimal whole-tones up and a pure 3:2 fifth.

Taking the generator of the scale as the object of "temperament," we
can determine the amount of temperament needed to make each category
pure, measuring our tempering (t) in fractions of the 1029:1024 comma
or schisma (analogous to the 81:80 syntonic comma as a measure of
meantone temperaments):

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)

An historical aside: it is fascinating that the fractions of our
1029:1024 ratio required to make our categories other than 8:7 or 7:4
pure happen to coincide with those of the three meantone temperaments
discussed by Zarlino (1558, 1571) and Salinas (1577): 1/4-comma (pure
5:4); 1/3-comma (pure 6:5); and 2/7-comma (5:4 and 6:5 equally impure,
25:24 pure).

Here, however, tempering our near-8:7 generator by 1/4 of our
1029:1024 comma gives a pure septimal _minor_ third at 7:6, or major
sixth at 12:7; tempering at 1/3-comma gives a pure 3:2 fifth; and
tempering at 2/7-comma gives a pure 9:7 major third.

Using a bit of calculus -- with me, a _bare_ minimum, since refreshing
myself on the Chain Rule might permit a more elegant set of steps to
the solution -- we can find the amount of temperament resulting in
least sum of squares error or variation from pure.

My crude calculus yields such a sum of squares at 7/25-comma, or a
temperament of the ~8:7 generator at ~2.36116 cents, or a generator
for least sum of squares on 12:14:18:21 at ~233.53526 cents.

Th generator's sum of squares is ~8.5332925438385603114, the smallest
I can recall among tunings other than those combining just ratios of 3
and 7.

This 7/25-comma solution involves a fraction very close to the
7/26-comma meantone solution advocated by Woodhouse and explicated
here by Paul. Since in both problems one interval category is pure
with no temperament, another with 1/4 of the relevant comma, and
another with 1/3 of this comma, this resemblance might not be so
surprising. Here the factor of a fourth interval (the 9:7) pure at 2/7
of this comma may pull the optimization toward a slightly larger
amount of temperament for the generator.

Note also that this least sum of squares generator is very close to
the 36-tET septimal major second or whole-tone at 7/36 octave, or
precisely 233-1/3 cents.

Thus the tuning might very nicely be implemented in 36-tET, just as
the "Miracle Tuning" with a generator of around 3/2^1/6 can be very
nicely implemented as 7/72 octave in 72-tET.

-------------------------------------
2. The optimized tuning: a Scala file
-------------------------------------

Here is a Scala file for this offshoot of Manuel Op de Coul's tuning,
here carried (like Manuel's scale) to 31 notes, a set reported here by
Dave Keenan to be a strictly proper MOS (Moment of Symmetry), and
which Scala reports to have Myhill's property:

! opt23354.scl
!
Generator of ~233.54 cents, 8/7 + 1029/1024^7/25, least squares 12:14:18:21
31
!
71.91664
104.24036
136.56408
168.88781
201.21153
233.53526
305.45189
337.77561
370.09934
402.42306
434.74679
467.07051
538.98715
571.31087
603.63459
635.95832
668.28204
700.60577
772.52240
804.84612
837.16985
869.49357
901.81730
934.14102
1006.05766
1038.38138
1070.70510
1103.02883
1135.35255
1167.67628
2/1

From a neo-Gothic perspective, by the way, this scale like 36-tET
offers not only near-pure 12:14:18:21 sonorities, but also "17-flavor"
sonorities near 14:17:21 (0-336-702 cents) -- or often, here, a
variant with a superminor third at ~330.51 cents, close to 23:19
(~330.76 cents), reminiscent of 29-tET.

The problem of how a computer application such as "Rusty" might
approach optimizations of the kind favored in neo-Gothic music,
especially for larger tuning sets (typically 24 notes), might be an
interesting one.

For example, we might ideally seek 3:2 fifths, regular 14:11 major
thirds, augmented seconds and diminished fourths at 17:14 and 21:17,
and close approximations of 12:14:18:21 in at least 8 positions of a
24-note tuning.

The targets for the major third, and for the augmented second and
diminished fourth, suggest a temperament with a fifth somewhere in the
region around 704 cents (e.g. 46-tET), where 13, 14, and 15 fifths or
fourths provide the best approximations of our 7-based ratios
(e.g. 9:7, 7:6, 7:4). Now it is a question of weighing the balance
between these intervals -- and also the fifth, which ideally we would
like to temper as gently as possible, or as efficiently as possible
for our desired effect.

Here I have limited the general type of problem to one involving a
chain of identical generators, whether conventional fifths or quite
unconventional (as in Manuel's 233.985-cent tuning and its
233.536-cent offshoot).

If one considers solutions involving _two_ 12-note chains of identical
generators spaced at any desired interval, for example, then the
possibilities are multiplied, including schemes combining some aspects
of regular tunings and multi-prime just intonation systems.

Again, I would warmly like to thank Paul for educating me on the sum
of squares optimization concept, Michael for discussions of "Rusty"
and optimization which invite many fascinating scenarios, and Manuel
for a scale at once inspiring in itself and offering an opportunity
for this List's typical dialogues on "fine-tuning."

-----
Notes
-----

1. Michael Saunders, "Pitch, Scale and Tuning in _Rusty_: A Fuzzy
Logic Approach to the Tuning Problem," _Xenharmonikon 17_ (Spring
1998), pp. 116-119. Maybe I might add that reading about the problem
of wanting two 9:8 major seconds to add up to 5:4 major third rather
than the 81:64 major third actually obtaining, I smiled to reflect
that I find the Pythagorean third ideal for much of my music, and am
very happy to have two 9:8 steps add up this way -- but thus
presenting a less interesting situation from the viewpoint of your
application's optimization process.

2. From one standpoint, the 1029:1024 is the difference between two
commas, the 64:63 (~27.26 cents) distinguishing, for example, a
Pythagorean major third at 81:64 (~407.82 cents) from a pure 9:7
(435.08 cents); and the 49:48 (~35.70 cents) distinguishing a pure 8:7
(~231.17 cents) from a pure 7:6 (~266.87 cents). Typically the
difference of two commas is called a schisma. However, in our tunings
with a generator of around 8:7 or 3:2^1/3, 1029:1024 might also be
regarded as the "comma" defining the difference between a pure 3:2
fifth and three pure 8:7's up. In the context of his scale, Manuel Op
de Coul calls this 8.43-cent interval a "gamelan residue," defining
the amount by which four pure 8:7's plus a pure 7:6 exceed a 2:1
octave.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
>
> Here we approach the problem of optimizing one of the favorite
> sonorities of neo-Gothic style, a septimal "quad" or tetrad with
> ratios of 12:14:18:21 (a rounded 0-267-702-969 cents) or 14:18:21:24
> (a rounded 0-435-702-933 cents).
>
> In seeking a least sum of squares variation from pure ratios for this
> sonority, we must consider four intervals:
>
> 1. The 3:2 fifth or 4:3 fourth;
> 2. The 9:7 major third;
> 3. The 7:6 minor third or 12:7 major sixth; and
> 4. The 8:7 major second or 7:4 minor seventh.

As you know, the 14:18:21:24 is an example of a "saturated" chord as I identified them, and an
anomalous saturated suspension, or ASS, in Graham Breed's terminology. You've correctly
identified the intervals in this chord. However, in the optimization (i.e., in the error function), I
would include the 3:2 _twice_, and the 7:6 _twice_, since those intervals each occur twice in the
chord in question. If you look at my 9-limit MIRACLE optimization, you'll see that I included 3:2
twice, since 3:2 occurs twice in a 9-limit complete pentad -- as 3:1 and as 9:3.

I'll look at the math later, if math is still permitted on this list.

as a begginner with cents could you please explain themcheersrob>

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--- In tuning@y..., "Robert Greco" <robgreco@h...> wrote:
> as a begginner with cents could you please explain them
>cheers

Hi Robert!

Cents are merely hundredths of an equal-tempered semitone. 1 ET semitone = 100 cents.

So the formulas from cents to ratios, and back, are

ratio = 2^(cents/1200)

cents = log(ratio)/log(2)*1200

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
> Interestingly, Manuel Op de Coul's Scala program uses a
"Pythagorean"
> category for any tuning built from a single repeated generator, and
> terms this generator the "formal fifth," regardless of the size of
the
> interval.

Yes Manuel, isn't it time to change that, or at least provide aliases?
How about:
Pythagorean -> Linear
Formal octave -> Interval of equivalence
Formal fifth -> Generator

-- Dave Keenan

Well Margo, I think more credit is due to you for working it all
out as elaborately as you did. Then I propose to keep your suggestion
for the name "Wonder scale". It is appropriate because of the analogy
with the Miracle scale. A generator about twice the size, an approximating
equal division of 36 instead of 72. The number of 14:18:21:24 chords in
your optimization is 24, with the inversion that makes a total of 48,
also considerably more than the 31 tones.

>Here our generator is around the size of the 8:7, or of 3:2^1/3, so
>that our first step is to consider the number of generators required
>to produce each of our four categories of intervals, in the case of
>the 9:7 subtracting an octave:

> 1 generator up = ~8:7
> 3 generators up = ~3:2
> 4 generators up = ~12:7
> 7 generators up = ~9:7

>Generator of ~233.54 cents, 8/7 + 1029/1024^7/25, least squares
12:14:18:21

Let me demonstrate how to calculate this result using Scala.
Type CALCULATE/LEASTSQUARE or an abbreviation thereof.
The first question is "Enter size:". Here one should enter the
desired number of tones in the temperament. This might seem a bit odd,
because the result shouldn't depend on it, but it is useful because
then the program can suggest how many generators up or down could be
taken to arrive at any given interval. The smaller of the two is
usually preferable. So we enter 31.
Then we enter the formal octave: 2/1.
The next question is "Enter fifth degree (0 for nearest degree)".
Here you can preselect the scale degree for the generator (formal fifth)
in advance, but entering a 0 will do fine too.
The next question is "Enter number of approximants including fifth".
We have four (8/7, 3/2, 12/7 and 9/7), so enter: 4.
Then "Enter pitch 1", we must enter 8/7 because this is the interval
the generator will approximate.
Then "Enter number of steps (1,-30)". We want 1 up so we enter 1 (or only
a <return> to take the default 1). A negative value means going down.
Then "Enter weight". One can give each interval a relative weight; in this
case we'll choose the default 1.0. We could use this feature to bias the
result towards 3/2 by choosing a higher weight there if we'd want to.
Subsequently we enter:
3/2
3 (default)
1.0 (default)
12/7
4 (default)
1.0 (default)
9/7
7 (default)
1.0 (default)
And voila 233.535255 cents is the result.
This result is automatically stored in pitch memory 0, so it
doesn't need to be retyped when actually creating the temperament
with the command PYTHAGOREAN, by using "$0" for the generator.

Manuel

Dave wrote:
>Yes Manuel, isn't it time to change that, or at least provide aliases?
>How about:
>Pythagorean -> Linear
>Formal octave -> Interval of equivalence
>Formal fifth -> Generator

Mwa, I always thought it was kind of clear. "Formal fifth" is
not my term, but Clough's. The reason that it doesn't need to
be near a fifth is exactly why the adjective "formal" is there.
Idem formal octave. The term linear is already in the help, and
putting the other terms there too won't hurt.

Manuel