Yahoo Tuning Groups Ultimate Backup tuning Re: towards a hyper MOS

Some more questions, and interesting patterns, no answers yet:

Is there any single two interval generator that makes arbitrarily large "Tryhill" scales?

Reason for asking this is that if one could make arbitrary large scales all for the same
generator, maybe one could discern patterns in the resulting sequence of scales that
could help understand what is happening. One might also
reasonably expect a scale that works to be somewhat fractal in nature.

I haven't found any that make arbitrarily large scales yet, but found some interesting finite sequences.

Found a 45 note one using 1/1 5/4 45/32, but then no more up to 1000 notes

This one at least has a pattern:

L M L M S

->
L S M S L S M S S

->
L S S M S S L S S M S S S

etc.

I.e. substitution rule

S -> S
L -> L S
M -> M S

But this is not what we are looking for as S intervals use up more and more of the L and M gaps, and ratio L/S and L/M get smaller all the time.

Another generator that does make large ones rather easily, up to a certain point:

1/1 4/3 7/4

1/1 7/6 4/3 49/32 7/4 2/1 (5 notes)
steps: 7/6 8/7 147/128 8/7 8/7

1/1 49/48 7/6 4/3 49/32 7/4 2/1 (6 notes)
steps: 49/48 8/7 8/7 147/128 8/7 8/7

1/1 49/48 7/6 4/3 343/256 49/32 7/4 2/1 (7 notes)
steps: 49/48 8/7 8/7 1029/1024 8/7 8/7 8/7

1/1 49/48 16807/16384 823543/786432 7/6 2401/2048 117649/98304 4/3 343/256 16807/12288 5764801/4194304 49/32 2401/1536 823543/524288 7/4 343/192 117649/65536 2/1 (17 notes)

steps: 49/48 1029/1024 49/48 131072/117649 1029/1024 49/48 131072/117649 1029/1024 49/48 1029/1024 131072/117649 49/48 1029/1024 131072/117649 49/48 1029/1024 131072/117649

Also 27, 37, 47, and 57 notes all work.

But then, no more scales up to 1000 notes.

However pattern is a little more complex than before:

M L L S L L L (7)
->

35.7 8.43 35.7 187.0 8.43 35.7 187.0 8.43 35.7 8.43 187.0 35.7 8.43 187.0 35.7 8.43 187.0
M - SML - SML - S - MSL - MSL - MSL (17)

->
M - S - M - SML - S - M - S - MSL - M - S - MSL - M - S - MSL - M - S - MSL (27)

I.e. the M and S stay as they are, but the L gets substituted for by SML (first one) or MSL

Eventually at 57 notes the L is smaller than the M, and one ends up with the scale
MSMSMSMSMSML SMSMSMSMSML SMSMSMSMSMSL MSMSMSMSMSL MSMSMSMSMSL

1/1 9/8 5/4 goes up to 87 notes at

M L M S L M S L M S L M S L M S M L S M L S M L S M L S M
L M S L M S L M S L M S L M S M L S M L S M L S M L S M L M
S L M S L M S L M S L M S M L S M L S M L S M L S M L S

which is most interesting so far. But the way it is reached is by gradually filling up large gaps
as before, though in a more complex fashion. The L/S and M/S ratios still keep varying.

Nothing like the Tribonacci sequence yet, but one wonders if one could find it with suitable choice of the two intervals for the generator?

Robert

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Another interesting failed attempt, this time trying for a Tribonacci sequence, without trying to work out how the MOS will fit in as yet - hoping one can work back to it later once one has the sequence of LMS.

Just try using the L M S as step sizes, e.g. in n-tet

L M S

= 3 2 1
->
M L - S L - L

2 3 1 3 3 in 12 tet = Tryhill

->
S L - M L - L - M L - M L

1 3 2 3 3 2 3 2 3 in 22 tet = Tryhill
(Paul - should interest you!)

->
L - M L - S L - M L - M L - S L - M L - S L - M L

DOESN'T WORK!

LLLMS LLMMS LLMSS LLLMM

Is this an effect of the truncation of sequence at the octave?

How about trying an infinite scale:
L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L M ~ L S ~ L M ~ L ~ L M ~ L S ~ L M ~ L S ~ L
(expanding a number of levels in FTS)

Looking for 5 step sequences, one gets

LLLMS, LLMMS, LLMSS LLLLMM, again.

One also finds those same sequences in the other three re-orderings of the maps for L and M.

Also, the patch of the infinite sequence just shown will occur at some level in the expansion from any starting point.

So that definitely rules out the
L -> L M
M -> L S
S -> L
map in any of its permutations for making arbitrarily large Trihill sequences.

Just one map though amongst many possiblities for the Tribonacci sequence. Other candidate maps could be tested in the same way.

Robert

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

I wrote,

>right off the bat, it looks like we could add 1 to each of the step sizes
-- 2 4 3 4 4 3 4 3 4 -- and have a Tryhill scale in 31-tET, right?

I checked it -- it's Tryhill. The scale is
0 2 6 9 13 17 20 24 27 (31)
and the harmonic lattice is

,' `. .'
9--------27
\ /
\ /
\ /
\ /
17
,'
6--------24
,' `. ,'
13---------0
,'
2--------20
,' `. ,'
9--------27
\ /
\ /
\ /
\ /
17
,'

Certainly looks promising for some kind of generator explanation (with the
5-limit minor triad appearing as an incidental feature of the temperament),
and musically too! The mirror version would have a 5-limit major triad
instead. In 22-tET the mirror scale (still Tryhill, of course) is

0 3 5 8 10 13 16 18 21

and the harmonic lattice is

10
/|\`.
/ | \ 5--------18
. . .21 \ | `. ,' `. 10
. /,' `.\| 0--------13 /|\`.
3--------16 `. / | \ 5--------18
8--------21 \ | `. ,'
`. /,' `.\| 0--. . .
3--------16

where again the ratios involving 5 are by-products of the temperament.

Interestingly, the 5-tone Tryhill scale you mentioned, 2 3 1 3 3 in 12 tet ,
can be cast in 22-tET by adding 2 to each step size: 4 5 3 5 5, leading to a
mirror scale of

0 5 10 13 18

10
`.
5--------18
`. ,' `.
. 0--------13

which is clearly a "generator" of the 9-tone scale above. (0 13 18 10 also
approximates an otonal 1:3:7:11 tetrad). I'll have to play with these
things! Thanks, Robert!

🔗D.Stearns <STEARNS@CAPECOD.NET>

Paul Erlich wrote,

<< No I don't already have the answers, but I was thinking of you
specifically as someone who might be interested in this idea, and
thought you might enjoy working this out. You, and also Carl Lumma who
once asked what a 2-d generalization of MOS might be like. I came up
with the idea while waking up Friday morning, when I discovered
that the pattern of very important scale sizes 3, 7, 12, 22, 41 is a
Tribonacci-like sequence, and one thought led to another >>

OK, if you look at a two-term series as a two-stepsize index, [a,b]
where the alphabetized variables are small to large stepsizes, then
a+b is the subset or scale of b+(a+b) where the size -- not to be
confused with the amount or index -- of a is 1, the size of b is 2,
and b+(a+b) is an equal division of the octave.

This of course would be true at any point in the sequence... witness
the famous Yasser example.

So a three term "Tribonacci" series is analogous to the Fibonacci
series in this sense where a+b+c is the corresponding subset of the
sum of the 4th and 6th terms and the size -- again, not to be confused
with the amount or index -- of a is 2, b is 3, and c is 4, and the sum
of the 4th and 6th terms is an equal division of the octave.

So if the "3, 7, 12, 22, 41" example you gave was backed up so to
speak so that the three-stepsize index were [2,2,3] you'd have 2, 2,
3, 7, 12, 22, ...

This is the generalized fib to tri (and higher term) parallel in the
Yasser sense.

What's missing from the three-term series is the built-in ordering
rule supplied by the single generator of the two-term series... once I
can adequately fill in that gap (I thought I had "solved" it today at
work only to have it not quite pan out once I got home and had a
chance to try it out), the "scaled by P" method will be complete for
n-term scale indexes.

Any ideas?

--Dan Stearns

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Sorry, a few corrections to previous post.

Also meant to add ref to the post with the SCALA results for the 17 tone scale:
http://www.egroups.com/message/tuning/16054

First, I forgot about poss. of the like of

L -> M, S -> L, M -> S, i.e. in reverse order.

But that's the same as -M -> -L -> -S where you think of -M, -L, and -S as subtracted
from an interval with one extra step.

E.g. LLS -> LMS -> LLM -> LLS
= LLMS-M -> LLMSM-L -> LLMS-S -> LLMS-M again.

Clearly, once more, if you change M to L everywhere, or M to S, or S to L,
you will still have two interval classes, and the scale will be Myhill.

So this doesn't affect the conclusiont.

Then at the end:

I wrote:

> 2 4 3 4 4 3 4 3 4
> S L M L L M L M L. (I left out final L here, sorry)

> We find the exceptional case straight away using two step intervals:
> SL LM LL

Sorry, mistake there.

That case is fine.
- it's got by S -> M -> L, so it is okay.

Also a single S is no good for the type of scale I was looking for with an
interval class obtained using LL -> LM -> MM -> LM -> LL, instead of
L -> M -> S, one needs at least two Ss, equally spaced, if it is going to work.

So back to the scale to see what has happened.

0 1 2 3 4 5 6 7 8
S L M L L M L M L

looking at class 3 we have
LMS, LLM, LMM, LLS
LLS = 4+4+2 = 10
LLM = 4+3+3 = 10.

So this scale manages to be Tryhill by using two compositions
of the intervals that happen to add up to the same note size.

This is something that can't happen to a Myhill scale, as Carey and Clampitt show
early in their proof.

My result in previous post about scales with prime numbers of notes won't work for
scales like this, and one needs to add an extra condition that all the interval sizes for
each interval class have to have a unique composition in terms of L, M and S.

Result then should be okay, I think.

Robert

🔗D.Stearns <STEARNS@CAPECOD.NET>

I wrote,

<< So a three term "Tribonacci" series is analogous to the Fibonacci
series in this sense where a+b+c is the corresponding subset of the
sum of the 4th and 6th terms and the size -- again, not to be confused
with the amount or index -- of a is 2, b is 3, and c is 4, >>

Whoops, that should've read "where a+b+c is the corresponding subset
of the sum of the 3rd, 4th, and 5th terms"...

So the two-stepsize [2,5] 7-out-of-12 two-term series is:

2, 5, 7, 12, 19, ...

And a+b is the subset of the sum of the 2nd and 3rd terms where "a" is
1 and "b" is 2.

The three-stepsize [2,2,3] 7-out-of-22 three-term series is:

2, 2, 3, 7, 12, 22, 41, ...

And a+b+c is the subset of the sum of the 3rd, 4th, and 5th terms
where "a" is 2, "b" is 3, and "c" is 4.

A four-stepsize [2,2,1,2] 7-out-of-43 four-term series would be:

2, 2, 1, 2, 7, 12, 22, 43, 84, ...

Where a+b+c+d is the subset of the sum of terms 4, 5, 6, and 7, and
"a" is 4, "b" is 6, "c" is 7, and "d" is 8.

--Dan Stearns

🔗D.Stearns <STEARNS@CAPECOD.NET>

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

🔗David Clampitt <david.clampitt@yale.edu>

I've just joined the list, and have been looking through the archives
following this thread, reading as many of the posts as I can find. I don't
know exactly when this thread originated, so have not exhausted it. My
dissertation, "Pairwise Well-formed Scales: Structural and
Transformational Properties," (SUNY at Buffalo, 1997; UMI no. 9801282)
explores hyper MOS, at least under one possible definition.

Pairwise well-formedness extends the concept of well-formedness, studied
by Norman Carey and me in a series of papers ("Aspects of Well-Formed
Scales," Music Theory Spectrum 11, 1989; "Self-similar Pitch Structures,
Their Duals, and Rhythmic Analogues," Perspectives of New Music 34, 1996;
and "Regions: A Theory of Tonal Spaces in Early Medieval Treatises,"
Journal of Music Theory 40, 1996). A scale or pitch-class set is
<well-formed> if it is generated by an interval of constant size and span,
that is, if all of the notes of the scale may be linked together in a
chain, where the links are intervals of the same size that span the same
number of scale steps. For example, the usual pentatonic scale
(e.g.,FGACD) is generated by the perfect fifth, and all perfect fifths
span three pentatonic scale steps. Under this definition, all
equal-divisions of the octave (or modular interval) are well-formed,
so we make the distinction between <degenerate well-formed scales> (equal
divisions of the octave) and <non-degenerate well-formed scales> (all
others satisfying the definition).

Carey and I proved that non-degenerate well-formedness is equivalent to
Clough and Myerson's Myhill property (two sizes for every non-zero generic
interval: 2nds, 3rds, etc.). These scales are also Wilson's MOS. At
the time of our 1989 paper, Carey and I were unaware of MOS, Xenharmonicon
not being in the holdings of the major research libraries we were
inhabiting. In our 1996 articles and in our respective dissertations we
have cited John Chalmers' 1975 Xenharmonicon article, "Cyclic Scales," and
his reference to Wilson's 1964 letter.

Now the relevant concept for this discussion (a form of hyper MOS): A
<pairwise well-formed scale> is one with three distinct step-interval
sizes which is non-degenerate well-formed when any two step-interval sizes
are taken to be equivalent. Examples of pairwise well-formed scales would
include forms of the Japanese In-scale such as hira-joshi (EFABC) [one of
the posts refers to the koto scale], Vieru's "Romanian major" (octatonic
minus one note), the so-called Hungarian or gypsy minor, in wide-spread
use in world music, and the just major scale (in one of its usual forms):

do re mi fa sol la ti (do)

9/8 10/9 16/15 9/8 10/9 9/8 16/5

If we consider the three sizes S=16/15, M=10/9, and L-9/8, then taking as
equivalent M and L (=X) , we have the pattern of the usual diatonic scale
(in equal-temperament, Pythagorean tuning, or some other tuning where the
pattern of step intervals is XXSXXXS). This is non-degenerate well-formed,
with perfect 4th (XXS) or perfect 5th as generator. If S and L are
equivalenced, similarly we have the pattern of step intervals XMXXMXX, a
rotation (or mode) of XXMXXXM, the diatonic pattern again. Taking the
remaining pair M and S as equivalent, again a non-degenerate well-formed
scale pattern results, LXXLXLX, with generating interval the third LX, (or
its complementary 6th, XLXLX).

To be clear, I should say that the equivalencing in the definition and in
the example above is to be construed as an abstract operation that only
involves the given pair of step intervals, not a tuning or temperament
operation that might by chance result in equivalencing all the intervals
of the scale. For example, in a mod 14 chromatic universe (14-tet), the
scale with the step-interval sequence <2132123> is pairwise well-formed,
despite the fact that one might understand equivalencing the 1 and 3 steps
to mean, in effect, tempering these intervals to size 2, thereby yielding
the degenerate well-formed scale <2222222>, thereby excluding it. The
step-interval size resulting from taking a pair of step intervals to be
equivalent is always, in my conception, to be considered distinct from the
remaining interval. (Thus, there is a close connection but not
identification with Stearns's scaling operation.)

Pairwise well-formed scales have a number of interesting properties. In
the context of the current thread, perhaps the one to mention first is the
trivalence property that generalizes Myhill property: not only do step
intervals come in three sizes, but so do all non-zero (octave-equivalent)
generic intervals. (I also initially called this "the Trihill property"
before settling on the more academic "trivalence.") Unlike Myhill (MP),
however, trivalence is not as logically powerful a property: whereas MP is
mathematically equivalent to well-formedness, trivalence is only an
entailment of pairwise well-formedness. A counterexample is Vieru's
Bacovia mode, the complement of the pairwise well-formed Hungarian or
gypsy minor: e.g. C E F Aflat B. This has trivalence but is not
pairwise well-formed.

The Hungarian minor (C D Eflat Fsharp G Aflat B) is notable as an instance
of a unique class that I call "singular pairwise well-formed scales."
These have the step-interval pattern <abacaba>, (and the same pattern in
some rotation for all generic interval-cycles). There are pairwise
well-formed scales (or sets) for every odd cardinality greater than or
equal to 3, but while in these cardinality 7 scales the multiplicities for
the step intervals (and for the varieties within any generic interval
class) are 1, 2, and 4, for all non-singular pairwise well-formed scales
of cardinality 2N-1, the multiplicities are X, X, and Y (where of course
2X+Y=2N-1).

These singular scales are the only pairwise well-formed ones that are
inversionally symmetric, and the only ones that are generated (obtained by
stacking up intervals modulo the octave, a la well-formed scales).

To underline the oddity of this situation, again, 7 is the only
cardinality that supports these possibilities.

A class of examples of singular type are the <golden scales> generated by
F(n+2) mod F(n+4), where the step intervals are the Fibonacci numbers Fn,
F(n-1), and F(n+1), in the sequence <Fn F(n-1) Fn F(n+1) Fn F(n-1) Fn>,
modulo F(n+4) (i.e., F(n+4)-tet). [Are these the Tribonacci scales I have
seen referred to in some of the posts? I will follow this up. I didn't
make the connection when I was browsing.] In the limit, these would be
generated by golden number phi+1 modulo 3phi+2. (See Clampitt 1997, 113.)

There is much more that I think is of interest, especially the
transformational aspects alluded to in my diss title. Of course, mine is
only one possible generalization of MOS to hyper-MOS that may be of
interest. I will try to follow up on some of those suggested on this list.

I will continue to study the archives to see if there are results I have
not been aware of. If there is a key word or message number that will
enable me to locate the bottom of the thread that would be useful to me.

David Clampitt

(A minor correction to an earlier post of John Chalmers: Norman Carey did
his dissertation at Eastman, under Robert Morris. I did mine in Buffalo,
under John Clough.)

🔗David Clampitt <david.clampitt@yale.edu>

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

🔗jon wild <wild@fas.harvard.edu>

🔗D.Stearns <STEARNS@CAPECOD.NET>

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

🔗David Clampitt <david.clampitt@yale.edu>

Paul, (et al.)

To follow up, a singular pairwise well-formed scale, (with step-interval sequence <abacaba>), is generated if and only if the size of the unique step interval is equal to the size of the other two combined, i.e., c=a+b. The interval size c+a generates the scale. (This is a little informal, but I'm sure you get the idea.)

On your query regarding Xenharmonikon, I agree with what Daniel Wolf wrote. I can confirm that 1/1 is much more readily available in music libraries. When Carey and I cited an article in 1/1, however, there was a query about the title from the editor, who evidently thought 1/1 referred to a volume and issue number. When I was at SUNY at Buffalo, I believe the librarian obtained issues of XH for us from Frog Peak Press. As for securing a wider audience for your vol. 17 paper, I would suggest reworking it for a more widely distributed academic journal. Alternatively, self-publish on the web. There is a long thread on issues of publication--including questions about peer review, timeliness, and distribution--on the SMT list over the past year, that you may find interesting.

David

>David Clampitt wrote,
>
>>A correction to para. 9 in the post I just sent: a scale of the
>>singular pairwise well-formed type may be generated by a single
>>interval, but not all such sets (such as Hungarian minor--harmonic
> >minor with sharp scale degree 4) are generated.

>Whew, thanks! I was beating my head against the wall trying to generate the
>Hungarian minor with a single interval!
>

🔗D.Stearns <STEARNS@CAPECOD.NET>

OK, I finally got it...

EUREKA:

By using a one- to two-dimensional conversion it is possible to
convert any given [a,b] index into an [a,b,c] index that is a
three-term "MOS"/"trivalence" within a given periodicity.

SILVER-WEIGHTED:

In the past I've referred to the scales created by a silver-weighted
generator as a sort of generalized Pythagorean scale. I think this is
a useful frame of reference for converting [a,b] indexes into [a,b,c]
indexes.

Here's the generalized formula for deriving a silver-weighted
generator for any given [a,b] index within a given periodicity.

X = P/((a+S*b))*(z+S*y)

Where:

"P" = any given periodicity

"S" = the sqrt(2)+1 silver constant which can be expressed by the
series 1/2, 2/5, 5/12, 12/29, 29/70, ...

"z"/"a", "y"/"b" = the adjacent fractions of a given [a,b] index

and "X" = the resulting silver-weighted generator

THE SCHISMATIC SILVER FOLD:

Multiplying "X" by the forth term of the two-term [a,b] index and
scaling it by "P" gives a silver comma -- or by way of the analogy, a
generalized Pythagorean comma.

Converting the one-dimensional silver chain into a two-dimensional
schismatic silver fold will give the two to three stepsize
cardinality.

SOME "SIMPLE" EXAMPLES:

These are all the 7-tone scale index conversions:

[1,6] = [1,5,1]
[2,5] = [2,2,3]
[3,4] = [3,3,1]
[4,3] = [3,1,3]
[5,2] = [2,3,2]
[6,1] = [1,1,5]

Here are the 7-tone silver schismic two-dimensional, two generator,
three-term MOS conversions.

[1,6] = [1,5,1]

420---232----45
/ \ / \ / \
/ \ / \ / \
0--1013---826---639

0 45 232 420 639 826 1013 1200
0 187 374 593 780 968 1155 1200
0 187 406 593 780 968 1013 1200
0 219 406 593 780 826 1013 1200
0 187 374 561 607 794 981 1200
0 187 374 420 607 794 1013 1200
0 187 232 420 607 826 1013 1200

[2,5] = [2,2,3]

376--1079---582
/ \ / \ / \
/ \ / \ / \
0---703---206---909

0 206 376 582 703 909 1079 1200
0 171 376 497 703 874 994 1200
0 206 326 532 703 824 1029 1200
0 121 326 497 618 824 994 1200
0 206 376 497 703 874 1079 1200
0 171 291 497 668 874 994 1200
0 121 326 497 703 824 1029 1200

[3,4] = [3,3,1]

1066---742---419
/ \ / \ / \
/ \ / \ / \
0---876---553---229

0 229 419 553 742 876 1066 1200
0 190 324 513 647 837 971 1200
0 134 324 458 647 781 1010 1200
0 190 324 513 647 876 1066 1200
0 134 324 458 687 876 1010 1200
0 190 324 553 742 876 1066 1200
0 134 363 553 687 876 1010 1200

[4,3] = [3,1,3]

898---534---169
/ \ / \ / \
/ \ / \ / \
0---836---471---107

0 107 169 471 534 836 898 1200
0 63 364 427 729 791 1093 1200
0 302 364 666 729 1031 1137 1200
0 63 364 427 729 836 898 1200
0 302 364 666 773 836 1137 1200
0 63 364 471 534 836 898 1200
0 302 409 471 773 836 1137 1200

[5,2] = [2,3,2]

194---855---316
/ \ / \ / \
/ \ / \ / \
0---661---122---783

0 122 194 316 661 783 855 1200
0 72 194 539 661 733 1078 1200
0 122 467 590 661 1006 1128 1200
0 345 467 539 884 1006 1078 1200
0 122 194 539 661 733 855 1200
0 72 417 539 610 733 1078 1200
0 345 467 539 661 1006 1128 1200

[6,1] = [1,1,5]

570---428---285
/ \ / \ / \
/ \ / \ / \
0--1057---915---772

0 285 428 570 772 915 1057 1200
0 143 285 487 630 772 915 1200
0 143 344 487 630 772 1057 1200
0 202 344 487 630 915 1057 1200
0 143 285 428 713 856 998 1200
0 143 285 570 713 856 1057 1200
0 143 428 570 713 915 1057 1200

SOME "EXOTIC" EXAMPLES:

The Bohlen-Pierce Lambda scale can be seen as [5,4] index where P =
1:3. Converting this into a silver schismic two-dimensional, two
generator, three-term MOS results in a [5,4] = [4,1,4] index
conversion and the following scale:

1535--1092---649---206
/ \ / \ / \ / \
/ \ / \ / \ / \
0--1459--1016---573---130

0 130 206 573 649 1016 1092 1459 1535 1902
0 76 443 519 886 962 1329 1405 1772 1902
0 367 443 810 886 1253 1329 1696 1826 1902
0 76 443 519 886 962 1329 1459 1535 1902
0 367 443 810 886 1253 1383 1459 1826 1902
0 76 443 519 886 1016 1092 1459 1535 1902
0 367 443 810 940 1016 1383 1459 1826 1902
0 76 443 573 649 1016 1092 1459 1535 1902
0 367 497 573 940 1016 1383 1459 1826 1902

Using Olivier Messiaen's "concept of limited transposability" as a
umbrella term under which I group all symmetric type scales, let's
look at a couple of silver schismic "Trihill" (after David Clampitt)
conversions within a given P.

Here's the [8,2] static symmetrical decatonic (after Paul Erlich)
where P = 1:2^(1/2):

[4,1] = [3,1,1]

281---187
/ \ / \
/ \ / \
0---506---413

0 187 281 413 506 600 787 881 1013 1106 1200
0 94 226 319 413 600 694 826 919 1013 1200
0 132 226 319 506 600 732 826 919 1106 1200
0 94 187 374 468 600 694 787 974 1068 1200
0 94 281 374 506 600 694 881 974 1106 1200

Here's the [6,3] "Tcherepnin scale" where P = 1:2^(1/3):

[2,1] = [1,1,1]

181
/ \
/ \
0---309

0 181 309 400 581 709 800 981 1109 1200
0 128 219 400 528 619 800 928 1019 1200
0 91 272 400 491 672 800 891 1072 1200

THE WRONG COMMA -- ETC:

So there it is... a generalized one- to two-dimensional Myhill/MOS
conversion formula!

Last night I was attempting to generalize the syntonic comma, turns
out that this was the right idea but the wrong comma... it was the
generalization of the Pythagorean comma that allowed any given [a,b]
index to morph into a "Trihill"/"MOS" [a,b,c] index.

While the c/b, b/a proportions could perhaps be better addressed, I
think the silver-weighting works well enough as it is for now...

--Dan Stearns

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

🔗D.Stearns <STEARNS@CAPECOD.NET>

Paul Erlich wrote,

<< Dan -- congratulations on your "EUREKA", but I'm afraid I can't
make heads or tails of what you've posted. I'll try again on Monday,
if not sooner -- >>

It's a little "excited" for sure, and a few minor things are still
undefined; specifically what to set the internal c/b and b/c
proportions to... some of the examples I gave have the second
dimension in a "wrong" (or reversed) commatic position as well... but
I had it all bottled up inside at work today, so when I finally got
home I just flew in the door and went at it... even forgot to take my
coat off for three hours!

It should all be easy enough to follow though I would think?

<< meanwhile, a nice goal might be to get it into a form that someone
like David Clampitt (i.e. intelligent but new to the list) might
understand. >>

Unfortunately (or whatever), I can only do things pretty much the way
I do them... "informally".

When we were kids, probably seven or eight or so, we had a game called
dice baseball. And I remember that I used to figure out all the
imaginary players earned run and batting averages, strikeouts per nine
innings and whatnot... the other kids were suspicious and thought I
must've "made it up"! Anyway, I had no interest in "math", but I could
get around with what I knew if it was tied close enough to something I
was interested in. Not that any of this is any big deal or anything,
but what I'm trying to say is that what I do with tuning and numbers
and whatnot now is exactly the same thing really... I have no real
math interest or skill outside of whatever narrow sort of thing I also
happen to be interested in... it's all sort of like creative problem
solving given a limited range of skills and options or something...
Suffice it to say that a more formal and charitable narrative will
have to come from somebody else. I simply can't do it!

I do enjoy trying to better explain something specific though...

--Dan Stearns

🔗D.Stearns <STEARNS@CAPECOD.NET>

I wrote,

<< a few minor things are still undefined; specifically what to set
the internal c/b and b/c proportions to... >>

Here's a normalized conversion rule that I like:

If a<b then [a,b]=[a,(b+c)], and c/b = (sqrt(2)+1)/2 and b/a =
sqrt(2).

If a>b then [a,b]=[(a+b),c)], and both c/b and b/a = sqrt(2).

Here's the Bohlen-Pierce Lambda scale I gave yesterday with the
normalized conversion rule.

[5,4] = [1,4,4]

1642--1199---756---313
/ \ / \ / \ / \
/ \ / \ / \ / \
0--1459--1016---573---130

0 130 313 573 756 1016 1199 1459 1642 1902
0 184 443 627 886 1070 1329 1513 1772 1902
0 260 443 703 886 1146 1329 1589 1718 1902
0 184 443 627 886 1070 1329 1459 1642 1902
0 260 443 703 886 1146 1275 1459 1718 1902
0 184 443 627 886 1016 1199 1459 1642 1902
0 260 443 703 832 1016 1275 1459 1718 1902
0 184 443 573 756 1016 1199 1459 1642 1902
0 260 389 573 832 1016 1275 1459 1718 1902

Here's all the 5-tone conversions.

[1,4] = [1,1,3]

431---159
/ \ / \
/ \ / \
0---928---656

0 159 431 656 928 1200
0 272 497 769 1041 1200
0 225 497 769 928 1200
0 272 544 703 975 1200
0 272 431 703 928 1200

[2,3] = [2,2,1]

1016---573
/ \ / \
/ \ / \
0---757---313

0 313 573 757 1016 1200
0 260 443 703 887 1200
0 184 443 627 940 1200
0 260 443 757 1016 1200
0 184 497 757 940 1200

[3,2] = [1,2,2]

893---370
/ \ / \
/ \ / \
0---677---153

0 153 370 677 893 1200
0 217 523 740 1047 1200
0 307 523 830 983 1200
0 217 523 677 893 1200
0 307 460 677 983 1200

[4,1] = [3,1,1]

561---374
/ \ / \
/ \ / \
0--1013---826

0 374 561 826 1013 1200
0 187 452 639 826 1200
0 265 452 639 1013 1200
0 187 374 748 935 1200
0 187 561 748 1013 1200

This would seem to tie up any loose ends from yesterday. The one
remaining thing that could perhaps be altered to better overall effect
would be the weighting... I like the silver-weighting myself, but
perhaps a strict syntonic model would be worth interesting as well?

--Dan Stearns

🔗D.Stearns <STEARNS@CAPECOD.NET>

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

🔗D.Stearns <STEARNS@CAPECOD.NET>

Paul Erlich wrote,

<< I'm still not following you, but I suspect that the Tribonacci
constant would play the same role for Trihill scales as the Golden
Ratio plays for Myhill scales. >>

Sure, in as much as the results always agree with this:

If you look at a two-term series as a two-stepsize index as [a,b]
where the alphabetized variables are small to large stepsizes, then
a+b is the subset (or scale) of b+(a+b) where the size -- not to be
confused with the amount or index -- of a is 1, the size of b is 2,
and b+(a+b) is an equal division of the octave.

This of course would be true at any point in the sequence; witness
Joseph Yasser's famous example.

So a three-term Tribonacci series is analogous to the two-term
Fibonacci series in this sense where a+b+c is the corresponding subset
of the sum of the 3rd, 4th, and 5th terms and the size -- again, not
to be confused with the amount or "index" -- of a=2, b=3, c=4, and the
sum of the 3rd, 4th, and 5th terms is an equal division of the octave.

This is the generalized fib to tri (and higher term) parallel in the
Yasser sense.

So a two-stepsize [2,5] two-term series example is:

2, 5, 7, 12, 19, ...

And a+b is the subset of the sum of the 2nd and 3rd terms where a=1
and b=2. This is the familiar 7-out-of-12.

A three-stepsize [2,2,3] three-term series example would be:

2, 2, 3, 7, 12, 22, 41, ...

And a+b+c is the subset of the sum of the 3rd, 4th, and 5th terms
where a=2, b=3, and c=4. This would be the familiar syntonic diatonic
as a 7-out-of-22.

So the Tribonacci constant would play the same role for as the Golden
Ratio in terms of the series that result.

I'm still working on a way to get the syntonic, two generator model to
be "Trihill" for all scales with even numbered [a,b] indexes (almost
there), but other than that it all works really well I think... I'm
quite pleased.

I suspect that a Tribonacci constant weighting will have some meaning
if a three-term (three fraction) analogue to adjacent fractions and
seeding a Stern-Brocot Tree is established; BTW, I wrote Pierre
Lamothe regarding this, and here's part of his response...

"Just a word about Stern-Brocot tree. It corresponds point-to-point to
space of modular matrix 2 x 2 where determinant ad-bc = 1. Summing
columns give 2D vector (ratio) of Stern-Brocot tree. (And summing
columns is like apply matrix to vector (1,1). Summing lines give 2D
(contra)vector of Euclid tree.

Generalizing to nD implies n(n-1) components. In 2D, tree has 2
branchs at each nodes, in 3D, tree has 6 branchs at each node, in 4D,
tree has 12 branchs at each nodes, etc. Besides, elements are not
always 2D vectors assimilable to ratios but nD vectors assimilable to
chords.

I investigate few strategies to extend minimally Euclid algorithm (the
problem seems complex) but I will look anew later for I found a new
application of Stern-Brocot tree in lattices.

Sorry, yet, not to be helping. I hope to do better next time..."

--Dan Stearns

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Carl Lumma <clumma@yahoo.com>

Wow, I wish I understood this. Dan, any chance you could write
up a version for someone who doesn't know what an [a,b] index
or silver-weighted generator are? What framework are you
working in here... for exmaple, how do you know the thirds
of "[1,6] = [1,5,1]" will be 420 cents?

-Carl

--- In tuning@yahoogroups.com, "D.Stearns" <STEARNS@...> wrote:
> OK, I finally got it...
>
> EUREKA:
>
> By using a one- to two-dimensional conversion it is possible to
> convert any given [a,b] index into an [a,b,c] index that is a
> three-term "MOS"/"trivalence" within a given periodicity.
>
> SILVER-WEIGHTED:
>
> In the past I've referred to the scales created by a silver-weighted
> generator as a sort of generalized Pythagorean scale. I think this
> is a useful frame of reference for converting [a,b] indexes into
> [a,b,c] indexes.
>
> Here's the generalized formula for deriving a silver-weighted
> generator for any given [a,b] index within a given periodicity.
>
> X = P/((a+S*b))*(z+S*y)
>
> Where:
>
> "P" = any given periodicity
>
> "S" = the sqrt(2)+1 silver constant which can be expressed by the
> series 1/2, 2/5, 5/12, 12/29, 29/70, ...
>
> "z"/"a", "y"/"b" = the adjacent fractions of a given [a,b] index
>
> and "X" = the resulting silver-weighted generator
>
> THE SCHISMATIC SILVER FOLD:
>
> Multiplying "X" by the forth term of the two-term [a,b] index and
> scaling it by "P" gives a silver comma -- or by way of the analogy,
> a generalized Pythagorean comma.
>
> Converting the one-dimensional silver chain into a two-dimensional
> schismatic silver fold will give the two to three stepsize
> cardinality.
>
> SOME "SIMPLE" EXAMPLES:
>
> These are all the 7-tone scale index conversions:
>
> [1,6] = [1,5,1]
> [2,5] = [2,2,3]
> [3,4] = [3,3,1]
> [4,3] = [3,1,3]
> [5,2] = [2,3,2]
> [6,1] = [1,1,5]
>
> Here are the 7-tone silver schismic two-dimensional, two generator,
> three-term MOS conversions.
>
> [1,6] = [1,5,1]
>
> 420---232----45
> / \ / \ / \
> / \ / \ / \
> 0--1013---826---639
>
> 0 45 232 420 639 826 1013 1200
> 0 187 374 593 780 968 1155 1200
> 0 187 406 593 780 968 1013 1200
> 0 219 406 593 780 826 1013 1200
> 0 187 374 561 607 794 981 1200
> 0 187 374 420 607 794 1013 1200
> 0 187 232 420 607 826 1013 1200
>
> [2,5] = [2,2,3]
>
> 376--1079---582
> / \ / \ / \
> / \ / \ / \
> 0---703---206---909
>
> 0 206 376 582 703 909 1079 1200
> 0 171 376 497 703 874 994 1200
> 0 206 326 532 703 824 1029 1200
> 0 121 326 497 618 824 994 1200
> 0 206 376 497 703 874 1079 1200
> 0 171 291 497 668 874 994 1200
> 0 121 326 497 703 824 1029 1200
>
> [3,4] = [3,3,1]
>
> 1066---742---419
> / \ / \ / \
> / \ / \ / \
> 0---876---553---229
>
> 0 229 419 553 742 876 1066 1200
> 0 190 324 513 647 837 971 1200
> 0 134 324 458 647 781 1010 1200
> 0 190 324 513 647 876 1066 1200
> 0 134 324 458 687 876 1010 1200
> 0 190 324 553 742 876 1066 1200
> 0 134 363 553 687 876 1010 1200
>
> [4,3] = [3,1,3]
>
> 898---534---169
> / \ / \ / \
> / \ / \ / \
> 0---836---471---107
>
> 0 107 169 471 534 836 898 1200
> 0 63 364 427 729 791 1093 1200
> 0 302 364 666 729 1031 1137 1200
> 0 63 364 427 729 836 898 1200
> 0 302 364 666 773 836 1137 1200
> 0 63 364 471 534 836 898 1200
> 0 302 409 471 773 836 1137 1200
>
> [5,2] = [2,3,2]
>
> 194---855---316
> / \ / \ / \
> / \ / \ / \
> 0---661---122---783
>
> 0 122 194 316 661 783 855 1200
> 0 72 194 539 661 733 1078 1200
> 0 122 467 590 661 1006 1128 1200
> 0 345 467 539 884 1006 1078 1200
> 0 122 194 539 661 733 855 1200
> 0 72 417 539 610 733 1078 1200
> 0 345 467 539 661 1006 1128 1200
>
> [6,1] = [1,1,5]
>
> 570---428---285
> / \ / \ / \
> / \ / \ / \
> 0--1057---915---772
>
> 0 285 428 570 772 915 1057 1200
> 0 143 285 487 630 772 915 1200
> 0 143 344 487 630 772 1057 1200
> 0 202 344 487 630 915 1057 1200
> 0 143 285 428 713 856 998 1200
> 0 143 285 570 713 856 1057 1200
> 0 143 428 570 713 915 1057 1200
>
> SOME "EXOTIC" EXAMPLES:
>
> The Bohlen-Pierce Lambda scale can be seen as [5,4] index where P =
> 1:3. Converting this into a silver schismic two-dimensional, two
> generator, three-term MOS results in a [5,4] = [4,1,4] index
> conversion and the following scale:
>
> 1535--1092---649---206
> / \ / \ / \ / \
> / \ / \ / \ / \
> 0--1459--1016---573---130
>
> 0 130 206 573 649 1016 1092 1459 1535 1902
> 0 76 443 519 886 962 1329 1405 1772 1902
> 0 367 443 810 886 1253 1329 1696 1826 1902
> 0 76 443 519 886 962 1329 1459 1535 1902
> 0 367 443 810 886 1253 1383 1459 1826 1902
> 0 76 443 519 886 1016 1092 1459 1535 1902
> 0 367 443 810 940 1016 1383 1459 1826 1902
> 0 76 443 573 649 1016 1092 1459 1535 1902
> 0 367 497 573 940 1016 1383 1459 1826 1902
>
> Using Olivier Messiaen's "concept of limited transposability" as a
> umbrella term under which I group all symmetric type scales, let's
> look at a couple of silver schismic "Trihill" (after David Clampitt)
> conversions within a given P.
>
> Here's the [8,2] static symmetrical decatonic (after Paul Erlich)
> where P = 1:2^(1/2):
>
> [4,1] = [3,1,1]
>
> 281---187
> / \ / \
> / \ / \
> 0---506---413
>
> 0 187 281 413 506 600 787 881 1013 1106 1200
> 0 94 226 319 413 600 694 826 919 1013 1200
> 0 132 226 319 506 600 732 826 919 1106 1200
> 0 94 187 374 468 600 694 787 974 1068 1200
> 0 94 281 374 506 600 694 881 974 1106 1200
>
> Here's the [6,3] "Tcherepnin scale" where P = 1:2^(1/3):
>
> [2,1] = [1,1,1]
>
> 181
> / \
> / \
> 0---309
>
> 0 181 309 400 581 709 800 981 1109 1200
> 0 128 219 400 528 619 800 928 1019 1200
> 0 91 272 400 491 672 800 891 1072 1200
>
> THE WRONG COMMA -- ETC:
>
> So there it is... a generalized one- to two-dimensional Myhill/MOS
> conversion formula!
>
> Last night I was attempting to generalize the syntonic comma, turns
> out that this was the right idea but the wrong comma... it was the
> generalization of the Pythagorean comma that allowed any given [a,b]
> index to morph into a "Trihill"/"MOS" [a,b,c] index.
>
> While the c/b, b/a proportions could perhaps be better addressed, I
> think the silver-weighting works well enough as it is for now...
>
> --Dan Stearns

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗daniel_anthony_stearns <daniel_anthony_stearns@yahoo.com>

yikes, this is an old post, so old i don't even remember it! Anyway,
just got back from a week in a cabin in the woods in New Hampshire so
i'm just catching up on a pile of posts and emails and whatnot...
anyway, i did have this bit saved and always wondered if anyone did
any more with this :

http://www.robertinventor.com/tuning-math/s___2/msg_1875-1899.html

http://www.myspace.com/danstearns

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Wow, I wish I understood this. Dan, any chance you could write
> up a version for someone who doesn't know what an [a,b] index
> or silver-weighted generator are? What framework are you
> working in here... for exmaple, how do you know the thirds
> of "[1,6] = [1,5,1]" will be 420 cents?
>
> -Carl
>
> --- In tuning@yahoogroups.com, "D.Stearns" <STEARNS@> wrote:
> > OK, I finally got it...
> >
> > EUREKA:
> >
> > By using a one- to two-dimensional conversion it is possible to
> > convert any given [a,b] index into an [a,b,c] index that is a
> > three-term "MOS"/"trivalence" within a given periodicity.
> >
> > SILVER-WEIGHTED:
> >
> > In the past I've referred to the scales created by a silver-weighted
> > generator as a sort of generalized Pythagorean scale. I think this
> > is a useful frame of reference for converting [a,b] indexes into
> > [a,b,c] indexes.
> >
> > Here's the generalized formula for deriving a silver-weighted
> > generator for any given [a,b] index within a given periodicity.
> >
> > X = P/((a+S*b))*(z+S*y)
> >
> > Where:
> >
> > "P" = any given periodicity
> >
> > "S" = the sqrt(2)+1 silver constant which can be expressed by the
> > series 1/2, 2/5, 5/12, 12/29, 29/70, ...
> >
> > "z"/"a", "y"/"b" = the adjacent fractions of a given [a,b] index
> >
> > and "X" = the resulting silver-weighted generator
> >
> > THE SCHISMATIC SILVER FOLD:
> >
> > Multiplying "X" by the forth term of the two-term [a,b] index and
> > scaling it by "P" gives a silver comma -- or by way of the analogy,
> > a generalized Pythagorean comma.
> >
> > Converting the one-dimensional silver chain into a two-dimensional
> > schismatic silver fold will give the two to three stepsize
> > cardinality.
> >
> > SOME "SIMPLE" EXAMPLES:
> >
> > These are all the 7-tone scale index conversions:
> >
> > [1,6] = [1,5,1]
> > [2,5] = [2,2,3]
> > [3,4] = [3,3,1]
> > [4,3] = [3,1,3]
> > [5,2] = [2,3,2]
> > [6,1] = [1,1,5]
> >
> > Here are the 7-tone silver schismic two-dimensional, two generator,
> > three-term MOS conversions.
> >
> > [1,6] = [1,5,1]
> >
> > 420---232----45
> > / \ / \ / \
> > / \ / \ / \
> > 0--1013---826---639
> >
> > 0 45 232 420 639 826 1013 1200
> > 0 187 374 593 780 968 1155 1200
> > 0 187 406 593 780 968 1013 1200
> > 0 219 406 593 780 826 1013 1200
> > 0 187 374 561 607 794 981 1200
> > 0 187 374 420 607 794 1013 1200
> > 0 187 232 420 607 826 1013 1200
> >
> > [2,5] = [2,2,3]
> >
> > 376--1079---582
> > / \ / \ / \
> > / \ / \ / \
> > 0---703---206---909
> >
> > 0 206 376 582 703 909 1079 1200
> > 0 171 376 497 703 874 994 1200
> > 0 206 326 532 703 824 1029 1200
> > 0 121 326 497 618 824 994 1200
> > 0 206 376 497 703 874 1079 1200
> > 0 171 291 497 668 874 994 1200
> > 0 121 326 497 703 824 1029 1200
> >
> > [3,4] = [3,3,1]
> >
> > 1066---742---419
> > / \ / \ / \
> > / \ / \ / \
> > 0---876---553---229
> >
> > 0 229 419 553 742 876 1066 1200
> > 0 190 324 513 647 837 971 1200
> > 0 134 324 458 647 781 1010 1200
> > 0 190 324 513 647 876 1066 1200
> > 0 134 324 458 687 876 1010 1200
> > 0 190 324 553 742 876 1066 1200
> > 0 134 363 553 687 876 1010 1200
> >
> > [4,3] = [3,1,3]
> >
> > 898---534---169
> > / \ / \ / \
> > / \ / \ / \
> > 0---836---471---107
> >
> > 0 107 169 471 534 836 898 1200
> > 0 63 364 427 729 791 1093 1200
> > 0 302 364 666 729 1031 1137 1200
> > 0 63 364 427 729 836 898 1200
> > 0 302 364 666 773 836 1137 1200
> > 0 63 364 471 534 836 898 1200
> > 0 302 409 471 773 836 1137 1200
> >
> > [5,2] = [2,3,2]
> >
> > 194---855---316
> > / \ / \ / \
> > / \ / \ / \
> > 0---661---122---783
> >
> > 0 122 194 316 661 783 855 1200
> > 0 72 194 539 661 733 1078 1200
> > 0 122 467 590 661 1006 1128 1200
> > 0 345 467 539 884 1006 1078 1200
> > 0 122 194 539 661 733 855 1200
> > 0 72 417 539 610 733 1078 1200
> > 0 345 467 539 661 1006 1128 1200
> >
> > [6,1] = [1,1,5]
> >
> > 570---428---285
> > / \ / \ / \
> > / \ / \ / \
> > 0--1057---915---772
> >
> > 0 285 428 570 772 915 1057 1200
> > 0 143 285 487 630 772 915 1200
> > 0 143 344 487 630 772 1057 1200
> > 0 202 344 487 630 915 1057 1200
> > 0 143 285 428 713 856 998 1200
> > 0 143 285 570 713 856 1057 1200
> > 0 143 428 570 713 915 1057 1200
> >
> > SOME "EXOTIC" EXAMPLES:
> >
> > The Bohlen-Pierce Lambda scale can be seen as [5,4] index where P =
> > 1:3. Converting this into a silver schismic two-dimensional, two
> > generator, three-term MOS results in a [5,4] = [4,1,4] index
> > conversion and the following scale:
> >
> > 1535--1092---649---206
> > / \ / \ / \ / \
> > / \ / \ / \ / \
> > 0--1459--1016---573---130
> >
> > 0 130 206 573 649 1016 1092 1459 1535 1902
> > 0 76 443 519 886 962 1329 1405 1772 1902
> > 0 367 443 810 886 1253 1329 1696 1826 1902
> > 0 76 443 519 886 962 1329 1459 1535 1902
> > 0 367 443 810 886 1253 1383 1459 1826 1902
> > 0 76 443 519 886 1016 1092 1459 1535 1902
> > 0 367 443 810 940 1016 1383 1459 1826 1902
> > 0 76 443 573 649 1016 1092 1459 1535 1902
> > 0 367 497 573 940 1016 1383 1459 1826 1902
> >
> > Using Olivier Messiaen's "concept of limited transposability" as a
> > umbrella term under which I group all symmetric type scales, let's
> > look at a couple of silver schismic "Trihill" (after David Clampitt)
> > conversions within a given P.
> >
> > Here's the [8,2] static symmetrical decatonic (after Paul Erlich)
> > where P = 1:2^(1/2):
> >
> > [4,1] = [3,1,1]
> >
> > 281---187
> > / \ / \
> > / \ / \
> > 0---506---413
> >
> > 0 187 281 413 506 600 787 881 1013 1106 1200
> > 0 94 226 319 413 600 694 826 919 1013 1200
> > 0 132 226 319 506 600 732 826 919 1106 1200
> > 0 94 187 374 468 600 694 787 974 1068 1200
> > 0 94 281 374 506 600 694 881 974 1106 1200
> >
> > Here's the [6,3] "Tcherepnin scale" where P = 1:2^(1/3):
> >
> > [2,1] = [1,1,1]
> >
> > 181
> > / \
> > / \
> > 0---309
> >
> > 0 181 309 400 581 709 800 981 1109 1200
> > 0 128 219 400 528 619 800 928 1019 1200
> > 0 91 272 400 491 672 800 891 1072 1200
> >
> > THE WRONG COMMA -- ETC:
> >
> > So there it is... a generalized one- to two-dimensional Myhill/MOS
> > conversion formula!
> >
> > Last night I was attempting to generalize the syntonic comma, turns
> > out that this was the right idea but the wrong comma... it was the
> > generalization of the Pythagorean comma that allowed any given [a,b]
> > index to morph into a "Trihill"/"MOS" [a,b,c] index.
> >
> > While the c/b, b/a proportions could perhaps be better addressed, I
> > think the silver-weighting works well enough as it is for now...
> >
> > --Dan Stearns
>