Generalized Myhill's property

I actually read all the way to the end of the "MOS Generalization" thread, but unfortunately failed to find what I'm looking for, which is a characterization of, or algorithm to produce, scales with the Generalized Myhill's property.

Definition: A scale has the Generalized Myhill's property of order N (NGMP) if every generic interval is represented by no more than N specific intervals.

1GMP is equal temperament.
2GMP is the ordinary Myhill's property, a.k.a. MOS.

Finding all the scales in a rank-2 temperament, e.g. meantone, that satisfy 2GMP is something everyone reading this should know how to do. The possible numbers of notes are the convergents, 2, 3, 5, 7, 12, 19..., and for each of those cardinalities there is exactly one scale (the MOS) which satisfies 2GMP. Its pitch classes form a consecutive block in the 1-dimensional pitch class space of meantone.

In other words, a Fokker block of a rank-2 temperament (1-dimensional pitch class space) has only one possible shape, and that shape always has 2GMP.

Also note that 2GMP is a property of the abstract scale (e.g. LLsLLLs), which does not depend on the specific temperament at all! It is a special property of the sequence of symbols "LLsLLLs", which is not shared by, e.g. "LsLLLLs".

Here's an example of what I want to do for a rank-3 temperament, e.g. marvel:

(1) Find all the possible cardinalities of scales with 3GMP, and for each of those cardinalities the appropriate numbers of small, medium, and large steps. (For example, you could have a 9-note scale with 3L + 4m + 2s, but it took me a while to figure that out. Is there a simple formula, like some generalization of continued fraction convergents?)

(2) Given a cardinality and numbers of steps, e.g. 3L + 4m + 2s, find all possible arrangements of these that actually satisfy 3GMP. The brute force way to do this would be simply to enumerate all permutations of LLLmmmmss and then explicitly check 3GMP for every generic interval, but come on, there must be a simpler way.

Another way to think of the problem here is that, for rank-3 and higher, a Fokker block ***has different possible shapes*** depending on how you position the unit cell relative to the lattice. What's worse, many of these possible Fokker blocks ***do not have 3GMP***. How can we characterize the ones that do?

Keenan

I just wrote a long thoughtful post about this, but it seems Yahoo ate it, so I'll just ask my main question again in case the other post doesn't turn up:

Is there a characterization of, or non-brute-force algorithm to produce, scales of rank-3 or higher temperaments that have the generalized Myhill's property, 3GMP (or NGMP where N is the rank)?

(There was a long thread about this, "MOS Generalization", but I couldn't seem to find a conclusion...)

Keenan

>(2) Given a cardinality and numbers of steps, e.g. 3L + 4m + 2s, find
>all possible arrangements of these that actually satisfy 3GMP. The
>brute force way to do this would be simply to enumerate all
>permutations of LLLmmmmss and then explicitly check 3GMP for every
>generic interval, but come on, there must be a simpler way.
>
>Another way to think of the problem here is that, for rank-3 and
>higher, a Fokker block ***has different possible shapes*** depending
>on how you position the unit cell relative to the lattice. What's
>worse, many of these possible Fokker blocks ***do not have 3GMP***.
>How can we characterize the ones that do?
>
>Keenan

...Two problems that are open as far as I know. Mike's MODMOS
are a class of scales obtained by transposing tones of a rank 2
Fokker block by the chromatic unison vector... -Carl

Keenan wrote:

>(There was a long thread about this, "MOS Generalization", but I
>couldn't seem to find a conclusion...)

There was also a 'rank 3 scales' thread here recently...

-Carl

Keenan wrote:
>In other words, a Fokker block of a rank-2 temperament (1-dimensional
>pitch class space) has only one possible shape, and that shape always
>has 2GMP.

Why do you call it 1-D? I think it's still 2-D in pitch classes.

And aren't there numerous equivalent bases for the temperament,
each corresponding to a different block (all with 2GMP)? -Carl

Carl Lumma <carl@lumma.org> wrote:
> Keenan wrote:
> >In other words, a Fokker block of a rank-2 temperament
> >(1-dimensional pitch class space) has only one possible
> >shape, and that shape always has 2GMP.
>
> Why do you call it 1-D? I think it's still 2-D in pitch
> classes.

Where "pitch classes" entail octave equivalence, it's 1-D.
The number of pitches is proportional to the number of
generator steps. The number of periods to the octave is
constant.

> And aren't there numerous equivalent bases for the
> temperament, each corresponding to a different block (all
> with 2GMP)? -Carl

All the blocks are the same shape -- a string of generators
modulo the period.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Where "pitch classes" entail octave equivalence, it's 1-D.
> The number of pitches is proportional to the number of
> generator steps. The number of periods to the octave is
> constant.

Right, that's what I meant when I said "pitch class". The pitch classes of an EDO are a finite set; the pitch classes of a rank-2 temperament are infinite in 1 dimension; the pitch classes of a rank-N temperament are infinite in (N-1) dimensions.

> > And aren't there numerous equivalent bases for the
> > temperament, each corresponding to a different block (all
> > with 2GMP)? -Carl
>
> All the blocks are the same shape -- a string of generators
> modulo the period.

Right. This is exactly what makes MOSes so simple, and what makes generalizing them to higher ranks so far from straightforward.

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Keenan wrote:
>
> >(There was a long thread about this, "MOS Generalization", but I
> >couldn't seem to find a conclusion...)
>
> There was also a 'rank 3 scales' thread here recently...

I'm having trouble finding this thread. I can't seem to find anything other than the "MOS Generalization" thread and some other brief mentions.

Keenan

My current search term for the math related to this is "rational Beatty sequence".

Keenan

Graham wrote:

>Where "pitch classes" entail octave equivalence, it's 1-D.
>The number of pitches is proportional to the number of
>generator steps. The number of periods to the octave is
>constant.

I was thinking of the mapping to JI in the 5-limit case.

-Carl

>> There was also a 'rank 3 scales' thread here recently...
>
>I'm having trouble finding this thread. I can't seem to find anything
>other than the "MOS Generalization" thread and some other brief mentions.
>
>Keenan

Sorry, that's the thread I was thinking of (and it looks like
"Testing for block status" was interwoven). -Carl

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> My current search term for the math related to this is "rational Beatty sequence".
>
> Keenan

Okay, now that I've looked into it a little deeper I think rational Beatty sequences are a red herring.

I got pretty excited about them because at the bottom of the first page of this paper: http://dx.doi.org/10.1016/0097-3165(86)90015-4 there is an actual example of a scale that satisfies 3GMP:

aabacab

(where a, b, and c represent different sizes of steps, like L, m, and s).

However, although I'm fairly sure that every disjoint covering system of rational Beatty sequences yields a scale that satisfies NGMP (where N is the number of Beatty sequences), the converse must only be true for the rank-2, 2GMP case. The first counterexample is actually quite small:

aabcb

This scale satisfies 3GMP because there are only three kinds of each interval:
steps: a, b, c
"thirds": 2a, a+b, b+c
"fourths": 2a+b, a+b+c, 2b+c
"fifths": 2a+b+c, a+2b+c, 2a+2b

However, it does not correspond to a system of rational Beatty sequences because the distribution of the 'a's is too uneven. The locations of the 'a's (1,2,6,7,11,12,16,17...) is not a rational Beatty sequence.

Another way to say basically the same thing is that, if you temper out the difference between 'b' and 'c', you get

aabbb

which is not a MOS. So it is not true that every NGMP scale becomes an (N-1)GMP scale upon tempering out a comma to reduce the rank by 1.

So, having no other way to generate 3GMP scales, I wrote a few dozen lines of Python to find them by exhaustive brute force search. Here's what it spit out:

['abc']
['abac']
['aabac', 'aabcb', 'ababc']
['abacbc', 'aabaac', 'abcabc']
['abababc', 'aabacab', 'abacabc', 'aaabaac']
['abacabac', 'abacbabc', 'aaabaaac']
['abcabcabc', 'abacbacbc', 'aaaabaaac', 'aabacabac', 'ababababc']
['aabcbaabcb', 'ababacbabc', 'abacbacabc', 'aaaabaaaac', 'ababcababc', 'aabaacabac', 'aabacaabac']
['abababababc', 'abacbabcabc', 'abacabacabc', 'aaaaabaaaac', 'aabaacaabac']
['aaaaabaaaaac', 'ababacbababc', 'abacbcabacbc', 'aabaacaabaac', 'abcabcabcabc', 'abacabacabac', 'abacbacbcabc', 'abacbacbacbc']
['aaaaaabaaaaac', 'ababababababc', 'abacbacabcabc', 'ababcabacbabc', 'aaabaacaabaac', 'aabacabacabac']
['abacbacbabcabc', 'aaaaaabaaaaaac', 'abababcabababc', 'abacabacbacabc', 'abacabcabacabc', 'aabacabaacabac', 'aaabaaacaabaac', 'abababacbababc', 'aabacabaabacab', 'aaabaacaaabaac']
['ababcababcababc', 'ababacbabacbabc', 'abacabacabacabc', 'abacbacbcabacbc', 'abacbacbacbacbc', 'ababacbabcababc', 'aaabaaacaaabaac', 'abababababababc', 'aaaaaaabaaaaaac', 'aabcbaabcbaabcb', 'aabaacabaacabac', 'abcabcabcabcabc', 'aabaacabacaabac', 'abacbacbacbcabc', 'aabacaabacaabac']
['abacabacabacabac', 'aaaaaaabaaaaaaac', 'abacbacbacabcabc', 'abacbabcabacbabc', 'aabaacabaacaabac', 'aaabaaacaaabaaac', 'abababacbabababc']

The numbers of distinct scale patterns for each cardinality (1, 1, 3, 3, 4, 3, 5, 7, 5, 8, 6, 10, 15, 8...) is not in the OEIS, and the superseeker reply had no content in it, which is surprising (usually it turns up a bunch of false matches).

There are two pairs of mirror-image scales with 15 notes:

'ababacbabcababc' <-> 'ababacbabacbabc'
'aabaacabacaabac' <-> 'aabaacabaacabac'

This is fascinating because (1) for the rank-2 case it is impossible to have such mirror images (every MOS is symmetrical), and (2) all of the 3GMP scales are symmetric up to 14 notes, but with 15 notes distinct mirror images suddenly appear.

If you consider mirror-image scales as equivalent, then the number of scales with 15 notes gets reduced from 15 to thirteen, but that sequenceu (1, 1, 3, 3, 4, 3, 5, 7, 5, 8, 6, 10, 13, 8...) also gets an empty superseeker reply.

And to answer the specific question I asked in my first post, it is *impossible* to have a 3GMP scale with 4 'a's, 3 'b's, and 2 'c's. There simply is no such sequence. Is there some simple formula that tells us this is impossible? If so, it is currently unknown.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> There are two pairs of mirror-image scales with 15 notes:
>
> 'ababacbabcababc' <-> 'ababacbabacbabc'
> 'aabaacabacaabac' <-> 'aabaacabaacabac'
>
> This is fascinating because (1) for the rank-2 case it is impossible to have such mirror images (every MOS is symmetrical), and (2) all of the 3GMP scales are symmetric up to 14 notes, but with 15 notes distinct mirror images suddenly appear.

It occurs to me that this could be easily misinterpreted. Of course if you have a scale "Lms", that is different from its mirror image "smL". For example one could be a "major arpeggio" and one could be a "minor arpeggio". But if you're allowed to change which intervals are which, then "Lms" and "smL" both have the same very general structure: they're just "abc". With the two examples above, however, it is not possible to do that.

Keenan

I modified my average harmonic entropy calculator to do these kinds of scales. I just did a few to see that it works; later I'll do a comprehensive survey and post the results.

As expected, some of the patterns, e.g. aabac, are optimized when they degenerate to a MOS because two steps become equal. In this case b=c and the optimum is the pentatonic scale.

I already found one interesting minimum though, this lovely scale in marvel temperament:

1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1

This is LsmsLss where L = 7/6, m = 9/8, and s = 16/15 = 15/14. (The abstract pattern happens to be exactly the one given as an example in that Beatty sequence paper.)

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> I modified my average harmonic entropy calculator to do these kinds of scales. I just did a few to see that it works; later I'll do a comprehensive survey and post the results.

I'm very interested in such a survey. How large a scale do you plan to survey? I think this might go somewhere on the Xenwiki when you are done.

> I already found one interesting minimum though, this lovely scale in marvel temperament:
>
> 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1

Marvel's answer to the diatonic scale! As with the diatonic scale, it might be worth noting what larger structures you can embed it in.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > I already found one interesting minimum though, this lovely scale in marvel temperament:
> >
> > 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1
>
> Marvel's answer to the diatonic scale! As with the diatonic scale, it might be worth noting what larger structures you can embed it in.

The 72et version of this is exactly a mode of "Xenakis Byzantine Liturgical Soft Chromatic", which is xenakis_schrom.scl.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Marvel's answer to the diatonic scale! As with the diatonic scale, it might be worth noting what larger structures you can embed it in.

Scala tells me I can embed it in the following 10 note scales:

smithgw72b.scl
smithgw72j.scl
smithgw_qm3a.scl

Also, the following 12 note scales:

Unimarv[12]/Prodigy[12] (Hobbits become identical in 72)
lumma72.scl

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> I'm very interested in such a survey. How large a scale do you plan to survey? I think this might go somewhere on the Xenwiki when you are done.

I wasn't going to go much past 12 notes, maybe 13 or 14. Personally I think of such scales as "extremely large", though I'm sure many people don't share my opinion.

Keenan

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > > 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1

> The 72et version of this is exactly a mode of "Xenakis Byzantine Liturgical Soft Chromatic", which is xenakis_schrom.scl.

If you take the above scale and project it to the marvel 5-limit by replacing 7 with 225/32, you get this:

1/1 75/64 5/4 45/32 3/2 225/128 15/8 2/1

One property it has is that it consists entirely of overtones, which would make some people happy, apparently. It's also a mode of a Scale of Many Names:

diff7b: Difference diamond on [1, 8/5, 3/2]
exptriad2: Two times expanded major triad
helmholz: Helmholtz's Chromatic scale and Gipsy major from Slovakia
jobbit7_5: 5-limit 7-note JI hobbit
mavchrome1: First 25/24&135/128 scale
tartini7: Tartini (1754) with 2 neochromatic tetrachords
trab7: transformed Euclidean ball {1,3,15} diamond

It's also clear how to embed this into larger structures: take a suitable Euler genus, such as Genus(15^n) for n>1, marvel temper it, and there you are.