Question re scales

Suppose G is a finite connected subgraph of the n-dimensional grid. Suppose g1, g2, ... gn are n generators, and we form a scale by taking all the products of the generators, in the given order, with exponents in G, and reducing by a period P. If the resulting scale has n sizes of steps, is there a name for such a thing? Should there be one? The point is, this is a rank n generalization of the rank 2 concept of "MOS" (not that MOS *must* be rank 2, but they are epsilon away from rank 2 at worst, same with these.) Is there a better idea around for a generalization of MOS?

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

Re this business, I've been looking for Fokker blocks with the number of step sizes equal to the rank of the p-limit, and it seems this is a pretty restrictive condition. Cataloging such things would be a place to start, or perhaps figuring out what makes them tick.

Gene wrote:

> Suppose G is a finite connected subgraph of the n-dimensional grid.
> Suppose g1, g2, ... gn are n generators, and we form a scale by taking
> all the products of the generators, in the given order, with exponents
> in G, and reducing by a period P. If the resulting scale has n sizes
> of steps, is there a name for such a thing? Should there be one? The
> point is, this is a rank n generalization of the rank 2 concept of
> "MOS" (not that MOS *must* be rank 2, but they are epsilon away from
> rank 2 at worst, same with these.) Is there a better idea around for a
> generalization of MOS?

Gene, I haven't been reading the list to see any more context, but someone just forwarded this to me so I'll reply. One such generalisation is Carey and Clampitt's "pairwise well-formed scale". This is a scale with three step sizes such that the pattern you get by equivalencing any two of the three steps is characteristic of a MOS. For example, the syntonic diatonic is <abcabac> which becomes <xxcxxxc>, <xbxxbxx> or <axxaxax>, each of which is a MOS. Quite a lot is known about these scales, and Clampitt's dissertation lists some other scales from world music that have this feature, and also shows some transformational properties of PWWF collections (a particular voice-leading cycle always occurs). There is only one template possible where the three steps have different multiplicities: the heptatonic <abacaba> (for example, the "Hungarian gypsy minor" C Db E F G Ab B C). No other PWWF template is inversionally symmetric.

I discovered that apart from scales with that exceptional template, all other PWWF scales can be found as convex sets on two adjacent rows of a lattice of two generators (octave equivalence is assumed). A PWWF scale of 2n+1 notes can be constructed as a chain of generators g1 with one different generator g2 inserted in the middle of the chain (think for example of the syntonic diatonic as a chain of 3:2s with one 40:27 in the middle of the chain).

(And the symmetrical PWWF is always constructible as the chain g1 g2 g1 g1 g2 g1)

In the course of investigating this I found that on the usual 5-limit lattice, where there are 3652 connected configurations of 7 hexagons (A001207 at the OEIS), there are 63 configurations that give scales with 3 step sizes, of which 21 are PWWF.

Regards - Jon Wild

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:

> Gene, I haven't been reading the list to see any more context, but someone
> just forwarded this to me so I'll reply. One such generalisation is Carey
> and Clampitt's "pairwise well-formed scale". This is a scale with three
> step sizes such that the pattern you get by equivalencing any two of the
> three steps is characteristic of a MOS.

It seems to me that among all the zoology of lattice animals, those which are rectangles, or rectangles with corners nibbled off (eg, the syntonic diatonic scale) are of special interest. Even for large edos, it seems quite fast to compute all the rectangles giving both exactly three sizes of steps and a proper scale. Something to bear in mind especially n connection with rank 3, I think.

Gene wrote:

>Suppose G is a finite connected subgraph of the n-dimensional grid.
>Suppose g1, g2, ... gn are n generators, and we form a scale by taking
>all the products of the generators, in the given order, with exponents
>in G, and reducing by a period P. If the resulting scale has n sizes
>of steps, is there a name for such a thing? Should there be one? The
>point is, this is a rank n generalization of the rank 2 concept of
>"MOS" (not that MOS *must* be rank 2, but they are epsilon away from
>rank 2 at worst, same with these.) Is there a better idea around for a
>generalization of MOS?

Seems like the natural generalization of MOS. Do they need a name,
other than 'scales of rank-n temperaments' or "periodicity blocks"
(in some cases)?

There have been efforts to generalize MOS before (look up "tribonacci"
or 3MOS or 3DES on the tuning list) but I forget the details.

Incidentally, the LLLSLLLLSLLL scale you mentioned may be Paul's
hexachordal scale in 22, analogous to the pentachordal scale that
he recommended over strict pajara[10] in his paper. I believe his
motivation for finding them was improving the symmetry-at-3:2 of
straight pajara[10].

-Carl

I wrote:
>There have been efforts to generalize MOS before (look up "tribonacci"
>or 3MOS or 3DES on the tuning list) but I forget the details.

NMOS was the most developed of these. See for instance this
message:

>>From: "Gene Ward Smith" <gwsmith@svpal.org>
>>Date: Thu, 01 May 2003 14:02:54 -0000
>>Subject: [tuning-math] Re: n in T[n]
>>Reply-To: tuning-math@yahoogroups.com
>>
>>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>> Gene, et al;
>>>
>>> So MOS aren't the only good n. Did we ever get a method
>>> for defining all the "good" n? Paul, how does Miracle[22]
>>> compare to Blackjack?
>>
>>Miracle[22] is a 2MOS, but the interesting thing is that there are
>>regular, "good" scales which aren't either MOS or NMOS. I don't have a
>>theory for them.
>

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> There have been efforts to generalize MOS before (look up "tribonacci"
> or 3MOS or 3DES on the tuning list) but I forget the details.

NMOS was my suggestion. If 7 notes of meantone is a MOS for meantone, 14 notes will be a 2MOS, etc.