Clough & Myerson

Reprints of two papers by Clough and Myerson appeared in my mailbox
today from Down Under, courtesy of Gerry Myerson. One is "Variety and
Multiplicity of Diatonic System", the other (from the Math
Monthly) "Musical Scales and the Generalized Circle of Fifths".

Considering d note scales in c-equal, they define the genus as the
step sizes of the gaps between the notes of the chord, modulo
transposition, in terms of d, and the species as the same in terms of
c. In other words, in C major CEG is 0 2 4 in diatonic terms and
0 4 7 in chromatic terms, so the genus is 223 and the species is 435.
Theorem 5 from Variety helps explain why Eytan was hipped about
certain kinds of scales:

Theorem 5
For any reduced scale, if every chord species is unambiguous in its
generic membership, then c = 2d-1 or c = 2d-2. The converse is also
true.

The fine print here is "reduced scale"; they reduce a scale with
Myhill's Property by lowering the size of the et c so that the sizes
of the scale steps are 1 and 2. This turns out to be slick way of
proving things, but it doesn't seem to be more than that.

A nice fact which is proven as well is that if (c, d)=1 then there is
a unique scale with Myhill's Property and it can be obtained very
simply as floor(kc/d), 0 <= k < d. This can be generalized
immediately by considering periods rather than octaves, allowing the
elimination of the condition that (c, d)=1. This does all the "white-
key black key" stuff I was doing a while back, (and Graham too, in
his own way?) in a slicker way.

That's not all but the papers are worth reading and done right, which
we cannot always count on to be the case.

Gene Ward Smith wrote:

>Considering d note scales in c-equal, they define the genus as the >step sizes of the gaps between the notes of the chord, modulo >transposition, in terms of d, and the species as the same in terms of >c. In other words, in C major CEG is 0 2 4 in diatonic terms and >0 4 7 in chromatic terms, so the genus is 223 and the species is 435. >Theorem 5 from Variety helps explain why Eytan was hipped about >certain kinds of scales:
>
>Theorem 5
>For any reduced scale, if every chord species is unambiguous in its >generic membership, then c = 2d-1 or c = 2d-2. The converse is also >true.
>
The reason Eytan gives for using them is that they have his kind of efficiency (each chromatic interval is represented at least once), and are proper with no more than one ambiguous interval pair (the tritone). I proved this in

http://x31eq.com/proof.html

You can reduce scales without affecting propriety by ensuring Rothenberg's alpha matrix (showing the ordering of intervals, rather than their sizes) is preserved. I believe that every strictly proper MOS has a maximally even counterpart with the same alpha matrix, but haven't proved it yet.

>The fine print here is "reduced scale"; they reduce a scale with >Myhill's Property by lowering the size of the et c so that the sizes >of the scale steps are 1 and 2. This turns out to be slick way of >proving things, but it doesn't seem to be more than that.
>
Do they insist on gcd(c,d)=1 as well? The reduction is a necessary condition for Agmon's efficiency. It does, however, mean that a lot of strictly proper scales lose their strictness.

>A nice fact which is proven as well is that if (c, d)=1 then there is >a unique scale with Myhill's Property and it can be obtained very >simply as floor(kc/d), 0 <= k < d. This can be generalized >immediately by considering periods rather than octaves, allowing the >elimination of the condition that (c, d)=1. This does all the "white-
>key black key" stuff I was doing a while back, (and Graham too, in >his own way?) in a slicker way.
> >
Yes, those are maximally even scales. See the link above.

>That's not all but the papers are worth reading and done right, which >we cannot always count on to be the case.
>
Oh, then I might look them out one day.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> >The fine print here is "reduced scale"; they reduce a scale with
> >Myhill's Property by lowering the size of the et c so that the
sizes
> >of the scale steps are 1 and 2. This turns out to be slick way of
> >proving things, but it doesn't seem to be more than that.
> >
> Do they insist on gcd(c,d)=1 as well?

Yes, always. Nonoctave periods do not seem to have been considered.