Well n-formed scales

Suppose S is a quasiperiodic monotone increasing function from the

integers to the reals with period a positive integer N and an "interval

of equivalence" O, meaning that the functional equation is S[i + N] =

S[i] + O. I propose to call S a "well n-formed scale" if it satisfies

the following:

There exist positive real numbers t1, ... tn, positive integers e1,

..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N,
e1*t1 + ... + en*tn = O, and ak < 1 so that

S[i] = floor((e1/N)*i + a1)*t1+...+floor((en/N)*i + an)*tn

with the result that S has exactly n sizes of step. The idea is that

this generalizes MOS and pairwise well-formed scales.

Comments?

On 26 May 2010 04:10, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> S[i] = floor((e1/N)*i + a1)*t1+...+floor((en/N)*i + an)*tn
>
> with the result that S has exactly n sizes of step. The idea is that
>
> this generalizes MOS and pairwise well-formed scales.
>
> Comments?

It looks like what I've coded as a generalization of distributionally
even scales. It may even be what distributionally even scales were
always intended to be. One day it might be how my website generates
scales for regular temperaments of arbitrary rank.

Graham