Efficient MOS

Those with good memories may recall the notation n+m;s I introduced to
describe MOS on ets. Here the n+m part defines the generator and et;
if we assume n>m and d = gcd(n,m) then if q/r is the next element of
the nth row of the Farey sequence after m/n, so that if u = m/d and v
= n/d then u*r-v*q=1, the generator Gen(n,m) = (q+r)/(u+v), the
period is 1/d, and the equal temperament is n+m. Then n+m;s means the
s-element scale in the n+m-et with generators Gen(n,m) and 1/d.

In terms of this notation, Eytan is considering 2*n+1+2*n-1;2*n+1.
This suggests how to put things in a wider context, by looking at
n+m;n for
various values of n-m togeter with n mod n-m.

Difference of one

We get scales n+1+n;n+1, with et 2*n+1 and generator 2/(2*n+1)

Differences of two

(1) Scales 2*n+2+2*n;2*n+2 with generators [1/2, 1/(2*n+1)] in the
4*n+2 et

(2) Scales 2*n+1+2*n-1;2*n+1 with generator (2*n+1)/(4*n) in the 4*n
et; if you add to this the very strange requirement that the generator
maps from 3/2, you get Eytan's diatonic systems

Differences of three

(1) Scales 3*n+3+3*n;3*n+3 with generators [1/3, 1/(2*n+1)] in the
6*n+3 et

(2) Scales 3*n+2+3*n-1;3*n+2 with generator (2*n+1)/(6*n+1) in the
6*n+1 et

(3) Scales 3*n+1+3*n-1;3*n+1 with generator (4*n)/(6*n-1) in the
6*n-1 et

And so forth...

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Those with good memories may recall the notation n+m;s I introduced to
> describe MOS on ets. Here the n+m part defines the generator and et;
> if we assume n>m and d = gcd(n,m) then if q/r is the next element of
> the nth row of the Farey sequence after m/n, so that if u = m/d and v
> = n/d then u*r-v*q=1, the generator Gen(n,m) = (q+r)/(u+v), the
> period is 1/d,

This should be Gen(n,m) = (q+r)/(n+m)