--- In tuning-math@y..., genewardsmith@j... wrote:

> One thing you might note about it is that it is the 22-31 linear
> temperament, with tunings calculated by Graham's minimax condition.
> If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we could
> also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this
basis
> we approximate rational numbers q in the 11-limit by
> U^h31(q) V^h22(q).

Another thing to note is how easy it is to find the generators and
MOS scales if you don't proceed all the way down to the PB. In the
22-31 case, we have a continued fraction [1,2,2,4] for 31/22; the
penultimate convergent is 7/5, and our generator is U^7 V^5. For the
41-31 temperament, we have a continued fraction [1, 3, 10], which
leads to a penultimate continued fraction 4/3, and a generator
W^4 X^3 which is the miracle generator; here W and X are to be
selected by some optimality condition, such as minimax or least
squares on a tonality diamond (if I have the jargon right.)

This is easy enough that I've been meaning to suggest that Manuel
consider putting into Scala a routine to calculate Gen(m, n, p) and
Mos(n,m,p) for two ets m and n and a prime limit p; in case m and n
are not relatively prime this needs to be adjusted by working inside
of the interval of repetition. Of course one can also think of this
in terms of the ets generated by linear combinations of hm and hn, as
for instance h53 = h22 + h31 and h72 = h31 + h41.