--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> http://groups.yahoo.com/group/tuning-math/message/18
>
> > > Did we ever take a serious look at 11-odd-limit approximations
> > > in the MIRACLE family?
> >
> > Oh yes . . . Dave Keenan has been thinking 11-limit all along.
> > He posted some 7-limit and 11-limit optimization results, and
> > I posted a 9-limit one, fully worked out step-by-step
> > (remember?). We've talked about the hexads in Canasta, and
> > these are 11-limit hexads, of course . . . etc. etc..
>
> Of course... duh! I knew all this. Guess it's just
> information overload.
>
>
> > > I'd like to include 11 if you have no preference.
> >
> > So shall we call our integer limit 12?
>
>
> Sure! Guess what?... that ties this in nicely with
> Schoenberg's alleged integer-limit of 12 in his
> _Harmonielehre_ (the explanation disparaged by Partch).

Umm . . . I thought that explanation used a _prime-limit_ of 13, not an
_integer-limit_ of 12. In
particular, Partch showed that Schoenberg's two derivations of the note C# -- as
the 11th
harmonic of G and as the 13th harmonic of F -- hence as 33/32 and 13/12 --
differed by virtually
an entire semitone (i.e., Schoenberg assumed a "unison vector" of 143:128).