Re: paul why 12-out-of-N

> Bob Valentine wrote,
>
> >There is certainly a limitation to 12-out-of-N,
> >since it is really just chipping off a portion
> >of the lattice, and the interesting 'tonal rules'
> >in a new tuning (where the puns are), may be broken
> >in a misleading way.
>
> When I diagrammed Jon Wild's 11-of-31 scale, the resulting lattice is more
> like a "sea" than a "chip". In terms of "puns", clearly they are in the same
> place as in the full lattice. In what way are the 'tonal rules' of 31 broken
> in the subset? Clearly if you expect all 12 notes to function exactly the
> same way, 12-equal is your only option. So what did you mean?
>

Generally, the lattice diagrams show the consistent connections. But
there is a connection from any note to any note, and some of the punning
comes from various higher order 'aliases' using the same connection (as
an example, how many JI ratios are mapped onto the 12tet tritone).

A crystal structure is probably required to show the paths that all puns
can take, and without the whole crystal, but rather some collection
of collected vertices, there may be functions in the tuning that are not
accessible.

As an example, I was working through some 12-out-of-N tunings in 31tet,
trying to maximize the existance of certain intervals in the tuning. I
noticed that

( 7 * 7 ) mod 31 = 18

which in this tuning means that a chain of 7 "7/6"'s is equal to "3/2"
(after octave reduction). This is present in a full lattice for 31tet
since 7-limit intervals are uniquely represented. I suspect that a
working tonal system in 31tet will take advantage of this in much the
same way augmented triads, diminished sevenths, and various symmetric
scales take advantage of such equivalencies in 12tet.

Producing a 12-out-of-N scale demonstrating this property can be done,
but selects a different subset of the vertices than one might
choose for having nice major triads, or 4:5:67 chords.

Bob Valentine