Re: 152-EDO as adaptive-JI

In-Reply-To: <a3cbnf+5q2u@eGroups.com>
Monz:
> > And since 1 step of 152-EDO is ~1/3-comma, you only need *3*
> > chains of 19-EDO each separated by 1 step of 152-EDO, in order
> > to represent the entire infinite JI lattice! And you don't
> > get comma problems! *WAY* cool !!!!!!

Paul:
> Umm, something slipped up somewhere. Each 19-tET chain itself
> represents the entire JI lattice already. But if you want larger and
> larger chords in JI, you need more and more of the 19-tET chains.
> Nothing special happens when you have 3 chains.

One period/generator mapping is

[(19, 0), (30, 1), (44, 1), (53, 3), (66, -2)]

This is accurate to 1.7 cents of 11-limit JI. You need 5 non-consecutive
chains to carry one 11-limit otonality. But that means you get 19 of them
into the bargain, and you can do infinite modulations along 19-equal, but
not an infinite lattice. You do need all 6 chains for that.

Another mapping is

[(19, 0), (30, 1), (44, 1), (53, 3), (65, 6)]

Tuned to 152-equal, that's 2.2 cents out, but it does mean the 6 should
map to 0. It now means the 11-limit otonality can be got from 4
consecutive chains, but isn't so close to JI.

> > So the 57-tone subset of 152-EDO which is three 19-EDOs starting
> > on the 1st, 2nd, and 3rd 152-EDO degrees is really some sort of
> > magical tuning for us adaptive-JI fans!
>
> Nope. You need as many 19-tET chains as the piece dictates.

You don't need more than 6 to cover 152-EDO.

The CNMI connected to London Guildhall University have built a trumpet for
19-equal. I was told that it can handle 7-limit JI fairly easily by pitch
bending, and the method is to think of multiple chains of third-comma
meantone. That looks to me pretty much like what you're doing, but in the
11-limit. So this might all be more practical than you realized.

Graham