Re: 22, 11tet in ji

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Oddly enough, Robert's first table does not include 3/2, even though it's
> only 7 cents away from 13/22 oct.
>
> Robert,
>
> I see what you mean now. There's something weird about your algorithm,
> especially the "relaxed" version. 10/22 oct. and 12/22 oct. are not good
> approximations of 4:3, 7:5, 10:7, or 3:2. 4:3 is 7 cents off 9/22 oct., 3:2
> is 7 cents off 13/22 oct. and 7:5 and 10:7 are each 17 cents off 11/22 oct.
> I don't know why your algorithm assigned these ratios to the wrong scale
> degrees.

The algorithm is exceedingly dumb. It starts
with calculating the et value (2^10/22 for instance). Then it loops
running the denominator up and seeing if the appropriate numerator for
the denominator improves upon the current error.

If I relax the error function completely, (for instance, initially you can
be off by 100) then the "3/2" neighborhood in 22 tet looks like

10 : =1/1 =3/2 =4/3 =7/5 =11/8 =26/19 =37/27
11 : =1/1 =3/2 =4/3 =7/5 =17/12 =24/17 =41/29 =99/70
12 : =1/1 =3/2 =10/7 =13/9 =16/11 =19/13 =35/24 =54/37
13 : =2/1 =3/2 =62/41 =65/43 =68/45 =71/47 =74/49 =77/51

So this should show clearly how the algorithm works. (Now I have to
go back and find if its producing different results than when this
thread started...)

As I said, I should refine it with a "most likely to be heard as"
function, using some of the complexity functions that have been
mentioned in the past. The variation from that thread that
I'd use is

1) reduce and factor numerator and denominator
2) sum all odd primes
3) subtract 1 if denominator is a power of 2

All that aside, the important thing that came out is the compositional
issue of the importance of context. This shouldn't be too surprising,
since in 12tet CDE can be 7:8:9 or 8:9:10 or 9:10:11 depending on
the context. But it was interesting to see it in this community
where those three sets would generally strive for uniqueness. Any
et will end up having some punning, and its probably a good thing to
play with. For a sufficiently high et, an opposite sort of punning
may be useful. A context may call for an approximation to a certain
ji but compositionally, a different tempered interval may be
more appropriate.

Bob Valentine