Re: [crazy_music] Digest Number 52

Hi Brian,

Was interested in your post.

I'm working on an option in FTS to find closest
approximations to a scale.

E.g. for quarter comma meantone, finds it to within
six cents as

1/1 n(1/15)~ 19/17~ 6/5~ 5/4 4/3~ 7/5~ 3/2~ n(9/14)~ 5/3~ n(5/6)~ 15/8~ 2/1

where n(9/14) means the 9th degree of 14-tet, shorthand for 2^(9/14).

To within 1 cent it is:

1/1 23/22~ 19/17~ n(8/31)~ 5/4 n(13/31)~ n(14/29)~ n(18/31)~ 25/16 n(20/27)~ 34/19~ 43/23~ 2/1

Numbers like 43/23 are often going to be beyond the prime limit one is working with,
so it can also check up to a prime limit

As 13 limit, to within 1 cent it is:

1/1 117/112~ 189/169~ 140/117~ 5/4 234/175~ 88/63~
175/117~ 25/16 117/70~ 315/176~ 144/77~ 2/1

or in terms of factorization into the primes:

1/1 3^2*13/(2^4*7)~ 3^3*7/13^2~ 2^2*5*7/(3^2*13)~ 5/2^2
2*3^2*13/(5^2*7)~ 2^3*11/(3^2*7)~ 5^2*7/(3^2*13)~ 5^2/2^4
3^2*13/(2*5*7)~ 3^2*5*7/(2^4*11)~ 2^4*3^2/(7*11)~ 2/1

This part is rather slow at present but I have ideas for
possibly speeding it up.

The whole area is quite amenable to programming.
That part still needs a fair amount of work though.

There is an html page in the FAQ that does this
kind of thing, just for the ratios and not for
the n-tet ones.
http://members.nbci.com/_XMCM/tune_smithy/tree/on_site_tree/robertwalker/ratios_with_factors.html
(available at this url till end of the month when
I need to find a new site for the FAQ).

It's reasonably fast at present if one is only
searching for small ratios, e.g quotient < 1000.

Robert