RE: [tuning] Multidimensional Continued Fractions

Kraig Grady wrote,

>Paul1
> Not quite what you asked for but appears to tackle this prolem in some
way > http://www.xs4all.nl/~huygensf/doc/realm.html
<http://www.xs4all.nl/~huygensf/doc/realm.html>

Thanks Kraig! I had not seen this web page before. But yes, this is the
inverse problem of what I am interested in -- Fokker is approximating JI
with ETs, while I'd like to approximate irrational ratios with JI. It is
interesting that Fokker chooses Brun's algorithm. As I recall from
Mandelbaum's book, Brun's algorithm does seem closest to generalizing the
Euclidean algorithm (which would get all the right Farey approximants in the
dyad case) to ratios of three or more terms. . . .

Manuel wrote,

>This reminds me that the method FIT/HARMONIC in Scala may be similar
>to what you want. See http://www.tiac.net/users/xen/scala/help.html#FIT
>and go down to /HARMONIC. This algorithm doesn't use continued fractions
>at all. It works like a combined Farey search for fractions with a common
>denominator.

For x:y:z, is it checking only x:y and y:z, or x:z as well? But remember,
what I'm looking for is not necessarily the closest approximation -- for
that, the voronoi cell approach would have already been perfect. In the 1-d
order 10 case, the mediant between 10/9 and 1/1 is 11/10, so an interval
like 150¢, even though it's closer to 10/9 than to 1/1, is pulled into the
field of attraction of 1/1 since it's below 11/10.

Kraig Grady wrote,

>Paul!
>You may not have noticed on the scale tree that the convergences toward
irrationals (of the zigzags) can be >found at the bottom attached to the
dotted lines.

Those irrationals are the so-called noble numbers, numbers whose continued
fraction expansions end in all 1's, hence corresponding to an alternation of
zigs and zags.

What does that have to do with what we're discussing?

Kraig, I wasn't saying the noble numbers don't end in all 1's. Any number
ending in all 1's would have to be rational, not irrational. I was saying
their continued fraction expansions end in all 1's (that is, an infinite
number of 1's). For example,

phi = 1 + 1
-----
1 + 1
-----
1 + 1
-----
1 + 1
......

Any irrational number can be expressed as an infinite continued fraction:

irr = a + 1
-----
b + 1
-----
c + 1
-----
d + 1
......

For the noble numbers, which are the convergents of the zigzags, the
expression

[a,b,c,d,...]

ends in an infinite number of consecutive 1s.

Any rational number can be expressed as a finite continued fraction.

Some of the merits of continued fractions vs. conventional decimal
expansions are discussed at http://www.inwap.com/pdp10/hbaker/hakmem/cf.html
<http://www.inwap.com/pdp10/hbaker/hakmem/cf.html> .

>Formed by "freshman sums' converging. JI towards Irrationals.

Yes, this convergence of the zig-zags in the Farey tree toward noble numbers
is discussed in Schroeder, _Number Theory in Science and Communication_, p.
86.

Although the Farey tree is nice with these zig-zag patterns, unfortunately
the ratios at a given "level" of the tree do not correspond to a Farey
series of order N for any N

Anyone have access to this paper:

@article{djg:fn,
author ={David J. Grabiner},
title = {Farey Nets and Multidimensional Continued Fractions},
journal = {Monatshefte fur Mathematik},
volume = 114,
year = 1992,
pages = {35--60}}

?

Manuel, your message was empty but contained an attachment reading:

>It's checking only x:y and x:z. But it isn't possible to replace the
decision
>criterion with an error function for all intervals based on harmonic
entropy
>or some such? And let x run from 1 to some number?

I'm not really following, but since what I'm looking for is a step in
calculating harmonic entropy, using harmonic entropy seems circular (maybe
not; please elaborate). The problem is, in general, a point in the Farey web
will have 4-7 "nearest neighbors" -- how do you draw the borders
appropriately?