Re: [tuning] Multidimensional Continued Fractions

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

"Paul H. Erlich" wrote:

> There was some discussion by John Chalmers and others a while ago about
> multidimensional continued fractions (such as ternary continued fractions)
> in the context of approximating a set of JI intervals with an ET.
> Multidimensional continued fractions could also be useful in approximating
> an irrational chord with JI. I was thinking that, since for dyads the Farey
> mediants could probably be derived by using an appropriately truncated
> continued fraction expansion for every interval in the continuum, perhaps
> the triadic harmonic entropy problem could be approached with a similar
> approach. Discouragingly, I found the following website:
> http://www.mathematik.uni-bielefeld.de/~sillke/NEWS/continued-fractions
> which says,
> "Alas, it turns out that in higher dimensions there are MANY competing
> algorithms, all equally disappointing ;)"
> although my particular problem might escape these disappointments. Anyone
> know anything about the Jacobi-Perron algorithm? Or if any of the algorithms
> discussed by Barbour might be appropriate?
>
>
>
>
>
>
>
>
Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Paul wrote:
> 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.

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. Doesn't this achieve the same, albeit in a slow and simple way?
I suppose I should mention the connection to Farey in the help text.
By the way the commands FIT/MODE and EUCLID achieve the opposite: approximating
JI with ETs.

Manuel Op de Coul coul@ezh.nl

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. These have only been
worked out to 12 dec.places. Yasser and Kornerup attempted to scoop each other to the one that
began 5,7,12,19,31 etc.

"Paul H. Erlich" wrote:

> 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
North American Embassy of Anaphoria island
www.anaphoria.com

Paul!
I think you are misunderstanding something here as I see only one of these numbers thart
ends in all 1's. They (can) show the convergence of JI interval toward irrationals. One can
see phi for instance between 1/1 and 2/1. Formed by "freshman sums' converging. JI towards
Irrationals.
"Paul H. Erlich" wrote:

> 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 Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Paul!
I See what you are saying here. But you should see how JI can converge towards
irrationals. It would also be possible to see ET MOS's as covergings toward irrationals. In
the scale tree we can see the continuum implied by an infinite series of JI, or ET's or
Irrationals!

"Paul H. Erlich" wrote:

>
>
> 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.
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

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> 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).

Aha, well you could base n-ary harmonic entropy on binary harmonic entropy.
Create a function based on the binary harmonic entropy of all pairs of pitches.
That would be similar to C. Barleau's "specific harmonicity" which is an
average of the harmonicity of all intervals.

> The problem is, in general, a point in the Farey web
> will have 4-7 "nearest neighbors" -- how do you draw the borders
> appropriately?

I don't know, that's why I formulated the suggestion as a question.
Perhaps you can think further along this line. Once you have a
satisfactory optimality criterion you could try standard techniques
like backtracking with alpha-beta pruning. These are used in chess programs.

Manuel Op de Coul coul@ezh.nl