Re: [harmonic_entropy] Digest Number 30

Carl: The book I just returned on multidimensional continued fractions
mentioned Ferguson-Forcade only in passing. It did say that Brun's
algorithm is equivalent to Barbours "mixed expansion" though it is a
subtraction, rather than division, algorithm. I'm sorry to say that I
know very little about the F-F algorithm.

There are some specialized algorithms that converge faster for some
problems, but I don't know if it is worthwhile trying to use them for
such as simple problem. Speed of convergence and high accuracy isn't
terribly important as we want small-integer solutions.

Personally, I find Brun's algorithm the best as it is simplest to use or
program and yields many small-number convergents and semi-convergents.
Selmer's version (subtract the smallest term rather than the next to
largest) oscillates a lot and sometimes yields a few more convergents.
It is equally easy to use.

Pipping's ramified version, IIRC, doesn't generalize easily beyond 3
terms, but I haven't looked at it recently. Basically, one choses either
a Brun or Selmer step depending upon the relative sizes of the intervals.

The closest relation between "freshman sums," aka "mediant averages,"
etc. I know of is that Erv Wilson's zig-zag algorithm across the
Stern-Brocot tree is equivalent to a simple continued fraction or Brun's
algorithm for 2 terms. The zig's and zag's correspond to times each new
interval is subtracted from the other (or as a CF, the quotient of the
remainder generated in the preceding step).

--John

Hello John,

I'm very interested in that book you red about
multidimensional continued fractions.
Can you give the reference?

John Chalmers:
> The closest relation between "freshman sums," aka "mediant averages,"
> etc. I know of is that Erv Wilson's zig-zag algorithm across the
> Stern-Brocot tree is equivalent to a simple continued fraction or
> Brun's algorithm for 2 terms. The zig's and zag's correspond to times
> each new interval is subtracted from the other (or as a CF, the
> quotient of the remainder generated in the preceding step).

Question
Can one call the number of zigzags (or the number of layers in a CF) a
degree of irrationality?
Hopefully you can receive the attachments.

Peter

Peter Mulkers
P.Mulkers@...
Belgium

Carl Lumma wrote,

>I would like to start a discussion of the Ferguson-Forcade algorithm,
>and related integer relation algorithms, and their application in
>finding a generalized mediant function, applicable to chords of more
>than two tones.

John Chalmers wrote,

> There are some specialized algorithms that converge faster for some
> problems, but I don't know if it is worthwhile trying to use them
for
> such as simple problem.

John, the question Carl brings up is not a simple problem, nor is it
a question of speed of convergence. According to Pierre's quote from
the now-censored Wolfram site, Brun's and all other algorithms known
before 1991 failed in certain cases to act as a generalization of the
continued fraction algorithm for two-term ratios, with all relevant
properties thereof. While the point is moot for the problem of simply
approximating a certain proportion with a given restriction on the
integers involved, since a brute-force computer search is fast enough
these days, for the much larger problem of partitioning the 2-d plane
according to a generalized mediant function.

> Speed of convergence and high accuracy isn't
> terribly important as we want small-integer solutions.

It sounds like you aren't addressing Carl's question at all but
perhaps thinking of a very different problem, perhaps that of finding
equal temperaments?

I wrote,

> While the point is moot for the problem of simply
> approximating a certain proportion with a given restriction on the
> integers involved, since a brute-force computer search is fast
enough
> these days, for the much larger problem of partitioning the 2-d
plane
> according to a generalized mediant function.

Clearly I didn't finish that sentence! It should conclude, "a much
more sophisticated approach is needed."

--- In harmonic_entropy@egroups.com, Peter Mulkers <P.MULKERS@G...>

> Question
> Can one call the number of zigzags (or the number of layers in a
CF) a
> degree of irrationality?

This doesn't look very promising from a musical point of view, since
it says that, for example, 38/37 is more "rational" than 5/3. And
what about those true irrationals which have an infinite "degree of
irrationality" by your definition but are infinitesimally close to a
rational of low "degree of irrationality"?

Peter:
>> Question
>> Can one call the number of zigzags (or the number of layers
>> in a CF) a degree of irrationality?

Paul:
> This doesn't look very promising from a musical point of
> view, since it says that, for example, 38/37 is more "rational"
> than 5/3. And what about those true irrationals which have
> an infinite "degree of irrationality" by your definition
> but are infinitesimally close to a rational of low "degree
> of irrationality"?

Peter:
Right, I have to review my definition of irrationality.
Maybe the number of layers and low integer simplicity in
the CF-expansions have to play together in a certain way.
I'll think about it. Thanks.

Peter wrote,

>Right, I have to review my definition of irrationality.
>Maybe the number of layers and low integer simplicity in
>the CF-expansions have to play together in a certain way.
>I'll think about it. Thanks.

I prefer to think of the low integer simplicity as the _only_ valid measure
of rationality, and then use harmonic entropyu, as Pierre said, to obtain
discordance as a continuous function of interval width.