Re: [harmonic_entropy] Digest Number 31

The book I looked at is "Multidimensional Continued Fraction Algorithms"
by A. J. Brentjes, published by the Mathematisch Centrum, Amsterdam,
1981. I also perused Brezinski, "History of Continued Fractions and
Pade' Approximants."

The failure of the Brun algorithm in certain cases is discussed by Brentjes.

Nothing as baroque as the generalized CF's on the MathSoft page (thanks
for the URL, Paul) were discussed, IIRC.

Erv Wilson also looked at some extended summation series of the Fib and
Tribonacci type. I don't know if Kraig has posted any of this to the
Anaphoria site.

What exactly was Carl's question or the musical application?

--John

--- In harmonic_entropy@egroups.com, John Chalmers <JHCHALMERS@U...>
>
> Erv Wilson also looked at some extended summation series of the Fib
and
> Tribonacci type.

Tribonacci! I just posted on that to the tuning list. Actually,
that's quite a coincidence -- continued fractions are to the MOSs of
the Scale Tree as the Ferguson-Forcade algorithm is to the 3-step-
size scales I brought up in the "hyper-MOS" thread over on the tuning
list -- which no one has attempted to tackle yet (perhaps we should
send the questions to Erv?)

> What exactly was Carl's question or the musical application?

It has to do with harmonic entropy, of course! In the dyad case, the
mediants partition the interval axis into non-overlapping, exhaustive
pieces, one for each ratio in the Farey, Mann, or Tenney series of a
given order, in a way that seems most "natural" in that the ratios of
the next higher-order series always fall exactly on the borders --
the mediants. This partition is then used as the probability function
in the harmonic entropy calculation. The question is, how do we
generalize this to triads?

I think we can live without the solution for now, since:

a) in the dyadic Tenney case I found that one could approximate the
partition sizes rather accurately with the inverse geometric mean of
the numbers in the ratio -- I verified that the weighted sum of these
remains essentially constant as one moves one's Gaussian window along
the interval axis -- and renormalize these to define one's
probability function, and get essentially the same results as using
the mediant-defined partition. In the triadic case, we don't know the
partition sizes but if we can verify that the weighted sum of the
inverse geometric means remains constant as one moves one's Gaussian
window around the triadic plane, we can probably live with that
approximate solution and never have to worry about the "real" one.

b) I recently brought up the idea of using this inverse geometric
mean idea with _all_ ratios, not just those in lowest terms. Clearly
the probability function no longer corresponds to a non-overlapping
partition of the interval axis, but continuing to use the inverse
geometric mean of each "ratio" as its probability (up to
normalization) yields a harmonic entropy function that looks like the
_exponential_ of the one obtained from just the lowest-terms ratios.
This is very interesting to me as it reflects the acoustical reality
that virtual fundamentals can sometimes correspond to ratio-
interpretations that are not in lowest terms, though the rule for
probabilities remains unjustified in this case . . .