Farey in Schroder

Paul Ehrlich recommended to me the book _Number Theory in Science and
Communication: With Applications in Crypotography, Physics, Digital
Information, Computing, and Self-Similarity_, by Manfred R. Schroeder. The
Third Edition was just published (1997) by Springer. The back blurb says it's
"an introduction for non-mathematicians". I've just read a little bit so
far. There seems to be a lot of material here. It is not in the all too common
sadistic style of most math books, but the mathematics here is very real all
the same.

Section 5.10, pp. 82-89, is on Farey Fractions.

Jim


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From: mr88cet@texas.net (Gary Morrison)
Subject: Re: Microstock 3 and ski fest
PostedDate: 02-11-97 06:16:51
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>>It seems that if you wrote a short book to explain this
>>and other things about acoustics to the average musician, you'd be doing a
>>service to mankind.
>
>An amazing thing happened last night. On the bookshelf of some Berklee
>students I was visiting, there was a copy of Hall's _Musical Acoustics_ which
>is their textbook. I had not read it before, but chapter 17 confirms
>virtually every claim I made in my long post here, and chapter 18 (the last
>chapter) is quite a nice introduction to tuning and temperament, doing a much
>better job of stressing the historical importance of meantone tuning than any
>other acoustics textbook I've seen.
>
>So get yourself a copy of this book (make sure it's the second, 1991
>edition)!


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From: mr88cet@texas.net (Gary Morrison)
Subject: Re: FW: Happy Halloween, Engineers!
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To: Manuel.Op.de.Coul@ezh.nl
From: mr88cet@texas.net (Gary Morrison)
Subject: Re: FW: Happy Halloween, Engineers!

On a completely unrelated note, I got my copies of Neil Haverstick's new
CD (I helped him out a little financially, so he sent me a couple copies).
I haven't had a chance to hear it yet, but I will soon.




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Subject: David Hykes in NY
PostedDate: 04-11-97 12:56:38
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Fans of Paul Erlich will no doubt already have read his paper available
through http://www-math.cudenver.edu/~jstarret/microtone.html . A
brave attempt at a theory of diatonic, 7-limit harmony.

Now, back to picking his posts apart paragraph by paragraph:

>The harmonic entropy curve, like the Plomp-Levelt roughness curve for tones
>with harmonic partials, has local minima at small-integer ratios. It is
>standard in physics (and justified by calculus) to approximate any local
>minimum with a parabola. Therefore, near a just ratio, the change in
>dissonance is proportional to the squared detuning.

I don't have Plomp & Levelt's curve to hand, but I do have Kameoka &
Kuriyagawa's. The minima don't look much like parabolas to me. I
question this generalisation for physics as well, but that needn't
bother the rest of you.


A Farey series would be like a Partch even limit, right?

> These curves look remarkably like many of
> the Helmholtz/Plomp curves that were derived from completely different
> assumptions,

I'd guess any function that bears some relation to the overtone series
would produce a qualitatively equivalent curve. The entropy method
then looks like a good way of going from functions of integer ratios
to a continuous curve. There are lots of the former, of course -- the
Partch limit, LCM, generalised harmonic distances, or counting the
"filled" partials of the virtual pitch.

> How to
> weigh the various subsets' contributions to the probabilities of
> particular fundamentals in an overall analysis is unclear. Even without
> the consideration of subsets, there appears to be no mathematical theory
> of ratios of three of more numbers analogous to Farey theory, and no
> easy way to create one. Unlike roughness, tonalness is not merely
> concerned with pairwise interactions of tones but three-way and higher
> interactions as well. A mathematical model for it is out of my grasp at
> the moment.

It's crucial that a concordance theory be made to work for more than
dyads. I think here that the entropy method will become rapidly
computationally intensive the more notes are involved. I would try
this myself, but I've got software to write, and I might get a job
soon, so I don't really have the time ...

As a composer, I feel a theory of chords is more useful than one of
dyads. Most useful of all, though, is a theory of chord sequences.
This is where lattices come into their own. As a programmer, speed
and memory are important, so I'm planning to go for harmonic
distances if I ever get around to dynamic tuning. I think a
Euclidian distance on a parallelogram lattice should include
approximations to both Partch limits and LCMs.


BTW, I had a flanger on when I did my listening experiment before,
which explains the unpleasentness of the 31TET 4:6:7.


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From: William Sethares
Subject: consonance calculations
PostedDate: 04-11-97 21:11:56
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>>I don't have Plomp & Levelt's curve to hand, but I do have Kameoka &
>>Kuriyagawa's. The minima don't look much like parabolas to me.
>
>Are they pointy, then? Or flattened? If they are round (i.e., finite, nonzero
>second derivative), then a parabola having the same second derivative as the
>function itself at the minimum will approximate the function quite closely
>near that minimum.
>
>>A Farey series would be like a Partch even limit, right?
>
>The "integer limit" of a ratio (just the larger number) is order of the
>lowest-order Farey series in which the ratio appears.


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From: John Starrett
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