# Rosser's Algorithm - Musical Scales and Generalized Ternary Continued Fractions

Rosser's algorithm is typically invoked during discussions of equal temperament scales, and is a way to obtain good approximations for multiple irrational numbers simultaneously. Is there a nice, accessible modern treatment? Rosser's paper was published in 1950.

Some background:

In an equal temperament scale, the ratio of frequencies of adjacent notes is a constant. This is a desirable feature, in that it allows you to transpose a piece from one key into any other. Now, for various reasons (psychological, historical) we'd like have have certain ratios of frequencies, or close approximations. If we require the unison ratio (2:1) to be exact, then we require that our frequency ratio R satisfies R^n = 2 for some integer n. Thus, R is an integral root of 2. Next, if we want a good approximation to the perfect 5th (3:2), we conclude that we need R^r to be a good approximation to log2(3), for some integer r. This leads us into continued fractions. Now, if we'd also like a good approximation to the perfect 3rd (5:4), we would like R^s to be a good approximation to log2(5) for some integer s. We've now left the realm of continued fractions, as we're seeking to approximate two irrational numbers with powers of the same root of 2.

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## 2 Answers

Your question reminds me of Edward Dunne's AMS post on Pianos and Continued Fractions, and I wonder if you'd find it useful.

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The link should now be oeis.org/DUNNE/TEMPERAMENT.HTML. –  I. J. Kennedy Oct 24 '12 at 14:16

This is a relatively recent list of references, originally posted by Chris Hillman on sci.math. I've also attached another sci.math posting from Chris Hillman on the subject of multidimensional continued fractions. It appears that this subject would generally be classified under the topic "geometry of numbers".

A recent book not included in the below on this topic is "Multidimensional Continued Fractions" by Fritz Schweiger (2000). Doug Hensley's "Continued Fractions" (2006) also covers this topic in chapter 6.

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From: Chris Hillman Newsgroups: sci.math Subject: Re: Hyperfractions ? : approximants to sqrt(-1) Date: Thu, 14 May 1998 18:32:09 -0700

On Thu, 14 May 1998, Peter Jack wrote:

Hyperfractions ? : approximants of sqrt(-1)

I have a problem I'm working on. Maybe someone can help.

The basic idea is to find a representation of the hypercomplex numbers in terms of a sequence of rational numbers.

[... snip ...]

Has anyone attempted such a construction before?

I don't know about the particular construction you outline, but you are looking for a variety of "multidimensional continued fraction algorithm" and there is an enormous literature on such things, mostly on continued fractions in R^n but some concentrating on various hypercomplex numbers, so it is quite possible someone has tried the approach you outline before.

Here are a few recent references which should give some idea of the variety of approaches recently taken to this problem:

@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}}

@article{n:pcfa, author = {A. Nogueira}, title = {The Three-Dimensional Poincare Continued Fraction Algorithm}, journal = {Israel Journal of Mathematics}, volume = 90, year = 1995, pages = {373--401}}

@book{s:fs, author = {Fritz Schweiger}, title = {Ergodic Theory of Fibered Systems and Metric Number Theory}, publisher = {Clarendon Press}, address = {Oxford}, year = 1995}

@article{iko:jpa, author = {S. Ito and M. Keane and M. Ohtsuki}, title = {Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm}, journal = {Ergodic Theory and Dynamical Systems}, year = 1993, volume = 13, pages = {319--334}}

@unpublished{l:skp, author = {Giles Lachaud}, title = {Sails and {K}lein Polyhedra}, journal = {Contemporary Mathematics}, note = {to appear}}

@unpublished{l:kpgf, author = {Giles Lachaud}, title = {{K}lein Polygons and Geometric Diagrams}, journal = {Contemporary Mathematics}, note = {to appear}}

@article{l:gmcf, author = {J. C. Lagarias}, title = {Geodesic multidimensional continued fractions}, journal = {Proc. London Math. Soc.}, volume = 69, year = 1994, pages = {464--488}}

Hope this helps!

Chris Hillman

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From: hillman@math.washington.edu (Christopher Hillman) Newsgroups: sci.math Subject: Re: Multidimensional Continued Fractions Date: 1 Jul 1997 11:49:40 GMT Organization: "University of Washington, Mathematics, Seattle"

noadd@nowhere.com (No Chance) writes:

A few months ago, someone posted a reply to an article about the continued fraction expansion of pi. At the end of the article, the poster wrote that research was being done on mulitdimensional continued fractions. I was wondering if anyone could tell me anything about this subject and give me some refrences.

You are probably thinking of an article I posted (I didn't save a copy).

The ordinary continued fraction algorithm provides a way to expand a real number in a way quite different from a "decimal" expansion wrt to some base, one which reveals some algebraic/number theoretic structure much better. By truncating the expansion after n, n+1, n+2, ... terms we obtain a sequence of rational approximations.

A multidimensional CFA is just some algorithm which gives a sequence of rational approximations to a d-tuple of real numbers. The best known such algorithm is the Jacobi-Perron algorithm.

One would naturally hope to be able to find an algorithm with a theory which works out just as nicely as the one dimensional algorithm, and which yields not only a definite sequence of approximations which is in some sense optimal but which also yields an "expansion" which reveals something about the number theoretic properties of the d-tuple, in particular whether the various components are rationally independent.

Alas, it turns out that in higher dimensions there are MANY competing algorithms, all equally disappointing ;) Well, if not all equally disappointing, certainly disappointing for one reason or another. Algorithms which are good from one standpoint are often quite bad according to another way of thinking. Yet in one dimension there is essentially only one algorithm which is at all reasonable, and this one turns out to be good for many purposes.

References: two books

A. J. Brentjes, Multidimensional Continued Fraction Algorithms, Amsterdam: Mathematisch Centrum, 1981.

Fritz Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford U Press, 1995.

(Has a chapter on multidimensional CFA's and many references.)

Some papers:

Giles Lachaud, Sails and Klein Polyhedra, Contemporary Mathematics, to appear. (According to Vershik, the notion of a sail is the best to come along in this field for years.)

David J. Grabiner, Farey Nets and Multidimensional CFA's, Mh. Math. 114 (1992) 35-60

J. C. Lagarias, Geodesic Multdimensional Continued Fractions, Proc. Lon. Math. Soc 69 (1994) 464-488.

A. Nogueira, The Three-Dimensional Continued Fraction Algorithm, Is. J. Math. 90 (1995) 373-401.

Shunji Ito and Makoto Ohtsuki, Parallelogram Tilings and Jacobi-Perron Algorithm, Tokyo J. Math. 17 (1994): 33-58.

(The reason for my interest in this is that these CFA's turn out to be relevant for studying the combinatorial properties of certain types of tilings which are idealized models of quasicrystals.)

This should give a quick impression of the scope of current research--- there are ALOT of ideas out there!

Chris Hillman

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