USENET post from 1995 by Gene Smith on homomorphisms and kernels

From: [gwsmith@cats.ucsc.edu] (Gene Ward Smith)

Newsgroups: sci.math

Subject: Re: Music

Date: 6 Nov 1995 08:05:08 GMT

In article <DHDL98.39r@dsbc.icl.co.uk>, Roy Lakin <roy@dsbc.icl.co.uk> wrote:

>The "cycle of 53" is more accurate: split the octave into 53 equal divisions.

>There have been 53-note keyboards invented for this temperament but they never

>caught on, probably because modulation was so difficult.

That's only part of the reason. Another part is that any such division

can be viewed as a homomorphism from a finitely-generated subgroup of

the positive rationals under multiplication to a rank-one free abelian

group (the "keyboard"), and the kernel of this map related crucially

to the structure of the harmony. If the "diatonic comma = 81/80 is not

in this kernel, things will happen that you may not want. This means

that 19 and 31 tones are not only easier to handle than 41 or 53, they

are also closer to the system we now use, and so easier to work with.