USENET post from 1995 by Gene Smith on homomorphisms and kernels

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From: [email protected] (Gene Ward Smith)
Newsgroups: sci.math
Subject: Re: Music
Date: 6 Nov 1995 08:05:08 GMT

In article <[email protected]>, Roy Lakin <[email protected]> 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.