USENET post from 1990 by Gene Smith on homomorphisms and kernels

Xref: utzoo sci.math:11756 comp.music:1622

Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!usc!ucsd!ucbvax!agate!ronzoni!gsmith

Newsgroups: sci.math,comp.music

Subject: Re: A Mathematical Theory of Music

Message-ID: <1990Jul20.035340.27688@agate.berkeley.edu>

Date: 20 Jul 90 03:53:40 GMT

References: <1268.268f84e0@gp.govt.nz> <25556@unix.cis.pitt.edu> <1269.26909383@gp.govt.nz>

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Reply-To: [gsmith@ronzoni.UUCP] (Gene Ward Smith)

Followup-To: comp.music

Organization: Bosco Gang Chocolate Center

Lines: 35

In article <1269.26909383@gp.govt.nz> [philip@gp.govt.nz] (Philip

Dorrell) writes:

>>>All [...] intervals in music can be expressed in terms of integral powers of

>>>2, 3 and 5. Taking logarithms, this gives a 3-dimensional vector space with

>>>basis vectors log 2, log 3 and log 5.

What you should say is that music based on 2,3 and 5 forms a

subgroup (free of rank 3) of Q*, the multiplicative group of

positive rationals, and that the log map takes this in a natural

way to a lattice in R3. If we assume a mean-tone type system, the

kernal of the homomorphism from this rank 3 abelian group to the

rank 2 group which covers mean-tone systems is generated by

81/80, which is the well-known diatonic comma: the difference

(9/8)/(10/9) between the two just major tones of the just

diatonic scale.

>It seems that some approximation error always occurs somewhere, and this is in

>fact a consequence of my theory - according to it, every tune contains a 'proof'

>that 80 = 81. For example : suppose there existed some tune played on an exactly

>tuned scale that did not somewhere come up against the problem that an interval

>in the tune was out by a factor of 81/80 from what you wanted it to be -

>according to my theory, such a tune would not be a tune.

Your theory is wrong. Lots of music can be made on just the

diatonic major scale. Make everything triadic, and so long as the

minor supertonic (ie d minor over C major) is avoided, you are

home free.

ucbvax!brahms!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720

Fifty flippant frogs / Walked by on flippered feet

And with their slime they made the time / Unnaturally fleet.

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