USENET post from 1990 by Gene Smith on homomorphisms and kernels

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Xref: utzoo sci.math:11756
Path: utzoo!utgpu!!!usc!ucsd!ucbvax!agate!ronzoni!gsmith
From: [email protected] (Gene Ward Smith)
Newsgroups: sci.math,
Subject: Re: A Mathematical Theory of Music
Message-ID: <[email protected]>
Date: 20 Jul 90 03:53:40 GMT
References: <[email protected]> <[email protected]> <[email protected]>
Sender: [email protected] (USENET Administrator;;;;ZU44)
Reply-To: [email protected] (Gene Ward Smith)
Organization: Bosco Gang Chocolate Center
Lines: 35

In article <[email protected]> [email protected] (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.