# USENET post from 1990 by Gene Smith on homomorphisms and kernels

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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 From: [email protected] (Gene Ward Smith) Newsgroups: sci.math,comp.music 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) Followup-To: comp.music 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.