source file: mills2.txt
Date: Fri, 17 Nov 1995 16:48:41 -0800

Subject: Re: minimum chords

From: Gary Morrison <71670.2576@compuserve.com>

> Thanks for the explanation of LCMs, but why is this important musically?

   I thought I mentioned that in the message before my last one on that topic,
but perhaps not.  

   Marion McCoskey views the LCM of the numbers in the extended ratio for a
chord as a good indicator of the harmonic complexity of the chord.  So the LCM
of 60 for either 4:5:6 major or 10:12:15 minor chords is a fairly low LCM,
implying that it's a fairly simple chord, harmonically speaking.  

   If you turn that into a septimal dominant seventh chord (4:5:6:7), the LCM
goes up by a factor of seven to 420, suggesting that it's a much more
harmonically complex - tense - chord.  If you tune the upper interval of that to
the usual 6:5 minor third instead, the LCM goes way up again 20:25:30:36 to 900,
again, correctly predicting that that chord has a yet more complex sonority.  

   Some people on the list have questioned whether LCM is the best choice of
harmonic complexity, which is after all, a somewhat subjective thing.  But I get
the impression that most listers accept it as a reasonable model.  

   Speaking for myself, it strikes me as reasonable, but I suspect that it might
be the numbers might be a little more appropriate if they were the log of the
LCM.  Using the common log for example, the 4:5:6 would have a complexity of
1.8, the 4:5:6:7 a complexity of 2.62, and the 20:25:30:36 dominant seventh
chord a harmonic complexity of 2.95.  That strikes me as a little more in
proportion to the complexity as I hear them.  Or perhaps even better would be
the square root: 7.75, 20.5, and 30 respectively.  


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