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Minimal complexity for a given equal division

🔗Gene Ward Smith <gwsmith@svpal.org>

5/8/2005 7:35:48 PM

Given a fixed EDO, the tuning is fixed, and hence temperaments may be
judged in terms of complexity. If we employ Graham complexity, we
sometimes get more than one temperament of the same complexity. For
example, in the 5-limit, both magic and porcupine give the same
complexity for 22, and both diaschismic and kleismic for 34. In the
7-limit, both meantone and myna have the same graham complexity for
31, and for 171 ennealimmal has the same complexity as another
microtemperament I've mentioned, but which has recieved very little
attention: tertiaseptal, the 140&171 system.

Using 9-limit rather than 7-limit Graham complexity completely changes
things, of course. Now meantone, with its generator of a fifth, blows
myna away, and ennealimmal crushes tertiaseptal, which has a high
complexity for the fifth. Instead of getting diminished for 12, in the
9-limit we get a tie between dominant and "augie", the 9&12 system.