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Rule of 36

🔗Gene Ward Smith <gwsmith@svpal.org>

3/23/2003 11:00:17 PM

Here is a curiosity--25/24 is very nearly three Pythagorean commas,
the difference being the monzizma, a microcomma. Hence a temperament
with pure or almost pure fifths as generator, which closes or nearly
closes
around a circle of N fifths, will have a "wolf" fifth about 25/24
sharp, making an augmented triad [1,5/4,25/16], for a temperament a
multiple of N-36 notes, and one about 25/24 flat, making a diminised
"wolf" triad [1,6/5,36/25], for a temperament a multiple of N+36
notes. The multiple aspect arises since the temperament may not have
octave period. Taking the continued fraction for log2(3/2), we get
41-36=5, leading to diaschismic, 53-36=17, leading to schismic,
41+36=77 and
53+36=89 note scales; then 306+-36, 665+-36 and so on.