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optimal diatonic tuning again

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

5/10/1999 2:55:54 PM

It turns out (according to Mandelbaum) that Wesley Woolhouse used the
criterion, RMS (major third, minor third, perfect fifth), in his 1835 study.
He finds that 50-tone equal temperament is nearly optimal, but advocates
19-tone instead for its simplicity and nearness to the optimal tuning. As I
recently proved, the actual optimal tuning according to this criterion is
7/26-comma meantone. 31-tone is closer than 19-tone to this value, and
despite the simplicity of 19-tone, I would definitely choose 31-tone if I
had to choose one, since the possibilities of expanding beyond diatonicity
in a musically meaningful way (including 7-limit and 11-limit harmony and
Arabic scales) seem far more promising in 31-tone than in 19-tone.