Re: Islamic Scalar variations, 12-tet

I might also point out that Ptolemy reported a scale 1/1 12/11 6/5 4/3
3/2 18/11 9/5 2/1 (Diatonon homalon) around 150 CE in Alexandria and
said it sounded "rather foreign and rustic." What he might have heard
could have been the prototype of the 3/4 tone scales associated later
with Zalzal. Earlier Greek scales also had 3/4 tones -- the hemiolic
chromatic of Aristoxenos where it was split (75 + 75 +350 cents) and the
Spondeion, where it was not originally, according to Winnington-Ingram.

As for tuning variations in performance, David Rothenberg deals with
these points under the names "range" and "blur," as well as "proper
modification." The range is the extent to which any one tone can vary
without destroying the propriety of the scale, the blur to which they
all can. Rothenberg also deals with the range of variation that
preserves "equivalence" class and with those that are perceptible to to
a given listener (for example, one who can ignore comma sized
deviations). However the math is somewhat difficult. See the microtonal
bibliography for references.

I think it is clear that Chinese theorists had workable approximations
to 12-tet before Europe. The earliest is about the 4th centure CE.
Prince Tsai Yu computed the 12th roots of 2 accurately a few years
before Simon Stevin and the other Europeans. So, 12-tet is not exactly
the evil invention of dead white guys.

Whether Chinese music ever made much use of it and whether Japan got it
from China or from Europe is unknown to me.

--John