Approximations, Mongolian music

Some good sources for historical methods of approximating
roots and 12-tet are these, especially the first, for both
geometrical and arithmetic techniques.
Barbour also suggest choosing a simple ratio for the
tempered major third or minor sixth and taking square roots and
quotients. Suggested ratios are 100/63, 27/17 and 19/12. Barbour
also states that Johann Faulhaber (1630) was the first
person to use logs to calculate the 12th roots of 2. The first to
publish ET in Europe was Simon Stevin (1586).

Barbour, J. Murray. 1951. Tuning and Temperament, Michigan
State College Press, East Lansing, reprinted 1953, 1961.

Heath, Sir Thomas. 1921. A History of Greek Mathematics,
Volumes I and II. Dover Edition, 1981. Dover Publications, Inc.
New York.

See also Neubauer , Otto (?, Neuberger, etc.) The Exact Sciences in
Antiquity. Sorry I don't have a better reference to this book.

I may be wrong, but I had thought that the historical Mongolian scale
was diatonic and Chinese pentatonic with the pien-tones added later. BTW, Contemporary Mongolian music sounds like C&W (I'm not kidding, a troop
visited Berkeley in the late 1980's) and played a mixture of musics,
some diatonic, some pentatonic. The vocal style was largely male falsetto
with heavy tremolo rather than vibrato.

--John

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