Numbers, cont.

My apologies for not knowning the number of persons in a minyan, I don't
know where I got the idea 13 were needed.

I think some numbers do have psychological significance. George Miller
wrote about this in his classic, "The Magical Number 7 Plus or Minus 2," in
which
he states that limitations on human information handling capacity allows
only
5-9 categories in each perceptual dimension. Most scales have 5 or 7 tones,
a few have 6, 8 or 9. Except in Western music, the whole chromatic set, of
whatever size and tuning, is seldom presented in one melody or section of a
piece.

Historically, 7 tone scales almost certainly evolved from pentatonics. This
is quite clear both from the Sumero/Bablylonian names of the strings and
the expansion of the major-third/semitonal "Olympian" pentatonic to the
heptatonic enharmonic of Greek theory (and practice) by splitting the
semitone into two
microtones. There is some evidence that a hexatonic was an intermediate
stage in which the semitone was split in only 1 tetrachord.

Similarly in China, the basic scale was pentatonic and the two added tones
were called pien-tones (neighbor?).

As for 9-tone scales in 31-tet, David Rothenberg and Connie Chan studied
this one: 5 3 3 3 3 5 3 3 3, generated by a chain of 14 degrees of 31-tet.
This scale is an MOS, is strictly proper, has "stability" of 1.0 and
"efficiency" of .7407. It is non-diatonic and has relatively few
approximations to 7-limit harmonic intervals.

One might want to continue the series to 11 and 20 tones.

I think recreational numerology of the type pioneered by Martin Gardner and
his literary alter-ego, the late Dr. Irving Joshua Matrix (and his Eurasian
daughter, Iva Toshiyori) is great fun. Hooey, yes, but FUN hooey...


--John