source file: m1496.txt Date: Thu, 6 Aug 1998 13:16:14 -0700 (PDT) Subject: Numbers, cont. From: John Chalmers 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