Greetings, Margo Schulter has posted another of her enlightening(for those that want it) posts on the Tuning list. I include it below. >>Objectively defined, the syntonic comma is simply the difference, for example, between a major sixth formed from a series of three pure 3:2 fifths (e.g. C-G-D-A), or 27:16, and a major sixth of the simplest possible ratio of 5:3. When we take the difference between the two intervals by dividing 27:16 by 5:3, we get 81:80, the syntonic comma. Likewise, a major third built from four pure fifths, at 81:64, will be a syntonic comma wider than the simplest ratio for this interval at 5:4 or 80:64. Using the early 17th-century tool of logarithms, we often measure such musical intervals in cents, there being 1200 cents in a pure 2:1 octave, and 100 cents in an equal semitone of 12-tone equal temperament or 12-tet. The syntonic comma at 81:80 is about 21.51 cents, or a bit more than 1/5 of a 12-tet semitone. However, the syntonic comma simply measures the difference between 27:16 and 5:3, or between 81:64 and 5:4; it doesn't tell us which interval is more "in tune" or "appropriate" for a given piece of music. If 81:64 or 27:16 is the preferred tuning for a major third or sixth, as in much medieval European music, then there is no problem, only an interesting mathematical observation. These thirds and sixths of Pythagorean tuning -- tuning in pure fifths or fourths -- are quite active, and nicely resolve to the stable fifths and fourths favored in the 13th and 14th centuries. However, by around the middle of the 15th century, the simpler 5:4 and 5:3 have come into vogue -- so what was nicely "in tune" a century earlier is now "out of tune," and the syntonic comma has indeed become the leading "problem" of keyboard tunings. The usual solution was to make each fifth slightly narrower than a pure 3:2, so that four of these narrowed or tempered fifths would exactly or approximately equal a 5:4 major third. Maybe what I'm trying to suggest is a kind of Intonational Theory of Relativity: "in-tuneness" or "out-of-tuneness" are relative to a given style of music. A "real major sixth" is 27:16 in medieval Pythagorean tuning, but at or close to 5:3 for most 16th-century music. There are various experiments you could try to illustrate this very important point about mathematics and music, for example taking musical pieces from different eras, and tunings from these eras, and seeing if people can match which tuning historically is most likely to go with each piece. Also, will people prefer 81:64 major thirds for a medieval piece, but a 5:4 major third for a Renaissance piece -- or might they prefer another tuning such as 12-tone equal temperament which might be more familiar to some of them? As others have mentioned, the research of William Sethares on tuning and timbre might also be very interesting, and lead into comparisons not only between different styles of European music, but between various world musics. Most respectfully, Margo Schulter mschulter@value.net
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