Re: Science Fair, syntonic comma, and musical diversity

Hello, there, and the Science Fair question seems to be an invitation
for us to demonstrate ourselves how music, mathematics, and diverse
cultural traditions can interact to produce very different judgments
of "in-tuneness" or "out-of-tuneness."

For example, 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