Re: TD 750 -- Paul Erlich on tetrads

Hello, Paul and everyone.

Thank you for a fascinating survey of tetrads. I'm not too surprised that
9:12:16 would be an optimal triad, but something close to 22-tet a nice
choice for a tetrad based on three fourths, with an outer interval close
to 7:3, and this may confirm your original remark about 22-tet "quartal
harmony" in your famous paper which got me interested in this question.

One question might be how much the septimal schisma or "beta-2" (~3.80
cents) affects the consonance rating of Xeno-Gothic tetrads approximating
12:14:18:21 or 14:18:21:24. These are unstable sonorities with an effect
at once "coloristic" and often cadentially oriented toward a stable trine
or fifth.

Anyway, thanks for another fascinating survey. Incidentally regarding the
not-so-consonant qualities of the minor sixth: it is often ranked by
13th-century theorists along with the minor second, major seventh, and
tritone or diminished fifth as a strong dissonance. I wonder if the
tension between the third partial of the lower note and the second partial
of the upper note could contribute to this result, not only with a more
complex ratio like the Pythagorean 128:81, but also with a pure 8:5.

Most appreciatively,

Margo Schulter
mschulter@value.net