Re: TD 929 -- "triadic" vs. "polydyadic" harmonic entropy

Hello, there, Paul, and thank you for reminding me that for a medieval
European theorist of around 1300 to note that the major ninth at 9:4
"seems to concord better" in a three-voice sonority of 4:6:9 with its
two concordant fifths is a statement about the consonance of triads,
but not necessarily an example of the more specific approach of
"triadic harmonic entropy."

As you very rightly point out, one can conclude that 4:6:9 has a
greater degree of concord than a bare 9:4 simply by looking at the
constituent dyads, and thus is what Coussemaker's Anonymous I and
Jacobus do, pointing to the two 3:2 fifths, ideal stable concords in
Gothic music.

In fact, when I first approached this question around the early
1980's, I found it easy to show that given the scale of intervallic
concord/discord in Gothic practice and theory, sonorities like 4:6:9
or 9:12:16 are among the most consonant unstable three-voice
combinations -- and also 6:8:9 or 8:9:12. They include two ideally
euphonious intervals of the fifth/fourth category, plus a relatively
tense M2, m7, or M9.

The other kind of mildest or most consonant unstable sonority consists
of a _quinta fissa_, or ideally euphonious fifth "split" into two
unstable but relatively blending thirds, 64:81:96 or 54:64:81.

To explain the pleasant effect of a sonority like 4:6:9 in a Gothic
setting (e.g. Perotin or Machaut), we might also note the factor of
"stylistic congruity." That is, in a texture where 2:3:4 or 3:4:6
trines set the standard of rich stable concord and euphony, a 4:6:9 or
9:12:16, etc., sounds like a provocative and energetic variation on a
trine, sharing the same overall quintal/quartal flavor.

Maybe a term like "polydyadic" harmonic entropy or concord/discord or
whatever would convey the idea that we're indeed looking at
multi-voice sonorities, but analyzing them in terms of combinations of
dyads rather in terms of the more intricate methods of "triadic
harmonic entropy."

One of the things that really caught my imagination two decades ago is
how combinations like 6:8:9 or 4:6:9 occur over a wide range of world
musics where fifths or fourths are treated as the main vertical
concords. Your "pizza" maybe provides another view of this pattern.

Most appreciatively,

Margo Schulter
mschulter@value.net