Re: Sonorities with fifths or fourths -- a Gothic tuning

Hello, there, and speaking from a medievalist viewpoint, I'd like to
suggest that for three-voice medieval sonorities featuring two fifths
or fourths, the historical choice of Pythagorean tuning (3-limit just
intonation) nicely optimizes the consonance of these sonorities.

Thus, using C4 for middle C and higher numbers for higher octaves:

| A4 | F4
| 3:2 | 4:3
9:4 | D4 16:9 | C4
| 3:2 | 4:3
| G3 | G3

(M9|5-5) (m7|4-4)

Below each example, I've indicated a kind of "neo-medieval" notation
for such three-voice combinations, showing first the outer interval,
then the lower and upper adjacent intervals. Thus an outer ninth
divided into two fifths by the middle voice -- 4:6:9 -- is shown as
(M9|5-5); a minor seventh divided into two fourths -- 9:12:16 -- is
shown as (m7|4-4).

Pythagorean tuning -- 3-limit JI, or "chain tuning in pure fifths" if
you like -- fits nicely with the statements of two theorists around
1300 that the major ninth "concords well" if a middle voice is added
to produce two fifths.

For example, there's a late 14th-century piece from an English
manuscript which has lots of very pleasing combinations of this kind
with major ninths and fifths; the combination of a minor seventh and
two fourths is likewise approved by Jacobus of Liege (c. 1325), and
has a prominent role in some music around 1200.

In a medieval setting, these kinds of sonorities are _relatively_
blending but definitely unstable; in the 20th century, of course, they
may be treated as full concords.

This is isn't to say, I'll add just for the sake of caution, that
3-limit JI is the _only_ way to tune these combinations; some people
may like meantone, and 12-tone equal temperament (12-tet) gives a
reasonably close approximation to 3-limit.

Incidentally, while a sonority with _three_ fourths (e.g. D3-G3-C4-F4)
goes a bit beyond usual medieval practice, Jacobus does mention it as
one possible combination with an outer minor tenth.

Here a Pythagorean tuning would give pure 4:3 fourths and 16:9 minor
sevenths, plus an outer minor tenth at 64:27 -- not too far from 19:8,
for those who regard 19:16 as an ideal minor third.

In Japanese _gagaku_, the traditional orchestral music, some very
complex sonorities are reportedly built from superimposed fifths and
fourths, and I wonder if a pure 3-limit tuning is used.

Most respectfully,

Margo Schulter
mschulter@value.net