Re: Question on Pythagorean

Hello, there.

Recently Fred Reinagel <violab@wny.com> remarked:

> I have always been strongly convinced that Medieval chant historically
> used strictly Pythagorean tuning (Margo?), which would make all the 4ths
> and 5ths pure, but 3rds and 6ths far from consonant.

Indeed I would agree that Pythagorean tuning (3-limit just intonation)
is generally the basis for this music, duly noting that any precise
tuning scheme applies more strictly for fixed-pitched instruments than
for others (including voices).

In plainsong, Pythagorean intonation lends a pleasing contrast between
whole-tones at a pure 9:8 (~203.91 cents) and diatonic semitones at a
compact 256:243 (~90.22 cents).

In medieval polyphony, this tuning, as you note, makes fifths and
fourths pure (3:2, 4:3). However, if we follow 13th-14th century
theorists as well as the practical music of the era, major and minor
thirds (81:64, 32:27) might be described as unstable but _relatively_
concordant. To borrow Ludmila Ulehla's useful terminology, they are
neither stable concords nor clear discords, but "dual-purpose"
sonorities.

With major and minor sixths (27:16, 128:81), a typical 13th-century
viewpoint regards M6 about on a par with M2 (9:8) and m7 (16:9) as
_relatively_ tense (more so than M3 or m3) but somewhat "compatible."
However, m6 is often classed with m2 (256:243), M7 (243:128), and A4
or d5 (729:512, 1024:729) as a sharp or "perfect" discord.

In the typical "modern" theory of the 14th-century Ars Nova, major
sixths and eventually minor sixths also tend to be grouped together
with the thirds as "imperfect concords," while M2 and m7 are regarded
as clearly "discordant."

A 14th-century composer such as Guillaume de Machaut sometimes seem to
follow the "modern" view in his free treatment of sixths as well as
thirds, and the "ancient" (i.e. 13th-century) view in his free
treatment of M2, m7, and M9.

Note generally that Pythagorean or 3-limit JI, like many JI systems,
tends to optimize the concord of intervals recognized as richly stable
(here fifths and fourths), while making some other intervals _more_
tense than they might be in some compromise system such as 12-tone
equal temperament (12-tet). Likewise, in 5-limit, the rather tense m7
at 9:5 contrasts with the Pythagorean 16:9 and the 7-limit 7:4.

> Also, a JI diatonic scale has a wolf 5th between the 2nd and 6th
> degrees, so it is not possible for _all_ the notes of the scale to
> be consonant with the reverberations of previously sounded notes.

At first blush, I might be tempted to argue that such a tuning
often suggests more than 12 notes per octave -- but this would not
address your issue of what might be termed "diagonal dissonances"
between notes of different sonorities, which the acoustical setting
can in effect "verticalize" through the effect of prolonged tones.

Of course, quite apart from Wolf intervals, any prolongation of this
kind will inevitably lead to dissonances such as minor seconds whether
melodic lines move by a semitone -- except maybe in pieces such as
Bartok's "Major Sevenths, Minor Seconds," where these vertical
intervals are treated as concords.

Most respectfully,

Margo Schulter
mschulter@value.net