Re: JI, Pythagorean and otherwise

Hello, there, and I'd like briefly to reply to a question about the
"color" of 1/3-comma or 1/4-comma meantone as compared with
Pythagorean and other systems of just intonation (JI). Please note
that although the term "just intonation" often implies 5-limit or
above, in my view Pythagorean tuning is a system of just intonation
not only in its basing all intervals on integer ratios, but in some of
its musical qualities.

First of all, I'd like to offer what may be the most important point
in this article: comparisons between Pythagorean and 1/4-comma
meantone, for example, are likely to reflect the differences between
the styles of music conducive to either tuning.

In a European concept, this difference might be summed up by the
adjectives "trinic" and "triadic." In trinic music (c. 1200-1420), the
three-note combination of octave, fifth, and upper fourth
(e.g. d-a-d') sets the standard of stable saturation, and Pythagorean
tuning permits a just tuning for such sonorities. In triadic music
(c. 1520-1900, with 1680 or marking the transition for stringed
keyboard instruments from meantone toward well-temperaments), the
combination of fifth, major third, and upper minor third similarly
sets the standard of complete harmony, and 1/4-comma meantone
optimizes the prime interval of the major third.

Note that the period of around 1420-1520 is transitional, although
meantone evidently starts coming into widespread use around
1450-1480. This is one historical case where tuning and musical style
seem to coevolve in a rather closely linked way.

Anyway, Pythagorean intonation or 3-limit JI seems to me parallel to
5-limit JI (approximated to some extent by 1/4-comma meantone) in
these ways:

(1) The most favored intervals, in this case fifths (3:2) and fourths
(4:3), are pure, optimizing the complete trine (2:3:4), just as
5-limit JI optimizes the most favored intervals of the major third
(5:4) and minor third (6:5), and thus the complete triad (4:5:6).

(2) In Pythagorean JI, the mildest unstable intervals, "imperfectly
concordant" or _relatively_ blending major thirds (81:64) and minor
thirds (32:27) are somewhat more acoustically tense than the 5:4 and
6:5 ratios of 5-limit. Similarly, in 5-limit JI, what might be
considered among the mildest of the unstable intervals, the minor
seventh (9:5), is likewise rather more tense than the 7:4 tuning of
7-limit.

(3) In Pythagorean JI, a diminished fourth or "schisma major third" at
8192:6561 (e.g. e-ab) is curiously only a schisma of 32805:32768 or
about 1.95 cents from a pure 5:4; and likewise an augmented second or
"schisma minor third" at 19683:16384 (e.g. db-e) in relation to a pure
6:5. In 1/4-comma meantone, likewise, an augmented sixth (e.g. eb-c#')
is only about 3.04 cents from a pure 7:4.

To state these features in more aesthetic terms, JI systems
(Pythagorean and other) often tend to accentuate the contrast between
stable and unstable intervals: Pythagorean thirds and 5-limit minor
sevenths are somewhat less "concordant" than in 5-limit and 7-limit
tunings respectively. Music conceived for these systems often makes
the most of this contrast: 13th-14th century cadences exploit the
tension of thirds leading to unisons or fifths, for example, while
16th-17th century music intended for 5-limit JI or meantone often
exploits the tension of sevenths, whether in the more subtle
suspensions of usual 16th-century style or in the bolder seventh
sonorities coming into vogue around the end of the century
(e.g. Monteverdi, Gesualdo).

In addition to this kind of obvious contrast, the schisma or similar
intervals can also invite "special effects" in certain epochs. Thus in
early 15th-century music, realizing major thirds as diminished fourths
leads to a strikingly new sound for at least some sonorities involving
sharp. My own tendency at the keyboard is to play cadential thirds and
sixths in the usual tuning (e.g. e-g#-c#'), but prolonged noncadential
thirds as schisma intervals (e.g. e-ab-b). This is just one of the
possible interpretations, and at any rate _could_ be the way that some
performers might have made use of the 17-note Pythagorean tuning
described by some theorists of the epoch.

Likewise, as Dave Hill has demonstrated with pianos tuned in 1/4-comma
meantone, the near-7:4 augmented sixth (e.g. eb-c#') can be a valuable
resource for certain Blues settings, for example. There's some
evidence, I hear tell, that certain 17th-century composers may have
also experimented with this effect.

These kinds of intonational possibilities, of course, can invite new
tunings devised to make such "special effects" routinely available at
most or all scale positions. Thus it appears that the schisma thirds
of the early 15th century whetted an appetite which could only be
satisfied by something like the "5-limit-like" meantone temperaments
of the later part of the century and after. Likewise, although I'm not
sure of a historical example, I can imagine how someone experimenting
with the augmented sixths on a 1/4-comma meantone keyboard might
develop a craving for 7-limit JI.

In European music, at least, this kind of evolution may correlate with
a general axiom: "One era's dissonant interval or combination will be
regarded as mildly unstable in the next, fully concordant in the one
after that, and `rather incomplete and empty' in the next yet era."

Note that one could call this either "progress" or "degeneration," but
I would simply call it artistic change.

Most respectfully,

Margo Schulter
mschulter@value.net