Re: TD 742 -- Thanks, John deLaubenfels (36:45:54:64)

Hello, there, and thank you for pointing out that there is a very
important tuning of the tritonic seventh that I omitted from my
earlier article.

> In discussing dominant 7ths, you give two tuning possibilities,
> 20:25:30:36 and 4:5:6:7. The first of these puts the 7th degree at
> 9/10 of the root above, the second puts it at 7/8. What about the
> intermediate tuning, 8/9? To my ear this last is discordant but
> tolerable, where the 9/10 is VERY high. Is 9/10 more "historical"?

In 1/4-comma meantone, one characteristic temperament of the era
around 1600 when tritonic seventh combinations came into vogue (the
later dominant sevenths of Rameau's key system), the tuning would
actually be about midway between 20:25:30:36 and the alternative to
which you very rightly call attention: 36:45:54:64. Specifically, the
meantone minor seventh is precisely the mean of 16:9 (as in this last
tuning) and 9:5.

When I cited the version with 9:5, I may have had in mind Vincenzo
Galilei's statement remarking that the minor seventh is a dissonance
with a ratio of 9:5. Galilei (father of the astronomer Galileo),
endorsing the bold tritonic seventh combination and its cadential use
during the 1580's, may have been the first theorist to describe an
idiom which came into vogue in the composers of the next decade or so
(e.g. Monteverdi, Gesualdo).

However, the minor seventh in 5-limit JI can be either 9:5 or 16:9.
The first value would occur when this seventh is built from a pure
fifth plus a minor third, for example in this progression from
Monteverdi:

C4 B3 C#3
A3 G#3 A3
D3 E3 A2

Here, if the fifth D3-A3 of the first sonority is a pure 3:2, and the
minor third A3-C4 a pure 6:5, then D3-C3 will be 9:5.

For an orthodox 16th-century example, let's consider a 7-6 suspension
with a 9:5 minor seventh:

D4 C#4 D4
G3 F#3
... E3 D3

If E3-G3 is a pure 6:5, and G3-D4 a pure 3:2, then E3-D4 is 9:5.
Incidentally, this example illustrates the kind of tritone resolution
(here augmented fourth to minor sixth between the two upper voices)
routine in the 16th century and described by theorists including
Vicentino (1555) and Zarlino (1558). This resolution combines with a
typical M6-8 between the outer voices. The innovation near the end of
the century was boldly combining the tritone resolution with the kind
of cadential motion of the bass down a fifth or up a fourth which
you'll see in the next example of 16:9.

The 16:9 minor seventh would typically occur in suspensions, where
this interval arises from two pure 4:3 fourths:

F4 E4 F4
C4 C4
G3 A3
... C3 F3

In this typical 4-3 suspension formula, the suspended sonority
C3-G3-C4-F4 includes the pure 4:3 fourths G3-C4 and C4-F4; therefore
G3-F4, equal to two such fourths, is 16:9.

Incidentally, the highest minor third of the 36:45:54:64 sonority is a
Pythagorean 32:27; this interval plus the pure 3:2 fifth (54:36) form
the 16:9 minor seventh.

Historically, in the era around 1600 or 1720 for that matter, I would
suggest that either 9-based tuning of the minor seventh in this
sonority seems a more likely approximation than a 7-based tuning;
meantone basically splits the difference, exactly what singers or
players of non-fixed-pitch instruments might do without
fixed-pitch-instruments in the ensemble remains an open question.

While conventional theorists such as Zarlino took 5-limit JI as the
model for voices, Vincenzo Galilei argued that singers actually tend
toward a regular temperament not too far from 2/7-comma meantone,
Zarlino's favorite keyboard tuning, but with the fifths somewhat
closer to those on the common lute tuning of 12-tone equal temperament
(12-tet).

There is possibly some humor here, by the way, because Galilei, at
one time a student of Zarlino, often engaged in heated controversies
with his former instructor. Specifically, he argues that unaccompanied
singers tend toward a regular rather than pure tuning -- proposing
Zarlino's own keyboard temperament as one fairly close approximation.

(Your adaptive tuning, of course, could be seen as a synthesis of the
two views.)

Some odd final comments for now. Maybe one could argue that if we
follow a JI rather than meantone approach, then 9:5 might be an ideal
tuning for a minor seventh conventionally treated as a suspension, but
that singers might shade this somewhat toward 16:9 with the new and
bolder seventh combinations.

At any rate, composers such as Monteverdi evidently meant it to be a bold
and audacious sonority, a figure of musical "rhetoric" as it were; one
theorist defending Monteverdi's technique says that the seventh in
place of the octave is a kind of poetic liberty, like a simile or
metaphor.

By the 18th century, keyboard temperaments other than 1/4-comma
meantone tend to shift the minor seventh mostly toward 16:9, I'd
guess. By this era the tritonic seventh combination -- now the
dominant seventh -- has shifted from a "special effect" to a routine
feature of the key system. Maybe the tuning in 1/6-comma meantone,
~1003.26 cents (16:9 is ~996.09 cents, and 9:5 is ~1017.60 cents),
gives some idea of this trend; in well-temperaments, of course, the
size of this interval would vary depending on the transposition.

Most appreciatively,

Margo Schulter
mschulter@value.net