Re: TD 668 -- Response for Joseph Pehrson

Hello, there, Joseph Pehrson.

First, please let me express my appreciation for such kind words, and
give a bit of credit where credit is due: Todd McComb, the Early Music
FAQ editor and maintainer of http://www.medieval.org, played a central
role in formatting my "Pythagorean Tuning FAQ" for HTML and in
offering various editorial suggestions and corrections.

To my remarks on the syntonic comma of Pythagorean tuning as a
positive "feature" in a Gothic musical setting, which you nicely
summarize as a very important theme of the document, I might mainly
add that to focus on this comma may reflect something of a comparative
perspective.

That is, from the viewpoint of 13th-century Continental Western
European polyphony, a major third at 81:64 might simply be taken as
the norm, "how a major third sounds in everyday music." Similarly,
around 1600, a minor seventh at 9:5 (or maybe 16:9 -- or actually the
mean of these two ratios in 1/4-comma meantone) would be taken as "our
minor seventh," without any necessary comparison to other possible
tuning systems.

From a comparative point of view, of course, we can say that a
medieval 81:64 third differs from a 5:4 (another rendition of the
major third) by a syntonic comma of 81:80; and likewise that a
16:9 differs from a 7:4 (another flavor of minor seventh) by a
septimal comma of 64:63, a 9:5 from a 7:4 by almost 50 cents, and a
meantone minor seventh from a 7:4 by close to a meantone diesis of
128:125 (~41 cents).

From this viewpoint, we can also say that the greater complexity of a
medieval 81:64 in comparison to a 5:4, or of a Renaissance 9:5 or
meantone minor seventh in comparison to a 7:4, is a "feature" rather
than a bug: these complex intervals are meant to be heard as active
and unstable, not as points of repose (as they may become in 5-limit
and 7-limit music respectively).

Indeed, the theorist Jacobus of Liege (c. 1325) took note of both the
syntonic comma (81:80) and the septimal comma (64:63) in demonstrating
that the 81:64 major third and 16:9 minor seventh differ from tunings
of 5:4 and 7:4 which might hypothetically be adopted for certain
instruments, but which could not be "well formed" from Pythagorean
whole-tones and semitones.

However, I wonder how often musicians composing or performing the
Continental Gothic repertory of the 13th and 14th centuries reflected:
"Isn't it interesting that we're tuning this keyboard with a major
third of 81:64 as opposed to 5:4?" Or, for that matter, how often did
harpischordists around 1600 reflect: "Isn't it worth noting that this
minor seventh is closer to 9:5 or 16:9 than to 7:4?"

In both cases, of course, the "anomalously" tuned intervals or their
near approximations are in fact lurking right there within the usual
tuning system: the Pythagorean diminished fourth or schisma third very
close to 5:4, and the meantone augmented sixth (which I might call a
"diesis seventh") very close to 7:4. In the keyboard music of the
early 15th century, and in Dave Hill's performances of the Blues on
meantone piano, these "odd" intervals come out and take center stage.

Having presented the syntonic comma of the medieval Pythagorena tuning
as a musically positive "feature" (active thirds and sixths), I might
add that various features discovered by users in software, this one
wasn't necessarily included by design.

Pythagorean tuning focuses on pure 3:2 fifths and 4:3 fourths, and
meantone likewise on pure or near-pure thirds (5:4 and 6:5) and sixths
(5:3 and 8:5). It just so happens that Pythagorean produces the active
Pythagorean thirds and sixths so effectively used in Gothic music, and
that meantone likewise produces the rather tense minor sevenths
fitting the Renaissance and Manneristic styles. At yet another level,
it happens that the same systems also produce almost optimally smooth
near-5:4 schisma thirds or near-7:4 "meantone diesis sevenths."

Maybe I should reply to your well-taken question about medieval
sources from conclusions about tuning by saying that I regard both the
music itself and the treatises as vital guideposts in reaching
conclusions, and that both practice and theory require interpretation,
which inevitably implies taking a viewpoint.

Authors such as Marchettus of Padua in 1318, and Prosdocimus of
Beldemandis and Ugolino of Orvieto in the early 15th century, do
discuss such points as cadential aesthetics: the idea, for example,
that major thirds and sixths should expand to stable fifths and
octaves using narrow semitones (for Marchettus, evidently, even
narrower than conventional Pythagorean). When such texts are placed
in the context of actual 14th-century music, and the music also placed
placed in the setting of such intonational ideals, a better artistic
understanding may emerge.

Similarly, Mark Lindley has impressively drawn connections between theory
and music during the era of around 1380-1450 to show a shift in aesthetics
first toward a deliberate use of schisma thirds as a "special effect," and
then toward the adoption of meantone tuning as a solution for making such
smooth thirds an intonational norm.

Most appreciatively,

Margo Schulter
mschulter@value.net