Yahoo Tuning Groups Ultimate Backup tuning Re: 81:64, 5:4, 9:7 as M3 "allophones"

Hello, there, and thanks to Paul Erlich, Dante Rosati, and others for
a recent discussion on the relationship between the Pythagorean 81:64
and other just intonation versions of the major third, specifically
5:4 and 9:7.

First of all, I should note that 81:64 at ~407.82 cents is
interestingly situation a syntonic comma from 5:4 or 80:64 at ~386.31
cents (a difference of 81:80, about 21.51 cents), and a septimal comma
from 9:7 or 81:63 at ~435.08 cents (a difference of 64:63, about 27.26
cents). Thus 81:64 is somewhat closer to 5:4 than to 9:7.

As a medievalist who regularly plays 13th-14th century European pieces
in Pythagorean tunings, I might say musically that 81:64 has its own
character nicely described by some writers of the period: it is a
_relative_ or "imperfect" concord. While it is certainly possible to
substitute a major third of 400 cents (12-tet), or even 5:4, without
radically disrupting the "grammar" of the harmony, I would say that
81:64 is just right: concordant and at the same time active.

In contrast, a 9:7 (or a Pythagorean approximation built from 16
fifths, precisely a Pythagorean comma or ~23.46 cents wider than an
81:64, or around 431 cents) is considerably more "discordant." In a
medieval context, it's usable specifically as a cadential interval
expanding to a fifth, the kind of idiom suggested by Marchettus of
Padua. However, I'd regard this as a "special effect," in contrast to
the concordant and usual 81:64.

Another way to put this is that Gothic polyphony assumes an active
major third less concordant than a fifth and somewhat more concordant
than a pure major second (9:8) or minor seventh (16:9), intervals also
having some degree of "compatibility." The 81:64 nicely fits this
role.

In contrast, many modern listeners, at least, perceive that a major
second or minor seventh is _more_ concordant than a 9:7. If
appropriate Gothic perceptions are similar, then it becomes clear why
using 9:7 as a usual major third would upset the medieval balance.
Indeed, if presented with _only_ a choice between 5:4 and 9:7, I would
go with 5:4, because like an 81:64 it seems less concordant than a 3:2
and more concordant than a 9:8 or 16:9. This isn't to cheer about the
loss of historical color in the harmony and the wide diatonic
semitones that would result, only to say that it would seem preferable
to a pervasive use of 9:7.

As an aside, I would say that if asked to perform Gothic music using
some well-known tuning for the major third other than 81:64, I might
indeed go with 400 cents, much closer to the ideal than either 5:4 or
9:7. In fact, 12-tet may be a closer approximation of Pythagorean
tuning than of 5-limit just intonation, for example. It's humorous
that some critics of 12-tet attack it as a reversion to "medievalism,"
and as a medievalist, I don't necessarily see that as a flaw
<grin>. Seriously, I would say that the real flaw is not 12-tet (one
temperament compromise of many), but a worldview which treats it as
the _only_ interesting or valuable tuning.

Getting back to Pythagorean -- for medieval music, why go with 12-tet
or another approximation if Pythagorean is available? -- I find that
playing some two-part clausulae from around 1200 with lots of
prominent thirds brings out the qualities of 81:64 (and 32:27). These
intervals can be quite "sweet," but at the same time active; they move
the music forward, even while also serving as momentary diversions.

Of course, the fact that 13th-14th century music is quite
comprehensible (at least to me) even in a Renaissance meantone where
things seem intonationally "backward" -- slightly compromised fifths
and pure major third -- suggests the importance of categorical
perception of intervals, another theme of this thread.

In fact, one might say that the usual "phoneme" for a major third has
varied historically in European compositional practice, along with the
stylistic use of this interval, but within a certain range. The narrow
end of this range might be about 1/3 syntonic comma narrower than 5:4
(as in 1/3-comma meantone and 19-tet), the middle range somewhere
between 1/6-comma meantone and 12-tet (or 1/11-comma), and the wide
end around 81:64. It isn't surprising that we find the wide end the
norm in the Gothic polyphony of Continental Europe, and the narrow to
middle range in later music with stable major thirds.

Two intonational systems, the modified Pythagorean tuning in vogue
around the early 15th century and the unequal well-temperaments of the
late 17th to mid-19th centuries, feature a mixture of major thirds
ranging from around 5:4 (or Pythagorean 8192:6561) to around 81:64.
Such systems, acceptable not only near the Gothic-Renaissance
transition but also in the Baroque with its definitely tertian
harmonies, indicate that 5:4 and 81:64 have both been tolerable
"allophones" of the major third.

In contrast, I'm not aware of a regular intonational practice
documented during these eras with 9:7 used as a common "allophone" for
M3. Another way of putting this is that apart from the Marchettan
context I mention above, 9:7 would be considered a "Wolf"; in
contrast, even in tertian music of a Baroque or Classic variety, 81:64
has been considered at least marginally within (or at the edge of) the
range of "playability."

Here I should discreetly add that "marginal playability" is not the
same as a good choice for the pervasive realization of an interval.
Thus to play typical Renaissance keyboard music in Pythagorean tuning,
or even 12-tet, is to make the thirds too "restless," a flaw perhaps
accentuated by the delicate shading of concord/discord in the 16th
century.

In short, the various choices for tuning the "major third" phoneme in
a European context -- 81:64, 5:4, 400 cents, 9:7, etc. -- nicely
exemplify such perennial themes as stylistic evolution, categorical
perception, and subjective taste.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: 81:64, 5:4, 9:7 as M3 "allophones"

🔗M. Schulter <mschulter@xxxxx.xxxx>

🔗Daniel Wolf <DJWOLF_MATERIAL@xxxxxxxxxx.xxxx>

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