Re: Clarification on 9:7 and importance of context

Hello, there, and thanks to Daniel Wolf and Kraig Grady for inviting a
clarification of my statement that for _Gothic_ music, it would seem
unwise to attempt to substitute a 9:7 for the usual 81:64 major third
of Pythagorean tuning (3-limit just intonation).

Mainly, I'd like to emphasize that my statements on this point and
relating to associated consonance/dissonance concepts were intended to
apply specifically to _medieval_ harmony, and to other "conventional"
styles (e.g. Renaissance), but by no means to 20th-century musics in
JI or other tunings which treat 9:7 or an approximation as an
independent concord, whether deemed an "allophone" of the major third
or something else.

My message actually had something of an "implicit agenda": to respond
to the rather common notion that Gothic composers regarded Pythagorean
thirds as "strong dissonances," or even (according to one recent
textbook whose author shows much knowledge of medieval theory) as
"howling thirds." The latter description might be applied in a
conventional Renaissance viewpoint to a meantone diminished fourth
such as c#-f at around 427 cents (1/4-comma tuning), not far from 9:7,
but hardly to the Gothic interval of around 408 cents which leading
13th-century theorists describe as an "imperfect concord."

Of course, these perceptions are stylistically and historically
colored, and 20th-century composers have very effectively "rebutted"
any claim they might have to universality. Just as Bartok's "Major
Sevenths, Minor Seconds" demonstrates that these intervals need not
always be treated as strong discords, so composers such as Gary
Morrison with his 88-cet systems have demonstrated that a "supermajor
third" at or near 9:7 (5 scale steps or 440 cents in 88-cet) can
indeed be felt as a concord.

My point was only that while many people have recognized the
_unstable_ nature of the major third in a 13th-century context (which
might be better realized by either 81:64 or 9:7 as opposed to 5:4),
the _relatively concordant_ nature of this interval points to 81:64 as
opposed to 9:7. In a different artistic context -- say 11-limit JI --
Gothic norms of concord/discord are no longer necessarily relevant,
and 9:7 could easily become an independent concord.

Incidentally, I might note in passing that two theorists of the
Renaissance-Manneristic era do specifically assert that 81:64 would be
a "dissonance," not a valid concord, as opposed to 5:4. Both Zarlino
(1558) and Lippius (1612) make this point in declaring that modern
practice depends on the syntonic diatonic tuning of Ptolemy with its
pure 5-based ratios for thirds and sixths, as opposed to the
Pythagorean or "ditonal diatonic" with its more complex ratios.

As Bill Alves has rightly emphasized, Vicentino (1555) makes a similar
point in reviewing the Pythagorean theory of Boethius, but then
championing Ptolemy as the basic for the modern practice of "mixed and
tempered music" which approximates the ideally blending ratios for
thirds at 5:4 and 6:5.

On the topic of categorical perception, Lippius offers especially
striking (and controversial) remarks against some of the alleged
misconceptions of his contemporaries. Specifically, he rebuts the view
of those who say that a Pythagorean major third is "a dissonance to
the intellect, but a consonance to the senses." Such a stance could
reflect both Lippius's ardent allegiance to Ptolemy's system of pure
5-based ratios (whose summa is the _trias harmonica_), and his own
perception that 81:64 is not a "reasonable facsimile" of 5:4.

A wild hypothesis: is it possible that in a relatively "heterogenous"
harmonic texture based on bold contrasts between stable and unstable
sonorities (Gothic or Baroque), an 81:64 may seem more "concordant"
than in the uniquely homogenous tertian texture of the Renaissance? If
so, we may have an one explanation of why this tuning for some major
thirds is more acceptable to Werckmeister than to Zarlino or Lippius.

Most respectfully,

Margo Schulter
mschulter@value.net

Let me clarify once more. La Monte Young's 9/7 works in the string quartet
largely because of the high tessitura (the lowest 9/7 is between f" and a")
and the natural harmonic timbres. In _The Well Tuned Piano_ , the same
interval appear in lower octaves but always first in the context of a
larger chord or sonority. The 9/7 interval is extracted from this complex
over a long period of time.

A more important aspect of TWTP is, perhaps, the simple fact that for much
of the work, a 9/7 is the only Major third available. Ironically, this
ends up a bit analogous to a temperament where one accepts a given complex
interval playing the role of a less complex one. I think this is not the
case in _Chronos Kristalla_, however, because the 9/7 is really heard as
the interval between the 7th and 9th partials. Incidentally, portions of
_Chronos_ sound very much like sho (mouth organ) playing in left side
Gagaku, which have traditionally used wider Major thirds.

I agree with Margo Schulter's posting about the unlikeliness of the 9/7 in
medieval music. The tessitura of gothic polyphony seems to me to be just
too low to project the interval (unlike Young's harmonics). On the other
hand, use of the interval in mannerist cadences, e.g. those resolving to
the outer fifth, while probably not practiced historically, would be well
within the mannerist 'project'. (I offer this as a challenge to Brian
Ferneyhough, who considers his music -- with its essentially arbitrary use
of quartertones -- to be mannerist. The historical mannerists were often
exotic but never arbitrary).