Re: Intonational politics and 9:8, etc.

Hello, there, and I'd like very briefly to support Daniel Wolf's point
that 81:64 can occur quite naturally in tunings as the sum of two pure
9:8 major seconds.

The medieval hexachord system, based on these just 9:8 intervals, may
have made the ditone at 81:64 a widespread feature of vocal
intonations in plainsong and polyphony alike.

Of course, experimental data may clarify at least how modern
performers actually tune these intervals. However, the "intonational
politics" of this dialogue often tends to focus on vertical intervals,
rather than on melodic factors.

As someone who likes to point out the positive vertical role an 81:64
can play in medieval polyphony, I should emphasize that the ditone
wasn't the creation of some polyphonic composer or theorist who said,
"Let's make those thirds nice and unstable so that we can have dynamic
cadences!" Rather, as the name "ditone" itself suggests, this interval
was originally the melodic product of two pure 9:8 whole-steps.

Thus it wasn't a question of vocal coaches exhorting monastic or
courtly singers: "Don't make those thirds too pure or blending!"
Rather it might have been a question simply of singing pure 9:8 steps
(or in reality close approximations, of course), with something close
to 81:64 major thirds (melodic or vertical) resulting.

While keyboard intonations do not bind voices -- the ditone at 81:64
on a Pythagorean keyboard was originally a result of simple
mathematics based on the powers of 3:2, as opposed to an aesthetic
decision, "Let's have active thirds!" -- the change in keyboard
intonations around 1400-1470 may reflect a shift in musical tastes
which could have affected vocal intonation styles as well.

Also, not only can (and _must_) singers vary from fixed-pitch schemes;
they may vary in musically meaningful ways. For example, one of the
treatises of the Berkeley Manuscript (part dating to 1375) suggests a
threefold division of the whole-tone which Oliver Ellsworth has read
as 19-tet -- but further specifies that while a usual mi-fa is
2/3-tone, at cadences one sings an ascending semitone of only
1/3-tone.

Like the much more extensive (and explicitly polyphonic) discussion of
tuning by Marchettus of Padua (1318), this source provides a clear
record of the concept of _variable_ intonation in the 14th century.

In fact, while Marchettus's system has received much discussion on
this newsgroup and elsewhere as we attempt to interpret his divisions
of the tone, I would say that his indisputable and often undervalued
accomplishment was to propose that the same intervals may be tuned
differently depending on the musical context. Certainly I've done my
fair share of neglecting this vital point <grin>. He likes very wide
major thirds and sixths at cadences -- whether one guesses that they
might be realized near 7-limit or otherwise.

With due diplomacy, maybe I should briefly comment on some of the
"intonational politics" which can color discussions such as this one,
especially as perceived (not necessarily objectively) by some of us
who call ourselves "medievalists."

For most of the past 200 years -- and indeed as early as the
Renaissance -- medieval music has been typically viewed as
"intolerably dissonant" or "the barbarian counterpoint of the Gothic
era" or actually "morally impossible." This last comment was offered
by a 19th-century writer who found the parallel fifths and fourths of
early polyphony almost literally unthinkable, and questioned whether
the treatises were _really_ describing such a practice.

Even in recent textbooks, some of them very serious histories of
undoubted merit, we find passages describing the full quintal/quartal
harmonies of the 13th-century as "open and hollow." Maybe I could say
"rich and spacious," but hardly "hollow" -- unless one is listening
using the wrong framework. Indeed Bach's saturated triadic harmonies
might sound "plain and simple" when contrasted with the ninth and
eleventh chords of 20th-century pop harmony, but music history texts
hardly offer such a description as an objective one.

Please let me emphasize that the question of tuning rarely comes up in
older and newer texts of this kind. The music is treated the way it is
not because it may have been performed in 3-limit rather than 5-limit
or 7-limit just intonation, but because it uses fifths and fourths as
the prime concords. It is almost as if anyone who would prefer these
intervals to thirds and sixths must be either acting under the
compulsion of some extramusical authority, or preferring purely
mathematical models to "what sounds good." The fact that many
innovative 20th-century composers have written in quintal/quartal
styles does not seem to modify such judgments -- although other
authors have eloquently noted the parallels (pun intended).

Interestingly, it was one of the leaders of the Alternate Tunings
Movement, Joseph Yasser, who wrote in defense of _Medieval Quartal
Harmony_ a bit more than 60 years ago.

Indeed, the interest in Just Intonation as well as other alternative
tunings should ideally promote a great interest in the range of world
musics, composed and otherwise, be they quintal/quartal, tertian, or
otherwise.

The expanded interest in performing historical European musics with
historical tunings and temperaments -- a trend to which I'm a newcomer
by comparison with the many veterans here -- represents a real
fulfillment of this vision, however partial.

For tertian music such as the vocal and instrumental polyphony of the
Renaissance, that movement rightly champions 5-limit just intonation
or the historical approximation of meantone for keyboards. When it
comes to this music, I would gladly join in accolades to the pure
third, and in practice enthusiastically favor meantone (which not only
makes major thirds pure but compromises my beloved fifths and fourths).

However, some views propounded in the name of "JI advocacy" seem to me
uncomfortably like the old anti-medievalism I describe above. Pure
major thirds at 5:4 are taken not as one historical ideal, not even as
a widespread tendency of many world musics, but as a universal
standard. The nature of quintal/quartal musics in many world
traditions, including European and related musics both of the medieval
era and of the 20th century, do not quite get perceived as equally
"natural."

Since Pythagorean (3-limit) tuning is in fact one of the oldest and
most widespread JI systems in the world, and possibly the one such
system with a significant historical practice on keyboards, one might
expect JI advocacy to include this as well as other systems. Yasser's
writings (although often focused on other kinds of tunings) show how
such a JI approach might put the familiar 5-limit into perspective
vis-a-vis both 3-limit and 7-limit or higher.

Unfortunately, such approaches do not always prevail. Medieval
polyphony remains a relatively obscure area, and the unstable musical
treatment of thirds (at whatever tuning) tends to be seen still as a
kind of curiosity, if not a flaw. In contrast, I have rarely seen
people propose here that Renaissance music was unusual because it
didn't use concordant sonorities with minor sevenths, or that
Zarlino's vocal tuning must have been purely theoretical because real
singers couldn't resist those just 7:4's.

All this being said, I would like again to applaud the kind of
empirical analysis of vocal intonation now being conducted and
discussed on this Tuning List -- indeed a worthy tribute to
Aristoxenes.

To place the results in perspective, I would like again to emphasize
the distinction between keyboard and vocal intonations. How close
singers may actually come to Pythagorean intonation (especially
without the influence of a Pythagorean keyboard or similar instrument)
is an interesting question, and traditional theories may be vindicated
or modified depending on the results. Of course, these results should
ideally reflect various performances of medieval plainsong and
polyphony, in a range of acoustical settings.

However, just as I suspect that Ed Foote would not base his advocacy
of Baroque-Romantic well-temperaments on the assumption that
unaccompanied singers of 18th-19th century music actually approximate
a Werckmeister or Kirnberger, so those of us advocating Pythagorean
keyboard tunings for medieval music need not be discouraged if
vocalists tend to diverge from such tunings.

Just as the well-temperaments bring both utility and beauty to the
music of their eras, so Pythagorean is at once practical and beautiful
for playing medieval plainsong and polyphony. There are, of course,
exceptions; some compositions of the well-tempered era might actually
fare better with 12-tet (forgive me, Ed, and remember that I only said
_might_), and some medieval English polyphony may do best with an
approximation of 5-limit (how about a full 53-tet keyboard, for
schisma thirds wherever desired and pure fifths?).

At any rate, discussing the "intonational politics" involved in
medieval tuning issues may suggest how the emotions may be a bit more
complex than when someone proposes doing Beethoven in 5-limit or
7-limit JI, for example. If our dialogue ultimately serves to make
Perotin or Machaut as familiar as Beethoven, then much of the
potential tension may be defused.

Most respectfully,

Margo Schulter
mschulter@value.net

After reading Margo's (yet another) profoundly insightful
post (as referenced in the subject line of this one), I
believe that I overstepped the bounds of my knowledge of the
subject in my previous post. My apologies.

David