Re: Marchettus and 5-limit?: Response to Joe Monzo

Hello, there, and I'd like to thank Joe Monzo for focusing so much
creative attention on the theories of Marchettus of Padua, whose
_Lucidarium_ (1318) is indeed an intriguing document from a
xenharmonic point of view.

However, I must question the following assertion that Marchettus is to
be numbered among medieval theorists who described or advocated the
use of thirds and sixths closer to 5-limit ratios than to their
conventional 3-limit (Pythagorean) ratios. Indeed, I shall argue that
in cadential contexts, he was doing quite the opposite.

At the outset, a disclaimer is in order. When interpreting a theorist
such as Marchettus, few analysts can resist being influenced by their
own musical tastes and perceptions, and I certainly must admit to at
least my share of such influences (sometimes called "biases") in what
follows.

Also, I would like to emphasize that alternative interpretations such
as those of Joe Monzo are vital to the progress of medieval musicology
and xenharmonic theory alike. Such interpretations may indeed lead to
new paradigms, and meanwhile force all of us to reexamine and more
closely scrutinize the "conventional wisdom."

> The first mention in European theory treatises of singers altering
> the pitch of Pythagorean '3rds' and '6ths', so that they more
> closely approximate 5-limit JI, is in the late 1200s/early 1300s
> (Marchetto was one of them).

First, as Joe Monzo has also observed, it may be helpful to recognize
that composed polyphony of the 13th-14th centuries does not exhaust
the diversity of traditional musics practiced in the Europe of these
times. I have expressed the viewpoint that various intervals and
intonations, possibly reaching at least to "11-limit" (e.g. bagpipes
with "neutral thirds"), may have been practiced in various regions.

Further, English music of the 13th century which has actually come
down to us, as well as the statements of Theinred of Dover (13th c.?)
and Walter Odington (c. 1300), show both that thirds were indeed
treated as principal and even stable concords in some English styles,
and that two English theorists described these intervals as being
close to 5:4 and 6:5. Odington, in particular, has a passage which
can be very reasonably read to suggest that singers tended to make
subtle adjustments in their pitches so as to approximate these
simple ratios.

However, I know of no such passages in Marchettus, and indeed his
teachings seem to me rather the contrary. Note that here I translate
Marchettus's terminology (with its "diatonic," "enharmonic," and
"chromatic" semitones) into more familiar terms.

In his system, the usual diatonic semitone (i.e. mi-fa, e.g. e-f)
seems close to the traditional Pythagorean semitone of 256:243 (~90.22
cents). If we read his "equal division" of the tone to use five equal
parts, then this normal semitone is 2/5-tone (as in 29-tone equal
temperament or 29-tet), somewhat _narrower_ than the Pythagorean. If
we read his division to be based on five segments built from nine
equal parts (the ninefold division as in 53-tet), then this semitone
is likely 4/9-tone, virtually identical to Pythagorean.

While the idea of dividing the whole-tone into _equal_ (geometric)
parts is indeed radically opposed to the Pythagorean concept of
integer ratios, his musical result for noncadential contexts is thus
either virtually identical (the 4/9-tone interpretation), or actually
a slight accentuation of the Pythagorean norm of narrow diatonic
semitones and rather wide major thirds and sixths and narrow minor
thirds and sixths (the 2/5-tone interpretation).

For cadential intervals (e.g. M3-5, M6-8, m3-1), as has been discussed
here, his approach might be characterized as "super-Pythagorean."
Reading his theory literally, the leading-tone should be an
ultra-narrow "diesis" equal to something in the neighborhood of
1/5-tone or 2/9-tone. This would actually result in something
not too far from 24-tet: a cadential major third of ~450 cents before
a fifth, and a cadential major sixth of ~950 cents before an octave,
etc. Personally, I'm tempted to guess that these superwide intervals
might have been tuned around 9:7 and 12:7.

In any case, the important point here is that Marchettus advocates
something close to usual Pythagorean intonation for noncadential
intervals, and favors the use of supernarrow cadential semitones and
superwide major thirds and sixths expanding to fifths and octaves
respectively. However, he says nothing of which I am aware about major
thirds narrower or "smoother" than Pythagorean, or about 5:4 and 6:5
as ratios or approximations for concords -- quite unlike Theinred of
Dover or Walter Odington.

One piece of evidence which might support this more traditional
reading of Marchettus -- as opposed to Joe Monzo's interpretation of a
"5-limit" system -- is that certain theorists of the Renaissance
borrowed or adapted Marchettus's "fivefold division of the tone" to
the new musical realities of their own era. When they did so, they
_reversed_ the divisions of Marchettus so that the diatonic semitone
(mi-fa, e.g. e-f) was now _larger_ than the chromatic (e.g. f-f#).
Rather than 2/5-tone or 4/9-tone, it was now 3/5-tone (as in 31-tet)
or 5/9-tone (as with the large semitone in 53-tet).

An appealing explanation is that Marchettus was proposing a system
which in cadential contexts actually exaggerated the usual Pythagorean
features of narrow semitones and active thirds and sixths, while the
Renaissance theorists were seeking to adapt his idea of a precise or
at least approximate division of the tone into five or nine parts so
that it would meet the needs of a 5-limit system (or its meantone
approximation on a keyboard instrument).

As Mark Lindley has stated, Marchettus was issuing a welcome call for
high leading-tones, even if other theorists quarreled with his
unorthodox mathematics, or preferred the conventional Pythagorean
tuning of cadences with narrow but not quite _this_ narrow semitones.
However, once 5-limit intervals had in fact become an artistic ideal (as
happened in the course of the first half of the 15th century in the
composed music of Continental Western Europe), wider leading-tones were
one inevitable consequence, as the adaptation and "reversal" of
Marchettus's scheme shows.

Incidentally, it has recently been proposed that Marchettus's "equal"
division of the tone actually refers to an _arithmetic_ division, as
when a monochord has a whole-tone ratio of 81:72 (9:8) divided into
nine intermediate segments. If this was indeed the intended meaning,
then Marchettus was widely misunderstood by Prosdocimus (also at
Padua, about a century later) and various others.

The idea of an equal _arithmetic_ division of the tone would hardly be
described as "impossible" by Pythagorean theorists: both Boethius and
Jacobus of Liege give the arithmetic division of the tone into 18:17:16
as an example showing that such a division produces _unequal_ ratios,
not a (geometrically) equal division (in fact impossible, quite unlike
this type of arithmetic division, if we use only integer ratios).

However, if we do choose to adopt such an "arithmetic" reading, then
an interpretation of 81:77 for the diatonic semitone (~87.68 cents)
would again give a quite close approximation to Pythagorean, and in
fact be slightly narrower than Pythagorean (and thus associated with
slightly _wider_ major thirds and sixths, etc.).

Most appreciatively,

Margo Schulter
mschulter@value