For Alison Monteith -- Dufay, Ockeghem, Sermisy

Hello, there, Alison, and the Pythagorean/meantone question for the
15th century can be a controversial one, with a general tendency at
least for Mark Lindley and I to suggest some form of Pythagorean for
music up to around 1450, and often meantone for later compositions.

For a later composer such as Sermisy, who flourished around 1530, it's
easy to suggest some form of meantone. I might lean toward 1/4-comma
with pure 5:4 major thirds, which is what Pietro Aron appears to
suggest in 1523, not excluding some scope for "irregularities"
(e.g. some fifths maybe closer to 2/7-comma, or 1/5-comma?).

Since Zarlino, in 1558, is the first to define such a tuning in
precise mathematical terms (2/7-comma), the evidence concerning
earlier meantones is more qualitative than quantitative, so to speak.

When Aron recommends tuning the major third C-E as "sonorous and just"
as possible, making the interval a "unity," a pure ratio of 5:4 seems
suggested, certainly supporting although not specifically mandating a
tuning of the type later termed "1/4-comma."

Some other sources, such as Schlick (1511) and Lanfranco (1533),
suggest a kind of meantone where the major thirds are somewhat wider
than just, maybe around 1/5-comma.

Zarlino's choice of 2/7-comma in 1558 might suggest that tunings of
this kind, with major thirds narrower than just and minor thirds
closer to just, may have also had some currency in previous decades.

If in doubt, I might go with 1/4-comma for Sermisy, but you might want
to experiment with other meantone tunings -- and, as Lindley says, we
should not assume that early 16th-century tuners generally aimed at
absolute regularity.

As Lindley comments, 2/7-comma might be especially pleasing for some
pieces with a slower kind of texture, generally, and lots of
sonorities with minor thirds above the lowest part, as with some later
16th-century keyboard music from Venice.

He finds 1/5-comma "sprightly" for certain English pieces around 1600
where there's lots of melodic motion, since the diatonic semitones are
a bit narrower.

In the 16th century, at least, meantone seems an obvious general
choice for most repertories. Now we come to the trickier matter of the
15th century, where there's lots of room for differing opinions.

For the early Dufay epoch, say 1420-1440, I might ideally recommend
the model of a 17-note Pythagorean tuning (Gb-A#), with Gb-G# as the
most frequently used part of this range.

The basic idea is to have a choice for sonorities with thirds
involving sharps of either a usual active Pythagorean form of these
intervals (e.g. A3-C#4-F#4 before G3-D4-G4 in a cadence), or the
"schisma" form with ratios very close to a simple 5:4 or 6:5
(e.g. A3-Db4-E4 for a prolonged noncadential sonority written as
A3-C#4-E4).

This approach calls for lots of judgment: sometimes you may have to
decide both whether an accidental inflection not present in a
manuscript itself should be added -- e.g. A3-F4 or A3-F[#]4 -- and, if
so, whether to take the inflection as A3-F#4 (usual Pythagorean major
sixth) or A3-Gb4 (Pythagorean diminished seventh, ~5:3). You can take
editorial accidentals as one guide to the first kind of question, your
mileage varying with the editor's taste <grin>, but still have the
second kind of question to deal with.

Here I sometimes have curious intuitions. For example, a sonority with
a major tenth like D3-A3-F#4 might especially invite a schisma third
(D3-A3-Gb4) in order to get a "sweet" near 5:2 between the outer
voices.

Here I'm assuming a fixed-pitch instrument that can tune at least 15
or so notes per octave (D# and A# seem to me a bit less important for
most of this music, with the pairs C#/Db, F#/Gb, and G#/Ab the most
important ones for this kind of choice).

Thus when a cadence combines the usual 14th-century motions of major
third to fifth and major sixth to octave, I would be inclined to
regular Pythagorean thirds and sixths. For prolonged noncadential
thirds with written sharps, or also cadential sonorities with an outer
major tenth or the like, I might lean toward the near-pure schisma
thirds.

With a 12-note instrument, a simpler system is generally the Gb-B
tuning, with the Pythagorean "Wolf fifth" between B and Gb. As long as
a piece doesn't rely on the fifth B-F# -- often not used in music of
this era -- this popular keyboard tuning should give a nice contrast
between these varieties of thirds, although "automatically" rather
than as an exercise of calculated discretion.

As we move to around 1450, conventional wisdom tends to favor
meantone: as Lindley notes, this would include the later Dufay and
Ockeghem. However, there has recently been a tendency among some late
15th-century music lovers to favor Pythagorean for certain styles,
including some of the intricate polyphony of Ockeghem.

Here the argument is that the melodic features of the music might
often favor a traditional Pythagorean approach, although this can be
in tension with some of the vertical traits. Curiously, there's an
intonational fine point here to consider.

Clearly the meantone sonority with major third below and minor third
above (near-4:5:6, approximating a rounded 0-386-702 cents,
e.g. 0-386-697 in 1/4-comma) is much smoother in usual timbres than
the Pythagorean version at 64:81:96 (0-408-702 cents). By the early
16th century, say the epoch around 1520, this sonority is more and
more definitively setting the standard of musical euphony.

However, for the arrangement with minor third below major third, the
meantone rendering of thirds near 6:5 and 5:4 (0-310-697 cents in
1/4-comma) might not be much more stable than the Pythagorean thirds
at 32:27 and 81:64 (0-294-702 cents).

From a later perspective, the Pythagorean tuning of 54:64:81 is
actually not that far from 16:19:24 (0-298-702 cents), a form where
the "rootedness" and difference tones may give the sonority a special
stability.

Thus while meantone, by around 1480, may have been the prevailing
choice for keyboards, it is quite possible that singers were still
influenced by more Pythagorean habits, with the increasing
predominance of sonorities with _major_ thirds above the bass by
around 1520 making a meantone or 5-limit paradigm more generally
applicable.

One possible line of evidence here is that around 1500, when closing
sonorities with thirds become common in the vocal pieces of composers
such as Josquin and Isaac, often they involve minor thirds above the
bass, likely without the kind of performers' alterations to major
thirds which may have been an increasingly common practice by around
1523 or 1525, when Aron documents such tendencies.

For a fixed-pitch instrument, I might still lean toward meantone by
the Josquin-Isaac period; I mention this point only to note that
current views are mixed. For a vocal ensemble, of course, some kind of
adaptive tuning leaning toward Pythagorean melodic steps, but with
adjustments to get "meantone-like" sonorities with major thirds above
the bass, could be an optimal strategy.

An interesting implication of this kind of recent view is that the
continued use of Pythagorean paradigms in theory around 1480-1520,
although sometimes with mention of the fact that keyboard instruments
are no longer tuned in this way, could reflect an element of vocal
practice as well as conceptual conservatism.

If the ideal tuning in vocal music for a conclusive sonority with
minor third below major third around 1500 is indeed near 16:19:24,
then a Pythagorean model might serve some vertical as well as melodic
features of the music, given the adaptive potential for leaning toward
a 4:5:6 ratio for the converse arrangement with major third below.

To sum up, the later 15th century, if I might borrow a phrase from
Scottish-English history, is a "debatable land" for Pythagorean and
meantone. Meantone keyboards, and vocal practices sometimes leaning in
either direction, are one kind of rough impression.

I would still say that meantone for the Ockeghem-Josquin-Isaac era
seems the "safest" choice: keyboard instruments would have likely been
tuned in this way. However, it is not the only possible alternative,
especially for flexible-pitch ensembles.

Similarly, while some kind of modified or flexible Pythagorean seems
to me indicated for the early Dufay era, the possibility of a more
"5-limit" kind of tuning shouldn't be excluded. English ensembles may
have leaned toward this kind of tuning.

If an early 15th-century piece has lots of traditional 14th-century
cadences with major thirds expanding to fifths as well as major sixths
to octaves, then I would be drawn to Pythagorean. Also, fauxbourdon
(parallel sixth sonorities punctuated here and there by such cadences)
can be very pleasant in Pythagorean, as Mark Lindley notes. Here the
many thirds and sixths are points of motion or "floating" rather than
solidity.

If it has a more "static" tertian sound, like some of Dunstable's
motets, where a more "solid" kind of sound for the sonorities with
thirds might be desired in at least one type of interpretation, then a
"5-limit" kind of approximation might not be out of place.

In other words, along with the general advice "Pythagorean to around
1450, and meantone from there on," I'd like to stress that there is
much room for choice.

In approaching some early 16th-century lute music, where meantone,
Pythagorean, or equal semitones are all possibilities, Mark Lindley
takes the approach of playing the piece and drawing conclusions from
that. Getting to know the music in the "historically likely" keyboard
tunings or similar systems, but with lots of experimentation when it
seems attractive, can help in making more informed decisions.

Most appreciatively,

Margo Schulter
mschulter@value.net