Re: Solage: a medievalist approach

Hello, everyone.

Please forgive me for taking so long to write the second portion of my
article on intonation of the Landini cadence, but seeing the
discussion about Solage and about chromaticism in various
temperaments, I thought that maybe responding to that discussion first
might be most helpful. Here I focus on Solage, this article turning
out to be long enough on that subject alone.

For readers who might find this article (about 300 lines) more than
long enough, I've stated a summary of my views in the first section.

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1. Divergent views of Solage: a quick summary
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First of all, as a medievalist, I'd like to emphasize that it is a
positive and encouraging step when people such as Walter Mathieu
include medieval and Renaissance music in general discussions of
tuning. In such discussions, each author may take a different
viewpoint, and there's no requirement that such viewpoints must always
agree. Different viewpoints, analyses, and performance interpretations
give the reader or listener a choice, and invite further exploration
and dialogue.

In fact, Mathieu's book _The Harmonic Experience_ has given me the
idea of maybe writing a kind of intonational guide to medieval and
Renaissance/Manneristic music, with an emphasis on Pythagorean and
meantone tunings, including pieces using or (arguably) inviting more
than 12 notes per octave.

Mathieu, in contrast, seems to me to be focusing mainly on two
deliberately contrasting approaches to harmony and tuning: 5-limit or
higher just intonation (JI), with a lot of focus on the music of
India, and 12-tone equal temperament (12-tet).

While he does discuss the Pythagorean or 3-limit intervals, and also
medieval music, the medieval European practice and theory of
verticality seems to me to be somewhat peripheral to his main focus.
Inevitably, one takes a viewpoint, and my viewpoint on a piece such as
Solage's _Fumeux fume_ may be quite different from Mathieu's.

To resolve rather than prolong the suspense for people who might want
a quick statement of my own views on _Fumeux fumee_, I regard it as a
masterpiece of late Gothic 3-limit or trinic verticality, and also a
piece inviting extended Pythagorean tuning with 15 notes per octave
(Gb-G#). Such a work at once exemplifies and transcends the theory of
the 14th and early 15th centuries, just as Beethoven or Wagner at once
fulfills and transcends 18th-19th century theory.

Solage's piece has been familiar to me as a listener for about 30
years, and I would say that if I wanted to pick a piece of Continental
European music from around 1400 suggesting tertian or "5-limit"
leanings, _Fumeux fume_ would not be my obvious choice. I might
nominate a piece such as Vaillant's _Tres doulz amis_, which has a
smoothly flowing texture featuring lots of sonorities with thirds or
sixths.

In fact, Valliant's piece occurred to me when I read Oliver
Ellsworth's interpretation of a treatise in the Berkeley Manuscript or
Paris Anonymous (part of it dated 1375) as proposing 19-tet; the
treatise on singing semitones describes a division of the tone into
three parts, with a usual diatonic semitone of 2/3-tone and a narrow
cadential semitone of 1/3-tone. If one were going to try out a 19-tet
interpretation of the treatise on a late medieval piece, _Tres doulz
amis_ seemed to me a likely candidate.

In what follows, I'd like first to focus on Mathieu's approach to
Solage's piece, and on the more general issue of the relationship
between great musical compositions and music theory. Then I'll present
my own approach to this piece, which owes a great deal both to
medieval theorists and to Willi Apel (a favorite of my college years).
More recently, other scholars such as Richard Crocker and Sarah Fuller
have published analyses of 14th-century polyphony to which I am also
much indebted.

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2. Mathieu and Hindemith: 5-limit views of 3-limit music
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If we read Mathieu's discussion of this piece as a presentation on how
Solage's adventurous music _can_ be tuned in a 5-limit lattice system
and analyzed in 18th-century harmonic terms, then I have no problem
with such a "xenohistorical" viewpoint. It may be somewhat like a
presentation on how Beethoven or Chopin _can_ be tuned and analyzed in
7-limit JI terms.

However, I would urge that such an approach is no substitute for a
musical and intonational analysis premised on Solage's 14th-century
setting or Beethoven's 19th-century setting. This means approaching
Solage's music as a complex and sophisticated texture based on 3-limit
sonorities and progressions, and likewise Beethoven's in terms of
5-limit sonorities and progressions.

Please let me emphasize that Mathieu is by no means alone in analyzing
14th-century music as if it were based on a tertian or 5-limit system
(with thirds and sixths regarded as stable concords) rather than a
trinic or 3-limit system.

In trinic music, the "complete harmony" or richest possible stable
sonority is the trine, consisting of outer octave, lower fifth, and
upper fourth. Sonorities involving thirds and sixths -- treated in
theory and practice as "imperfect concords" which are relatively
blending but inherently unstable -- often resolve by directed
progressions to stable trines, and also frequently are treated in a
more free or "coloristic" manner.

Not only Mathieu in approaching Solage, but Hindemith in an analysis
of Solage's great predecessor Guillaume de Machaut (c. 1300-1377)
presented in the 20th-century classic _The Craft of Musical
Composition_, treats 14th-century trinic music as if it were tertian.
Specifically, this means describing complete stable trines in Machaut
or Solage as if they were incomplete versions of 5-limit triads. Thus
Hindemith, for example, concludes that Machaut's piece involves only
small fluctuations in tension between what might be called "triads in
root position" and "triads in first inversion."

From a 14th-century perspective, however, such an approach neutralizes
or disregards a critical contrast in this music: the contrast between
stable trines and mildly unstable sonorities with thirds and/or sixths
often inviting directed cadential resolutions to trines. This is
rather like regarding complete triads in Beethoven or Chopin as
incomplete seventh chords -- again neutralizing or disregarding the
element of cadential tension and action.

While such "xenohistorical" analyses can be entertaining and
intriguing, I would emphasize that they are not necessarily richer or
more satisfying that more historically grounded approaches. In fact,
theorists such as Johannes Boen writing in 1357, around the middle of
Machaut's career, and Prosdocimus of Beldemandis and Ugolino of
Orvieto in the early 15th century, provide many insights on
verticality and accidentalism quite relevant to Solage's piece.

For example, Johannes Boen discusses the role of parallel thirds and
sixths as the "forerunners and handmaidens" of stable concords such as
fifths and octaves, as in the passage which follows (C4 is middle C,
and higher numbers show higher octaves). While the F# inflection in
the middle voice is not expressly indicated, it is called for by the
14th-century norm that a third expanding by directed contrary motion
to a fifth should be major:

D4 C4 B3 C4
A3 G3 F[#]3 G3
F3 E3 D3 C3

Here we have a series of unstable sonorities with an outer sixth, a
lower third, and a fourth between the two upper voices, leading to
a cadential resolution in which a lower major third expands to a fifth
while an outer major sixth expands to an octave, arriving at the
satisfying goal of a complete trine (M6-8 + M3-5).

From this viewpoint, Solage's _Fumeux fume_ and similar pieces around
the end of the 14th century engage us both by sometimes fulfilling
this usual cadential expectation as described by Boen, and sometimes
by artfully delaying or diverting that expectation. In the late trinic
or 3-limit music of Solage, as in the late triadic music of Chopin or
Max Reger (the last comparison being Willi Apel's), expectations
delayed or diverted may be as significant as those promptly fulfilled.

A Pythagorean or 3-limit JI tuning brings out both the _relatively_
concordant quality of these thirds and sixths and what Mark Lindley
aptly terms their "lithesome tension." This tension underscores the
expected cadential progressions which actually occur while lending a
certain edginess to passages where one unstable sonority is followed
by another in a chain of prolonged or diverted resolutions.

Indeed, this "clouding" of usual directed trinic progressions might
add a level of musical meaning to the "smoky" text -- whether we take
this text to refer to hemp smoking (a theory popular during the
1960's) or to the "whims and humors" of the members of a poetic
society (in English one might speak of "vapours" or possibly "airs").

Of course, to say that the music Machaut or Solage fits and expands
upon the framework of 14th-century theory does not mean that such
great music can be _reduced_ to such a theory, any more than the music
of Bach or Mozart can be reduced to the propositions of 18th-century
counterpoint or harmony.

However, I would urge that 14th-century theory can provide a basic
framework for an understanding of Machaut and Solage at least as rich
as that provided by substituting the framework of some other era. As
with many composers from various eras, of course, we are concerned not
only with how late Gothic composers realize such a framework, but how
they elaborate upon it and sometimes stretch it to suit their
purposes.

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3. Solage: late 3-limit music and extended Pythagorean tuning
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From a medievalist viewpoint, Solage's _Fumeux fume_ is a splendid
example of extended Pythagorean intonation, with the qualification of
course that voices and other non-fixed-pitched instruments will
necessarily approximate rather than literally implement any fixed
tuning scheme.

In Pythagorean intonation, all regular whole-tones are a pure 9:8, and
diatonic semitones are 256:243, a compact 90 cents or so. Taking this
approach, we negotiate _Fumeux fume_ from an individual performer's
melodic viewpoint as a network of hexachords, each consisting of the
pattern T T S T T (T = 9:8, S = 256:243). From a vertical point of
view, we have a series of stable and unstable 3-limit sonorities, with
directed resolutions from unstable sonorities to stable trines
(complete or incomplete) serving as unifying cadential events.

The piece opens on a complete G trine (G2-D3-G3), and closes on an F
trine (F2-C3-F3), so that we might say that the overall center or
modal final is F. Interestingly, while G is the step above the final
(the "superfinal" if we wish to use this term), the first section of
the rondeaux concludes on Eb -- the trine a whole-tone _below_ the
final (Eb2-Bb2-Eb3).

From a vertical point of view, the steps G-F and F-Eb are related by a
kind of "cadential chain" of progressions involving M6-8 and/or M3-5
or M10-12 resolutions. Thus, noting the strikingly low range (for this
period) of Solage's piece, we find these directed progressions:

B3 C4 D3 F3 D3 Eb3
E3 F3 B2 C3 A2 Bb2
G2 F2 or G2 F2 F2 Eb2

(M10-12 + M6-8) (M3-5) (M6-8 + M3-5)

These are standard 14th-century cadences following the usual vertical
logic, although Eb is a rather uncommon cadential goal (and one
fitting the adventurous accidentalism of this piece). In contrast, the
emphasis on the superfinal G is common in F-centered pieces of the
14th century, although a more typical 14th-century scheme might have
some internal cadences on A involving descending rather than ascending
semitonal motion, e.g. in this low register

G3 A3
D3 E3
Bb2 A2

Returning to the structure of Solage's piece, with its sectional
cadence on an Eb trine, we find also at the beginning of the rondeau's
second section an artful passage moving from Eb through a prolonged
and ornamented unstable sonority on G to the momentary goal of F:

1 2 | 1 2 | 1 & 2 & | 1 ...
Bb3 C4 B3 B3 C4 B3 A3 C4
Eb2 G2 D3 E3 F3 E3 F3
Eb3 D3 G2 F2

Note the pleasant juxtaposition of Bb and B-natural, characteristic of
the hexachord system generally as well as Solage's extension of it
here to Gb-G#. Here the major tenth G2-B3 in the second measure
suggests a progression to a stable sonority including an M10-12
resolution -- and such a directed cadence does occur a measure later,
with some ornamentation in both upper voices. In the course of the
decoration, the middle voice touches on E3, the major sixth above G2,
contributing an M6-8 resolution to reinforce this M10-12 resolution.

In addition to these cadential progressions which _do_ occur, we also
encounter some invited or expected cadential resolutions which do not
occur as we might anticipate. In the second section, for example, we
have:

1 2 | 1 & 2 & |

C#3 D3 C3 B[b]2
F#3 G3 F3
A2 Bb2 D2

Here the first sonority suggests a standard cadential resolution to a
complete trine on G:

F#3 G3
C#3 D3
A2 G2

(M6-8 + M3-5)

While the upper two voices move as might be expected, the lower voice
moves not to G2 but to Bb2, forming another unstable sonority with an
outer major sixth.

Such "diverted" progressions give the texture a certain restless
quality which might be also be described as "floating" or "smoky" --
somewhat analogous to the floating seventh sonorities of Romantic
triadicism in the 19th century.

While an extended Pythagorean framework for this piece may bring out
both the directed cadences when they do occur, and the "restlessness"
of the many diversions and digressions, this is not to say that
performers cannot shade certain intervals in one direction or
another. For example, some major thirds and sixths might lean now and
then toward Pythagorean diminished fourths (~5:4) or sevenths (~5:3).

A Pythagorean keyboard instrument with 15 or more notes per octave
would not only permit an apt performance of this piece in a
fixed-pitch framework, but would also offer the choice of using some
of these variant thirds and sixths in sonorities involving written
sharps. As one moves into the early 15th century, and approaches the
epoch of Guillaume le Grant (another practitioner of daring
accidentalism) and the young Dufay, such a contrast between regular
and "smoothed" Pythagorean thirds and sixths might be especially
likely.

This has by no means been a full analysis of _Fumeux fume_, only a
sketch of one medievalist approach to the piece.

Most respectfully,

Margo Schulter
mschulter@value.net