Harmony, trinic and triadic -- a reply

Hello, there.

Recently there has been some discussion about the emergence of regular
three-voice and four-voice compositions in Western European music
around the end of the 12th century. Indeed the year 1998 may be an
ideal time for this dialogue, since one of the earliest known
documents referring to the use of such compositions (possibly
Perotin's four-part organa _Viderunt omnes_ and _Sederunt principes_)
dates to 1198 or so.

Please let me begin by dissociating this specific question from the
matter of cross-cultural responses to the European-style triadic
harmony of a later period, fraught with very some very sensitive
cultural issues. Interestingly, for example, the magazine _Early
Music_ some years ago had an article on a Japanese delegation which
visited Venice somewhere around the 1580's, an era of early triadic
music in Europe, and may have found this polyphony an "interesting
toy" or the like.

Anyway, two people have offered comments which invite further
discussion. First Bill Alves:

> Medieval polyphony was something perfectly suited to the aesthetics
> of gothic church music, so it is little wonder that contemporary
> writers praised it, just as later writers praised the symphony
> orchestra, Bach's counterpoint, Mozart's melodies, or Beethoven's
> emotionalism. Yet I would hardly conclude that other cultures if
> exposed to Bach, Mozart, or Beethoven would gladly don their
> styles. One might also point out that the Medieval music refered to
> in those accounts certainly did not have "full chordal harmony" as
> we think of it.

One complication here is that some of "us" may have different
conceptions of just what "full chordal harmony" implies. To me, it
implies a texture which, when judged by the standards of a given
period, is saturated with the most complex stable sonorities
possible. It also implies to me at least three voices, as opposed to a
two-voice texture with simple intervals but not "chords."

To many people, "chords" are implicitly built in thirds, or must
consist of three distinct "pitch classes," so that I personally prefer
to use the terms "combination" or "multi-voice sonority" in a medieval
context in order to minimize confusion.

Whatever terms we prefer to use, Gothic harmony from Perotin onward is
based on a unit of stable saturation which I call the _trine_ after
Johannes de Grocheio (c. 1300), who refers to a _trina harmoniae
perfectio_ or "threefold perfection of harmony" based on a complete
sonority of three voices and intervals: outer octave, lower fifth, and
upper fourth (e.g. d-a-d'). Note that the octave counts as a "real"
interval, and indeed a full trine might be described as a harmonic
division of the octave. In term of string-ratios, we have 12:8:6, with
the differences between adjacent terms -- (12-8):(8-6) or 4:2 or 2:1
-- having the same ratio as that between the outer terms, 12:6 or 2:1.

For people on the list with geometric inclinations, it's interesting
that a cube has 12 edges, 8 corners, and 6 faces.

Gothic music of the era 1200-1420 or so is thus based on a texture in
which complete trines ideally alternate with unstable sonorities
considered in a given epoch and style to be apt and pleasing, whether
as cadential sonorities used in directed resolutions, or as
"coloristic" sonorities (e.g. g-b-d' or g-c'-d' in a 13th-century
style). To me, this is "full trinic harmony," whether or not one wants
to introduce the adjective "chordal."

Starting around 1420, there is a transitional period of about a
century from the early Dufay through Josquin, say, when the sonorous
ideal shifts from an alternation of stable trines and unstable
sonorities to a pervasive tertian sound. By around the time of
Jannequin and the early 16th-century Italian madrigalists, say the
1520's, we might speak of a "triadic" texture where sonorities with a
fifth divided into two thirds, or a sixth divided into a lower third
and upper fourth (e.g. c-e-g, c-e-a) are predominant. By the middle
of the century, theorists such as Vicentino (1555), Zarlino (1558),
and Tomas de Santa Maria (1565) are emphasizing this standard of full
sonority.

Note that either trinic or triadic sonority, although "full" in its
own terms, might sound quite "bare" to an enthusiast of modern jazz
or 11-limit just intonation, for example.

Of course, on this list, it seems obligatory to add that Pythagorean
intonation is nicely suited to the trinic harmony of the Gothic, while
5-limit just intonation or meantone nicely fits the triadic style of
the Renaissance.

In another post, Daniel Wolf remarked:

> Other musicians, however, have come round to a triadic basis after a
> deep consideration of the consequences suggested by particular
> musical materials. I am quite fond, for example, of polyphonic music
> in pythagorean intonation (c.f. Margo Schulter's postings to this
> list) or of Javanese slendro (which is pythagorean in structure but
> not intonation). But neither of these systems lends itself to
> completely independent (i.e. non-parallel) part writing in more than
> two voices, and the introduction of an additional consonant tone
> within the space of the fifth is terribly convenient in this
> regard.

Here I might comment that "completely independent" part-writing may be
defined differently in different eras and styles. Thus parallel fifths
and fourths are not seen as compromising the essential independence of
the parts in 13th-century trinic harmony as long as contrary motion
remains the general rule, and the same may be said for parallel thirds
and sixths in a 16th-century triadic context.

If three or more voices all move at the same time, it's inevitable
that at least two of them must move in similar or parallel
motion. However, one index of maximum "harmonic efficiency" in a
directed cadential progression might be the number of unstable
intervals that get resolved by conjunct or near-conjunct contrary
motion. Thus consider these 13th-century progressions:

f'-g' d'-c'
e'-d' b -c'
c'-d' g -f
a -g e -f

(m6-8 + m3-5 + M3-1 + m2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Each of these sonorities includes among its six intervals _four_
unstable ones each resolving by conjunct contrary motion; there are
also some parallel fifths and fourths which enrich the texture, just
as parallel thirds and sixths are common in later triadic cadences.

Also, while of course it is true that we cannot create any _stable_
sonority by adding a third voice "within the space of a fifth," we can
create some _relatively_ concordant sonorities by precisely this
method, as can be seen in Perotin and various 13th-century
repertories. Following the concepts of Jacobus of Liege (c. 1325), I
propose a notation in which a sonority is shown as an outer interval
"split" by a third voice into lower and upper adjacent intervals,
outer|lower-upper:

d' d' d'
b a c'
g g g

5|3-3 5|M2-4 5|4-M2

Adding a third voice at a major or minor third, a major second, or a
fourth produces a mildly unstable sonority in which all three
intervals are to some extent "compatible," although the resulting
thirds or major seconds introduce tension requiring further music.

In sum, I would say that both Bill Alves and Daniel Wolf have shown how
concepts such as "full chordal harmony" or "independent part-writing" can
change in meaning depending on the style. While this issue may be somewhat
peripheral to questions of tuning systems proper, I would consider it not
irrelevant. Also, with Daniel, I would agree that characteristic meantone
tunings (e.g. 1/3-1/6 comma) do invite a triadic kind of sonority, unless
one is doing something a bit offbeat like using augmented sixths to
approximate 7:4's, as Dave Hill has done very effectively on piano -- or
diminished fourths to approximate 9:7's, etc.

Most appreciatively,

Margo Schulter
mschulter@value.net