# 1588

Topic No. 15

Date: Fri, 20 Nov 1998 20:03:13 -0800 (PST)
From: "M. Schulter"
To: Tuning Digest
Subject: Re: Harmony, trinic and triadic -- a reply
Message-ID:

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