Thirteenth-Century Polyphony

3. Multi-voice combinations

Music for three or four voices is at once a logical extension and a glorious expansion of the two-voice elements we have just considered. It involves combining intervals to create new multi-voice sonorities, and also combining or superimposing two-voice progressions to build unifying cadences.

Around 1300, Johannes de Grocheo tells us that three voices are required for complete harmony, and in fact three-voice compositions become the norm from the age of Perotin on.

In this section, we survey some of the most important categories of stable and unstable combinations for three or four voices. Then in Section 4, we focus on directed cadential resolutions, while in Section 5 we consider obliquely resolving sonorities.

3.1. The unit of stability: the complete trine (8/5 or 8/4)

In theory and practice, the unit of complete harmony in the 13th century is a combination with three voices and intervals: the trine (trina harmoniae perfectio, or "threefold perfection of harmony," as Johannes de Grocheio calls it). This sonority requires three voices, the foundation-tone, fifth, and octave, and it includes three intervals: an outer octave, a lower fifth, and an upper fourth:

  |  4
  |  5

(Notation graphic)

Throughout the 13th century, and well beyond, this combination represents ideal euphony and stable plenitude; it is a point of rest and the goal of unstable sonorities.

Using a much later but rather familiar form of notation, we may describe this combination as 8/5 (8 + 5 + 4). The "8/5" tells us that the intervals above the lowest tone are an octave and fifth, while the "(8 + 5 + 4)" identifies all three intervals, including the upper fourth.

In theory and practice, these same three intervals may be arranged conversely so that the fourth is below and the fifth above:

  |  5
  |  4

(Notation graphic)

This combination - 8/4 (8 + 5 + 4) -- is also common, especially in the period around 1200, but is very rarely conclusive. Around 1325, Jacobus of Liege expresses the likely view of 13th-century musicians that this sonority, while concordant, is less pleasing than the ideal arrangement of fifth below and fourth above. He suggests a general rule that a larger or more blending interval (here the fifth) should best be placed below a smaller or less blending interval (here the fourth).

In this guide, I shall use the term "trine" for both the 8/5 and 8/4 combinations, but with the 8/5 trine normally presumed unless the context indicates otherwise.

3.2. Mildly unstable combinations (5/3, 9/5, 7/4, 5/4, 5/2)

While the trine represents complete and stable harmony, two families of mildly unstable combinations add sheer vertical color to the music as well as lending themselves to a variety of directed and decorative resolutions. Happily, we have an eloquent witness: Jacobus of Liege mentions the pleasing qualities of these combinations, and indeed the music speaks for itself.

3.2.1. The "split fifth" (5/3)

In the quinta fissa or "split fifth" of Jacobus, an outer fifth is "divided" by a third voice into two thirds: 5/M3 (5 + M3 + m3) or the "converse" arrangement of 5/m3 (5 + m3 + M3). Here the fifth is ideally blending, while the two thirds are unstable but relatively blending (being the mildest unstable intervals).

Incidentally, Jacobus prefers the form with the major third below and minor third above, but notes that the converse is also permissible, citing the opening of a 13th-century motet preserved in the Bamberg Codex.

  | d'           | e'
  |   m3         |    M3
5 | b          5 | c'
  |   M3         |    m3
  | g            | a

(Notation graphics: 1, 2)

The 5/3 combination often resolves by directed contrary motion (Section 4.1), and has a featured role in many internal and final cadences. Additionally, throughout the century it is often treated more freely, as we might expect for one of the mildest unstable combinations, and in fact the only one to consist exclusively of stable or relatively blending intervals.

3.2.2. Energetic quintal/quartal fusion (9/5, 7/4, 5/4, 5/2)

Jacobus also tells us about another favorite kind of mildly unstable combination common in practice from Perotin on. Two fifths, two fourths, or a fifth and a fourth combine with a relatively tense M2, m7, or M9 in a kind of energetic blend or fusion.

In his monumental Speculum musicae or "Mirror of Music," Jacobus enthusiastically recommends the three-voice sonority of a major ninth "split" into two fifths by a third voice, i.e. 9/5 (M9 + 5 + 5). He also observes that it is pleasant if a minor seventh is "split" into two fourths, i.e. 7/4 (m7 + 4 + 4).

   | g'                | f'
   |    5              |    4
M9 | c'             m7 | c'
   |    5              |    4
   | f                 | g

(Notation graphics: 1, 2)

Additionally, Jacob mentions another very common type of sonority, in which an outer fifth is "split" into a fourth below and a major second above, or the converse:

  | d'                 | d'
  |    M2              |    4
5 | c'               5 | a
  |    4               |    M2
  | g                  | g

(Notation graphics: 1, 2)

These four sonorities, like 5/3, represent the mildest unstable combinations possible: here two of the intervals are ideally blending fifths and/or fourths, while the third interval (M2, m7, or M9) is relatively tense but not sharply discordant.

The treatise of Jacobus suggests that to 13th-century ears, as to modern ones, the overall impression was one of an energetic variant on 8/5 or 8/4, with the unstable major second or ninth or minor seventh lending a sense of excitement and motion.

While these combinations sometimes participate in directed cadential progressions (Section 4.2), they often lend themselves to resolutions by oblique mention (Section 5.2) - or, like 5/3, to freer treatments.

3.3. Tenser cadential combinations

In addition to stable trines and mildly unstable combinations, composers of the 13th century deploy some strikingly tense cadential combinations resolving very effectively to a complete trine or a fifth (the prime interval of an 8/5 trine). These combinations fall into two major families.

3.3.1. Sixth combinations (6/3, 6/5, 6/5/3, 6/2, 6/4)

Combinations with an outer sixth characteristically resolve to a complete trine, with the sixth expanding to the octave of this trine. We shall focus on this group of cadences in Section 4.3.

For now, it may suffice to give some examples of the most common forms: 6/3, 6/5, 6/5/3, 6/2, and 6/4. These sonorities, although all on the tense side, may vary considerably in their degree of tension. We should recall that M6 is relatively tense, roughly on par with M2 and m7, while m6 is often regarded as sharply discordant (like m2, M7, tritone). Forms involving m6, m2, or tritonic fifths or fourths heighten the level of tension, and along with their somewhat gentler relatives are very effectively employed by Perotin and other composers. The following examples may give a sampling of these possibilities:

                                                | f'    |
                                                |    m2 |
     | c'                   | e'                | e'    | 4 |
     |   4                  |    M2         m6  |    M3 |   |
  m6 | g                 M6 | d'                | c'    |   | 5
     |   m3                 |    5              |    m3     |
     | e                    | g                 | a         |

    m6/m3                  M6/5                   m6/5/m3
(m6 + m3 + 4)         (M6 + 5 + M2)     (m6 + 5 + m2 + m3 + M3 + 4)

     | f'                   | e'
     |   d5                 |    M3
  m6 | b                 M6 | c'
     |   M2                 |    4
     | a                    | g

   m6/M2                  M6/4
(m6 + M2 + d5)        (M6 + 4 + M3)

(Notation graphics: 1, 2, 3, 4, 5)

The fact that these sixth sonorities include a large number of unstable intervals means from a cadential perspective that they invite some very dynamic resolutions. As we shall see, the 6/3, 6/5, and 6/5/3 combinations can expand to a complete trine in an especially efficient manner which makes them among the most favored of cadential sonorities.

Additionally, the 6/5 combination often resolves by oblique motion to a simple fifth (the highest voice descending a step while the others remain stationary), as is discussed in Section 5.3.

3.3.2. Seventh combinations (7/3, 7/5, 7/5/3)

In the other leading family of more tense cadential combinations, an outer minor seventh characteristically contracts by stepwise contrary motion to a fifth. In Section 4.4, we shall explore these standard progressions.

For now, let us briefly look at these sonorities themselves. As with the sixth combinations, they are all decidedly on the tense side, but more so in the case of forms including M7 or a tritonic fifth. Here we consider 7/3, 7/5, and 7/5/3, sampling some of these possibilities:

                                            | d'   |
                                            |   m3 |
     | e'               | a'                | b    | 5 |
     |   5              |   M3           m7 |   M3 |   |
  M7 | a             m7 | f'                | g    |   | 5
     |   M3             |   d5              |   m3     |
     | f                | b                 | e        |

    M7/M3             m7/d5                    m7/5/m3
(M7 + M3 + 5)     (m7 + d5 + M3)      (m7 + m3 + M3 + m3 + 5 + 5)

(Notation graphics: 1, 2, 3)

As in the case of our sixth combinations, a preponderance of unstable intervals means a wealth of opportunities for directed cadential action. Additionally, the 7/5 combination lends itself to an oblique resolution where the upper voice ascends stepwise (Section 5.4).

3.4. Summary of combinations

Here we have by no means covered the full range of combinations appearing in 13th-century music: Jacobus lists a catalogue of such sonorities with outer intervals ranging from a major third to a twelfth (about the practical limit, given the typical range of voices in this period). However, attuning ourselves to some of the most prevalent and important families of sonorities is a large step toward appreciating and understanding this music.

The stable trine (8/5, with its variant form of 8/4) is the unit of complete harmony, and the ultimate goal of unstable combinations.

The "split fifth" with its two thirds (5/3), and mildly unstable combinations featuring a preponderance of fifths or fourths (9/5, 7/4, 5/4, 5/2), are relatively blending. They add pleasant vertical color to the music, and lend themselves either to standard resolutions or to a freer treatment.

Other, tenser, combinations strongly invite directed cadential resolutions where an outer sixth expands to the octave of a complete trine (6/3, 6/5, 6/5/3, 6/2, and sometimes 6/4), or an outer seventh contracts to a fifth (7/3, 7/5, 7/5/3).

Having considered the harmonic vocabulary of 13th-century music, we now turn to its dynamic grammar: the ways in which an unstable sonority may effectively resolve to a stable one.

To Section 4 - Directed cadences for three or four voices.

To Table of Contents.

Margo Schulter