Thirteenth-Century Polyphony

4. Directed cadences for three or four voices

In two-voice writing, as we have seen (Section 2.2), the most compelling cadential events involve progressing from an unstable to a stable interval by stepwise or near-stepwise contrary motion. Music for three or four voices permits a logical and glorious expansion of this pattern. In the most favored cadences, an unstable combination artfully resolves by directed contrary motion to a richly stable sonority, ideally a complete 8/5 trine.

Such cadences, whether involving mildly unstable combinations or dramatically discordant ones, tend to follow patterns suggesting two main "guidelines":

(1) The unstable combination as a whole should resolve to a richly stable sonority, ideally an 8/5 trine, and next most preferably a fifth (the prime interval of 8/5), with 8/4 or a fourth as alternate goals.

(2) Each unstable interval should ideally resolve by stepwise contrary motion, and next most preferably by near-stepwise contrary motion (one voice moving by step, and the other by a third).

An actual example may make these points clearer:

    a -g

(m6-8 + m3-5)

(Notation graphic)

In this progression, very common throughout the 13th century, a rather tense m6/m3 combination (m6 + m3 + 4) expands to a complete 8/5 trine (8 + 5 + 4). Both unstable intervals resolve by stepwise contrary motion (m6-8, m3-5), and these two resolutions reinforce each other, together making possible our arrival at a complete trine. All three voices move by step, each contributing to the total effect of directed cadential action.

Our notation below the example indicates that an m6/m3 sonority resolves to 8/5, and then identifies the mutually reinforcing two-voice resolutions (m6-8 + m3-5).

Note that while either m6-8 or m3-5 serves as an effective cadence in two-part writing, music for three or voices opens the new possibility of combining both progressions at once, and introduces the vital new element of the complete trine as an ideal cadential goal.

More generally, we will often find it useful to analyze a multi-voice cadence as a mutually reinforcing union of directed two-voice resolutions. Given the variety of such resolutions (Section 2.2), we might expect to find a diverse range of multi-voice cadences; and so we do.

Before exploring this universe of directed vertical progressions, we may do well to consider the broad meaning of "cadential" action in our present context. While the progressions we are about to discuss are indeed favored at final closes or internal cadences in the narrower sense, they may occur more generally at almost any point where there is a "change of harmony" - that is, where the lowest voice or foundation-tone (fundamentum) changes.

In this broad sense, cadential events occur constantly in most 13th-century pieces, often making possible smoother melody as well as satisfying vertical contrasts (Section 6).

4.1. The versatile "split fifth" (5/3)

The 5/3 combination or "split fifth" (Section 3.1) is much favored in 13th-century music not only because it is one of the mildest unstable combination, but also because it invites some very efficient resolutions. Three such resolutions are favored throughout the century:

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

    5/M3-5            5/M3-5          5/M3-8/5
(M3-1 + m3-5)     (m3-1 + M3-5)        (M3-5)

(Notation graphics: 1, 2, 3)

In the first two progressions, the two unstable thirds both resolve by stepwise contrary motion: one contracts to a unison while the other expands to a fifth, as all three voices move stepwise. The outer voices of the "split fifth" move together, either ascending or descending by a step, while the middle voice which "splits" the fifth into two thirds moves in the opposite direction, neatly resolving both unstable intervals.

In the third progression, the lower third of 5/3 expands to the fifth of a complete 8/5 trine (here M3-5), while the outer fifth of 5/3 expands to the octave of the trine.

Each of these efficient resolutions involves a slight compromise of our ideal cadential guidelines. In the first two cases, each unstable interval resolves by stepwise contrary motion (3-1 + 3-5), but we arrive at a simple fifth rather than a complete trine.

In the third case, only the lower third resolves by stepwise contrary motion (3-5), the upper third resolving by similar motion to the upper fourth of the trine. Thus we gain greater sonority at a slight sacrifice of cadential efficiency.

Both minor compromises are quite acceptable, and indeed these progressions are among the most popular cadences from Perotin to Petrus de Cruce. While our last examples happen to involve 5/M3, Jacobus of Liege's favored form with major third below and minor third above, the same progressions also occur with his "converse" form 5/m3, and also sometimes with the discordant tritonic variant of d5/m3:

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

   5/m3-5            5/m3-5         5/m3-8/5         d5/m3-5
(m3-1 + M3-5)    (M3-1 + m3-5)       (m3-5)       (m3-1 + m3-5)

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

In four-voice writing, we encounter an interesting variation on this pattern: the 8/5/3 combination (8 + 5 + 4 + 3 + 3 + 6) resolving by stepwise motion of all voices to a complete trine. Note that this sonority may be somewhat more tense than 5/3 because of the major or minor sixth between two of the upper voices. In any case, all three unstable intervals resolve by stepwise contrary motion, and we arrive at a complete trine, so the popularity of this type of cadence in four-voice music is not surprising:

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

     8/5/M3-8/5               8/5/3-8/5
(M3-1 + m3-5 + m6-8)     (m3-1 + M3-5 + m6-4)

(Notation graphics: 1, 2)

4.2. Mildly unstable quintal/quartal sonorities (9/5, 7/4, 5/4, 5/2)

Like 5/3, these combinations serve as a great resource for harmonic color; however, they play a somewhat less prominent role as directed cadential sonorities. Let us first quickly consider the possibilities of 9/5 and 7/4, and then explore the more versatile role of 5/4 and 5/2, which do rather frequently resolve by directed contrary motion in the course of a piece, now and then providing material for final cadences.

The 9/5 combination (M9 + 5 + 5) seems to lend itself more to oblique than to directed resolutions, although I am aware of one curious resolution where the major ninth contracts to a fifth, each voice leaping by a third:

    a -c'
    d -f


(Notation graphic)

In another resolution by contrary motion, the major ninth expands to a twelfth while the lower fifth expands to an octave, arriving at a 12/8 sonority. This progression is probably rather rare because of the limited range of most 13th-century polyphony, and it is my impression that it occurs mostly in pieces from around 1300:

  f -d


(Notation graphic)

The 7/4 combination permits a resolution where the outer minor seventh contracts to a fifth, and this progression does occur now and then, although it is more characteristic of the tenser seventh combinations (7/5, 7/3, 7/5/3) to be examined in Section 4.4:

   g -a


(Notation graphic)

As we shall see (Section 5.2), 9/5 and 7/4 both invite very effective resolutions by oblique motion, as well as freer treatments.

In the case of 5/4 and 5/2, however, resolutions by directed contrary motion as well as oblique motion play a significant role in the cadential lexicon of the 13th century. In such directed progressions, the unstable major second of 5/4 or 5/2 expands to a stable fourth or fifth (M2-4 or M2-5):

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

 5/4-8/5     5/4-5    5/2-8/5     5/2-5
  (M2-4)    (M2-5)    (M2-5)      (M2-5)

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

As with 5/3 (Section 4.1), the outer fifth may expand to the octave of a complete trine; or the two outer voices may ascend or descend together by step, with the progression arriving at a simple fifth rather than a full trine.

These directed resolutions of 5/4 and 5/2 are quite common, and occasionally serve as final cadences.

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

Moving to tenser combinations, we now consider two groups of cadential sonorities par excellence: sixth and seventh combinations.

As already noted (Section 3.3), cadential sixth combinations characteristically resolve to a complete 8/5 trine, with the outer sixth expanding to the octave of the trine (M6-8 or m6-8).

This process is especially efficient in the case of the 6/3 and 6/5 combinations, the 6-8 resolution combining nicely with a 3-5 or 2-4 resolution:

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

   M6/M3-8/5                  M6/5-8/5
 (M6-8 + M3-5)             (M6-8 + M2-4)

(Notation graphics: 1, 2)

Note that in each case both unstable intervals resolve by stepwise contrary motion, and the progression arrives at an ideally sonorous 8/5 trine, satisfying our criteria for a superb cadence.

As it happens, these two examples feature the comparatively milder M6/M3 (M6 + M3 + 4) and M6/5 (M6 + 5 + M2) combinations, but more discordant permutations involving m6, m2, and tritonic fourths or fifths are also common.

In four-part writing, another powerful cadential sonority becomes available: 6/5/3 (6 + 3 + 3 + 2 + 5 + 4). For an example, let us take one of the more discordant forms, m6/5/m3, including m6 and m2 as well as two more mildly unstable thirds:

           a -g

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

(Notation graphic)

This memorable cadence occurs at the close of Vetus abit littera, an anonymous piece worthy of Perotin. Here we have no fewer than four unstable intervals, each of which felicitously resolves by stepwise contrary motion, bringing us to an 8/5 trine.

While 6/3, 6/5, and 6/5/3 are ideally efficient cadential sonorities, the 6/2 combination is somewhat less ideal, since it requires a leap of a third in the middle part in order to resolve to 8/5:

     a -c'
     g -f

(M6-8 + M2-5)

(Notation graphic)

The 6/4 combination seems yet less efficient as a cadential sonority resolving to 8/5 by way of a 6-8 progression, since the middle voice remains stationary, and the upper third thus resolves by oblique rather than contrary motion - a pattern not especially favored in this period for directed cadences:

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

 m6/4-8/5                        M6/4-8/5
  (m6-8)                          (M6-8)

(Notation graphics: 1, 2)

Interestingly, a progression in which both unstable intervals resolve by stepwise contrary motion is possible, but to the usually less conclusive 8/4 rather than 8/5:

     a -g

 (m6-8 + m3-5)

(Notation graphic)

In practice, the 6/3 and 6/5 forms are most important in three-part music, while the 6/5/3 form is very popular in four-part pieces. Final closes, internal cadences, and transient progressions from one sonority to the next provide frequent occasions for their use in directed vertical action.

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

Like the sixth combinations we have just considered, seventh combinations such as 7/3, 7/5, and 7/5/3 also invite compelling directed resolutions in which the outer seventh contracts to a fifth (m7-5 or M7-5). In 7/3 (7 + 3 + 5) and 7/5 (7 + 5 + 3), the milder unstable third likewise contracts to a unison in the most typical pattern:

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

     M7/M3-5                  m7/5-5
 (M7-5 + M3-1)             (m7-5 + m3-1)

(Notation graphics: 1, 2)

These cadences nicely meet our guideline that all unstable intervals should resolve by stepwise contrary motion (7-5 + 3-1), but represent a slight compromise in the department of full sonority: we arrive at a simple fifth rather than a full trine.

In four-part pieces, including Perotin's, we also meet an especially impressive seventh combination: 7/5/3 (7 + 5 + 5 + 3 + 3 + 3). Here is a milder form with m7 and without any tritonic fifths:

          b -c'
          g -f
          e -f

(m7-5 + m3-1 + M3-5 + m3-1)

(Notation graphic)

As in the typical resolution of 6/5/3 (Section 4.3), all four unstable intervals progress to stable ones by stepwise contrary motion. In this case, as with the other seventh combinations, we arrive at a simple fifth rather than a complete trine - a small and acceptable compromise of sonority.

Like the sixth combinations, these seventh combinations play a vital cadential role throughout the century, often with superb effect. A fuller discussion would cover related forms such as 8/7, 8/7/3, and 8/7/5, and also idioms which might be considered part-writing variations on the basic resolutions we have just surveyed.

To Section 5 - Multi-voice Obliquely Resolving Sonorities.

To Table of Contents.

Margo Schulter