13 century polyphony

8/3/2019 13 Century Polyphony

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Thirteenth-CenturyPolyphony

2. Two-voice intervals andprogressions

Rather as 3-D graphics can be seen as a logical extensionof 2-D drawing techniques, so music in any period forthree or more voices reflects certain artistic assumptions

about the nature of elementary two-voice intervals. The13th century is no exception.

2.1. A subtle continuum of tension

Two-voice intervals range along a subtle scale of tensionfrom the most purely blending to the most stronglydiscordant. While such distinctions are often relative, the

absolute distinction between stable and unstableintervals is vital:

Stable:

Purely blending (1, 8)

Optimally blending (5, 4)

Unstable:

Relatively blending (M3, m3)Relatively tense (M2, m7, M6)

Strongly discordant (m2, M7, tritone - and often m6)

Stable intervals, and especially the fifth, represent pointsof ideal concord and rest, often serving as goals ofdirected cadences. The more tense an unstable interval,the more urgently it suggests resolution to a stable


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concord - an expectation often but not alwaysimmediately fulfilled.

In two-voice music, the fifth is both optimally euphonious

and conclusive, while the fourth is relatively stable butsomewhat less conclusive. However, two-part piecesoften begin or end with a purely blending (although lesssonorous) unison or octave.

2.2. Directed cadences

The most powerful two-voice progressions in this period

involve moving from an unstable interval to a stable oneby stepwise or near-stepwise contrary motion. Either bothvoices move by step, or one moves by step and the otherby a third. Here are examples illustrating some of themost important resolutions:

m2-4 m2-5 M2-4 M2-5 m3-1 m3-5 M3-1 M3-5

--- ---- ---- ---- ---- ---- ---- ----

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

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

m6-8 m6-4 M6-8 M6-4 m7-5 M7-5

--- ---- ---- ---- ---- ----

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

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

(Notation graphics: m2-4, m2-5, M2-4, M2-5, m3-1, m3-5, M3-1, M3-5,m6-8, m6-4, M6-8, M6-4, m7-5, M7-5)

In two-voice music, these progressions give a sense of

directed and unifying motion. In music for three and fourvoices, they serve as the elementary building blocks ofmany powerful and beautiful cadences, as we shall see.

2.3. Obliquely resolving sonorities
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Additionally, unstable intervals have standard resolutionswhere one voice remains stationary while the othermoves by step or by a third (sometimes with the middletone filled in). Here are some of the most common cases:

m2-1 M2-1 m3-1 m3-5 M3-1 M3-5

--- ---- ------- --------- ------- --------

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

e - g - d - a - f - g -

m6-5 M6-5 m7-8 m7-5 M7-8 M7-5

--- ---- ---- --------- ---- --------

f'-e' d'-c' c'-d' f'-(e')-d' e'-f' e'-(d')-c'

a - f - d - g - f - f -

(Notation graphics: m2-1, M2-1, m3-1, m3-5, M3-1, M3-5, m6-5, M6-5,m7-8, m7-5, M7-8, M7-5)

2.4. Summary

The music of the 13th century boldly exploits the entirespectrum of intervals from the most blending to the mostaggressively discordant. While the distinction betweenstable and unstable intervals is absolute, there are

various degrees of tension among the unstable intervals.Thus M3 and m3 are relatively blending but have sometension, while M2 and m7 are rather tense but have some"compatibility."

Given this approach of flexibility and bold contrast, weshould not be surprised to find an amazing variety andrichness of multi-voice combinations and cadences.

Thirteenth-Century
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Polyphony

3. Multi-voice combinationsMusic for three or four voices is at once a logicalextension and a glorious expansion of the two-voiceelements we have just considered. It involves combiningintervals to create new multi-voice sonorities, and alsocombining or superimposing two-voice progressions tobuild unifying cadences.

Around 1300, Johannes de Grocheo tells us that threevoices are required for complete harmony, and in factthree-voice compositions become the norm from the ageof Perotin on.

In this section, we survey some of the most importantcategories of stable and unstable combinations for threeor four voices. Then in Section 4, we focus on directed

cadential resolutions, while in Section 5 we considerobliquely 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 inthe 13th century is a combination with three voices and

intervals: the trine (trina harmoniae perfectio, or"threefold perfection of harmony," as Johannes deGrocheio calls it). This sonority requires three voices, thefoundation-tone, fifth, and octave, and it includes threeintervals: an outer octave, a lower fifth, and an upperfourth:
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|g'

| 4

8|d'

| 5

|g

(Notation graphic)

Throughout the 13th century, and well beyond, thiscombination represents ideal euphony and stableplenitude; it is a point of rest and the goal of unstablesonorities.

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 arean octave and fifth, while the "(8 + 5 + 4)" identifies allthree intervals, including the upper fourth.

In theory and practice, these same three intervals may bearranged conversely so that the fourth is below and thefifth above:

|g'| 5

8|c'

| 4

|g

(Notation graphic)

This combination - 8/4 (8 + 5 + 4) -- is also common,especially in the period around 1200, but is very rarelyconclusive. Around 1325, Jacobus of Liege expresses the

likely view of 13th-century musicians that this sonority,while concordant, is less pleasing than the idealarrangement of fifth below and fourth above. He suggestsa general rule that a larger or more blending interval(here the fifth) should best be placed below a smaller orless blending interval (here the fourth).
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In this guide, I shall use the term "trine" for both the 8/5and 8/4 combinations, but with the 8/5 trine normallypresumed 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 sheervertical color to the music as well as lending themselvesto 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 fifthis "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 thirdsare unstable but relatively blending (being the mildestunstable intervals).

Incidentally, Jacobus prefers the form with the major thirdbelow and minor third above, but notes that the converseis also permissible, citing the opening of a 13th-centurymotet preserved in the Bamberg Codex.

| d' | e'

| m3 | M3

5 | b 5 | c'

| M3 | m3

| g | a

(Notation graphics: 1, 2)
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The 5/3 combination often resolves by directed contrarymotion (Section 4.1), and has a featured role in manyinternal and final cadences. Additionally, throughout thecentury it is often treated more freely, as we might

expect for one of the mildest unstable combinations, andin fact the only one to consist exclusively of stable orrelatively 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 Perotinon. Two fifths, two fourths, or a fifth and a fourth combinewith a relatively tense M2, m7, or M9 in a kind ofenergetic blend or fusion.

In his monumental Speculum musicae or "Mirror ofMusic," 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 itis 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


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

| d' | d'

| M2 | 4
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5 | c' 5 | a

| 4 | M2

| g | g


These four sonorities, like 5/3, represent the mildestunstable combinations possible: here two of the intervalsare ideally blending fifths and/or fourths, while the thirdinterval (M2, m7, or M9) is relatively tense but notsharply discordant.

The treatise of Jacobus suggests that to 13th-centuryears, as to modern ones, the overall impression was one

of an energetic variant on 8/5 or 8/4, with the unstablemajor second or ninth or minor seventh lending a senseof excitement and motion.

While these combinations sometimes participate indirected cadential progressions (Section 4.2), they oftenlend 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 unstablecombinations, composers of the 13th century deploysome strikingly tense cadential combinations resolvingvery effectively to a complete trine or a fifth (the primeinterval of an 8/5 trine). These combinations fall into twomajor families.

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

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

For now, it may suffice to give some examples of themost common forms: 6/3, 6/5, 6/5/3, 6/2, and 6/4. These

sonorities, although all on the tense side, may varyconsiderably in their degree of tension. We should recallthat M6 is relatively tense, roughly on par with M2 andm7, while m6 is often regarded as sharply discordant (likem2, M7, tritone). Forms involving m6, m2, or tritonic fifthsor fourths heighten the level of tension, and along withtheir somewhat gentler relatives are very effectivelyemployed 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 | M3m6 | b M6 | c'

| M2 | 4

| a | g

m6/M2 M6/4

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

(Notation graphics: 1, 2, 3, 4, 5)
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The fact that these sixth sonorities include a largenumber of unstable intervals means from a cadentialperspective that they invite some very dynamicresolutions. As we shall see, the 6/3, 6/5, and 6/5/3

combinations can expand to a complete trine in anespecially efficient manner which makes them among themost favored of cadential sonorities.

Additionally, the 6/5 combination often resolves byoblique motion to a simple fifth (the highest voicedescending a step while the others remain stationary), asis 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 cadentialcombinations, an outer minor seventh characteristicallycontracts by stepwise contrary motion to a fifth. InSection 4.4, we shall explore these standardprogressions.

For now, let us briefly look at these sonorities themselves.As with the sixth combinations, they are all decidedly onthe tense side, but more so in the case of forms includingM7 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 +
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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 fordirected cadential action. Additionally, the 7/5combination lends itself to an oblique resolution wherethe upper voice ascends stepwise (Section 5.4).

3.4. Summary of combinations

Here we have by no means covered the full range ofcombinations appearing in 13th-century music: Jacobuslists a catalogue of such sonorities with outer intervalsranging from a major third to a twelfth (about thepractical limit, given the typical range of voices in thisperiod). However, attuning ourselves to some of the mostprevalent and important families of sonorities is a largestep 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 ofunstable combinations.

The "split fifth" with its two thirds (5/3), and mildlyunstable combinations featuring a preponderance of fifthsor fourths (9/5, 7/4, 5/4, 5/2), are relatively blending.

They add pleasant vertical color to the music, and lendthemselves either to standard resolutions or to a freer

treatment.

Other, tenser, combinations strongly invite directedcadential resolutions where an outer sixth expands to theoctave of a complete trine (6/3, 6/5, 6/5/3, 6/2, andsometimes 6/4), or an outer seventh contracts to a fifth
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(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

Thirteenth-CenturyPolyphony

4. Directed cadences for three orfour voices

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

Such cadences, whether involving mildly unstablecombinations or dramatically discordant ones, tend tofollow patterns suggesting two main "guidelines":

(1) The unstable combination as a whole should resolveto a richly stable sonority, ideally an 8/5 trine, and nextmost preferably a fifth (the prime interval of 8/5), with 8/4or a fourth as alternate goals.
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(2) Each unstable interval should ideally resolve bystepwise contrary motion, and next most preferably bynear-stepwise contrary motion (one voice moving bystep, and the other by a third).

An actual example may make these points clearer:

f'-g'

c'-d'

a -g

m6/m3-8/5

(m6-8 + m3-5)

(Notation graphic)

In this progression, very common throughout the 13thcentury, a rather tense m6/m3 combination (m6 + m3 +4) expands to a complete 8/5 trine (8 + 5 + 4). Bothunstable intervals resolve by stepwise contrary motion(m6-8, m3-5), and these two resolutions reinforce eachother, together making possible our arrival at a completetrine. 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/m3sonority resolves to 8/5, and then identifies the mutuallyreinforcing two-voice resolutions (m6-8 + m3-5).

Note that while either m6-8 or m3-5 serves as aneffective cadence in two-part writing, music for three orvoices opens the new possibility of combining both

progressions at once, and introduces the vital newelement of the complete trine as an ideal cadential goal.

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

Before exploring this universe of directed vertical

progressions, we may do well to consider the broadmeaning of "cadential" action in our present context.While the progressions we are about to discuss areindeed favored at final closes or internal cadences in thenarrower sense, they may occur more generally at almostany 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 inmost 13th-century pieces, often making possiblesmoother 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 muchfavored in 13th-century music not only because it is oneof the mildest unstable combination, but also because itinvites some very efficient resolutions. Three suchresolutions 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 bothresolve by stepwise contrary motion: one contracts to a
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unison while the other expands to a fifth, as all threevoices 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 resolvingboth unstable intervals.

In the third progression, the lower third of 5/3 expands tothe fifth of a complete 8/5 trine (here M3-5), while theouter 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 firsttwo cases, each unstable interval resolves by stepwisecontrary motion (3-1 + 3-5), but we arrive at a simplefifth rather than a complete trine.

In the third case, only the lower third resolves bystepwise contrary motion (3-5), the upper third resolvingby similar motion to the upper fourth of the trine. Thuswe gain greater sonority at a slight sacrifice of cadentialefficiency.

Both minor compromises are quite acceptable, andindeed these progressions are among the most popularcadences from Perotin to Petrus de Cruce. While our lastexamples happen to involve 5/M3, Jacobus of Liege'sfavored form with major third below and minor thirdabove, the same progressions also occur with his

"converse" form 5/m3, and also sometimes with thediscordant 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.


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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 interestingvariation on this pattern: the 8/5/3 combination (8 + 5 +4 + 3 + 3 + 6) resolving by stepwise motion of all voicesto a complete trine. Note that this sonority may besomewhat more tense than 5/3 because of the major orminor 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 thepopularity of this type of cadence in four-voice music isnot 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/quartalsonorities (9/5, 7/4, 5/4, 5/2)

Like 5/3, these combinations serve as a great resource forharmonic color; however, they play a somewhat lessprominent role as directed cadential sonorities. Let usfirst quickly consider the possibilities of 9/5 and 7/4, andthen explore the more versatile role of 5/4 and 5/2, whichdo rather frequently resolve by directed contrary motionin the course of a piece, now and then providing materialfor final cadences.
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The 9/5 combination (M9 + 5 + 5) seems to lend itselfmore to oblique than to directed resolutions, although Iam aware of one curious resolution where the major ninthcontracts to a fifth, each voice leaping by a third:

e'-c'

a -c'

d -f

9/5-5

(M9-5)

(Notation graphic)

In another resolution by contrary motion, the major ninth

expands to a twelfth while the lower fifth expands to anoctave, arriving at a 12/8 sonority. This progression isprobably rather rare because of the limited range of most13th-century polyphony, and it is my impression that itoccurs mostly in pieces from around 1300:

g'-a'

c'-d'

f -d

9/5-12/8

(M9-12)

(Notation graphic)

The 7/4 combination permits a resolution where the outerminor seventh contracts to a fifth, and this progressiondoes occur now and then, although it is morecharacteristic of the tenser seventh combinations (7/5,

7/3, 7/5/3) to be examined in Section 4.4:

f'-e'

c'-e'

g -a

7/4-5

(m7-5)
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(Notation graphic)

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

treatments.

In the case of 5/4 and 5/2, however, resolutions bydirected contrary motion as well as oblique motion play asignificant role in the cadential lexicon of the 13thcentury. In such directed progressions, the unstablemajor second of 5/4 or 5/2 expands to a stable fourth orfifth (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 tothe octave of a complete trine; or the two outer voicesmay ascend or descend together by step, with theprogression arriving at a simple fifth rather than a fulltrine.

These directed resolutions of 5/4 and 5/2 are quitecommon, 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 twogroups of cadential sonorities par excellence: sixth andseventh combinations.
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As already noted (Section 3.3), cadential sixthcombinations characteristically resolve to a complete 8/5trine, with the outer sixth expanding to the octave of thetrine (M6-8 or m6-8).

This process is especially efficient in the case of the 6/3and 6/5 combinations, the 6-8 resolution combining nicelywith 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)


Note that in each case both unstable intervals resolve bystepwise contrary motion, and the progression arrives atan ideally sonorous 8/5 trine, satisfying our criteria for asuperb cadence.

As it happens, these two examples feature thecomparatively milder M6/M3 (M6 + M3 + 4) and M6/5 (M6+ 5 + M2) combinations, but more discordantpermutations involving m6, m2, and tritonic fourths orfifths are also common.

In four-part writing, another powerful cadential sonoritybecomes available: 6/5/3 (6 + 3 + 3 + 2 + 5 + 4). For anexample, let us take one of the more discordant forms,

m6/5/m3, including m6 and m2 as well as two moremildly unstable thirds:

f'-g'

e'-d'

c'-d'

a -g
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m6/5/m3-8/5

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

(Notation graphic)

This memorable cadence occurs at the close ofVetus abitlittera, an anonymous piece worthy of Perotin. Here wehave no fewer than four unstable intervals, each of whichfelicitously resolves by stepwise contrary motion, bringingus to an 8/5 trine.

While 6/3, 6/5, and 6/5/3 are ideally efficient cadentialsonorities, the 6/2 combination is somewhat less ideal,

since it requires a leap of a third in the middle part inorder to resolve to 8/5:

e'-f'

a -c'

g -f

M6/M2-8/5

(M6-8 + M2-5)

(Notation graphic)

The 6/4 combination seems yet less efficient as acadential sonority resolving to 8/5 by way of a 6-8progression, since the middle voice remains stationary,and the upper third thus resolves by oblique rather thancontrary motion - a pattern not especially favored in thisperiod 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)

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

f'-g'd'-c'

a -g

m6/4-8/4

(m6-8 + m3-5)

(Notation graphic)

In practice, the 6/3 and 6/5 forms are most important inthree-part music, while the 6/5/3 form is very popular in

four-part pieces. Final closes, internal cadences, andtransient progressions from one sonority to the nextprovide frequent occasions for their use in directedvertical 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 alsoinvite compelling directed resolutions in which the outerseventh contracts to a fifth (m7-5 or M7-5). In 7/3 (7 + 3+ 5) and 7/5 (7 + 5 + 3), the milder unstable thirdlikewise 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)


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

In four-part pieces, including Perotin's, we also meet anespecially impressive seventh combination: 7/5/3 (7 + 5+ 5 + 3 + 3 + 3). Here is a milder form with m7 andwithout any tritonic fifths:

d'-c'

b -c'

g -f

e -f

m7/5/m3-5

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

(Notation graphic)

As in the typical resolution of 6/5/3 (Section 4.3), all fourunstable intervals progress to stable ones by stepwisecontrary motion. In this case, as with the other seventhcombinations, we arrive at a simple fifth rather than acomplete trine - a small and acceptable compromise ofsonority.

Like the sixth combinations, these seventh combinationsplay a vital cadential role throughout the century, oftenwith superb effect. A fuller discussion would cover relatedforms such as 8/7, 8/7/3, and 8/7/5, and also idioms whichmight be considered part-writing variations on the basicresolutions we have just surveyed.

Thirteenth-Century
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Polyphony

5. Multi-voice Obliquely ResolvingSonorities

Having looked at directed cadential progressions for threeor four voices, we now turn to resolutions by obliquemotion. Some popular combinations lend themselvesreadily to both treatments: for example, 5/3, 5/4, 5/2, 6/5,and 7/5.

As we saw in Section 2.3, two-voice music frequentlyfeatures resolutions from an unstable to a stable intervalwhere one voice remains stationary while the othermoves by a step, or by a third (sometimes with themiddle step filled in), e.g. 7-8, 7-(6)-5.

Extending this technique to music for three or voices,

composers often apply these basic resolutions to theunstable intervals of various combinations, with one ormore voices moving by oblique motion while the othersremain stationary. Let us consider some of the mostcommon resolutions of this kind, with a definite cautionthat the following presentation is only a very partialsampling.

5.1. The "split fifth" unsplit (5/3, 8/5/3)

While the "split fifth" or 5/3 often resolves by directedcadential action (Section 4.1), it can also resolve byoblique motion of the middle voice which "splits" the fifthinto two unstable thirds:
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a - d' -

f-(e)-d b-(c')-d'

d - g -

5 - 5 -

m3-(M2)-1 M3-(4)-5

[m3-(M2)-1 + M3-(4)+5] [M3-(4)-5 + m3-(M2)-1]


The middle voice descends or ascends by a third, possiblymoving through the intervening step, to arrive at a unisonwith one of the outer voices and a fifth with the other.

Thus we are left with a stable "unsplit" fifth. From a

vertical point of view, we might say that the mild tensionof the two thirds has "evaporated" without impelling adirected progression. Of course, the motion of the middlevoice (often in quick note values) also adorns the musicmelodically.

In music for four voices, the 8/5/3 sonority likewise invitesthis kind of oblique resolution as well as the standarddirected resolutions. Here the resolution of the twounstable thirds - and also an unstable sixth with respectto the highest voice - leaves us with a complete 8/5 trine:

f' - d' -

c' - a -

a-(g)-f f-(g)-a

f - d -

8 - 8 -

5 - 5 -

M3-(M2)-1 m3-(M2)-1

[M3-(M2)-1 + m3-(4)-5 + m6-(m7)-8] [m3-(M2)-1 + M3-(4)-5

+ M6-(5)-4]


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

These relatively blending combinations very frequentlyresolve by oblique motion, sometimes with impressive

effect indeed. The 9/5, 5/4, and 5/2 combinations canneatly resolve to a stable trine or fifth by the stepwisemotion of a single voice, while 7/4 invites a beautifulresolution in which the two upper voices ascend stepwisein fourths to the octave and fifth of an 8/5 trine.

The 9/5 combination most typically resolves to an 8/5trine by the stepwise descent of the upper voice (M9-8),

and may alternatively resolve to an 8/4 trine by thestepwise descent of both upper voices in fifths:

g'-f' b'-a'

c' - e'-d'

f - a -

M9-8 M9-8

5 - 5 -4

[M9-8] [M9-8](Notation graphics: 1, 2)

As just mentioned, 7/4 often resolves by a stepwiseascent of the two upper voices to the octave and fifth of acomplete trine:

f'-g'

c'-d'

g -

m7-8

4 -5

[m7-8]

(Notation graphic)
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The energetic but relatively concordant qualities of the7/4 sonority, the pleasant motion of the upper voices inparallel fourths, and the arrival at an ideal 8/5combination make this a strikingly beautiful progression.

Especially in the music of Perotin's time, it sometimesserves as a very felicitous final or sectional cadence.

While the 5/4 and 5/2 combinations rather frequentlyresolve by directed cadential motion (Section 4.2), theyalso invite oblique resolutions rather similar to those of5/3 (Section 5.1). The middle voice, which here "splits"the outer fifth into a euphonious fourth and a relatively

tense major second, typically moves stepwise into aunison with the nearer outer voice (M2-1):

d' - c'-

c'-d' g-f

g - f -

5 - 5 -

4-5 M2-1

[M2-1] [M2-1](Notation graphics: 1, 2)

Thus we are left with an "unsplit" stable fifth.

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

Again we move to more tense combinations. In addition

to their vital role as directed cadential sonorities,combinations with an outer sixth also permit a variety ofresolutions by oblique motion. Here we consider only afew main possibilities.

The 6/5 combination, in addition to inviting one of the
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most effective directed progressions of the 13th century(Section 4.3), also has an alternative resolution byoblique motion in which the upper voice descends fromthe sixth to the fifth:

d'-c' c'-b

c' - b -

f - e -

M6-5 m6-5

5 - 5 -

[M6-5 + M2-1] [m6-5 + m2-1]


This motion of the upper voice produces twosimultaneous resolutions: a 6-5 resolution with the lowestvoice, and a 2-1 resolution with the middle voice. Ourexamples show the milder M6/5 form, where bothunstable intervals (M6 and M2) are relatively tense butnot sharply discordant, and one of the more intenseforms: m6/5 (with m6 and m2).

The 6/2 combination, when it does not resolve in adirected manner, sometimes resolves by the stepwisedescent of both upper voices, arriving at a stable fifth:

e'-d'

a -g

g -

M6-5

M2-1

[M6-5 + M2-1]

(Notation graphic)

Here both upper voices have standard oblique resolutionswith the lower voice: 6-5 for the highest voice, and 2-1 for
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the middle voice. The parallel fifths between the uppervoices add an element of texture and color.

5.4. Seventh Combinations (7/3, 7/5, 7/5/3)

While many oblique resolutions are possible for thesecombinations as alternatives to their superb directedresolutions (Section 4.4), we here consider an especiallyimportant form in which the seventh of 7/5 ascends to theoctave of a complete trine:

c'-d' e'-f'

a - c' -

d - f -

m7-8 M7-8

5 - 5 -

[m7-8 + m3-4] [M7-8 + M3-4]


The motion of the upper voice produces a 7-8 resolutionwith the lowest voice and a 3-4 resolution with the middlevoice. Both the milder form with m7/5 and thedramatically intense one with M7/5 are very common.

A complete discussion would look at various otherresolutions: for example, the seventh in the aboveexamples might descend to the fifth rather thanascending to the octave, and 7/3 can also resolve byoblique motion in various ways.

6. Concluding words

This first crude draft is, as cautioned, a very partial
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introduction. I have focused on patterns rather thanexceptions. The music itself presents patterns, variations,and exceptions - and any line between these categoriesmust also necessarily be arbitrary.

As in most polyphonic music, there is a constant andoften subtle interaction between the vertical and melodicdimensions. A rather transient tone may permit asmoother and more graceful melody at the same time asit results in a directed vertical progression. Almost anypoint where the lowest voice moves by step or by thirdprovides an opportunity for "cadential action" in this freer

sense, however momentary, and there are pieces wheresuch progressions seem to occur at most transitionsbetween measures in a modern score.

The sustained-tone organum passages of Perotin and hiscolleagues, like the "pedal harmonies" of other periods,present a special kind of vertical color. Typically suchpassages focus on a stable 8/5 or 8/4 sonority above thesustained tenor, but exploit the full range of combinationswe have explored. At times the upper two or three voicesmay engage in directed progressions (Section 4) above oraround the unchanging sustained note; a change in thetenor provides the opportunity for a decisive cadenceinvolving all voices.

For an excellent example, see (and hear or perform) theend of the first portion of Perotin's Sederunt principles for

four voices. To set the syllable "-runt", Perotin repeats aphrase twice. The first time, the vertical tensiondissipates; the second time, it is released in the cadencewith m7/5/m3 described in Section 4.4.

It is well to end on a note of humility: any modern
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explanation of this music must rest in good part on one'sown artistic perception and conceptual imagination,hopefully not inconsistent with the theory of the periodand the evidence of the music itself.

13 century polyphony

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