Surveying the Symphonial Era (c. 900-1300)
                  A perspective on four centuries of European polyphony

The Symphonial Era, extending from the Enchiriadis treatises to the epoch of Petrus de Cruce (c. 900-1300), is a very rich panorama of some four centuries of recorded music as well as documentation of techniques for improvised polyphony. The purpose of this article is to present a possible roadmap highlighting some points of interest. It might be well for me to emphasize that the rounded dates are symbolic: the Enchiriadis treatises may date somewhat earlier than 900; while the year 1300 symbolizes the end of the Ars Antiqua, but in fact the summa of this great era is the monumental Speculum musicae of Jacobus (c. 1330?).

In seeking a name and a theme to unify the first four long and fruitful centuries of recorded Western European polyphony, I draw on the existing but rarely used adjective "symphonial," which evokes a harmonious union of voices and also the Greek term symphonia referring in this era to the stable concords of the fourth, fifth, octave, and their octave extensions. Thus to improvise or compose music for two or more voices based on these stable consonances is to "symphonize," and an improvisation or composition produced in this way may be called a piece of "symphonizing," as also the process of creating it.

Here I am indebted to Jacobus, who says that the verb diapentizare or quinthiare may be applied to musicians who prefer fifths, and quartare or diatesseronare to those who prefer fourths (Book VII, Chapter 10), and to Sarah Fuller's use of "fifthing" as an equivalent in English. By analogy to "fifthing" or "fourthing," my "symphonizing" would refer to any polyphony favoring either or both of these intervals, especially in a 9th-13th century European style.

My special intent here is for the terms "symphonial" and "symphonizing" in a 9th-13th century context to correspond to and also contrast with "contrapuntal" and "counterpoint" in the 14th-16th century context that is the focus of Rob C. Wegman's germinal article "What is Counterpoint?". Each of these great eras or super-eras has its own interval aesthetics and patterns, and holding conceptual and artistic space for each to reveal its own art and perfection is a powerful motivation for seeking out language and descriptive tools that can help us appreciate 9th-13th century polyphony as a unique art unconstrained by the limitations and prohibitions of other periods.

Hallmarks of the Symphonial Era are the centrality of the symphoniae or stable concords, that is the 4:3 fourth, 3:2 fifth, and 2:1 octave; and the use, often quite bold, of a variety of unstable intervals to create a vertical texture often rich with contrasts in tension. For example, both Guido d'Arezzo early in this era (1025-1026) and Jacobus at its conclusion about three centuries later regard the 9:8 tone or major second as a kind of concord, with Guido especially recommending it for use in an occursus or convergence on the unison at the end of a phrase; and Jacobus including it both in his catalogue of two-voice interval resolutions (Book IV, Chapter 50), like Guido recommending its oblique resolution to the unison, and in his catalogue of acceptable multivoice sonorities (Book IV, Chapter 51)[1].

In practice, even the most acutely tense or "perfectly discordant" intervals (e.g. minor second, major seventh, tritone or diminished fifth, and often the minor sixth) are used boldly and effectively. The minor second, although excluded by Guido, makes a bold appearance in the Wincester organa as transcribed by Andreas Holschneider; and we find all of these acutely tense intervals in use, often boldly, by the Notre Dame School. In Ars organi, the Vatican Organum Treatise (early 13th century?), we likewise find all 13 or 14 diatonic species of intervals from the unison to the octave in use[2], including the earcatching Notre Dame idiom of a piece opening with a major seventh that resolves obliquely to the octave. Jacobus is evidently recognizing this practice when he remarks that the major seventh is placed among the perfectly discordant intervals if "it does not seek the octave, or unison" (Book II, Chapter 95).

Here I propose to divide the Symphonial Era into two periods of about equal length. The Diaphonic Era (c. 900-1100) begins with the Enchiriadis treatises, followed by two longer 10th-century compositions (Benedicta es and Sancte Bonifati martyr), and culminating in the age of the Wincester repertory and Guido. The Discantic Era (c. 1100-1300) begins with the expansion of the scope of variable interval organum to include the full range of vertical intervals from the unison to at least the octave, a development that leads to the various repertories of the 12th century, the advent by the end of that century of a regular practice of writing for three or four voices, and the flowering of multivoice polyphony through the 13th century.

It seems fitting that the terms diaphonia and discantus, in Greek and Latin respectively, refer literally to a "singing apart," that is, to a pleasing mixture of different simultaneous notes and voices. As in various other world polyphonies, so in Western European polyphony of the 9th-13th centuries, fifths and fourths as the main stable intervals, coupled with a wealth of unstable intervals sometimes rated on a subtle continuum of concord and discord, provide ample materials for this harmonious mixture of voices. In what follows, I attempt to trace out some main themes and patterns of this art as it develops through four centuries.

Table of Contents

 1. The Diaphonic Period (c. 900-1100)
    1.1. The Enchiriadis treatises: Polyphony has arrived!
    1.2. The intriguing 10th century
    1.3. The age of Winchester and Guido
    1.4. Summary: some interesting concordances

 2. The Discantic Period (c. 1100-1300)
   2.1. Two-voice interval space expanded: The 12th century
   2.2. The 13th century: Multivoice writing and trinic aesthetics
        2.2.1. Possible worlds: A piece from c. 1100 and 13th-century practice
        2.2.2. Mediating tones and sonorities
        2.2.3. Opposing tones and sonorities
        2.2.4. Connecting tones and sonorities

 3. Conclusion: Unity and Diversity

1. The Diaphonic Period (c. 900-1100)

This period extends from the Enchiriadis treatises to the advent and increasing practice of two-voice polyphony where vertical intervals from the unison at least to the octave are in use, and where contrary motion is generally preferred -- hallmarks of what I will here term "discant" in a generic sense.

1.1. The Enchiriadis treatises: Polyphony has arrived! 

An obvious but noteworthy thing about the Enchiriadis treatises (c. 900?) is that they announce that polyphony has arrived in Western Europe, complete with an aesthetic of consonance and a deliberate choice of intervals, and also of the ordering or arrangement of intervals when one or more voices of an organum is doubled at the octave.

Parallel motion, as in many other world polyphonies, is itself sufficient for the beautiful amplification of a melody. At the same time, both technical features of the dasia system, and possibly the influence of practical traditions favoring a more complex technique, result in an approach to organum at the fourth involving frequent oblique motion, and at times contrary motion also: thus the appearance of the oblique Maj2-1 resolution and the Maj3-1 resolution by stepwise contrary motion as ways to end phrases.

We also find a recognition of a complication that will get attention in later discant and counterpart theory also: the problems of the deuterus or second step of a dasia tetrachord sounding against the tritus or third step of the tetrachord below it where a concordant 4:3 fourth is intended (e.g. Bb2-E3, where E3 is the usual final for the Deuterus or Phrygian family of modes). In later terms, this will be known as the mi contra fa problem.

While the application of accidentals is one solution, the kind of technique documented in these treatises involving variable intervals and motions is another. Thus, whatever popular or otherwise undocumented styles of polyphony may have been current around 900, the Enchiriadis treatises document the art of vertical concord and also set some initial conditions out of which the styles of the 10th-11th centuries and beyond could fruitfully evolve.

1.2. The intriguing 10th century

Between the Enchiriadis treatises and the epoch of Wincester and Guido, there are some intriguing indications of the kind of practices that might have evolved through the 10th century, with the piece Sancte Bonifati martyr illustrating some of the creative choices available. Like the sequence Rex caeli, Domine from the Musica enchiriadis, this piece is in Deuterus on E, the family of modes that Guido will consider merely "apt" for diaphony, by comparison to the "more apt" Protus on D or A, and the "most apt" Tritus or Tetrardus on C, F, or G.

As Giovanni Varelli suggests, "Two Newly Discovered Tenth-Century Organa," Early Music History vol. 32 (2013), pp. 277-315 at 304-306, "Two Newly Discovered Tenth-Century Organa", some of the procedures in this piece such as phrases beginning on major or minor thirds may be constrained; tones or major seconds are also prominent, with one used to open a phrase (te quesumus). Phrase endings where a major or minor third resolves obliquely to a unison occur often, with the oblique resolution from D3-F3 to D3-D3 (ut nos) notably used as the final cadence between the lower pair of voices in Congaudeant catholici (c. 1140) about two centuries later. Varelli's analysis also raises open questions about what constraints musicians may felt in this epoch: for example, did they share Guido's preference to avoid vertical minor seconds, ranked by 13th-century theorists such as Garlandia, Franco, and Jacobus as among the most acutely tense or "perfectly discordant" intervals -- but sometimes tellingly used in 12th-13th century practice?

There may be a factor of selection operating here: musicians might be most likely to set a piece in writing if it involved the solution of difficulties not so easy for singers to negotiate extempore (the same reason that Pietro Aron gives in 1529 for his advice that composers explicitly indicate accidentals rather than leaving even the most seasoned performers to guess and sometimes inevitably produce unintended dissonances, at least on first reading). At the same time, such a solution might be valued as an artistic creation in its own right worth preserving, as well as a guide to similar situations.

A piece from the Parisian treatise De organo a setting of Benedicta es, has noteworthy and indeed exciting features also. The excerpts transcribed from Coussemaker's rendition of the original notation by Willi Apel in The Notation of Polyphonic Music 900-1600 show the expected leaning toward fourths in organum with variable intervals, but also feature use of the fifth, which Guido will exclude between the two basic voices. Apel also transcribes a momentary occurrence within a phrase of D3-E3 to C3-F4, Maj2-4 by stepwise contrary motion, a progression characteristic of the 12th-13th centuries. As with Benedicta es, elements such as conclusions of phrases with a resolution of oblique Maj2-1 or Maj3-1 by contrary motion suggest a continuity both with the Enchiriadis treatises and with Guido. Both pieces provide us with an opportunity to see how polyphony might explore a variety of intervals and motions in settings on a larger scale than the brief Enchiriadis examples.

1.3. The age of Wincester and Guido

The epoch around the early 11th century presents us with two windows on diaphony as a mature art in practice and theory: the copious repertory of Winchester, and the stylistic precepts and preferences of Guido d'Arezzo as expressed in his Micrologus (c. 1025-1026). Guido briefly addresses organum in simple parallels, with the second voice at a fourth below the chant, and with an option for doubling this voice at the octave, so that this outer interval at 2:1 is divided into a 4:3 fourth below and a 3:2 fifth above.

In addressing diaphony at the fourth with variable intervals, Guido sets a landmark in theory by taking an interest not only in the symphoniae or stable concords, but also the role of unstable intervals not merely as a conceded necessity, but as a positive resource and aspect of style. He excludes the semitone or minor second (256:243 in its diatonic form) from his technique, likely because of its acutely tense character; and also the fifth, which he permits only as a result of octave doubling of one or both main voices. His technique is thus based on "four concords" in addition to the unison: the tone or major second (9:8), semiditone or minor third (32:27), ditone or major third (81:64), and the diatessaron or fourth itself (4:3). Of these the fourth is preeminent, and the minor third least favored.

In focusing on the occursus or usual convergence of the voices on a unison at the end of a phrase, Guido notes that the major second is preferred (i.e. oblique Maj2-1), and the major third not so highly (Maj3-1 by stepwise contrary motion or oblique motion, both of which occur in his examples), while the minor third is never used in this connection. He adds that occursus is rarely (vix) made from the fourth, marking this as a place where vertical instability is not only accepted but highly desired. Guido also notes an option to forgo occursus and end a phrase instead on a fourth, for example when the unison would be approached from a minor third.

If Andreas Holschneider's transcriptions of the Wincester repertory are accurate enough to be fairly representative, they provide evidence that, as in the 10th century, tastes and techniques varied. Thus while the fourth is the most favored symphonia or stable concord, fifths do occur now and then (e.g. Alleluia Te martirum, with Benedicta sit and one version of the sequence Rex caeli Domine from the Musica enchiriadis as precedents). The semitone sometimes occurs as a vertical interval (e.g. Alleluia Te martirum, with the occursus of oblique min2-1, B2-C3 to C3-C3); and convergences from minor third to unison are found both by stepwise contrary motion (with ascending or descending semitonal motion, e.g. Alleluia Angelus Domini, B2-D3 to C3-C3; Cristus natus est nobis, D3-F3 to E3-E3); and by oblique motion, e.g. Sequentia Beatus vir, closing D3-F3 to D3-D3), with Sancte Bonifati martyr as a precedent. While Guido says that occursus from the fourth "scarcely" (vix) occurs, this is the final cadence for four Winchester pieces ending with 4-1 by contrary motion: Alleluia Die sanctificatus, Alleluia Video caelos apertos, Alleluia Ascendens Cristus and Alleluia Judicabunt sancti nationes.

Thus one sign of the maturity of the polyphonic art by around 1000-1050 is the diversity of practices and tastes. A common trait through the Diaphonic Era is that, in variable interval organum, the fourth is preferred; and either it or the fifth, in styles where the latter is also used from time to time, marks the widest vertical interval in use between the two main voices. Within these metes and bounds, an art of admirably diverse motions, intervals, and also preferences flourishes.

1.4. Summary: some interesting concordances

As one great tradition of world polyphony, music of the Diaphonic Era in Europe has correspondences to music of other times and places. Here it is prudent to say that correlation is not causation, but can lend perspective. Thus in seeking to base the technique of diaphony entirely upon "concords," including not only the stable unison and fourth, but the unstable tone and major and minor third, Guido takes an approach that Jacobus will follow about three centuries later in the Speculum musicae (c. 1330?), likewise concluding that all intervals used in recognized vertical structures should be concords (Book IV, Chapter 51), and recognizing the tone or major second (9:8) as an "imperfect concord" (Book IV, Chapter 37).

It is also noteworthy that the great philosopher and physician Ibn Sina (c. 980-1037, known to Latin Europe as Avicenna), a close contemporary of Guido (991/992-after 1033), describes a musical technique of tarqib, an Arabic word for a compound or combination, in which two notes are sounded simultaneously at the concord of the fourth or fifth, or also at the octave, and likewise expresses a preference for the fourth. In a rather different setting, Russia in the 16th-17th centuries (and possibly earlier), we find a polyphonic style described by Viktor Beliaev: "All compositions written in the two-voice style begin in the unison of the two voices that diverge at the accented sections by no more than a fourth, which is the basic interval of this two-voice polyphony. The basic interval is used both in alternation with other intervals and in successions of parallel fourths." (Beliaev, "Early Russian Polyphony," in Studia Memoriae Bela Bartok Sacra, 1959, 3rd ed., pp. 311-328 at 313-314). Studia Memoriae Bela Bartok Sacra

Thus we may celebrate European polyphony of the era 900-1100 as at once unique, and as situated within a larger context of world musics of various times and places favoring polyphony with fourths or fifths as the main concords. In the later 11th and 12th centuries, the metes and bounds of this art shift, producing improvised and composed settings first mostly for two voices, and by around 1200 for three or four voices also, which likewise fit within this larger context.

2. The Discantic Period (c. 1100-1300)

Sometime around 1100, or a bit earlier, the range of vertical intervals for variable organum expanded: from a fourth or sometimes a fifth, to an octave or more. This change dramatically increases both the variety of stable and unstable intervals available, and the scope for a variety of motions, and especially the contrary motion recommended by John Afflighem (or Cotton) at the beginning of this period, a preference made axiomatic by various theorists of the 12th-13th centuries (e.g. Franco, Anonymous IV, Jacobus).

A second incremental and yet revolutionary change, around 1200 or actually a bit earlier, is the advent of regular writing for three or four voices, an achievement associated with the great composer Perotin. Just as expanding the range of vertical intervals, or width of the "polyphonic envelope," increases the scope for the taste we see in diaphony for a variety of intervals and motions, so multi-voice compositions (and improvisations in similar styles) not only permit richer sonority, but open the way to an amazing diversity of stable and especially unstable combinations for three or four voices, and of satisfying multivoice progressions arising from the harmonious superimposition or alliance of two-voice resolutions. The works of Perotin and his contemporaries show how quickly and consummately this art was perfected, with musicians of the 13th century exploring its many ramifications.

Like the term "diaphony" in the 9th-11th centuries, "discant" is both a contemporary 12th-13th century term covering many forms of polyphony, or polyphony in general (e.g. Jacobus), and also a useful term in evoking a specific period and set of musical concepts and styles. The main caution is that "discant" continues to be used in the 14th-century Ars Nova, and indeed through the 16th century, as synonymous with what has become the new art of counterpoint, composed or improvised, with Thomas Morley (1597) using "descant" specifically to mean extempore counterpoint rather than a written "setting." Thus discant, as used here, is meant to contrast with 14th-16th century counterpoint, an art which moves in other directions as to interval aesthetics, with both genres deserving appreciation on their own merits.

2.1. Two-voice interval space expanded: The 12th century  

By around 1100, a new approach to two-voice polyphony had emerged with two basic parameters. First, the voices are free to explore a span of vertical intervals ranging from the unison out to the octave, and in some pieces out to around a 3:1 twelfth, which Jacobus later notes as close to the limit of intervals in practical use. Secondly, while embracing all kinds of motion (parallel, similar, oblique, and contrary), composers and theorists show a preference for frequent contrary motion, which at once accentuates the independence of the voices and promotes the variety of intervals also sought.

Whatever the origins of this new approach, its elements are highly compatible and mutually reinforcing. Expanding the vertical domain of polyphony to the whole realm from a unison out to an octave or twelfth certainly affords more scope for contrary motion. In variable organum from the Enchiriadis treatises through Wincester and Guido, 1-4 and 4-1 are the obvious progressions between stable intervals by contrary motion; but in the new style we also have 5-8 and 8-5; 4-8 and 8-4; and 1-5 and 5-1 (and sometimes 8-11 and 11-8; and 8-12 and 12-8), etc.

Within this enlarged territory for two-voice polyphony, a simple feature of intervallic geometry, as we might call it, operates to promote yet more variety and contrast. It is impossible to move from one symphonia to another by stepwise contrary motion. Progressions such as 4-1 and 1-4 (common in diaphony), and 5-8 or 8-5, come closest, calling for "near-conjunct" contrary motion with one voice moving by step and the other by a third. To put this another way: if we wish to arrive at a stable interval by stepwise contrary motion, we must approach it from an unstable interval (e.g. Maj3-1 or min3-1; Maj2-4 or min2-4; Maj3-5 or min3-5; Maj6-8 or min6-8; Maj6-4 or min6-4; Maj7-5 or min7-5). From such early sources as Chartres Bibliotèque municipale 109 on, these progressions rapidly proliferate in note-against-note and more ornamented textures alike, serving melodic smoothness and grace as well as the sense of contrast between instability and stability.

Some of these progressions, specifically Maj3-1 as a standard occursus and also in some styles min3-1 (Sancte Bonifati martyr, Winchester), are common to the old diaphony and what is emerging as the new discant. Apel's excerpts from Benedicta sit in his Notation of Polyphonic Music show a momentary Maj2-4. An excerpt from the middle of a phrase in the Winchester composition Psallite Domino, where the fifth is in use, provides a fortuitous example of what rapidly becomes a pervasive idiom of discant:

Ex. 1
         1   4   4   4   1  Maj2 1  min3  5  1
         D3  C3  C3  D3  D3  D3  D3  D3  C3  D3
         D3  F3  F3  G3  D3  C3  D3  F3  G3  D3

The brief portion of interest here begins with the voices at a unison on D3, with the chant then moving obliquely to the minor third D3-F3, and this interval expanding by stepwise contrary motion to the fifth C3-G3; the fifth then moves to a unison on D3, as the phrase continues. Although in the context of Winchester this seems an incidental happening, in the new discant various 1-3-5 or 5-3-1 figures become a commonplace, at salient cadences as well as within phrases. More specifically, this early example involves what I term a mediating tone or sonority, where the unstable interval (here the minor third) makes a progression between two stable intervals by contrary motion (here 1-5) more melodically smooth, as well as adding an element of tension and relaxation, however transient or accentuated. Some other examples may clarify this concept of a mediating sonority:

Ex. 2

 (a)            (b)           (c)
  1  Maj3-5      5  Maj3-1     5  min3-1 
  G3  B3  C4     A3      G3    A3  F3  E3
  G3      F3     D3  F3  G3    D3      E3    

In a kindred and also pervasive discant idiom, the two voices each proceed by stepwise contrary motion, 1-3-5 or 5-3-1:

Ex. 3

(a)            (b)           (c)           (d) 
 1  min3-5      1  Maj3-5     5  min3-1     5  Maj3-1
 E3  F3  G3     A3  B3  C4    A3  G3  F3    B3  A3  G3
 E3  D3  C3     A3  G3  F3    D3  E3  F3    E3  F3  G3

The interplay of the fifth and octave also offers fertile ground for this interaction of the melodic and vertical dimensions with its characteristic pattern of stability-instability-stability, as shown in this figure placed at two locations to illustrate the different shadings of tension available:

Ex. 4

(a)                   (b)
 8  Maj7-5  Maj6-8   | 8  min7-5  min6-8
 F4  E4  D3  E4  F4    G4  F4  E4  F4  G4
 F3      G3      F3    G3      A3      G3

In the first example (4a), the upper voice begins at the octave and moves obliquely to the major seventh, an interval which theorists of a 13th-century tradition seeking systematically to rank the intervals in terms of concord/discord (e.g. Garlandia, Franco, Jacobus) agree is one of the most acutely tense, or "perfectly discordant," with this tension resolved by Maj7-5 by contrary motion. Then the voice at the fifth moves to the major sixth, an interval often ranked in the same tradition as relatively but not acutely tense -- which resolves in turn back to the octave, again by stepwise contrary motion (Maj6-8). The second example (4b) follows the same pattern, but this time with a milder minor seventh (often deemed in the 13th century, like the major sixth, relatively tense but somewhat "compatible") mediating between octave and fifth and resolving min7-5; and then a minor sixth (often deemed in the 13th century as an acutely tense interval like the major seventh) expanding back to the octave (min6-8).

One idiom of this type is already present in both Enchiriadis treatises, and occurs in Benedicta es as well as Winchester as transcribed by Holschneider: the 4-3-1 figure. Given that the fourth is the preferred stable concord within phrases, and the occursus to the unison highly favored for ending phrases, the use of a mediating third is not surprising. While 4-Maj3-1 occurs in theoretical and practical sources alike, both this and min3-1 occur in Winchester, although Guido specifically disallows occursus from a minor third:

Ex. 5

(a)                    (b)                   (c)                             (d)
Musica enchiriadis     Scolica enchiriadis   De organo             Winchester          
Tu patris sempiterus   Nos qui vivimus       Benedicta es          Alleluia Video caelos apertos  

   4  Maj3-1           1   4  4 Maj3-1       4  4  Maj3---- 1                  1  1  4 Maj3-1
   F3  E3  D3          D3  F3 F3 E3  D3      F4 F4 E4 E4 E4 D3                 D3 D3 C3     D3
   C3      D3          D3  C3 C3 C3  D3      C3 C3    C3 C3 D3                 D3    F3 E3  D3
   fi- li- us          in  se-cu-lu- um                            [Vi-de-]o..._______________

(e)                                              (f)
                Winchester                                    Winchester   
         Cristus natus est nobis                        Alleluia Angelus Domini

4  4  4  4  4  4  4  4  4  4  4  4  min3-1       4  4  4  4  4  4 min3 1  4  4 min3-1
C3 D3 E3 D3 C3 D3 E3 E3 D3 E3 D3 C3  D3  E3      D3 D3 E3 D3 C3 D3 D3  D3 B2 A2 B3  C3
F3 G3 A3 G3 F3 G3 A3 A3 G3 A3 G3 F3  F3  E3      G3 G3 A3 G3 F3 G3 F3  D3 E3 D3 D3  C3    
a-       do-re------------------ mus_______      re-vol--------------------------- vit 

The second and fourth excerpts nicely illustrate how contrary motion is itself an element of diaphony, but in discant this presence becomes preeminence. The last two excerpts serve at once to illustrate the occursus from minor third to unison by stepwise contrary motion with either descending or ascending semitonal motion; and also to affirm the truism that "variable organum at the fourth" typically offers a fair share of the advertised fourths, often in parallel, as well as a variety of intervals and motions.

Contrary motion and the use of mediating sonorities (characteristically a 4-3-1 idiom) are already present in diaphony, but take on a new significance as the vertical playing field is enlarged by 1100 or so: if singers or composers simply seek to move between stable concords as smoothly as possible while exercising a general preference for contrary motion, these mediating progressions seem almost to sing or write themselves.

Just as diaphony already has within it the element of contrary motion and its use at times in the occursus, so discant continues to draw on the elements of parallel and oblique motion typical of diaphony. The use in some of Guido's examples and the Winchester pieces of what is in effect a pedal point, with one voice repeating a single note while the other moves above or around it, will be expanded in the 12th-13th centuries into sustained-note organum for two and eventually also three or four voices. And obliquely resolving sonorities, with the Maj2-1 occursus as especially characteristic of diaphony, continue to play a vital role, with Maj2-1 itself playing a role through the 12th and 13th centuries, as reflected by its canonical status in Garlandia, Anonymous IV, and Jacobus (Book IV, Chapter 50).

What is new is the plethora of such oblique resolutions now available: not only Maj2-1 (and in Winchester also min2-1), Maj3-1 (and min3-1 in Sancte Bonifati and Winchester), and Maj3-4 and min3-4; but also Maj3-5, min3-5; Maj6-5, min6-5; and Maj7-8, min7-8, etc. These unstable sonorities and resolutions may occur as fleeting ornaments, but also as prominent events definitely meant to be heard and savored.

From the Chartres organa marking the advent of this style to the 12th-century repertories known in my youth as St. Martial and Compostella, to Leonin and the early Notre Dame styles, the art of combining two voices within a greatly expanded ambitus of vertical motion, in sustained-note and note-against-note styles, takes shape in its amazing variety and vitality. In quasi-biological terms, we might say that variable organum or discant with intervals running from the unison to the octave and beyond opened a new evolutionary space, quickly populated by many species of intervals and interval resolutions involving either the newly preeminent contrary motion or oblique motion.

Sometime near the end of the 12th century, another revolutionary development would build on these foundations while again vastly expanding the possibilities of an already impressive art.

2.2. The 13th century: Multivoice writing and trinic aesthetics

Textures for three and four voices are as old as the Enchiriadis treatises, since either or both of the two basic voices in either simple parallel or variable organum could be doubled at the octave; and Guido allows this also.

2.2.1. Possible worlds: A piece from c. 1100 and 13th-century practice

Pieces with variable intervals where the third voice is not an octave doubling of another part are known from as early as around 1100, as noted by Manuel Pedro Ferreira in his article "Early Cistercian Polyphony: A Newly Discovered Source," Lusitania Sacra, 2nd series, 13-14 (2001-2002), pp. 267-313 at 294-295 and Ex. 9. "Early Cistercian Polyphony". His example transcribes the opening of a "previously unreported" setting of Miserere mei (c. 1100) based on Guido's setting in simple parallel organum for three voices, with the organal voice at a fourth below the chant doubled at the higher octave. Here it may be helpful to give the relevant excerpts first from Guido, and then from this "remake" as Ferreira styles it:

Example 6

Guido, Miserere mei

 C4 D4 E4 C4 D4 E4 D4 C4 C4

 F3 G3 A3 F3 G3 A3 G3 F3 F3
 Mi-se-re-re me----i  De-us

 C3 D3 E3 C3 D3 E3 D3 C3 C3

Variation, c. 1100

 C4 A3 C4 C4 A3 B3    C4 C4 

 F3 G3 A3 F3 G3 A3    F3 F3
 Mi-se-re-re me-i____ De-us 

 C3 D3 E3 D3 D3 E3 D3 C3 C3 

The lower pair of voices, in Guido illustrating simple organum at the fourth, remain mostly the same, but with one feature which Guido's precepts would exclude: the fifth D3-A3 between these voices near the end of the phrase. Both Guido and the composer of this offshoot show a preference that in a stable sonority dividing an outer octave into a fourth and fifth, the fourth be placed below the fifth rather than above, thus C3-F3-C4.

We can represent this sonority, using a notation I have devised based on some concepts of Jacobus, as 8|4_5, that is, "an outer octave split (Jacobus says fissa) into lower fourth and upper fifth." (Since he refers to the process of adding one or more extra voices to divide an outer interval into two or more adjacent intervals as partitio or "partition," we may call this a partitional notation.) Jacobus, while preferring the converse arrangement with the fifth below the fourth, 8|5_4, notes that some musicians such as the Galenses (whom Rob Wegman identifies as either Walloon or Welsh) prefer fourths to fifths (Book II, Chapter 36); and if such musicians followed this preference in improvised or composed three-voice polyphony, they might reach a result like this example. Here I will focus on a few general observations and on two especially noteworthy features.

While the lower two voices strongly favor the fourth -- as we would expect in an offshoot of Guido's simple parallel organum at that interval -- the highest voice, originally a simple doubling at the octave of the lowest and at a fifth above the chant, now ranges between a fifth and an octave above the lowest voice, and between a tone or major second and fifth above the chant in the middle voice of the texture. The Guidonian preference for 8|4_5 as a stable sonority shapes the writing: as in Guido's original version, we begin and end on this ideal three-voice sonority.

The first noteworthy feature is the occurrence as the second, fifth, and sixth sonorities of a sonority with an outer fifth split into a lower fourth (as in Guido's setting) and an upper tone or major second between the chant and the added voice, e.g. D3-G3-A3 on the second syllable of miserere, or 5|4_Maj2. The composer evidently likes this sonority, which has a relatively simple string ratio of 12:9:8, using two such sonorities in parallel at the syllables of mei, D3-G3-A3 followed by E3-A3-B3.

The intervallic ingredients of this confection, so to speak, are all familiar from the Enchiriadis treatises on, with the fifth and fourth as stable symphoniae and the tone in use as a vertical interval throughout the period of diaphony, being much favored by Guido and in the Winchester repertory alike. What is new is the mixture of the richly stable 3:2 fifth and 4:3 fourth with the 9:8 tone, somewhat tense in itself -- to produce a sound that is at once euphonious and, at least in this medieval Western context, also unstable. Beliaev describes this common sonority in early Russian polyphony as a "fourth-fifth consonance" ("Early Russian Polyphony," see Section 1.4 above, at 324-325), and it is also characteristic of some styles of Georgian polyphony today.

The second noteworthy feature is the cadence at the end of the phrase:

Ex. 7

  B3      C4
  A3      F3
  E3  D3  C3

Here the two outer voices start at the fifth E3-B3, with the lower voice moving to the major sixth D3-B3, which then resolves by stepwise contrary motion to the octave, a progression common to this piece and to 12th-13th century polyphony generally: a mediating idiom of 5-Maj6-8. Taking all voices into account, from E3-A3-B3, a relatively blending and mildly unstable 5|4_Maj2, we move to D3-A3-B3 or Maj6|5_Maj2, which resolves to a stable C3-F3-C4 or 8|4_5. This progression from D3-A3-B3 to C3-F3-C4 involves the two-voice resolutions of Maj6-8 between the outer voices, and Maj2-5 between the upper voices, the first involving stepwise contrary motion and the second "near-conjunct" contrary motion with one voice moving by step and the other by a third. We can summarize this progression as (Maj6-8 + Maj2-5) to show its union of these resolutions.

While we can only guess exactly how this composer might have ranked different unstable intervals in terms of concord/discord, we can say that following one 13th-century approach, the major sixth like the major second is relatively tense in itself, so the motion from E3-A3-B3 or more generally 5|4-Maj2 with ideally euphonious fourth and fifth and relatively tense major second to D3-A3-B3 or Maj6|5-Maj2 with the euphonious fifth but two relatively tense intervals, would be a building of cadential tension released in the resolution to C3-F3-C4.

In fact, this three-voice cadence uses elements all familiar in the actual practice that would come to prevail in the 13th century: the Maj5-6-8 idiom between the outer voices; the relatively blending 5|4-Maj2 and relatively tense Maj6|5-Maj2 (both described and approved by Jacobus, Book IV, Chapter 51); and the frequent use of the latter sonority in directed resolutions. Indeed, it would not be surprising to find this precise progression as an internal cadence in the music of Perotin around 1200, for example, who often uses 8|4-5 as a stable sonority (although not a conclusive one like 8|5_4). However, by making one modification, we can arrive at a form that typifies 13th-century style:

Ex. 8

  B3      C4
  A3      G3
  E3  D3  C3

Here the D3-A3-B3 sonority resolves by stepwise motion in all voices to a stable 8|5-4, with lower fifth and upper fourth, the arrangement almost always preferred at final closes. This superbly efficient progression features two resolutions by stepwise contrary motion, major sixth to octave between the outer voices (as in the setting around 1100) and Maj2-4 between the upper voices, or (Maj6-8 + Maj2-4).

The critical factor distinguishing these alternate universes of multivoice polyphony is the prevalence in the 12th-13th centuries of a decided preference for the fifth as more smooth and conclusive than the fourth, although both play an important role; thus 8|5-4 becomes the canonical standard for what Johannes de Grocheio terms consonantia perfectissima, or most perfect concord. He explains that the outer 2:1 octave is like a mother, the lower 3:2 fifth like a daughter, and the upper 4:3 fourth proceeds from both, drawing a parallel to the Trinity (where the Holy Spirit proceeds in the view of the Latin West from both Father and Son). Thus he notes that truly perfect concord requires three voices, ideally arranged in this manner, which he describes as expressing trina harmoniae perfectio, a "threefold perfection in harmony." See Excerpts and translations from Grocheio.

This concept of the trine, as we may call it in English, focuses on a three-voice sonority itself known from the Enchiriadis treatises on, and available in two variations. Guido prefers 8|4_5, while 8|5_4 is clearly prevalent from the age of Perotin on, although 8|4_5 provides welcome variety in many 13th-century contexts. A beautiful set of multivoice progressions evolved around the 8|5_4 trine as an ideal goal for unstable combinations, with directed progressions where each voice moves by step and each unstable interval resolves by stepwise contrary motion especially preferred: D3-A3-B3 to C3-G3-C4 (Maj6-8 + Maj2-4) is a paradigm case. Indeed, this progression, a 13th-century mainstay, appears as a final cadence in note-against-note writing as late as the Old Hall Manuscript (c. 1400?).

It is intriguing to ask if the setting of Miserere mei suggests some of the directions in which multivoice polyphony might have evolved if Guido's 8|4_5 had gained the preference rather than 8|5_4 -- or, as mentioned above, this piece may in fact point to a road taken in mostly unrecorded multivoice polyphony of a kind favored by musicians giving preeminence to the fourth, whose presence Jacobus helpfully documents around 1330.

Returning to the polyphonic tradition of composition that prevailed in the 13th century, we can describe this tradition as "trinic," with 8|5_4 as an organizing principle for multivoice composition, and 8|4_5 used for variety. The term "trine" in this context generally presumes 8|5_4, but can apply to 8|4_5 also, which we might style a "Guidonian trine," since Jacobus associates Guido with the placement of the fourth below the fifth (Book VII, Chapters 5-6); and the writer of a treatise called Compendium de musica, which has also been attributed to Jacobus, associates this sonority with the "Guidonians" (Guidonistae).

2.2.2. Mediating tones and sonorities

Here it may suffice to consider a few examples showing how the 13th-century world of polyphony involves not only the superimposition of intervals to build stable and unstable vertical combinations, and of two-voice resolutions to build cadences and other progressions by directed contrary motion or oblique motion, but also of melodic-vertical interactions between different pairs of voices. One way to approach this process is by adding a third voice to some of the two-voice discant idioms addressed in Section 2.1. For example, taking Example 4(a) as our starting point:

Ex. 9(a)

F4  E4  D4  E4  F4
C4      D4      C4  
F3      G3      F3

Here the outer voices are the ones of the earlier example, with a middle voice added. In this straightforward solution, our added voice simply moves in fifths with the lowest voice. This achieves complete 8|5_4 trines (F3-C4-F4) at the beginning and end of this example, and enriches the unstable mediating sonorities that occur as the outer voices move 8-Maj7-5-Maj6-8.

From the initial 8|5_4, the movement of the upper voice through E4 now creates a momentary Maj7|5_Maj3 sonority: an outer major seventh with its acutely tense nature, a fifth between the lower voices, and a mildly unstable major third between the upper voices, resolving stepwise F3-C4-E4 to G3-D4-D4. The impressively tense major seventh thus resolves to a fifth by stepwise contrary motion, and the upper major third to a unison (Maj7-5 + Maj3-1). While the outer voices illustrate the mediating 8-Maj7-5 idiom, here the upper voices feature a mediating major third: 4-Maj3-1, with C4-E4 resolving to a unison on D4.

From the stable fifth G3-D4-D4, the upper voice ascends to E4, generating a mediating sonority we met in the Miserere mei setting from around 1100: Maj6|5_Maj2, with a usual resolution to a complete 8|5_4, or (Maj6-8 + Maj2-4). Here, in addition to the mediating idiom 5-Maj6-8 between the outer voices, we have 1-Maj2-4 between the upper voices.

From another perspective, the three voices move from a complete trine F3-C4-F4 through the mediating seventh sonority F3-C4-E4 to a fifth G3-D4, then expanding back through the mediating sixth sonority G3-D4-E4 to the opening trine on F. Very often, the stable sonorities will occur on main beats, with unstable mediating sonorities much favored at the end of a rhythmic unit, as seen for example in many motets. While a complete trine is the ideal stable sonority, the simple fifth often occurs in its place as the next most preferred choice. From another perspective, the outer voices weave a typical pattern of 8-Maj7-5-Maj6-8, while the upper voices simultaneously realize 4-Maj3-1-Maj2-4. These patterns at once add an element of alternating tension and resolution, and facilitate smooth stepwise motion in all voices as they move in one typical pattern from trine to fifth to trine.

These patterns illustrate two aspects of mediating tones or sonorities. They decorate what would otherwise be a progression by contrary motion from one stable interval to another (8-5-8 in the outer voices, and 4-1-4 in the upper voices); and the voice producing the mediating sonority, here the highest, approaches and leaves this sonority in the same direction.

Ex. 9(b)

              F4  E4  D4  E4  F4
              C4      D4      C4  
              F3      G3      F3
Upper voices: 4  Maj3-1  Maj2-4              
Outer voices: 8  Maj7-5  Maj6-8

Moving the point of departure and return for this passage from F to G, with Example 4(b) as our starting point, will show how the same basic patterns, here 8-7-5-6-8 and 4-3-1-2-4, can be realized with varying nuances of tension and color, and also introduce a memorable cadence:

Ex. 10(a)

G4  F4  E4  F4  G4
D4      E4      D4  
G3      A3      G3 

Where the first mediating sonority in the previous example featured an acutely tense major seventh, here we have the relatively tense but somewhat compatible minor seventh as an outer interval split into lower fifth and relatively blending upper minor third, G3-D4-F4 or min7|5_min3; the minor seventh contracts to the fifth and the upper minor third to the unison by stepwise contrary motion (min7-5 + min3-1). Jacobus regards the minor seventh as an "imperfect concord" along with the major second and major sixth, recognizing and approving both the min7|5_min3 sonority and the two-voice resolutions of min3-1 and min7-5. Then the upper voice introduces another mediating F4, producing a sonority of A3-E4-F4 or min6|5_min2 where the outer minor sixth, like the upper minor second, is often deemed acutely tense by 13th-century theorists, and resolving to produce a most impressive cadence with the minor sixth expanding to the outer octave, and the minor second to the upper fourth, of the complete trine G3-D4-G4 from which we began.

Jacobus excludes the minor second and minor sixth, as "discords," from his catalogues of two-voice resolutions and multivoice combinations, but composers of the 12th-13th centuries are often more inclusive. For example, piece #142 in the Montpellier Codex, the motet Nus ne set/Ja Dieus ne/PORTARE, has a final cadence identical to that of our example, A3-E3-F4 to G3-D4-G4. As with the previous examples, this diagram shows the mediating idioms:

Ex. 10(b)

              G4  F4  E4  F4  G4
              D4      E4      D4  
              G3      A3      G3
Upper voices: 4  min3-1  min2-4
Outer voices: 8  min7-5  min6-8

A different solution for the middle voice produces other sonorities and shadings:

Ex. 11

G4  F4  E4    F4 G4
D4  C4  A3 B3 C4 D4  
G3      A3       G3

This time our mediating seventh sonority is G3-C4-F4, a minor seventh split into two euphonious fourths (string ratios 16:12:9), or min7|4_4, which Jacobus finds "is agreeable" (competit) (Book II, Chapter 93) and includes in his catalogue of multivoice sonorities (Book IV, Chapter 51): as the outer minor seventh contracts to the fifth A3-E4, the lower voices move from the fourth to the unison by contrary motion, with m7-5 as the directed resolution of instability to stability. From this fifth, the lower voice moves through a momentary B3, producing the transient sonority A3-B3-E4, a fifth split into lower major second and upper fourth (5|Maj2-4), with string ratios 9:8:6, to C4 and the cadential sonority A3-C4-F4, a minor sixth split into lower minor third and upper fourth or min6|min3_4: the outer minor sixth expands to an octave as in the previous example, and the lower minor third to the fifth of G3-D4-G4 (min3-5 + min6-8).

Both G3-C4-F4 or min7|4_4 and A3-B3-E4 or 5|Maj2_4 exemplify a kind of three-voice structure featuring two intervals of the richly stable type (fifths and/or fourths), and a relatively tense major second or minor seventh, as likewise the sonorities D3-G3-A3 and E3-A3-B3 in Miserere mei, or 5|4_Maj2. For 13th-century musicians such as Jacobus, these sonorities are relatively blending although unstable. As compared to the cadential sixth sonority A3-E4-F4 or m6|5_min2 in the last example with its impressively tense minor second, A3-C4-F4 or min6|min3_4 seems somewhat milder, with the tense minor sixth split into two milder intervals, the relatively blending minor third and the stable fourth. Like m6|5_min2, it occurs in 13th-century practice from the age of Perotin and his colleagues (starting a bit before 1200) on; bold contrasts between instability and stability are a trademark of this era.

An interesting nuance of this solution is that in the approach of the lower voices to the closing cadence, both B3 and C4 could be considered as mediating, or co-mediating, since they each help to smooth and make more conjunct an expansion from the unison A3-A3 to the fifth G3-D4 (1-5).

Ex. 12(a)

 1   Maj2 min3  5
 A3   B3   C4   D4
 A3             G4

Here the overall pattern is:

Ex. 12(b)

              G4    F4  E4      F4  G4
              D4    C4  A3  B3  C4  D4  
              G3        A3          G3
Outer voices: 8    min7-5       min6-8
Lower voices: 5     4   1  Maj2 min3-5

The last three examples can all be seen as embroideries on the common motion of trine-to-fifth-to-trine. In the first two, the added middle part moves in fifths with the lowest voice, while in the third it moves mostly in fourths with the highest voice. By taking our three examples of the mediating idiom 1-3-5 or 5-3-1, Examples 2(a)-(c), and adding a third voice in the highest position, we can explore other possibilities.

Ex. 13        (a)            (b)           (c)

               D4  E4  F4     D4  C4  D4    D4  C4  B3
               G3  B3  C4     A3      G3    A3  F3  E3
               G3      F3     D3  F3  G3    D3      E3    
Outer voices:  5  Maj6-8                    8  min7-5
Lower voices:  1  Maj3-5      5  Maj3-1     5  min3-1 

In the first example, 1-Maj3-5 in the lower voices is supplemented by the now-familiar 5-Maj6-8 in the outer voices to produce a fifth-to-trine progression with a cadential sonority of Maj6|Maj3_4 and resolutions of (Maj6-8 + Maj3-5). Jacobus includes this sonority with a relatively tense outer major sixth split into relatively blending major third and stable fourth in his catalogue of recognized combinations, and Maj6-8 as well as Maj3-5 in his catalogue of two-voice resolutions. Like the previous more tense minor sixth sonority A3-C4-F4 or min6|min3_4 resolving to G3-D4-G4 (min6-8 + min3-5), this combination has an outer sixth expanding to the octave and a lower third expanding to the fifth, with the upper parts moving smoothly in fourths.

In the second example, the original 5-Maj3-1 is enriched by the added highest voice to a trine-to-fifth progression. This voice does not add any mediating intervals, since it moves D4-C4-D4, descending and then ascending, while a mediating interval must be approached and left in the same direction, but rather engages in a different idiom to be discussed below (Section 2.2.3, Ex. 18). The unstable sonority F3-A3-C4 illustrates the quinta fissa or "split fifth" of Jacobus (Book IV, Chapter 51), where an outer fifth is split or partitioned by a third voice into a major third below and minor third above 5|Maj3_min3, as here, his preferred arrangement, or else the converse, e.g. A3-C4-E4 or 5|min3_Maj3 (Book II, Chapter 76). Here the lower major third resolves to a unison, and the upper minor third to a fifth. The split fifth of Jacobus is one of the mildest unstable multivoice sonorities, since it has a richly stable fifth plus a relatively blending major third and minor third. The overall effect may be regarded as mildly unstable or relatively blending, as also with combinations such as 5|4_Maj2, 5|Maj2_4, and min7|4-4, with two richly stable intervals and one relatively tense interval.

In the third example, the original 5-min3-1 idiom in the lower voices is supplemented with 8-min7-5 in the outer voices to produce a trine-to-fifth progression with an unstable sonority of D3-F3-C4 or min7|min3-5, with resolutions from outer minor seventh to fifth and from lower minor third to unison (min7-5 + min3-1). The intervals and resolutions are identical to those of the cadence G3-D4-F4 to A3-E4-E4 presented above, except that the adjacent intervals are "conversely" arranged, as Jacobus would say (e converso), with the minor third below the fifth rather than above it. He describes and approves both arrangements, which share a widespread practical use in the 13th century.

Our two-voice examples of 1-3-5 or 5-3-1 where both voices move throughout by stepwise contrary motion, note-against-note, Examples 3(a)-(d), also invite the addition of a third voice:

Ex. 14        (a)            (b)                                      (c)           (d)

               B3  A3  G3     E4  D4  C4                               D4      C4    E4      D4
               E3  F3  G3     A3  B3  C4                               A3  G3  F3    B3  A3  G3
               E3  D3  C3     A3  G3  F3                               D3  E3  F3    E3  F3  G3 
Upper voices:  5  Maj3-1      5  min3-1                  Outer voices: 8  min7-5     8  Maj7-5               
Lower voices:  1  min3-5      1  Maj3-5                  Lower voices: 5  min3-1     5  Maj3-1                                 

The first two progressions involve moving from one stable fifth to another by smoothly conjunct motion through an intervening split fifth sonority. While the complete trine of 8|5_4 sets the standard of ideal stable euphony, these fifth-to-fifth progressions are a textural resource providing welcome variety in three-voice conductus settings, for example. The outer voices move in fifths, with the middle voice proceeding in contrary motion.

In the first example, the split fifth D3-F3-A3 is the 5|min3_Maj3 type, resolving to C3-G3-G3, with the lower minor third resolving to the fifth and the upper major third to the unison (min3-5 + Maj3-1). This involves interwoven patterns of 1-min3-5 (lower voices) and 5-Maj3-1 (upper voices). In the second, the split fifth G3-B3-D4 has the converse 5|Maj3_min3, with resolutions of (Maj3-5 + min3-1), and interwoven patterns of 1-Maj3-5 (lower voices) and 5-min3-1 (upper voices).

The last two progressions, although equally amenable to this fifth-to-fifth treatment, are used to illustrate trine-to-fifth progressions involving higher levels of vertical tension. Above the opening fifth in the lower voices, the added voice takes and holds the octave of the trine and holds this note while the lowest voice moves obliquely to a mediating minor or major seventh, resolving to a fifth by stepwise contrary motion. First we have a milder E3-G3-D4 or min7|min3-5 resolving to F3-F3-C4 (min7-5 + min3-1), thus interweaving 5-min3-1 (lower voices) and 8-min7-5 (outer voices). Then we have the acutely tense F3-A3-E4 or Maj7|5-Maj3 with its bold major seventh, resolving to G3-G3-D4 (Maj7-5 + Maj3-1), with the interwoven 5-Maj3-1 (lower voices) and 8-Maj7-5 (outer voices).

The motion of all voices in the first two examples, and of the original lower two voices in the others also, illustrate what might best be called "quasi-mediating" idioms. In a mediating idiom, based on a framework of motion by contrary motion between two stable intervals (here 5-1 or 1-5), one voice remains stationary while the other moves from the first stable interval to an unstable one by oblique motion, then the continues in the same direction as both voices participate in a resolution to stability by contrary motion. The progression of the outer voices in the last two examples fits this model (8-7-5). In a quasi-mediating idiom, both voices move note-against-note in contrary motion, each keeping the same direction, as the opening stable interval moves to an unstable one which resolves in turn to stability, with 1-3-5 or 5-3-1 a classic case in these four examples.

2.2.3. Opposing tones and sonorities

Having considered some patterns of moving trine-to-fifth, fifth-to-trine, and fifth-to-fifth, we may now turn to the sonorous situation of trine-to-trine, which often invites a different type of melodic-vertical interaction. Let us begin with this idiom at its most basic:

Ex. 15(a)

A4   F3 | G4   E4 | F4
E4   C4   D4   B3   C4
A3        G3        F3

We begin at the trine A3-E4-A4, with the two upper voices each moving down by a third to form the unstable sonority A3-C4-F4 or min6|min3_4, and then each ascending by step as the lowest voice likewise descends to produce a standard resolution to G3-D4-G4 (min6-8 + min3-5). Thus we have, so far, 8-min6-8 between the outer voices and 5-min3-5 between the lower voices. From G3-D4-G4 we repeat the process, with the upper voices each moving in the same way down a third to form the unstable G3-B3-E4 or Maj6|Maj3_4, and up by step as the lowest voice descends in a usual resolution to F3-C4-F4 (Maj6-8 + Maj3-5). Here the pattern is 8-Maj6-8 and 5-Maj3-5.

Ex. 15(b)

              A4   F3 | G4   E4 | F4
              E4   C4   D4   B3   C4
              A3        G3        F3
Outer voices: 8    min6-8    Maj6-8
Lower voices: 5    min3-5    Maj3-5

Both upper voices exhibit a characteristic zigzag motion, approaching the unstable sonority in one direction and leaving it in the other. The term "opposing idiom" may invoke this pattern of opposite directions for approach and resolution. While mediating idioms serve to smooth motion between two stable intervals by contrary motion, opposing idioms decorate motion between stable intervals by parallel, similar, or oblique motion. Here, the simple trine-to-trine progressions from A3-E4-A4 to G3-D4-G4 to F3-C4-F4 involve parallel octaves between the outer voices and parallel fifths between the lower voices, 8-8 and 5-5, here embellished to 8-6-8 and 5-3-5 respectively. These patterns are repeated, with only the interval qualities varying, so as to form a kind of vertical sequence that often occurs in 13th-century motets, and also for example in the concluding portion of Adam de la Halle's three-voice rondeau Tant con je vivrai, with many variations and elaborations.

In one variation, we could let the lower voices begin by moving in simple fifths from A3-E4 to G3-D4, thus omitting the 5-min3-5 pattern between these voices, but illustrating another opposing idiom:

Ex. 16

Upper voices: 4    min2-4    4    4 
              A4   F4 | G4   E4 | F4
              E4        D4   B3   C4
              A3        G3        F3
Outer voices: 8    min6-8    Maj6-8
Lower voices: 5         5    Maj3-5

Here the first F4 in the highest voice forms simultaneous opposing intervals of a minor sixth with the lowest voice, and also a minor second with the middle voice, or 8-min6-8 and 4-min2-4. The resulting unstable sonority A3-E4-F4 or min6|5_min2, resolving as usual (min6-8 + m2-4), adds a moment of tension and color with its vertical minor second, and a creative element of asymmetry between this progression to G3-D4-G4 and the following one, as before, to F3-C4-F4.

While these zigzag or opposing idioms are especially typical for trine-to-trine progressions, they are also useful for fifth-to-fifth progressions:

Ex. 17

              C4   E4 | D4  F4 | E4
              C4        D4       E4
              F3        G3       A3
Upper voices: 1    Maj3-1   min3-1
Outer voices: 5    Maj7-5   min7-5

Like the first example of proceeding trine-to-trine, this is a vertical sequence, with the opposing idioms of 5-7-5 (outer voices) and 1-3-1 (upper voices) repeated. We first have the more tense major seventh sonority F3-C4-E4 or Maj7|5_Maj3, resolving (Maj7-5 + Maj3-1) and bringing us to G3-D4-D4; there follows the minor seventh sonority G3-D4-F4 or min7|5_min3, resolving (min7-5 + min3-1) to A3-E4-E4. The difference in interval qualities in these repeated patterns, especially the vertical contrast between the acutely tense 5-Maj7-5 idiom and the Maj7|Maj3_5 sonority it produces and the milder 5-min7-5 and min7|5_min3, adds an element of variety.

Our examples of opposing intervals have adorned parallel motion, as with patterns involving octaves (8-6-8) and either fifths (5-3-5) or fourths (4-2-4) in a trine-to-trine context; and also a pattern with fifths (5-7-5) as well as unisons (1-3-1) in a fifth-to-fifth context. The opposing or zigzaging melodic motion has to this point involved a leap of a third by oblique motion to arrive at the unstable interval, and then motion by step in the opposite direction to achieve a standard resolution by conjunct contrary motion. However, opposing intervals may also decorate progressions by similar or oblique motion, as we discover by revisiting a previous three-voice example based on the mediating idiom 5-Maj3-1 and moving trine-to-fifth:

Ex. 18

              D4     C4  D4
              A3         G3
              D3     F3  G3 
Upper voices: 4     min3-5 
Lower voices: 5     Maj3-1 

The two upper voices start on the fourth of the trine D3-A3-D4, with the highest voice then descending by step to C4, so that these voices form a minor third (A3-C4) within the split fifth sonority F3-A3-C4, and then ascending by step in a min3-5 resolution to G3-D4 so that the ensemble arrives at the fifth G3-G3-D4. Thus the obvious mediating pattern of 5-Maj3-1 is supplemented by the opposing pattern of 4-min3-5. The latter idiom thus adorns what would be a simple oblique progression between the upper voices from A3-D4 to G3-D4, or 4-5.

Progressions between two stable intervals by similar motion, also, can be adorned by an opposing interval, as here, where a simple 4-5 becomes 4-Maj2-5:

Ex. 19(a)

4   Maj2-5
D4   B3  C4
A3       F4

The Maj2-5 resolution by near-conjunct contrary motion is typical of the 13th century, and we can add a third voice to this example so as to produce another characteristic three-voice cadence:

Ex. 19(b)
D4   E4  F4
D4   B3  C4
A3       F4

The cadential sonority is the mildly unstable A3-B3-E4 or 5|Maj2_4, with the the unstable major second expanding to a fifth (Maj2-5) as the outer voices expand from a fifth to the octave of a complete trine F3-C4-F4. Thus while mediating intervals embellish progressions between stable intervals by contrary motion, opposing intervals can apply to any of the other types of motion between two stable intervals: parallel, oblique, or similar. It is only fair to add that while A3-B3-E4 to F3-C4-F4 and similar progressions are common, the approach by an opposing figure of 4-Maj2-5 may be less typical. However, this last example is one possible simplification of the highly ornamented final cadence in Montpellier Codex piece #101, the motet Dieus, je n'i os aler/Amors qui m'aprist/ET SUPER, where the motetus or duplum is in this passage the highest voice, and the triplum the middle voice.

Ex. 20 

Montpellier #101, final cadence

  1  +  2  +  3 | 1 2 3 ||
  D4 C4 B3 A3 B3  C4 
  D4 C4 D4    E4  F4
  A3              F3

2.2.4. Connecting tones and sonorities

Returning to our first example of trine-to-trine motion (Ex. 15) as an opportunity for opposing intervals and sonorities, we can further adorn the ending so as to bring into a play a new kind of melodic-vertical idiom:

Ex. 21(a)

A4   F3 | G4  F4  E4 | F4
E4   C4   D4  C4  B3   C4
A3        G3           F3

In the second measure, the leaps of a third down in each upper voice -- D4-B3 for 5-Maj3-(5) and G4-E4 for 8-Maj6-(8) -- are filled in with the intervening tones C4 and F4. This creates a momentary sonority of G3-C4-F4 or min7|4_4, which Jacobus describes as an outer minor seventh aptly split into two equal fourths. In the resulting melodic figures D4-(C4)-B3 or 5-(4)-Maj3 and G4-(F4)-E4 or 8-(min7)-Maj6, the added C4 or F4 serves as a smooth stepwise connection between notes of the opposing figure a third apart, suggesting the name "connecting" tone, interval, or sonority. Here C4 and F4 are connecting tones; the resulting intervals of the fourth G3-C4 and minor seventh G3-F4 are connecting intervals; and G3-C4-F4 or min7|4_4 is a connecting sonority. We can likewise speak of mediating or opposing tones, intervals, and sonorities (with the distinction between "intervals" and "sonorities" applying to multivoice contexts, these terms being synonymous in two-voice contexts).

Although connecting sonorities are often quite transient, they can contribute, as here, to interesting gradations of vertical tension. Thus we move from G3-D4-G4, a stable trine or 8|5_4, to our momentary G3-C4-F4 or min7|4_4, mildly unstable or relatively blending with its two richly stable fourths and relatively tense minor seventh; and then to G3-B3-E4 or Maj6|Maj3_4, which might be regarded as somewhat more tense with its major sixth relatively tense like the minor seventh (Garlandia, Franco, Jacobus), the lower major third relatively blending, and the fourth richly stable. There is a build-up of tension from stable trine to the mild instability of min7|4_4 to the relatively tense Maj6|Maj3_4, a tension very effectively released in the resolution to the trine F3-C4-F4 (Maj6-8 + Maj3-5).

Thus opposing idioms involving the leap of a third invite adornment by connecting tones and sonorities, as here with 8-min7-Maj6-8 and 5-4-Maj3-5:

Ex. 21(b)

              A4   F3 | G4  F4  E4 | F4
              E4   C4   D4  C4  B3   C4
              A3        G3           F3
Outer voices: 8    min6-8  min7 Maj6-8
Lower voices: 5    min3-5   4   Maj3-5

While connecting sonorities are often quite fleeting, they can also be prolonged so as to be heard as very distinct vertical events, adding much color to the music. The Montpellier motet Conditio/O Natio/MANE PRIMA SABBATI (piece #51) produces a fine example at its final cadence:

Ex. 22

              1   2   3   4   5   6 | 1
              E4  D4      D4  C4      D4
              B3  A3      A3  G3      A3
              E3                      D3
Outer voices: 8  min7    min7 min6----8
Lower voices: 5   4       4   min3----5

The basic opposing and connecting idioms are the same as in our last example: 5-(4)-3-5 and 8-(7)-6-8. The rhythmic context here gives the connecting sonority E3-A3-D4 considerable prominence both because of its duration, and by its reiteration on the relatively strong fourth beat of this transcription (following Yvonne Rokseth). Again there is a build-up of vertical tension as this min7|4_4 sonority is followed by the opposing sonority E3-G3-C4 or min6|min3-4, including the decidedly or even acutely tense minor sixth (ranked by Franco and Jacobus as a thorough discord like min2, Maj7, and Aug4 or dim5, but by Garlandia as somewhat milder), and then the release of this tension by a resolution to D3-A3-D4 (min6-8 + min3-5).

A fascinating dimension of 13th-century polyphony is the way that mediating, opposing, and connecting idioms may be combined or superimposed in many patterns, like different stitches brought harmoniously together in a tapestry. At times, the same note may serve as a connecting tone with one voice, and a mediating tone with another voice. To illustrate these points, we may begin with this common fifth-to-trine progression:

Ex. 23

              D4     E4 | F4
              D4     B3   C4
              G3          F3 
Outer voices: 5      Maj6-8
Lower voices: 5      Maj3-5

The mediating figure 5-Maj6-8 in the highest voice nicely combines with the opposing figure 5-Maj3-5 in the middle voice, with an interweaving of the smooth linear motion D4-E4-F4 and the zigzag motion of D4-B3-C4. A typical embellishment is the following:

Ex. 24

Upper voices: 1  Maj2-4    4
              D4      E4 | F4
              D4  C4  B3   C4
              G3           F3
Outer voices: 5       Maj6-8
Lower voices: 5   4   Maj3-5

Adding the connecting tone C4 to the middle voice brings about a momentary G3-C4-D5 or 5|4_Maj2, a relatively blending or mildly unstable sonority leading to a more tense sonority with an outer sixth, here Maj6|Maj3_4, and its directed resolution to a complete trine. The obvious understanding of C4 as a connecting tone in relation to the lowest voice in a 5-4-Maj3-5 figure fits with this cadential logic.

However, if we focus only on the pair of upper voices, we find a mediating idiom of 1-Maj2-4: these parts move from a unison on D4 to the mediating tone or major second C4-D4, which resolves to the fourth B3-E3, with the voices then moving in parallel to C4-F4 (the upper fourth of the trine F3-C4-F4). Thus C4 is at once a connecting tone in relation to the lowest voice, and a mediating tone in relation to the highest voice. A detail of this interplay is that the Maj2-4 resolution between the upper voices, which would mark a resolution of tension if these voices were heard alone, is coupled to an overall increase in tension for the three-voice ensemble or concentus, as the relatively blending 5|4_Maj2 moves to the relatively tense Maj6|Maj3_4 -- followed by a satisfying resolution to a complete trine.

Sampling a few of the main mediating, opposing, and connecting idioms is one way to appreciate the variety of intervals and multivoice sonorities available to 13th-century musicians, and also some common ways of alternating between rich vertical stability and apt instability, a contrast at once impelling directed resolutions and providing sheer textural variety and vertical color.

This pluralistic approach to intervals, combinations, resolutions, and melodic-vertical interactions fits nicely with a layered approach to 13th-century composition, where the voices of a new piece are conceived successively, or a third or fourth voice is added to an existing piece. Given the many choices available, one can compose in layers while seeking to maximize melodic grace and vertical beauty at each stage of the process. This is not to say that composers invariably followed such a layered procedure, but that such an approach seems quite congenial to the flexible and encompassing nature of 13th-century polyphony.

3. Conclusion: Unity and diversity

In this survey of the Symphonial Era, I have sought to emphasize the commonalities that knit together its four productive centuries, as well as to convey a sense of some of the intricate and diverse sonorities, progressions, and melodic-vertical interactions typical of this era.

Especially I would like to focus on how the Diaphonic Period of the 9th-11th centuries establishes patterns and procedures both of great interest in themselves, and helpful in understanding the music of the Discantic Period that follows in the 12th-13th centuries. For example, this final cadence from a Montpellier motet for two voices, Endurez/ALLELUYA, piece #248, brings home to me this connection:

  Ex. 25

  Montpellier #248, final cadence

  1   2   3  | 1   2   3 | 1  2  3 ||
  D3  E3       F3  E3      D3
  D3           C3          D3 
  1  Maj2  -   4  Maj3  -  1

The central role of the fourth in this passage, as in a number of two-voice motets from the Montpellier Codex, and the mediating 4-Maj3-1 figure at the end dating back to the Enchiriadis treatises, give a sense of continuity with the world of 900-1100. Yet the new has transformed the old: the first mediating figure of 1-Maj2-4, and the mirrorlike or palindromic symmetry between it and the following 4-Maj3-1 (each being a time-reversed version of the other) are elements that seem uncharacteristic of 10th-11th century technique. Likewise, another excerpt from this piece illustrates the oblique Maj2-1 resolution beloved of Guido for the occursus, but in the context of other 13th-century progressions, with "r" showing a rest in the lowest part:

  Ex. 26 

  Montpellier #248, mm. 5-8

  1   2    3  | 1   2   3 | 1   2   3  | 1   2   3 |
  G3  B(b)3     A3          G3  G3  F3   E3  D3 
  C3            D3          C3           D3      r
  5   min7  -   5                       Maj2-1

If we adopt Yvonne Rokseth's Bb inflection (which is notated in another source for this piece), then we have an opposing figure of 5-min7-5, typical of two-voice as well as multivoice polyphony, with the minor seventh comparable in its degree of tension to the major second, and the min7-5 resolution listed and approved by Jacobus along with Maj2-1 (Book II, Chapter 50). What this piece and much early variable interval organum share is a love of contrast between stable intervals and unstable intervals of various kinds.

In theory as well as practice, there are continuities as well as transformations from the Enchiriadis treatises and Guido to Jacobus. Thus Jacobus cites Guido's example of simple three-voice organum with the fourth placed below the fifth (i.e. 8|4_5) to refute the opinion of Johannes de Muris that the fourth in this position is not a concord (Book VII, Chapters 5-6). At the same time, he explores in depth (Chapter 8) why the fourth is more pleasingly placed above the fifth (i.e. 8|5_4), a 13th-century preference he notes elsewhere, as when observing that this sonority is "the best way one may end measured songs" for three or more voices, Hinc est quod tunc optime cantus mensuratos potest terminare (Book II, Chapter 36).

The solidarity Jacobus shows Guido in using an example from the Micrologus to support the traditional view that the fourth and fifth alike rank as stable concords or symphoniae, even while voicing and seeking the best explanation for the widespread 13th-century perception that between these intervals the fifth is definitely first among equals, is one sign of Ars Nova developments that represent a break from some vital aspects of this tradition.

By 1336, as Rob Wegman details in "What is Counterpoint?" Petrus dictus Palma ociosa has defined the rules of discantus simplex, soon to be known as counterpoint. These rules reflect a new interval aesthetics which, as it develops through the following three centuries or so, will lead music in a different direction than that of the Symphonial Era. The new constraints associated with the art of counterpoint are as follows:

     (1) Excluding the traditional symphonia of the 4:3 fourth from the list of simple and practical concords;

(2) Prohibiting parallel motion involving the other symphoniae or "perfect concords" of the fifth and octave;

(3) Permitting such parallels involving the "imperfect concords," major and minor thirds and sixths, which thus gain an increasingly privileged position; and (4) Relegating all other intervals as "discords" to a more or less ornamental role, including what are in symphonial terms the somewhat "compatible" or even "concordant," major second or ninth and minor seventh.

These rules, if consistently observed, exclude many of the most beautiful idioms of the 9th-11th and 12th-13th centuries alike, even while opening the way for others; such is the nature of shifting artistic fashions. With a composer such as Machaut, who often uses sonorities (e.g. Maj9|5_5, min7|5_min3, Maj6|5_Maj2) and resolutions (e.g. min7-5) typical of the 13th century and approved by Jacobus but in tension with the new rules, while at the same time showing a modern bent for more emphasis on thirds and sixths, understanding both the symphonial and contrapuntal outlooks is essential for a balanced perspective.

The study of each era deserves its own congenial set of concepts and analytical approaches, ideally borrowed from or at any rate not incompatible with period sources. Here I have sought tentatively to propose some approaches for describing and notating sonorities, resolutions, and melodic-vertical idioms (mediating, opposing, connecting). An important caution is that these approaches have mostly grown out of a focus on 13th-century polyphony, and are greatly influenced by two writers summing up the multivoice technique of that century, Johannes de Grocheo (c. 1300?) and Jacobus. How well they fit the 12th century is an open question, and for diaphony of the 9th-11th centuries, the theory of that period is an ideal starting point.

To conclude, I would like most warmly to thank the members of the Ars Antiqua and Ars Nova and related groups on Facebook, and especially Rob Wegman with his scholarly vision and humility, for much inspiration and encouragement leading me to arrive at the concept of a Symphonial Era, and to renew my focus on the writings of Jacobus and the challenge of analyzing medieval polyphony in its own terms. As always, the responsibility for any errors, oversights, or infelicities remains solely mine.


1. For the text and translation of these chapters on two-voice resolutions and multivoice sonorities, see Speculum musicae, Book IV, Chapters 50 and 51. Jacobus uses the term cadentia to describe the tendency and process by which an unstable or less concordant interval "seeks" (petit) resolution to a stable or more concordant interval, either by contrary or oblique motion. As discussed in Section 2.2.1 of this paper, Jacobus describes sonorities for three or more voices as resulting from a process of partitio or partition in which an outer interval is "split" (fissa) into two or more adjacent intervals by one or more added middle voices.

2. The 13 or 14 species of intervals from unison to octave, inclusive, addressed by theorists of the 13th and early 14th centuries are those arising within a seven-note diatonic mode. In Pythagorean intonation, these are the unison (1:1); limma, semitone, or minor second (256:243); tone or major second (9:8); semiditone or minor third (32:27); ditone or major third (81:64); fourth (4:3); semitritone or diminished fifth (1024:729); augmented fourth (729:512); fifth (3:2); semitone-plus-fifth or minor sixth (128:81); tone-plus-fifth or major sixth (27:16); semiditone-plus-fifth or minor seventh (16:9); ditone-plus-fifth or major seventh (243:128); and octave (2:1). Standard 13th-century theory based on 13 interval species recognizes the tritone, e.g. F-B, equal to three 9:8 tones or 729:512 (612 cents), but does not have a separate category for the diminished fifth, e.g. B-F, smaller by a Pythagorean comma at 1024:729 (588 cents). Marchettus (or Marcheto) of Padua in his Lucidarium of 1318 distinguishes between the dyapente imperfectum or diminished fifth and the tritonus or augmented fourth (Treatise 9, Chapter 1); and Jacobus in the Speculum (Book II, Chapters 81-83) recognizes this interval as the semitritonus, noting that it is a comma smaller than the tritone, and in his view not quite so discordant.

Margo Schulter
21 July 2016
Sacramento, CA