16 Theory and notation

Dolores Pesce

An overview of medieval theory involves three primary content areas: pitch, rhythm and counterpoint. Two far-reaching concerns of medieval theory surface time and again in this overview: early theorists continually attempted to understand an inherited repertory, chant, in terms of an evolving theory, and they tried to bring their own theory into congruence with venerated ancient Greek theory. Not unrelated, the writers searched for a satisfactory notation, both letters and music symbols, to transmit their repertory and theory about it. As we read these theorists’ exposés, differences emerge in how they formulate and present their ideas. With respect to audience, teaching in the Middle Ages shifted increasingly from monastic settings to cathedral school and university settings. We also witness certain differences in genre presentation, including dialogues and compendia. Finally, it is useful to think of early theoretical writings in terms of categories set up by Claude Palisca: precompositional, compositional, executive (performance) and critical. ¹ That is, theorists could engage in theory largely for its own sake, prescribe how to compose and perform, and describe/critique the music they were hearing in their respective societies.

Early Middle Ages

The writer who unquestionably exerted the greatest influence on medieval theorists was Boethius (AD 480–524). In De institutione musica , ² one of four treatises he devoted to the quadrivium or the four mathematical disciplines, Boethius foregrounded an understanding of music in terms of numerical ratios: he interpreted musical intervals as consonant or dissonant in accordance with the simplicity of their ratios. Through Boethius the emphasis on unison, octave, fifth and fourth as perfect consonances entered the consciousness of the medieval mind.

Although Boethius had taken his harmonic science from the Greek writer Nicomachus (first–second century), he contributed his own thoughts on the classification of music and the definition of a musician. He divided music into three types: musica mundana, musica humana and musica instrumentalis . Thus, music determines planet movements and other natural cycles, harmonizes the body and soul, and is created by man through his voice or an instrument. Music, in essence, is a microcosmic reflection of the macrocosmic experience. Boethius also emphasized music's potential ethical power to influence human behaviour, an idea he had inherited from Plato. As to the musicus or musician, Boethius asserted that this individual (as opposed to one who composed or performed) applied his knowledge of music's principles to judging music compositions and performances – in short, the musicus exercised critical judgement. Although many medieval theorists passed on Boethius's definition of a musician, some challenged the elevated status of the musicus as opposed to a singer or cantor whose practical musical skills aided in the Christian act of worship.

The final aspects of Boethius's theory that affected medieval theory concerned letter notation and scales. Using a monochord to derive his tonal system, Boethius designated points on the string with the letters A to P; although he may have intended these letters in a theoretical sense only, some tenth- and eleventh-century manuscripts actually notated chant using A to P. Boethius also explained the Greek tonoi , or transpositions of the entire two-octave tonal system, each of which carried a Greek name such as Dorian, Phrygian, Lydian. Later writers appropriated these tribal names as designations for the medieval modes.

Three other pre-ninth-century writers deserve mention. Martianus Capella (first part of the fifth century) predates Boethius; his significance rests in having supplied a widely read allegory about the seven liberal arts, in which music was included. ³ Cassiodorus (ca485–580) was pivotal, with Martianus, in establishing the number of liberal arts as seven; he also argued that music theory could be founded on a quadrivial approach yet remain applicable to Christian practice. ⁴ Isidore of Seville (559–636) provided an account of musical terminology in his encyclopaedia on all branches of knowledge. Most important, he described music in the divine office, offering us a glimpse of musical practice at a very early date. ⁵

From the seventh to the ninth century no new music theory appeared. Beginning in the ninth century manuscripts with some music notation survive. Charlemagne's politically driven educational and ecclesiastical reforms are discussed in Chapter 15, but the reforms’ importance must be reiterated in a discussion of how music theory grew and notation took shape around this time. To guarantee that his kingdom would be strong, Charlemagne followed on Alcuin's late-eighth-century educational initiative, and furthermore promoted a unified liturgy, both textually and musically. Charlemagne wanted to disseminate throughout the Frankish kingdom the chant heard by his emissaries in Rome, the so-called Gregorian chant, whose development was credited to Pope Gregory I (590–604). Hence among the Franks the impetus grew to develop a musical notation and theory through which singers could be trained to read and sing the all-important music of the church.

As we turn to the ninth-century treatises, we recognize that none of the specific notational solutions they proposed was widely adopted. But some surviving ninth- and tenth-century manuscripts reveal a first step in notational standardization: they use musical symbols called neumes, probably derived from the Greek system of accents, indicating from one to usually four notes; the neumes themselves may be diastematic, that is, they indicate the pitch of notes by their vertical placing on the page (in the absence of ruled horizontal lines), or non-diastematic (which requires the singer to remember the relative directional movement from one neume to another). A handful of manuscripts from the St Gall monastery uniquely preserve neumes with performance nuances such as lengthening of note values, microtonal inflections, acceleration, etcetera Surviving chant manuscripts in general reveal a fair amount of uniformity in their pitch content, despite the regional notational dialects that emerged. ⁶

Aurelian (fl. ?840), one of several ninth-century Frankish theorists, provides a view of plainchant repertory and its performance at this time. Unfortunately, he did not possess the nomenclature to describe with any accuracy what he experienced, so his treatise Musica disciplina is of limited use when we attempt to re-create his examples of modes, psalm formulas, and office and mass chants. ⁷ The slightly later anonymous Alia musica (ca900) deserves mention: it associated the medieval modes with the Greek tribal names and their associated octave species. ⁸ For some time, this appropriation was an isolated case, but use of the Greek tribal names eventually became the norm once a scalar understanding of mode took hold.

The third ninth-century writer is Hucbald (ca840–930), who wrote his Musica as a handbook for training young monks in psalmody. ⁹ Musica is organized in a cyclical manner, going through its subjects three times, each time elaborating them and compensating for the lack of a dedicated medieval music nomenclature: he asked the reader to recall chant melodies which illustrated his points; he referred to Boethius's A–P notation and even suggested adding some of the letters, in lower-case form, to neumes to pinpoint some pitches; and he proposed a kind of graphic notation in which syllables of chants would be placed between lines on a six-line staff, to which tone (T) and semitone (S) would be attached in the manner of clefs. Most important, Hucbald fruitfully began the process of discussing the tonal system that lay behind the chant repertory; provided with such a vocabulary, the singers could learn the chants more quickly. In essence, he introduced the idea of the tetrachord of the finals D, E, F, G as the primary building block of the medieval gamut, explaining it by reference to the Greek division of their tonal system by a different tetrachord; the medieval gamut is thus divided into 4 tetrachords whose pitches are separated by tone–semitone–tone (Hucbald's Greek names for letters are given below in their modern equivalent):

The final relevant ninth-century treatises are known as Musica enchiriadis and Scolica enchiriadis (reaching their standard form by the end of the century). ¹⁰ Both treatises serve as handbooks for training singers (hence enchiriadis ), while the Scolica additionally presents the quadrivial rudiments. The Scolica was written as a dialogue between master and pupil, modelled after Augustine's De musica . The Enchiriadis treatises provided the earliest instruction in improvising organum or polyphony: organum consisted of a vox organalis moving in parallel movement with the chant or vox principalis , at the perfect intervals of an octave, fifth or fourth. A second type of organum grew from the practice of sounding a drone under the chant: the vox organalis , at first stationary (that is, in oblique motion), moves into parallel motion with the chant, then reaches a unison with the chant at the end. The Enchiriadis treatises presented their exposés using a variation of the notation Hucbald had proposed, namely, text syllables are placed in between lines. But in this case, the lines are prefaced by a system of clefs derived from manipulations of the daseia of ancient Greek prosody, hence the term Daseian notation. While the concept of a staff, albeit with notes on the lines and spaces, was to become standard, the Daseian symbols presented orthographic difficulties that prevented their widespread adoption.

Medieval music theory experienced a breakthrough in the late tenth century with the treatise known as Dialogus de musica , most likely written in the province of Milan by an anonymous monk. ¹¹ Presented as a dialogue between master and pupil, the Dialogus codified the letters A B C D E F G with octave duplication as the standard nomenclature for the medieval gamut; the next octave was designated by lower-case a–g, culminating in aa to complete the two-octave span (below A lay Γ, hence the word ‘gamut’). The tone b was inflected: it could appear as ‘square’ or ‘hard’ b ( ) and ‘round’ or ‘soft’ b (♭ ). One of the writer's pedagogical innovations was to promote the use of the monochord, whereby a singer could learn a chant quickly by imitating its intervals sounded on the monochord. Significantly, he contributed to the classification of chants according to mode, which also helped singers master the vast repertory. Following on Hucbald's ideas about the importance of the finals D E F G within the gamut, the Dialogus author identified their importance for modal recognition in the first clear definition of mode: ‘A tone, or mode, is a rule which classes every melody according to its final.’ That is, the final of a chant identifies its mode. Each of the finals is designated by a Greek number, as shown in column 2 of Table 16.1,and each numerical modal category in turn can be subdivided into authentic and plagal, depending on the range of the melody: specifically, an authentic melody usually does not ascend more than an octave above and one tone below the final, while a plagal melody usually does not ascend more than a fifth, sometimes a sixth above, and a fifth below the final. (An alternative naming system shown in Table 16.1 simply assigned two numbers in order to each final for its authentic and plagal form.) Despite his spelling out of the ranges as quasi-octaves in order to distinguish authentic and plagal, the Dialogus author was not proposing a scalar concept of mode; clearly, a final could identify a melody's mode only by virtue of its distinguishing intervallic movement around that final tone: D protus moves a tone and a semitone up, a tone down; E deuterus moves a semitone and a tone up, a tone down; F tritus moves two tones up, a semitone down; G tetrardus moves two tones up, a tone down.

Table 16.1 Finals and their modes

Often accompanied by the Dialogus de musica , Guido d'Arezzo's treatises (1026–33) were the most widely circulated music writings of the Middle Ages. ¹² Since Guido devoted his writings to his thoughts on how to train a choir, probably in Arezzo, his works have little quadrivial content. The wide-ranging topics of his Micrologus include an introduction to the emotional qualities of the modes, a method for composing chant, the rhythmic performance of neumes, and a brief overview of organum. The latter suggests he witnessed the parallel organum described in the Enchiriadis treatises, but also a freer sort of parallel organum into which a higher degree of oblique motion had infiltrated.

The Micrologus also extended the discussion begun by Hucbald about the gamut's construction; although Guido did not use tetrachord terminology, he recognized that the interval set of the finals D E F G (tone–semitone–tone) was replicated at the fifth above on a c d and fourth below on A B C D, a relationship he referred to as affinitas . Guido recognized a practical aspect of this theory by stating that these related tones can serve as alternative finals or affinales (at least for a c) since they share the same configuration of surrounding intervals as the finals. In his Epistola ad Michahelem , Guido presented what is now a well-known pedagogical tool to aid singing, the hymn Ut queant laxis , whose lines’ opening syllables are ut re mi fa sol la , matching up with the pitches C D E F G a. Although Guido did not discuss a second site at a c d e f, one can assume that he recognized it, since the core of these two six-note segments are, respectively, the four finals and the four affinales or cofinales , whose relationship he did acknowledge. Eventually, the two segments came to be known as the natural and hard hexachords, while a third segment with b flat, F G a ♭ c d, was called the soft hexachord. Mi–fa signalled the semitone in any of the three locations. The fundamental importance of the hexachordal concept in medieval musicians’ thinking is reflected in the fact that from 1270 onward, some theorists, in discussing the mode of a chant, referred to a modal final by its pitch or by its vox or syllable, so that re signalled protus, mi deuterus, fa tritus , and sol tetrardus . ¹³

Guido's final pedagogical contribution involved notation. By the late tenth century some scribes had begun to scratch onto the parchment a single line in relationship to which neumes could be arranged diastematically. What Guido seems to have promoted for the first time was multiple lines separated by a third. He also prescribed using letter clefs before certain lines or spaces, but a surviving manuscript from Dijon suggests that letter clefs (without lines) may have predated 1031. His third suggestion, unique to his writings, involved adding coloured lines to certain lines or spaces, notably those signifying C and F, the two notes in the gamut distinguishable by the semitone that falls below them. This last innovation manifested itself in European manuscripts from the eleventh to fourteenth centuries. Of course, the multiple-line staff based on the separating interval of a third has continued to the present day.

Two non-Italian theorists deserve some mention in this overview of early theory. A contemporary of Guido, Hermannus Contractus (1013–54) was a Benedictine monk associated with Reichenau. Hermannus mentioned the principle of affinitas as he knew it through Guido as well as his own interpretation, which grew from an eleventh-century Germanic emphasis on species of fourth, fifth and octaves as basic building blocks of the tonal system: that is, D E F G and a c d are identical tetrachords, an identity retained when one extends the core by one tone in either direction, yielding two identical six-note segments. ¹⁴

The other non-Italian was Johannes Affligemensis (fl. 1100), most likely from southern Germany or northeast Switzerland. Johannes's De musica , essentially a reworking and expansion of Guido's Micrologus , diverged in its prescription for writing organum: Johannes favoured contrary motion in the newly composed voice. ¹⁵ Examples of free organum survive from the late eleventh century and became the norm in the twelfth century.

Early polyphony and mensural music

Whereas how to discuss, notate, and teach chant dominated Carolingian music writings, twelfth- and thirteenth-century theoretical writings turned to polyphony – how to improvise it, compose it and notate its rhythm. The theoretical endeavours were initially centred in Paris, whose cathedral of Notre Dame spawned a renowned body of polyphony and whose university attracted scholars and students to ponder man's achievements, music included. The scholasticism of Parisian thinkers at the time affected the cast of some music theoretical writings.

The music of the cathedral of Notre Dame survives in three main sources dating from considerably later than the repertory's conception between the 1150s and ca1200: W ₁ (1230s), F (1245–55), W ₂ (ca1260). ¹⁶ This gap between conception and transmission allows that the notation does not necessarily represent what was current at the time of conception. When we turn to the theorists for clarification, particularly of the rhythm, a similar disconnect occurs, since they too – Johannes de Garlandia (second quarter of the thirteenth century), the St Emmeram Anonymous (1279), Franco of Cologne (after 1279), Anonymous 4 (after 1279) – were looking retrospectively at the Notre Dame repertory.

Until recently Garlandia was thought to have written De musica plana and De mensurabili musica ¹⁷ as texts for the University of Paris; the most recent scholarship suggests they were probably the work of another, nameless author active about the middle of the thirteenth century. ¹⁸ Following the scholastic tendency to create classification schemata, this anonymous author presented a new consonance theory with three subdivisions: perfect (unison and octave); intermediate (fourth and fifth); imperfect (major and minor thirds). Dissonances too were imperfect (major sixth, minor seventh); intermediate (whole tone, minor sixth); perfect (semitone, tritone, major seventh). A similar classifying tendency led the author to present an elaborate working out of the rhythmic modes that underlay Notre Dame polyphony. The rhythmic modes, particularly relevant to the subspecies of polyphony called discant (where both voices are measured), were based on a common unit called the tempus . The tempus could appear singly (a short or breve ), in a note twice its length (a long or longus ), or three times its length (an extended long). Assuming a tempus of , the six rhythmic modes were:

A discant in a given mode reiterates the succession of values proper to that mode. A glimpse at the Notre Dame sources does not in fact reveal all of the modes nor all of the variations on them that the author proposed. ¹⁹

This same author explained the notational concept that underlay the rhythmic modes: the modal pattern is not shown primarily by individual note shapes, but instead by a combination of ligatures (neumes) and single notes of the ‘square’ plainchant notation that developed in France during the twelfth century. Each mode was identified by a particular succession of ligatures and single notes, as follows: first mode: 3–2–2…2; second mode: 2–2–2…3; third mode: 1–3–3…3; etc. ²⁰ This notation relying on ligature patterns was appropriate only for melismatic passages. Even then, the author recognized its ambiguity, particularly for more complicated rhythmic substitutions, and made some innovative suggestions. But these were codified and applied more generally only through the influence of Franco of Cologne's Ars cantus mensurabilis , ²¹ hence the expression Franconian notation. Franconian notation specified a distinct note shape for a short value and the various forms of a long, presented rules for ligatures in standard and modified form, and created a set of graphics for rests. Thus mensural notation provided a reasonably unambiguous way to present rhythm in syllabic contexts. Once such unequivocal symbols were available, composers could move outside the confines of the rhythmic modes and their underlying adherence to triple metre. But most music continued for some time to be written under the influence of the rhythmic modes, as witnessed by the primary motet manuscripts from the later part of the thirteenth and early fourteenth centuries: Montpellier, Bamberg, La Clayette and Las Huelgas. ²²

Mention of the motet links back to the issue of polyphonic genres and styles. The writing previously attributed to Garlandia described organum per se , copula, and discant, three styles that coexist within a two-part organum (still the generic term for polyphony in the thirteenth century). Whereas discant is measured in both voices, copula has sustained tones in the lower voice (tenor) presenting the chant, while the upper organal voice uses the rhythmic modes; organum per se is distinguished by sustained notes in the tenor and a rhythm not strictly modal (modus non rectus ) in the upper part (that is, the modal ligature patterns appear sometimes, but not always). Organum per se is thus not measured in the regular way that discant and copula are, and the treatise offered three not always compatible rules for distinguishing long and short notes in this species: they included judging the length of a note by its relative consonance with the tenor, and attending to its note shape. The explanations for interpreting organum per se given by the St Emmeram Anonymous ²³ and Franco of Cologne are no less ambiguous. Consequently, the rhythm of these sections in Notre Dame polyphony remains conjectural in modern-day performances. Some scholars argue that a flexible approach is consistent with the rise of Notre Dame polyphony from an improvisatory tradition.

The three main Notre Dame sources contain conductus and motets, in addition to organum. Franco of Cologne referred to the conductus as the one type of liturgical music that was not based on a pre-existing melody. He mentioned the motet without defining it; in short, it is a genre that arose from texting the upper voice(s) of discant sections of organum, while the lower voice (the chant-based tenor) remained untexted. Conductus notation is problematic in that this syllabic genre required the use of single notes, whose as yet unstandardized shapes could not convey an unequivocal rhythmic meaning. The few comments offered by the theorists, together with a verse analysis of the poetic text, make it possible to come up with reasonable guidelines for rhythmic interpretation of the genre. Similar problems plague the early motets transmitted in the Notre Dame sources, but the genre attained clarity through the use of Franconian notation to varying degrees in the main motet manuscripts from the end of the thirteenth and early fourteenth centuries.

One final rhythmic development of the thirteenth century concerned shorter note values. Franco extended the basic relationship that governed the long ( ) and breve (▄ ) to the breve (▄ ) and semibreve (◆ ), that is, the principle of ternary mensuration. Petrus de Cruce (fl. ca1290) took the next step of introducing a notation that could allow other subdivisions of the breve : he accepted up to seven ◆ within the breve , subdivisions that could be shown by placing a dot of division (punctus divisionis ) on either side of the grouping. ²⁴

Finally, the thirteenth-century treatise by Hieronymus de Moravia, written shortly after 1272, represents a new genre: a compilation made up of excerpts and a few entire treatises, ranging from Boethius to Hieronymus's contemporaries. Hieronymus was a member of the Dominican order, believed to have been active in Paris at the order's convent on the rue St-Jacques. He apparently compiled the work to help his fellow Dominicans to judge, compose and perform chant and polyphony, but he also addressed some precompositional concerns of quadrivial writers, including the science of harmonics. In Hieronymus's four positiones or theses on polyphony, he transmitted the mensural treatise formerly attributed to Johannes de Garlandia uniquely with chapters on three- and four-voice writing. ²⁵

The fourteenth century

France and England

Fourteenth-century theorists confronted two main challenges: how to expand the notational system to accommodate duple as well as triple metres and note values shorter than the semibreve , and how to contextualize the increasing number of accidental pitches appearing in polyphony. The latter concern at times took them into highly speculative realms, in which they expanded on the mathematical calculations of their Greek forebears. Some theorists also provided a critical, quasi-historiographical view of earlier music which was still being performed even as new repertory emerged.

The primary name associated with fourteenth-century rhythmic theory is Philippe de Vitry (b. ?Champagne, 31 October 1291; d. 9 June 1361), a composer and bishop who travelled in royal and princely circles from the 1320s onward. Although de Vitry was long considered the author of a treatise known as Ars nova , scholars now agree that while its teaching may be associated with Vitry and with the innovations of his compositions, its connection with him as author is tenuous. The treatise can be dated to around 1320 based on comparisons of its contents with other dated treatises. ²⁶

While the thirteenth-century motet repertory offered some isolated examples of duple metre, the fourteenth century witnessed its full-blown acceptance on an equal level with triple metre. This change in rhythmic profile is acknowledged by both the Ars nova and Johannes de Muris (b. diocese of Lisieux, ca1290–95; d. after 1344), the latter in his Notitia artis musicae of 1319–21. ²⁷ The French notational system now contained four graphically distinct values: long ( ), breve (▄ ), semibreve (◆ ), and minim ( ). The relationship of long to breve was termed modus , of breve to semibreve tempus , and of semibreve to minim prolatio . Each of these relationships could be ternary (perfect or major) or binary (imperfect or minor) and their combinations came to be known as mensurations. The signs for the four combinations of tempus and prolatio were attributed to de Vitry:

These signs did not attain much currency until late in the century. While the L had been the basic unit of the musical measure in the thirteenth century, the B took that role in the fourteenth century. Most upper-voice parts moved in a combination of tempus and prolatio as shown above, while lower-voice parts used modus and tempus .

The other notational innovation of the Ars nova treatise was the use of red colour to indicate a number of rhythmic changes. Where black notes were perfect, red signalled imperfect modus or imperfect modus and tempus ; the roles of black and red could also be reversed. Furthermore, red could be used to prevent individual notes from being perfect or altered (that is, to fix their value regardless of context). Later fourteenth-century treatises begin to discuss dots: dots of division on either side of a group of notes indicated how many fell within the breve , while a dot of addition added half again to the value of an imperfect note, a usage it retains to the present.

Outside the realm of rhythmic concerns, several fourteenth-century treatises deserve mention. Walter Odington and Jacobus de Liège wrote significant compendia. The Benedictine monk Odington (fl. 1298–1316), an English theorist and scientist, dedicated part of his Summa de speculatione musice ²⁸ to the quadrivial aspects of music theory, then turned to the practising musician, including sections on chant, a tonary, discant theory and notation, and a discussion of polyphonic genres. The Summa continued to be copied into the fifteenth century. Jacobus of Liège (born ca1260; died after 1330), a Franco-Flemish theorist, wrote his principal work, the Speculum musice , ²⁹ probably in Liège, not before 1330, after he had spent most of his life in Paris. The Speculum is the largest surviving medieval treatise on music, containing 521 chapters arranged in seven books, of which the first five treat speculative music largely according to Boethius; the sixth deals with ecclesiastical chant, and the seventh discant. In the seventh, Jacobus defended ‘ancient’ practice according to Franco of Cologne versus the more modern rhythmic and notational usages he was witnessing. He commented specifically on how there had been a slowing of performance tempo to accommodate the smaller note values (minims ) employed by the ‘moderns’, and he also weighed the relative perfection, subtlety, freedom and stability of the older and newer styles. Like Odington, Jacobus discussed various categories of polyphonic composition.

The issue of genres brings us another theorist who contributed a great deal to our understanding of medieval music and its performance contexts, Johannes de Grocheio (fl. ca1300) in his De musica . ³⁰ Grocheio is linked to Paris both because his observations focused most pointedly on the practices of that city and because he showed a deep knowledge of Aristotelian thinking, which may have been garnered while he was a Parisian master. Grocheio's significance lies in his abandonment of the Boethian taxonomy of music that had been handed down for centuries. First, he rejected celestial music (Boethius's musica mundana ), then set out his own classifications for Paris: musica civilis (music for laymen), musica canonica (music for clerics) and musica ecclesiastica (chants of the mass and offices). He rejected the dichotomy between measured and unmeasured music, saying that all music is measured to some degree. Whereas musica canonica included precisely measured polyphonic music such as the motet to be performed before individuals who could appreciate subtleties in the arts, musica civilis included a variety of monophonic music such as trouvère songs, rondeaux, epics, dances, and instrumental genres. Interestingly, when Grocheio described chants, he compared them to secular genres, perhaps suggesting the ‘affective power of plainchant over the minds of clergy and laity alike’. ³¹ By applying to his generic taxonomy the Aristotelian practice of describing each subject three times, Grocheio offered a highly detailed and perceptive account of musical experience in his time.

Italy

Marchetto da Padova wrote his Lucidarium in 1317 or 1318 and the Pomerium shortly thereafter but no later than 1319. ³² Marchetto is a valuable source of information on the rhythm not only of Italian music of the early fourteenth century but of contemporaneous French music as well. The earliest surviving Italian manuscripts date from the mid to late fourteenth century: Rossi/Ostiglia and Panciatichi ³³ .

Italians based their notational system on a breve which could encompass 8 divisiones as follows, wherein the first option is to divide the breve into 2 or 3 semibreves :

On the level of the divisiones secunda and tertia , the semibrevis carried an upward stem (we would call it a minim ). A given grouping of notes within one divisio was marked with a dot on either side, a feature suggesting that Italian notation was related to Petronian usage. In some manuscripts, a piece's divisio was indicated at the outset by an abbreviation: .q.: quaternaria ; .i.: senaria imperfecta ; .p.: senaria perfecta; .n.: novenaria; .o.: octonaria; .d.: duodenaria .

This notational system was responsive to the fluid, melodic character of Italian music, characterized by chains of short note values, often in sequential patterns. It also allowed for adjustments and variations within a divisio , both through additional note shapes and contextual rules such as via naturae (when less than the full complement of notes are present in a divisio , long notes appear at the end of a group) versus via artis (larger values at the beginning or middle of the group, indicated by an altered semibreve shape). On the other hand, the system basically precluded syncopation across the breve unit. Perhaps because of this feature and because of Italian contact with contemporary French music which showed a predilection for syncopation, a hybrid notation developed. It combined French and Italian features, which scholars today call ‘mixed’ notation. In fact, many of the surviving Italian manuscripts that preserve fourteenth-century repertory are primarily in mixed notation.

Marchetto da Padova deserves further mention because he developed an influential extended theory of mode for plainchant. According to Marchetto, a mode is either perfect, imperfect, pluperfect or mixed, depending on whether its range is respectively normal, narrow, wide in the direction away from the mode's authentic or plagal partner, or wide in the direction of that of the partner. A further term, ‘mingled’ (commixtus ), was applied when the mode in question showed qualities of a mode other than its authentic or plagal partner. In addition, Marchetto tackled the problem of explaining the increasing number of inflected pitches in Italian pieces by devising a division of the Pythagorean whole tone into five equal parts. His thinking represented a highly innovative, speculative extension of what had been passed down from the Greeks, and it opened the way to experiments in tuning and temperament in centuries to come.

The late fourteenth century

The end of the fourteenth century witnessed an extraordinarily complicated music and notation that has come to be known as mannered or Ars Subtilior, based on Philippus de Caserta's Tractatus de diversis figuris (ca1370), which referred to composers moving away from the style of the Ars Nova motets ‘post modum subtiliorem comparantes’ and developing an ‘artem magis subtiliter’; ³⁴ similarly Egidius de Murino (uncertain dating) referred to composition ‘per viam subtilitatis’ in his Tractatus cantus mensurabilis . ³⁵ The musical style demanded greater rhythmic complexities, primarily of syncopation and proportional relationships, to which end the two primary manuscript transmitters, Chantilly and Modena, ³⁶ are filled with new note shapes, many of which have varied meanings from piece to piece and even within a piece. Red coloration, both filled and void, is used generously, and a beautifully decorated circle and heart convey the technique of canon.

This striking style developed in the secular courts of southern France, Aragon and Cyprus during the period known as the Great Schism. From 1309 to 1377 the papacy had been exiled to Avignon, but now the schism (1378–1417) produced rival popes in Avignon and Rome. The creative impetus in the realm of sacred music first associated with the papal chapel and rival cardinals’ chapels now transferred to secular courts, which had available to them virtuosic singers and an intellectually sophisticated audience who were amused by both the notational and sounding complexities. The Ars Subtilior manuscripts include works by both French and Italian composers.

Mode and musica ficta

The theory of the eight church modes (protus, deuterus, tritus, tetrardus , each with an authentic and plagal member), conceived for plainchant, was regularly appended to medieval discant treatises although most did not establish any definitive link between composing counterpoint and modal theory. A continuous tradition applying mode to polyphony began only in 1476 with Tinctoris, ³⁷ who worked with Marchetto's extended theory of mode. Tinctoris touted the tenor as establishing the mode for the polyphonic complex as a whole, though scholars interpret this to mean the soprano–tenor pair which together form most of the principal cadences. As for polyphonic music before Tinctoris, often one or two tonal foci inherent in the pre-existing tenor may be emphasized as a piece unfolds. ³⁸ Sarah Fuller holds that a composer such as Machaut chose tenors not for their modal coherence, but for their distinctive tonal traits that could be developed through harmonic and linear means. ³⁹ One can describe many thirteenth- and fourteenth-century pieces not so much as being ‘in a mode’, but rather as having certain tonal emphases.

Modern scholars refer to the issue of pitch inflections in medieval music as musica ficta . As explained earlier, the medieval gamut, allowed for the inflection of the tone b as hard b ( ) and soft b (♭ ). The entire gamut, including the inflected b, came to be known as musica vera or recta . From the ninth century onward, medieval theorists began to discuss tones they could not notate within their tonal system, at first referring to them as semitones outside the gamut, in the thirteenth century as musica falsa , and in the fourteenth century as musica ficta or coniuncta . Such tones could arise in plainchant, but theorists tended to concentrate on their occurrence in polyphony where they were essential to consonant contrapuntal writing; this was particularly true in the fourteenth century when theorists placed greater stress on contrary motion and a controlled succession of imperfect to perfect intervals. A perfect interval is approached by a third or sixth, with a semitone movement in one part; thus, at cadences, an octave is preceded by a major sixth, a unison by a minor third, and a perfect fifth by a major third. Theorists linked ficta notes to producing such correct interval successions; for example, when the sixth B–G progresses to the octave A–A, either B or G would move to A by a semitone, requiring that one of them be inflected.

A few theorists from ca1300 onward distinguished the reasons for using semitones as causa necessitatis and causa pulchritudinis ; the former apparently referred to the essential correcting of vertical perfect consonances (for example, making a diminished fifth perfect) while the latter referred to the ‘colour of beauty’ that resulted when a tone was inflected in the imperfect to perfect progression. ⁴⁰ Beginning with the treatise formerly attributed to Johannes de Garlandia, the latter inflected tones are described as melodic ‘leading notes’ a semitone from their destination. More detail comes from Johannes de Muris who stated that lower returning notes (e.g. in the progression G–F–G) should be raised (G–F♯ –G); and that leading notes approached by any other means (for example, by leap) should be raised. This concept of leading tones in imperfect to perfect progressions resulted in double leading-tone cadences in fourteenth-century manuscripts; in essence, two interval progressions are combined: a sixth to octave and a third to fifth, each with inflected tone. ⁴¹

Medieval manuscripts do not always show pitch inflections in the very situations theorists described as requiring them. Most scholars accept that inflections were and should be applied according to a partly or largely unnotated tradition. ⁴² This viewpoint is supported by the late-fourteenth-century Berkeley treatise author: ‘But these are frequently present virtually in BfaBmi although not always notated’, ⁴³ as well as several others slightly later. When signs were used, they were the same ones, and ♭ , hard and soft b, that were applied to the tone b. Hard b signified mi and soft b fa , that is, they indicated where the semitone lies in relation to the sign, even when it occurs in a location other than in the three basic hexachords on C, F and G. By referring to inflected notes in this way, medieval writers revealed that the ut–la syllables served as the primary navigational tool for discussing their tonal system. Many scholars today hold that the syllables served the same purpose in practice. ⁴⁴