¹ Fuller, Sarah, ‘On Sonority in Fourteenth-Century Polyphony: Some Preliminary Reflections’, Journal of Music Theory, 30/1 (1986), 5–70CrossRefGoogle Scholar, at 38, discusses how the simpler discant of thirteenth-century counterpoint came to act as the foundation for more florid polyphony in the fourteenth century. The resultant distinction between levels of surface and essential structure noted by Fuller is represented in the present study by the analysis of single harmonies (the ‘essential consonances’) as active over a measured duration space. Leach, Elizabeth Eva, ‘Counterpoint and Analysis in Fourteenth-Century Song’, Journal of Music Theory, 44/1 (2000), 45–79CrossRefGoogle Scholar, provides a highly detailed heuristic for using basic medieval theoretical tenets, especially for assessing ficta and directed resolutions, as a starting premise for sophisticated modern analysis of Machaut's polyphony. Leach differs from Fuller in parsing three-voice music into simultaneous duets, reflecting the essentially dyadic nature of medieval counterpoint teaching. In portraying the grouping of sonorities at broad metrical levels, local surface misalignments resolve out and I pair the upper voices against the tenor.

² Karen Desmond's recent monograph, Music and the moderni, 1300–1350: The Ars nova in Theory and Practice (Cambridge, 2018), attributes the freedom to notate metrical irregularities in Ars Nova France to a fundamental shift in the conception of time.

³ Metrical organisation is here distinguished from mensural by its tracking of tonal articulation across spans of time independent of any singly notated rhythmic value, though measured in those terms. For example, a work's mensuration of the notated breve, be it perfect or imperfect, is here less pertinent than a sonority of n-breves’ duration that is judged perfect or imperfect at the next higher level (i.e., modus).

⁴ The practice of arranging contrapuncti in vertical alignment with their score helps penetrate melodic embellishments and clarify surface mensurations. See, for example, Hartt, Jared C., ‘Rehearing Machaut's Motets: Taking the Next Step in Understanding Sonority’, Journal of Music Theory, 54/2 (2010), 179–234CrossRefGoogle Scholar, at 206. However, such analyses themselves do not explicitly record the layered metrical play to be detailed here.

⁵ Boogaart, Jacques, Guillaume de Machaut: The Complete Poetry and Music, 13 vols. (Kalamazoo, 2018), 9: 15Google Scholar.

⁶ David Maw's seminal study, ‘“Trespasser mesure”: Meter in Machaut's Polyphonic Songs’, Journal of Musicology, 21/1 (2004), 46–126, by contrast, explicitly claims composer intent for the metrical groupings identified because they result directly from Machaut's notated mensural values. For example, Maw states that ‘the changes from imperfect modus to perfect modus are very much part of the original and indicate that Machaut was thinking not in terms of a consistent modus but in terms of an inconsistent one’ (76).

⁷ Fuller, Sarah, ‘A Motet Conceived in Troubled Times: Machaut's Motet 22’, in A Critical Companion to Medieval Motets, ed. Hartt, Jared C. (Woodbridge, 2018), 321–340Google Scholar, at 329–33.

⁸ Fuller, Sarah, ‘Modal Tenors and Tonal Orientation in Motets of Guillaume de Machaut’, Current Musicology, 45/47 (1990), 199–245Google Scholar, at 229–31, esp. Example 19.

⁹ Boogaart, Jacques, ‘Encompassing Past and Present: Quotations and their Function in Machaut's Motets’, Early Music History, 20 (2001), 1–86CrossRefGoogle Scholar, at 48. Further metrical analysis of Motet 17 is undertaken later.

¹⁰ Maw, ‘“Trespasser mesure”’, 85–7 and 106.

¹¹ See also Maw ‘“Trespasser mesure”’, 55, n. 21. Desmond, Music and the moderni, 216 and passim, makes a clear distinction between notated rhythm and the interpretation of grouping through tonal articulation. She writes that ‘a voice notated in imperfect modus could project perfect or imperfect modus, or even no particular modus at all’.

¹² Maw ‘“Trespasser mesure”’, 57–8. The distinction between practical mensuration and the analysis of metrical organisation is made evident in Jacques Boogaart's 2018 edition of Machaut's motets; for example, in his observation that ‘M20 has just one longa, so that the modus mensuration is irrelevant’. Guillaume de Machaut: The Complete Poetry and Music (Kalamazoo, 2018), 9: 20. From an analytical view, Machaut's directed progressions in that motet effect a regularly alternating pattern of perfect and imperfect modus groups (3+3+2+2+2), which thus alternate imperfect and perfect at the level of maximodus grouping, that is, two groups of 3s + three groups of 2s.

¹³ Friedrich Ludwig, ed., Musikalische Werke: Motetten, vol. 3 (Leipzig, 1954). Leo Schrade, ed., Guillaume de Machaut: Oeuvres completes, 2 vols. (Monaco, 1956). Boogaart, ed., Guillaume de Machaut, vol. 9.

¹⁴ In his critical notes, however, Boogaart acknowledges flexibility at the maximodus level. In Motet 4, for example, he notes that a talea of 17 longs cannot be divided regularly into twos or threes (imperfect or perfect maximodus). Boogaart, Guillaume de Machaut, 9: 196; see also 29.

¹⁵ Boogaart, Guillaume de Machaut, 9: 20.

¹⁶ This flexible, moment-by-moment perspective of medieval metrical organisation is informed by Christopher F. Hasty, Meter as Rhythm (New York, 1997), especially in the continuous reckoning of projected grouping durations with ‘a readiness to interpret emerging novelty in the light of what has gone before’ (69). Speaking more specifically about fourteenth-century notation, Kohn, Karl, ‘The Renotation of Polyphonic Music’, The Music Quarterly, 67/1 (1981), 29–49CrossRefGoogle Scholar, at 31, affirms that mensural notation as a means of quantitative measurement carries no inherent sense of metrical pulsation.

¹⁷ The Berkeley Manuscript implicitly excludes the possibility of syncopation at mensural levels deeper than modus. See Lawrence Earp, ‘Notation II’, in The Cambridge History of Medieval Music, ed. Mark Everist and Thomas Forrest Kelly (Cambridge, 2018), 674–717, at 706. Yet this omission may be a reflection of the practical range of notated values and a medieval focus on the three central mensural levels: modus, tempus and prolatio.

¹⁸ See the table in Strunk, Oliver, ed., Source Readings in Music History: The Early Christian Period and the Latin Middle Ages (New York, 1998), 267Google Scholar. A critical edition is found in Hans Ulrich Michels, ed., Johannes de Muris, Notitia artis musicae, Compendium musicae practicae; with Petrus de Sancto Dionysio, Tractatus de musica, Corpus Scriptorum de Musica 17 (Rome, 1972), 79. For a detailed discussion, see Desmond, Music and the moderni, 104 and 184–6. Similarly, Petrus frater dictus Palma ociosa's Compendium de discantu mensurabili of 1336 goes no deeper than modus in its treatment of practical composition.

¹⁹ Oxford, Bodleian Library, Digby 90, fols. 44v–45r. A translation can be found in Luminita Florea Aluas, ‘The Quatuor principalia musicae: a critical edition and translation, with introduction and commentary’, Ph.D. diss., Indiana University (1996), 673–5.

²⁰ Albert Seay, ed., Declaratio musicae disciplinae, 3 vols. (Rome, 1959), 2: 198–201. Busse Berger, Anna Maria, Mensuration and Proportion Signs: Origins and Evolution (Oxford, 1993)Google Scholar traces the convoluted history of signalling modus; see esp. 14. Most fourteenth-century French manuscripts, such as those of Machaut's motets, use no mensuration signs. Nevertheless, modus is sometimes discernible through the length of long rests, the long spanning either two or three vertical spaces on the staff for imperfect or perfect modus, respectively.

²¹ Leach's valuable Moodle course on reading fourteenth-century notation can be found at: http://diamm.nsms.ox.ac.uk/moodle/ (accessed 17 February 2020). Ruth I. DeFord, Tactus, Mensuration, and Rhythm in Renaissance Music (Cambridge, 2015), 36, observes of fifteenth-century repertoire that ‘minor modus is limited to relatively complex works and major modus is extremely rare’. All these authors refer to practical rhythmic notation in the repertoire, whereas the present analytical study refers to the musical organisation of those measured duration spaces, regarding notated values as idiomatic diminutions.

²² See Maw, ‘“Trespasser mesure”, 46–126. His compendious Table 7 (at 94–5) catalogues the lively motion between perfect and imperfect modus in the polyphonic songs of Machaut. The present study is in part a response to Maw's call for more formal analytical procedures to parse irregular modus in Machaut's output, with focus here on the motets rather than the songs (62).

²³ Maw, ‘“Trespasser mesure”’, 62.

²⁴ Desmond, Music and the moderni, 212–13, n. 38. See Maw, ‘“Trespasser mesure”’, esp. 77, for an argument against mechanical barring and 72 for a score excerpt of Machaut's Honte, paour (B25) barred according to a flexible reading of modus. Maw (at 74) puts it boldly: ‘If the barring [of a modern edition] seems arbitrary or subjective, that is because the rhythmic organization is so; and it is the composer's prerogative that this is how it should be.’

²⁵ Graeme M. Boone, ‘Marking Mensural Time’, Music Theory Spectrum, 22/1 (2000), 1–43, at 11.

²⁶ Ibid., 40.

²⁷ Harald Krebs, Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann (New York, 1999). For an analysis of metric displacement at the modus level in Machaut, see Maw, ‘“Trespasser mesure”’, 68–9. Maw's displacement occurs because of irregular grouping in one voice, as in my own examples.

²⁸ See Desmond, Music and the moderni, 202–9, for a recent review of the development of mensural theory. Desmond details a change in the conception of time from cyclic, in the modal perfections of Garland and Franco, to an unconstrained continuum in Jean des Murs. An important statistical account of the pairing of metrical perfections with consonant perfections in over sixty motets from Montpellier (Bibliothèque Interuniversitaire, Bibliothèque universitaire de médecine, H. 196) to Ivrea (Biblioteca Capitolare, 115) is given in Desmond's Figure 6.1 (206). An initially strong association, the curve trends towards creative substitutions of sonority without seriously challenging the norm of pairing perfect with perfect. Indeed, the stylistic encoding of that pairing enabled composers to play with expectation while maintaining coherence. Examples of this ingenuity in Ars Nova polyphony are detailed in my own analyses of Machaut in the following sections.

²⁹ Sarah Fuller ‘Theoretical Foundations of Early Organum Theory’, Acta Musicologica, 53/1 (1981), 52–84, at 77, notes how the regular grouping of consonances contributes to the perception of downbeats, manifestly so in the bar line practice of modern editors.

³⁰ Discussing twelfth-century polyphony, John Caldwell, ‘Rhythm and Meter’, in The Cambridge History of Medieval Music, ed. Everist and Kelly, 718–46, at 737, notes parenthetically that final syllables coincide ‘on a metrically strong (and ipso facto consonant) position’. Similarly linking metrical grouping with consonance, Fuller, ‘Theoretical Foundations’, 84, finds that ‘in Western medieval music theory a categorical and systematic distinction between consonance and dissonance lay dormant until the emergence of meter’. Desmond, Music and the moderni, 210–12, traces the arrivals of directed progressions on the coincidence of modus downbeats in Philippe de Vitry's Colla/Bona (and other examples throughout chapter 6). The motet's novel mensural design, featuring a tenor in imperfect modus against a perfect upper-voice pair, is highlighted by directed progressions marking the coincidence of their downbeats every six breves (three bars of the tenor against two of the upper voices). My own analytical notation to follow was developed to highlight and interpret such metrical misalignments. Whereas in this example Desmond resolves the modus misalignment into a single tonal rhythm as the driving ‘experiential mensuration’, reasonably based upon regular directed progressions, I wish to emphasise the push and pull of layered metrical dissonance as central to the musical experience and argue for how both tonal and poetic narratives reflect that tension. Desmond's reading of Vitry's Vos/Gratissima (see 222–8) provides a model wherein tonal-metric rupture is foregrounded as the central conceit of the motet.

³¹ Maw, ‘“Trespasser mesure”’, 65, describes how ‘a rhythm of simultaneous articulations can be abstracted from the texture as the durations between such points, and the sequence of these durations may form rhythmic groups that will suggest a metrical outline’.

³² Most inappropriate, for Kohn, was the ‘inevitable assertion of the bar line with its mechanical projection of regular metrical beats [that] generates a superimposed pattern of artificial and distorting pulsations’. See Kohn, ‘Renotation’, 30.

³³ Kohn, ‘Renotation’, 46–7.

³⁴ The interpretation of cadences as downbeats settles the ambiguity of the tenor's modus in Machaut's Motet 10, as described by Margaret Bent, ‘Machaut's Motet 10 and its Interconnections’, in A Critical Companion to Medieval Motets, ed. Jared C. Hartt (Woodbridge, 2018), 301–20, at 309–10. The question of whether the opening breve in the talea pattern BLLLBB is an upbeat (placing the first long on a downbeat) or a syncopating agent that displaces the three longs (completing its own perfection with the final two breves) can be decided by the cadence on that pattern's penultimate breve in breve 35. If that cadence is a downbeat, the first breve must be a pickup. Boogaart's 2018 edition (at 29 and 103) is set in this way.

³⁵ Because medieval notation is so demanding of judgements about accidentals, the chance to design the tonality of a work is ever dangled before the modern editor and analyst. To wit, whether I wish to raise some notated F to F♯ can depend upon whether I think the goal tone G is worth emphasising. Whatever their liberality or reservations about adding musica ficta, the analyst is obligated to first create the score they wish to analyse through a prefatory act of analysis. To recuse myself from such consequential assessments, I have accepted Boogaart's 2018 edition of Machaut's motets (from Mach A, c.1375–77) as my texts, reproducing his ficta above the staff in my figures and Mach A's accidentals in line with their pitches. On the ‘preconditions for analysis’, see Margaret Bent, ‘The Grammar of Early Music: Preconditions for Analysis’, in Tonal Structures in Early Music, ed. Cristle Collins Judd (New York, 1998), 15–59.

³⁶ Alice V. Clark, ‘Tracing the Tenor in Medieval Motets’, in A Critical Companion to Medieval Motets, ed. Jared C. Hartt (Woodbridge, 2018), 61–76, at 65–6, describes the sourcing and liturgical context of this chant tenor.

³⁷ See Sarah Fuller, ‘Tendencies and Resolutions: The Directed Progression in “Ars Nova” Music’, Journal of Music Theory, 36/2 (1992): 229–58, at 234–9, for further detail on the emergence of pitch hierarchies A→G→F in Motet 4.

³⁸ Leach, ‘Counterpoint and Analysis’, 63–5 (esp. Example 10), defines this common procedure as contrapuntal concatenation.

³⁹ I cannot help but suggest bb♭ early in the first two colores for contrapuntal reasons (the minor tenth resolving inward) and to bring them into line with color C's notated bb♭ in the manuscript (Mach A). In the analytical reductions to follow, a choice has been made to display directed over stable harmony when both are set against the same tenor pitch, even if the directed sonority represents only a fraction of the duration of the stable one, because of the defining importance of directed progressions to the mapping of downbeats in this study. For example, see the first directed progression to G in color C (breves 109–11), a quick contrapuntal resolution marking text breaks in the upper-voice poetry.

⁴⁰ Slurs have advantages over arrows, as used by Fuller and Desmond, in specifying which two voices are crucial to a directed progression and in their familiarity as inherently musical symbols.

⁴¹ Kohn, ‘Renotation’, 39–43, sounds an early call for a notation of Renaissance and medieval polyphony that more accurately depicts flexibly independent metrical groupings in each part. His examples include polyphony by Josquin, Dunstable, Landini and Machaut (B4).

⁴² Boogaart, Guillaume de Machaut, 9.

⁴³ The opening twelve breves of each talea are omitted here to focus the figure on isorhythmic boundaries.

⁴⁴ See the discussion of polymetre resulting from metrical displacement in Machaut's polyphonic songs in Maw, ‘“Trespasser mesure”’, 103–6.

⁴⁵ Leo Plantinga, ‘Philippe de Vitry's “Ars Nova”: A Translation’, Journal of Music Theory, 5/2 (1961), 204–23, at 218. Lawrence Earp, ‘Isorhythm’, in A Critical Companion to Medieval Motets, ed. Jared C. Hartt (Woodbridge, 2018), 77–102, at 96, attributes the innovation of red ink to illuminate grouping play to Vitry, as used in his Fauvel motet Garrit/In nova.

⁴⁶ All score examples of Machaut's motets come from Boogaart, Guillaume de Machaut.

⁴⁷ When this repeats in diminution during color C (breves 112–17), I interpret no mensural expansion (i.e., no modus perfection) because Machaut withholds the directed progression to F; recall the exclamation point in Example 2. This alternative tenor setting suggests a composer well-attuned to the interdependency of metrical and tonal emphases.

⁴⁸ The upper voices remain imperfect here so that the motetus's perfect long (on A4) in breve 53, consistent with breves 19 and 87, also initiates on a downbeat.

⁴⁹ While the tenor's metrical independence and overlapping poetry blur the aural strength of arrival here, the methodological focus is on contrapuntal alignment. In the corresponding diminution section, the broadening of modus is omitted but the directed progression to F is retained, arriving together in all voices.

⁵⁰ Alice V. Clark, ‘Listening to Machaut's Motets’, The Journal of Musicology, 21/4 (2004), 487–513, at 500, describes how departure from an established pattern can signify impending closure in Machaut.

⁵¹ I thank my anonymous reader for pointing out this revealing text–music relationship.

⁵² While the tenor/triplum pair does progress +6–8 (G/e–F/f), the triplum is in the middle of a word (‘despair’), it gives F no agogic accent, and the motetus's A does not participate in directed counterpoint.

⁵³ For all its contrapuntal strength, Machaut's text setting bridges across this cadence into the diminution section. The motetus's arrival on F is extended a breve by an illustrative post-cadential flourish on ‘plaisir’. As for the triplum, it is not revealed until the next line of poetry that the prior auspicious list of personifications (e.g., Grace, Love, Generosity) have in fact fallen asleep for the poor amant alone.

⁵⁴ Recent discussions are found in Bent, ‘Machaut's Motet 10’, 313–15 (including a modern transcription); Boogaart, ‘Encompassing Past and Present’, 54–5; Boogaart, Guillaume de Machaut, 196; and Jared C. Hartt, ‘Tonal and Structural Implications of Isorhythmic Design in Guillaume de Machaut's Tenors’, Theory and Practice, 35 (2010), 57–94, at 67–70. See Maw ‘“Trespasser mesure”’, 116–21, on the editorial decisions faced by Ludwig and Schrade at the conclusions of some of Machaut's ballades and his suggested solution. The difficulty lies in realising rests as general pauses or for their notated value, especially when faithful transcription would skew parts out of metric alignment for the end of a section.

⁵⁵ Bent, ‘Machaut's Motet 10’, 313–15.

⁵⁶ Ibid., 315.

⁵⁷ Boogaart, Guillaume de Machaut, 71. His directive to ignore the final rest in the tenor, though it is found in all manuscript sources (196), is unsupported by his edition (71). The tenor rest is required to counterpoint the full duration of the final notated long in both upper voices.

⁵⁸ This simple counting system is a flexible version of that established by Boone, ‘Marking Mensural Time’. A similar system is adopted by Desmond, Music and the moderni, 220.

⁵⁹ In the course of this study, it became necessary to coin a term for this next mensural level. Ultramaximodus, going beyond the medieval maximodus and outside the range of singly notated rhythmic values, is a capacious domain of metrical organisation in the modern reception of Ars Nova music.

⁶⁰ Following Maw, ‘“Trespasser mesure”’, 111, it is often ‘possible to imagine a version of [Machaut's] music in regular meter which was altered to create something less predictable with an irregular meter’. Indeed, an unmeasured contrapunctus represents just such a realignment of counterpoint by the analyst.

⁶¹ Insofar as the metric independence of one voice may suggest successive composition, adjusting other voices to bring the polyphony back into alignment would represent a later stage of composition. See the subtle sense of ‘successive composition’ in Bent, ‘The Grammar of Early Music’, 33, which refers to ‘the procedure uncovered by analysis and inherent in its grammar, and not necessarily to the order of working or compositional process of a native-speaking composer’.

⁶² Acknowledged by Bent, ‘Machaut's Motet 10’, 315. Typical of Machaut's motets generally, the lover and the beloved do not come together in unity at the conclusion of M4. There is, rather, a stubborn commitment to love faithfully until death despite all the suffering and disregard, interpreted in part as metric dislocations.

⁶³ Boogaart, Guillaume de Machaut, 230. His syncopation deviates from Ludwig and Schrade.

⁶⁴ Modern scholars have pointed to how the broadening of mensuration can achieve the effect of what we call a ritard before cadence. Kohn, ‘Renotation’, 41–2, comments upon a cadential retardation effected through mixed metres in the Dunstable motet Sancta Maria. Desmond, Music and the moderni, 217 (Example 6.5), shows a change in all voices from imperfect to perfect modus as cadential retardation to conclude the Fauvel motet Garrit/In nova. Maw, ‘“Trespasser mesure”’, esp. 64–5, locates modus perfections in an imperfect context leading up to cadences in Machaut's ballade Gais et jolis (B35), among other examples. He later (86) calls this effect ‘a reverse hemiola at the cadence’.

⁶⁵ Fuller, ‘On Sonority’, 45, describes the dual nature of the imperfect ⁵₋₃, which she marks with the symbol ł.

⁶⁶ Described in Boogaart, ‘Encompassing Past and Present’, 47–50.

⁶⁷ The contratenor's E sounds below the structural G⁺⁶₊₃ in breve 5 before skipping up to G in unison with the tenor. Insofar as this ‘inverted’ E⁵_-3 is directed towards the very same resolution on F⁸₅, the reduction in Figure 13b represents G⁺⁶₊₃ as the primary form that encapsulates the analogous sonority.

⁶⁸ In his critical notes to Motet 5, Boogaart, Guillaume de Machaut, 9:200, prefers B♭ for contrapuntal reasons.

⁶⁹ A helpful précis on colouration practice at the turn of the fifteenth century is Margaret Bent, ‘Principles of Mensuration and Coloration: Virtuosity and Anomalies in the Old Hall Manuscript’, in Le notationi della polifonia vocale dei secoli IX–XVII: Antologia–Parte Seconda, ed. Antonio Delfino (Pisa, forthcoming).

⁷⁰ Maw, ‘“Trespasser mesure”’, 97.