¹ Ragarding the transmission of the Paris repertory, see Roesner, E.H., ‘The Problem of Chronology in the Transmission of Organum Duplum’, Music in Medieval and Early Modern Europe: Patronage, Sources and Texts, ed. Fenlon, I. (Cambridge, 1981), pp. 365–99Google Scholar, to which the present paper is a companion study; and ‘“Peripherie” und “Zentrum” in der Geschichte der ein-und mehrstimmigen Musik des 12. bis 14. Jahrhunderts’, round-table discussion chaired by Arlt, W., Bericht über den internationalen musikwissenschaftlichen Kongress, Berlin 1974, ed. Kuhn, H. and NItsche, P. (Kassel, 1980), esp. pp. 24–135Google Scholar. On the change from a non-written to a written mode of composition, see Treitle, L., ‘Oral, Written, and Literate Process in the Transmission of Medieval Music’, Speculum, 56 (1981), pp. 471–91CrossRefGoogle Scholar.

² One aspect of the change in character of the theoretical writical writings of the thirteenth century is discussed in Fuller, S., ‘Theoretical Foundations Early Organum Theory’, Acta Musicologiac, 53 (1981), pp. 52–84 (esp. pp. 73–82)CrossRefGoogle Scholar.

³ Compare, for example, the mensural redactions of organum duplum and triplum in, respectively Berlin, Staatsbibliothek der Stiftung Preussischer Kulturbesitz, MS lat. 4°523, and Montpellier, Faculté de Médecine, MS h. 196, with their concordances in such modally written sources as Florence, Bebliothek, MS 628 Helmst., and Wolfenbüttel, Herzog August Bibliothek, MS 1099 Helmst. The character of the rhythmic designs not expressible in modal notation.

⁴ On Johannes de Garlandia see esp. Reimer, E. Johannes de Garlandia, De mensurabili musica; Kritische Edition mit Kommentar und Interpretation der Notationslehre, Beihefte zum Archiv fül Musikwissenschaft 10–11, 2 vols (Wiesbaden, 1972)Google Scholar; Rasch, R.A., Iohannes de Garlandia en de ontwikkeling van de voor-Franconische notatie, Institute of Mediaeval Music Musicological Studies 20 (Brooklyn, 1969)Google Scholar, a study that is not based on a critical text of Garlandia's treatise; and Waite, W. G., ‘Johannes de Garlandia, POet and Musician’ Speculum, 35 (1960), pp. 179–95CrossRefGoogle Scholar, many of the conclusions of which must be revised in the light of the work of Reimer and Rasch.

⁵ See, for example, Reckow, F., ‘Proprietas und Perfectio: Zur Geschichte des Rhythmus, seiner Aufzeichnung und Terminologie im 13. Jahrhundert’, Acta Musicologica, 39 (1967), pp. 115–43CrossRefGoogle Scholar, in which one aspect of the Garlandian influence is traced.

⁶ Reimer, , Johannes de Garlandia, De mensurabili musica, i, pp. 88–9 (Chap. xiii)Google Scholar.

⁷ The one significant exception, perhaps, is Johannes de Grocheio, who, however, was writing for a very different kind of audience from Garlandia and the other writers considered in this study.

⁸ This point has been made in Sanders, E. H., ‘Consonance and Rhythm in the Organum of the 12th and 13th Centuries’, Journal of the American Musicological Society, 33 (1980), p. 284CrossRefGoogle Scholar. See also the exchange of communications between Sanders, and Reckow, Fritz, Journal of the American Musicological Society, 34 (1981), pp. 589–90, 591CrossRefGoogle Scholar.

⁹ Franco of Cologne, Ars cantus mensurabilis, ed. Reaney, G. and Gilles, A., Corpus Scriptorum de Musica 18 (n.p., 1974); the chapter on organum is on pp. 80–2Google Scholar.

¹⁰ Reckow, F., Der Musiktraktat des Anonymus 4, Beihefte zum Archiv für Musikwissenschaft 4–5, 2 vols. (Wiesbaden, 1967), i, pp. 84–9Google Scholar.

¹¹ Sowa, H., ed., Ein anonymerglossierter Mensuraltraktat 1279, Königsberger Studien zur Musikwissenschaft 9 (Kassel, 1930), pp. 127–32Google Scholar.

¹² See esp. Apel, W., ‘From St. Martial to Notre Dame’, Journal of the American Musicological Society, 2 (1949), pp. 145–58CrossRefGoogle Scholar; Waite, W. G., ‘Discantus, Copula, Organum’, Journal of the American Musicological Society, 5 (1952), pp. 77–87CrossRefGoogle Scholar; the exchange of communications between Apel, and Waite, , Journal of the American Musicological Society, 5 (1952), pp. 272–6CrossRefGoogle Scholar; Waite, , The Rhythm of Twelfth-Century Polyphony: its Theory and Practice, Yale Studies in the History of Music 2 (New Haven, 1954)Google Scholar; Reckow, , Der Musiktraktat des Anonymus 4, iiGoogle Scholar; Reckow, , ‘Das Organum’, Gattungen der Musik in Einzeldarstellungen: Gedenkschrift Leo Schrade, ed. Arlt, W. and others (Berne, 1973), pp. 434–96Google Scholar; Reckow, , ‘Organum’, Handwörterbuch der musikalischen Terminologie, ed. Eggebrecht, H. H. (Wiesbaden, 1971)Google Scholar; Frobenius, W.. ‘Longa-brevis’, and ‘Modus (Rhythmische)’, Handwörterbuch der musikalischen Terminologie (1973, 1974)Google Scholar; Eggebrecht, H. H., ‘Organum purum’, Musikalische Edition im Wandel des historischen Bewusstseins, ed. Georgiades, T. (Kassel, 1971). pp. 93–112Google Scholar: Sanders. ‘Consonance and Rhythm’, pp. 264–86; and the exchange of communications between Reckow, and Sanders, , Journal of the American Musicological Society, 34 (1981), pp. 588–91CrossRefGoogle Scholar. See also Treitler, L., ‘Regarding Meter and Rhythm in the Ars antiqua’, The Musical Quarterly. 65 (1979), pp. 524–58CrossRefGoogle Scholar; and the exchange between Treitler, and Sanders, , Journal of the American Musicological Society, 33 (1980), 602–11CrossRefGoogle Scholar.

¹³ Cserba, S. M., ed., Hieronymus de Moravia O.P., Tractatus de musica, Freiburger Studien zur Musikwissenschaft 2 (Regensburg, 1935), pp. 194–230Google Scholar; see also Reimer, , Johannes de Garlandia, De mensurabili musica, i, pp. 91–7Google Scholar.

¹⁴ See the argument in Reimer, , Johannes de Garlandia, De mensurabili musica, ii, pp. 39–42Google Scholar. It is very likely to have been Jerome's text of Garlandia that Anonymous iv knew; see Reimer, . i. pp. 30–1CrossRefGoogle Scholar, and compare Reckow, , Der Musiktraktat des Anonymus 4, i, p. 23, ll. 3–5Google Scholar. with Cserba, , ed., Hieronymus de Moravia O. P., Tractatus de musica, p. 195, ll. 9–10Google Scholar.

¹⁵ Reimer, , Johannes de Garlandia, De mensurabili musica, i, pp. 88–9, and the critical commentary in ii, pp. 37–9Google Scholar. Compare the editions in de Coussemaker, E., ed., Scriptorum de musica medii aevi novam seriem a Gerbertina alteram, 4 vols. (Paris, 1864–1876), i, p. 114Google Scholar; Cserba, , ed., Hieronymus de Moravia O.P., Tractatus de musica, pp. 224–5Google Scholar; and Reckow, , Der Musiktraktat des Anonymus 4, ii, pp. 35–7Google Scholar. Compare the rather free paraphrase that follows (on pp. 134–6) with the translation in Johannes de Garlandia, Concerning Measured Music (De mensurabili musica), trans. Birnbaum, S. H., Colorado College Music Press Translations 9 (Colorado Springs, 1979), pp. 42–3Google Scholar.

¹⁶ Exclusive of real errors in the St Emmeram Anonymous, such as those described below, pp. 141–2. All citations from Sowa's edition have been checked with the manuscript preserving the treatise, Munich, Bayerische Staatsbibliothek, Clm 14523.

¹⁷ Reimer, , Johannes de Garlandia, De mensurabili musica, i, p. 35 (Chap, i, 1–3)Google Scholar; see also Reckow, , Der Musiktraktat des Anonymus 4, ii, pp. 36–7Google Scholar, and Reckow, , ‘Organum’, Handworterbuch, p. 10Google Scholar.

¹⁸ Reimer, , Johannes de Garlandia, De mensurabili musica, i, p. 88 (Chap. xii, 2)Google Scholar.

¹⁹ See Reckow, F., Die Copula: Über einige Zusammenhänge zwischen Setzweise, Formbildung, Rhythmus, und Vortragsstil in der Mehrstimmigkeit con Notre-Dame, Akademie der Wissenschaften und der Literatur. Mainz, Abhandlungen der geistes- und sozialwissenschaftlichen Klasse, 1972/13 (Wiesbaden, 1972), pp. 8–28Google Scholar; Reckow, , ‘Copula’, Handworterbuch (1972)Google Scholar; Sanders. ‘Consonance and Rhythm’, pp. 283–5; and Yudkin, J., ‘The Copula According to Johannes de Garlandia’, in Musica Disciplina, 34 (1980; forthcoming)Google Scholar.

²⁰ Compare the emendations to this phrase proposed by Reckow, Fritz (Der Musiktraktat des Anonymus 4, ii, p. 37)Google Scholar and the reading offered by the St Emmeram Anonymous, none of which alters the sense of Jerome's text (the correctness of which was first demonstrated in Reimer, , Johannes de Garlandia, De mensurabili musica, ii, pp. 37–8)Google Scholar.

²¹ Compare sentences 7–8. Jerome's relative pronoun ‘quae’, clearly erroneous, could be emended to its ablative form, the phrase then reading ‘recta [mensura], in which longs and breves …’. The sense would be the same as that in the St Emmeram Anonymous reading.

²² Apel, ‘From St. Martial to Notre Dame’, p. 149.

²³ See esp. Waite, , The Rhythm ofTwelfth-Century Polyphony, Chap, v, and his communication in the Journal of the American Musicological Society, 5 (1952), pp. 273–6CrossRefGoogle Scholar.

²⁴ Reckow, , Der Musiktraktat des Anonymus 4, ii, pp. 38–9Google Scholar.

²⁵ Sanders, ‘Consonance and Rhythm’, pp. 269–71.

²⁶ Ibid., p. 271.

²⁷ Sowa, . ed., Ein anonymer glossierter Mensuraltraktat 1279, pp. 127–8, 1. 5Google Scholar. The theorist notes that organum cum alio has been discussed previously, that is, in the chapter on copula (pp. 125–6). Regarding his particular view of copula, see Reckow, , Die Copula, pp. 45–50Google Scholar.

²⁸ Sowa, , ed., Ein anonymer glossierter Mensuraltraktat 1279, p. 125, 11. 18–20Google Scholar; Reckow, , Der Musiktraktat des Anonymus 4, i, p. 84, 11. 16–17Google Scholar. The Emmeram, St. Anonymous also cites an organum duplum, Judea et Jherusalem, as an example; Sowa, p. 125, 11. 21–2Google Scholar.

²⁹ Reimer, , Johannes de Garlandia, De mensurabili musica, ii, pp. 38–9Google Scholar; Reckow, , Der Musiktraktat des Anonymus 4, ii, p. 44Google Scholar; Arlt, W. and Haas, M., ‘Pariser modale Mehrstimmigkeit in einem Fragment der Basler Universitätsbibliothek’, Forum musicologicum: Basler Studien zur Musikgeschichte, 1 (1975), p. 257Google Scholar.

³⁰ Florence plut. 29.1, fol. 68^r–68^v; Wolfenbüttel 628 Helmst., fols. 18^v–19^r; Wolfenbüttel 1099 Helmst., fols. 48^V–49^V; London Egerton 2615, fols. 74^r–74^v, 94^r–94^v.

³¹ Reckow, , Der Musiktraktat des Anonymus 4, i, p. 82, ll. 21–5Google Scholar.

³² In his definition of discantus, for example, Garlandia does not allude to a cantus prius factus: ‘discantus est aliquorum diversorum cantuum sonantia secundum modum’; Reimer, , Johannes de Garlandia. De mensurabili musica. i. p. 35 (Chap. i.4)Google Scholar. Compare Franco of Cologne. Ars cantus mensurabilis, ed. Reancy, and Gilles, , pp. 73–4 (Chap. xii. 33)Google Scholar.

³³ Zaminer, F.. Der vatikanische Organum-Traktat (Ottob. lat. 3025): Organum-Praxis der frühen Notre Dame-Schule und ihrer I'orstufen, Münchner Veröffentlichungen zur Musikgcschichte 2 (Tutzing, 1959). p. 185, ll. 4–5Google Scholar: ‘Quia nihil ualet organum per se nisi aliquis cantus sit cum organo’. Compare 11. 1–2: ‘Organum est cantus subsequens precedentem’. Formulations similar to the latter occur in some twelfth-century treatises: see. for example. Eggebrecht, H. H. and Zaminer, F.. Ad organum faciendum: Lehrschriften der Mehrstimmigkeit in nachguidonischer Zeit, Neue Studien zur Musikuissenschaft 3 (Mainz. 1970). p. 187Google Scholar (Montpellier treatise. 1–3: ‘Diaphonia duplex cantus est. cuius talis est difrinitio. Organum uox sequens est praecedentem sub ecleritate diatesseron uel diapente …. Si quis ergo organum componere desiderat. duas ultimas uoces clausule prius eligat. et eis competenter cum cantu iungat. ut ex alia pane cum cantu veniant’.). In a very different vein Anonymous iv observes that ‘quandoque simplex organum dicitur ut in simplicibu conductis’: Reckow, , Der Musiktraktat des Anonymus 4, i, p. 70, 1. 31Google Scholar. See also the meticulous terminological investigation by Reckow, Fritz, ‘Organum-Begriffund frühe Mehrstimmigkeit, zuglcich ein Bcitrag zur Bcdeutung des “Instrumentalen” in der spätantiken und mittelalterlichen Musiktheorie’. Forum Musicologkum. 1 (1975). pp. 31–167Google Scholar. In connection with viewing an organum as a single (added) voice, Marion Gushee has kindly drawn my attention to the practice in such musical sources as Cambridge, Corpus Christi College, MS 473 (the Winchester troper), and Rome. Biblioteca Yaticana, Reg. lat. MS 586, of transmitting the organal voice apart from the cantus firmus it embellishes.

³⁴ Sowa, . ed., Ein anonymer glossierter Mensumltraktat 1279, p. 129. 1. 35–p. 130,1. 3Google Scholar: ‘Nunc per se (siue organum speciale et hoc supra burdonem in tenore) dabitur nunc cum (siue cum discantu) reliquo repentur/Cum per (id est quociens per se ponitur supra burdonem tenoris) se tantum uult (id est tune relicta mensura stricta uult melice se proferi) iure suo dare cantum’.

³⁵ Reimer, , Johannes de Garlandia. De mensurabili musica, i, p. 88 (Chap. xii, 3–4)Google Scholar.

³⁶ Sowa, , ed., Ein anonymer glossierter Mensuraltraktat 1279, p. 125, ll. 15–16Google Scholar. The large ‘U’ at ‘uniuerso’ in the manuscript (fol. 157 ) suggests placing the punctuation before ‘uniuerso’ rather than after it. Compare Reckow, , Die Copula, p. 46Google Scholar.

³⁷ Sowa, , ed. Ein anonymer glossierter Mensuraltraktat 1279, p. 127, ll. 31–5Google Scholar.

³⁸ Ibid., p. 125, 11, 22–3, p. 126, II. 21–2. Compare Reimer, , Johannes de Garlandia, De mensurabili musica, i, p. 88 (Chap. xii, 5, 7)Google Scholar.

³⁹ Sowa, , ed., Ein anonymer glossierter Mensuraltraktat 1279, p. 127, 27–8Google Scholar.

⁴⁰ For example, see note 21. above, and note 78, below.

⁴¹ Waite, , The Rhythm of Twelfth-Century Polyphony, p. 17Google Scholar. It may be that the anonymous writer, seeing Garlandia's remark that organum in speciali can be considered in a twofold manner, and supposing per se and cum alio to be distinct types, set out to produce a description of organum cum alio for his own purposes, doing so by taking the organum non rectum passage and altering it by substituting ‘cum alio’ for ‘non rectum’ and removing each instance of the word ‘non’.

⁴² Reimer, , Johannes de Garlandia, De mensurabili musica, i, p. 35 (Chap. i, 4 )Google Scholar: ‘Discantus est aliquorum diversorum cantuum sonantia secundum modum et secundum aequipollentis sui aequipollentiam’.

⁴³ Ibid., i, pp. 35–6 (Chap. i, 5): ‘Sed quia in huiusmodi discantu consistit mancries sive modus, in primis videndum est, quid sit modus sive maneries, et de specicbus ipsius modi sive maneriei et gratia huiusmodi maneriei ac specierum eius plura alia videbimus’.

⁴⁴ Ibid., i, pp. 74–5 (Chap, xi, 3): ‘Unde discantus est aliquorum diversorum cantuum sonantia secundum modum et secundum aequipollentis sui aequipollentiam per concordantiam’.

⁴⁵ Ibid., i, p. 37 (Chap. i. 16–17).

⁴⁶ See. for example, Ibid., i. p. 88 (Chap. xii, 5: the redefinition of ‘punctus’), and pp. 37–8 (Chap, i, 22: the definition of ‘tempus’). In fact, Garlandia has already used ‘modus rectus’ in passing in its broader sense, in connection with the interpretation offigurae imperfectae in discantus; Ibid., i, p. 51 (Chap, iii, 20).

⁴⁷ Ibid., i, p. 38 (Chap, i, 27). See also Ibid., p. 36 (Chap, i, 7) and Discantuspositio vulgaris, in Cserba. ed., Hiemnymus de Moravia O.P., Tractatus de musica, p. 190, 11. 5–8.

⁴⁸ Even when a ligature has the same values in two different modes, the actual ‘measure’ (in a more recent sense of the term) may be different. Modes 1 and 2 both include the values brevis recta-longa recta, expressed in notation as a binaria; in the first mode, however, the long falls at the beginning of the ternary ‘beat’ and the breve functions as an ‘upbeat’, while in the second mode it is the breve that falls on the ‘beat’.

⁴⁹ Compare the views of Reckow, Fritz in Der Musiktraktat des Anonymus 4, ii, p. 34Google Scholar (in which he sees Chap, VII as one in which Anonymous iv has ‘zwei ofiensichtlich unvereinbare Rhythmisierungsmethoden nebeneinandergestellt’) and p. 62 (in which he affirms the ‘seltsame Gegensatz’ and ‘der grundsatzliche Widerspruch’ between the different parts of the chapter); also those of Ernest Sanders in ‘Consonance and Rhythm’, p. 270, who argues that ‘[Anonymous iv] and modern scholars have expanded or misunderstood Garlandia's rule’.

⁵⁰ Reckow, , Der Musiktraktat des Anonymus 4, ii, pp. 56–7Google Scholar.

⁵¹ Ibid., i, pp. 84, 1. 14– p. 85, 1. 17.

⁵² Ibid., i, p. 85, 11. 18–22: “Et iuxta septem dona spiritus sancti est Septimus modus nobilissimus et dignissimus, magis voluntarius et placens. Et iste modus est modus permixtus et communis et est de omnibus duobus supradictis et de omnibus tribus et de omnibus quatuor etc. Et proprie loquendo denominatur organum purum et nobile etc’ Wolf Frobenius recognised the relationship between Anonymous iv's seventh mode and Garlandia's modus non rectus (‘Modus (Rhythmische)’, Handwörterbuch, pp. 4–5), but accepts Reckow's view of the latter and of the role of the principle of consonance in generating its rhythmic flow.

⁵³ This is preceded by material on the performance of rests and on the consonances, topics common to both copula and organum in speciali; Reckow, , Der Musiktraktat des Anonymus 4, i, p. 85, 1. 27–p. 86, 1. 12Google Scholar.

⁵⁴ This is further confirmed by Anonymous iv's concluding remarks on the irregular modes; Ibid., i, p. 85, ll 23–7. Compare Reckow's opposing view inIbid., ii, pp. 29–30.

⁵⁵ Discussed in Roesner, E. H., “The Performance of Parisian Organum”, Early Music, 7 (1979), pp. 184–7CrossRefGoogle Scholar.

⁵⁶ Hence the proverb, ‘Septenarius numerus significat septem dona Spiritus sancti’, cited in Meyer, H. and Suntrup, R., ‘Zum Lexikon der Zahlenbedeutungen im Mittelalter: Einführung in die Methode und Probeartikel—die Zahl 7’, Frühmittelalterliche Studien, 11 (1977), p. 13CrossRefGoogle Scholar.

⁵⁷ Reckow, , Der Musiktraktat des Anonymus 4, ii, p. 59Google Scholar.

⁵⁸ See esp. Meyer, H., Die Zahlenallegoriese im Mittelalter: Methode und Gebrauche, Münstersche Mittelalter-Schriften 25 (Münster, 1975), pp. 133–9Google Scholar, and Meyer and Suntrup, ‘zum Lexikon der Zahlenbedeutungen im Mittelalter’, pp. 18–73.

⁵⁹ Moralium libri, sive expositio in librum b. Job, in Sancti Gregorii papae I … opera omnia, ii, ed. Migne, J.-P., Patrologiae Cursus Completus, ser. Latina, 76 (Paris, 1878), cols. 757–60Google Scholar. See also Westhoff, F., Die Lehre Gregors des Grossen über die Gaben des Heiligen Geistes (Hiltrup, 1940), pp. 130–6Google Scholar.

⁶⁰ Summa theologicae, a2ae, quest. 68, art. 7.

⁶¹ Sententiarum libri quatuor, in Lombardi, P. … opera omnia, ii, ed. Migne, J.-P., Patrologiae Cursus Completus, ser. ii, 192 (Paris, 1880), cols. 900–5Google Scholar. See also Reynolds, R. E., ‘“At Sixes and Sevens” — and Eights and Nines: the Sacred Mathematics of Sacred Orders in the Early Middle Ages’, Speculum, 54 (1979), pp. 675, 681–3CrossRefGoogle Scholar.

⁶² Meyer and Suntrup, ‘Zum Lexikon der Zahlenbedeutungen im Mittelalter’, pp. 13, 26.

⁶³ For an early use of these ideas see Bede, , Explanatio Apochalypsis, in Venerabilis Bedae … operaomnia, IV, ed. Migne, J.-P., Patrologiae Cursus Completus, ser. Latina prior, 93 (Paris, 1862), cols. 135, 183Google Scholar.

⁶⁴ De septem dona spiritus sancti, included as Chap. 5 of Hugh's De quinque septenis, in Hugonis de S. Victore … opera omnia … editio nova, ed. Migne, J.-P., Patrologiae Cursus Completus, ser. Latina prior, 175–6, 2 vols. (Paris, 1879–1880), I, col. 411Google Scholar.

⁶⁵ Hugh of St Victor comments on the allegorical uses of numbers in De scripturis et scriptoribus sacris, Opera omnia, ed. Migne, , i, cols. 9–28. Hugh's system is discussed in Meyer and Suntrup, “Zum Lexikon der Zahlenbedeutungen im Mittelalter” pp. 2–9Google Scholar, and in Hopper, V. F., Medieval Number Symbolism: its Sources, Meaning, and Influence on Thought and Expression, Columbia University Studies in English and Comparative Literature 132 (New York, 1938), pp. 100–3CrossRefGoogle Scholar.

⁶⁶ Opera omnia, ed. Migne, , ii, cols. 807–8Google Scholar. See also Smalley, B., The Study of the Bible in the Middle Ages (2nd edn, Oxford, 1952), p. 94Google Scholar.

⁶⁷ Roesner, ‘The Problem of Chronology’ pp. 379–81. See also Trietler, Leo, ‘Regarding Meter and Rhythm in the Ars antiqua’ pp. 545–6Google Scholar. In a remarkably similar manner the discussion of verse types by the grammarian Johannes de Garlandia represents a synthesis of a broader practice, lines of rhymed poetry being classified as either iambic or spondaic according to whether the penultimate syllable is long or short; see Lawler, T., ed. and trans., The ‘Parisiana poetria’ of John of Garland (New Haven, 1974), pp. 160–1, 265Google Scholar. Compare, J. J. Murphy, Rhetoric in the Middle Ages: a History of Rhetorical Theory from Saint Augustine to the Renaissance (Berkeley, 1974), p. 135Google Scholar.

⁶⁸ See Reimer, , Johannes de Garlandia, De mensurabili musica, i, p. 36Google Scholar.

⁶⁹ Compare the comments on the development of the modal system in Michalitschke, A. M., ‘Studien zur Entstehung und Friihentwicklung der Mensuralnotation’, i: ‘Der Ursprung der formal differenzierten Einzelnoten’, Zeitschrift fur Musikwissenschaft, 12 (1929/1930), pp. 257–79Google Scholar; Waite, , The Rhythm of Twelfth-Century Polyphony, pp. 58–77Google Scholar; and Sanders, E. H., ‘Duple Rhythm and Alternate Third Mode in the 13th Century’, Journal of the American Musicological Society, 15 (1962), pp. 282–6CrossRefGoogle Scholar, none of which accounts entirely satisfactorily for how and why the modal system came to take the form it did, or, indeed, for why the beginnings of a Western rhythmic language came to be conceived in ‘modal’ terms at all. In a future paper I hope to offer a hypothetical model describing the origins and evolution of modal rhythm.

⁷⁰ Compare the observation by Anonymous iv that the mode could change during the course of a single phrase, ‘sometimes with the help of rests, sometimes without it’, a procedure that he calls ‘admixture’; Reckow, , Der Musiktraktat des Anonymus 4, i, p. 22, ll. 22–4Google Scholar: ‘Et nota, quod quandoque fiunt admixtiones mediante pausatione, quandoque sine, quod videtur esse et non esse.’

⁷¹ Compare the states of Alleluia, Nativitas in Florence plut. 29.1, fol. 130r (in which only the discantus phrases are concordant) and Berlin lat. 4° 523, fol. A^v. In the latter the ternaria that introduces the copula phrase is drawn as a single long with somewhat elongated head followed by a binaria. a reading that bears an intriguing resemblance to the copula ligata described in Franco of Cologne, Ars cantus mensurabilis, ed. Reaney, and Gilles, , pp. 75–6 (Chap, XII, 3–5)Google Scholar. The quaternaria in the final phrase of the Berlin text is drawn cum opposita proprietate while the final binaria is written sine proprietate and with the first note elongated; see the facs. in Roesner, ‘The Performance of Parisian Organum’, p. 181.

⁷² Indeed, it must not be assumed that Garlandia based his classification of polyphonic idioms on the repertory of responsorial chant settings that we tend to associate with the word ‘organum’. His classification appears to stand apart from considerations of genre, ignoring as it does such factors as text setting and the presence or absence of a cantus prius factus (see note 32, above, and the careful distinction made by Reckow between Garlandia's treatment of ‘Satz’ and, for example, Grocheio's description of a ‘Gattung’, in ‘Das Organum’, pp. 435–6). Anonymous iv indirectly supports this conclusion when he mentions conducti ‘in quibus continentur puncta finalia organi’; Reckow, , Der Musiktraktat des Anonymus 4, i, p. 82, ll. 12–14Google Scholar. He cites two specific conducti in this regard, Salvatoris hodie and Relegentur ab area, both of which have melismatic caudae with elaborate currentes flourishes against a sustained tone in another voice. (These conducti are found, among other places, in Wolfenbüttel 628 Helmst., fols. 95^r–96^v and 96^v–98^r, at the beginning of the conductus collection that makes up fascicle ix of the manuscript; on the striking fact that these and other works cited by Anonymous iv occur in Wolfenbüttel 628 at major points of division in the collection, see Roesner, E. H., ‘The Origins of W ₁,’ Journal of the American Musicotogical Society, 29 (1976), pp. 379–80Google Scholar.) But it is not inconceivable that Anonymous iv had other pieces in mind as well when he made his allusion to conducti with sustained-tone passages. In Wolfenbüttel 628 Salvatoris hodie, Relegentur ab area and two other conducti are followed by three Benedicamus domino organa in sustained-tone style (fols. 103^v–104^v ), and then, after a Gloria paraphrase, Gloria in excelsis deo redemptori meo, by a fourth (fols. 105^v–106^r), also in organumpurum. Friedrich Ludwig regarded these four works as supplements to the collection of Office organa in fascicle in of the manuscript (Repertorium organorum recentioris et motetorum vetustissimi stili, i/l (Halle, 1910), p. 40)Google Scholar, and indeed the first three are found in Florence plut. 29.1 among the Office settings, and in Wolfenbüttel 1099 at the end of the organum collection, if not directly appended to the Office pieces. But their placement in Wolfenbüttel 628 among the conducti of fascicle ix seems odd; the same scribe copied both these works and the organa in fascicle m, and there would have been ample room for the four Benedicamus domino settings in the organum collection (which was copied into a ternio instead of the more customary quaternio because there were not enough Office pieces to fill a gathering of the standard size). It appears more likely, then, that the four Benedicamus domino pieces belong with fascicle ix, and that in the institution for which Wolfenbüttel 628 was copied, at least, they were intended to be used with the conducti. One of the conducti preceding them in the fascicle, Naturas deus regulis (fols. 98^v–101^r), concludes its third strophe with a caudaon the words ‘Benedicamus domino’, a musical and textual refrain that occurs at the ends of both strophes 2 and 3 in the copy in Florence plut. 29.1, fols. 211^v–214^r and 286^v–287^v, and at the end of strophe 2 only in the redactions in Wolfenbüttel 1099, fols. 96^r–99^r, and Madrid, Biblioteca National, MS 20486, fols. 107^r–109^v (strophe 3 is not included in these two copies). This refrain on ‘Benedicamus domino’ recurs in Wolfenbüttel 628, moreover, in the copy of Leniter ex merito on fol. 16^r–16^v, but not in the redaction of the same piece on fols. 81^v–82^r. Furthermore, the lower voice of this refrain cauda incorporates a segment of plainchant. (On this and similar situations see Bukofzer, M. F., ‘Interrelations between Conductus and Clausula’, Annales Musicologiques, 1 (1953), pp. 76–88.)Google Scholar I suspect that the four organa were intended to function in a manner similar to this refrain, to convert liturgically flexible conducti into more clearly defined Benedicamus domino substitutes through the addition of a ‘Benedicamus domino’ refrain — in these instances, one conceived in organum purum. These refrains could have been applied to any conductus in the manuscript with a text suitable for use as a Benedicamus substitute, and their use would produce works corresponding closely with the description by Anonymous iv cited above. On the relationship between the Benedicamus domino and the conductus see esp. Arlt, W., Ein Festoffizium des Mittelalters aus Beauvais in seiner liturgischen undmusikalischen Bedeutung, 2 vols. (Cologne, 1970), Darstellungsband, pp. 160–217Google Scholar.

⁷³ Sanders, ‘Consonance and Rhythm’, pp. 269–71.

⁷⁴ See the communication from Reckow, Fritz. Journal of the American Musicological Society. 34 (1981). pp. 588–9CrossRefGoogle Scholar.

⁷⁵ See Anonymous iv's interesting observation that the sixth mode can end with either a breve or a long; Reckow, , Der Musiktraktat des Anonymus 4, i. p. 33. 1. 15 – p. 34. 1. 19Google Scholar.

⁷⁶ Ibid., i, p. 86, ll. 14–17.

⁷⁷ Ibid., i, p. 86, ll. 19–22: ‘Item omnis punctus ultimus erit longus et concordans. Item omnis punctus paenultimus ante longam pausationem sicut in fine puncti vel clausulae est longus. Item omnis punctus paenultimus similitudinarie perceptus longus per modum. sive fuerit concordans sive non.’

⁷⁸ The St Emmeram Anonymous, who shares Anonymous iv's second rule, offers a corrupt — or, at least, modified — version of his rule 3: ‘whatever is notated as a long in the manner of organum before a perfect concord is said to be long’; Sowa, , ed., Ein anonymer glossierter Mensuraltraktat 1279, p. 130, ll. 34–5Google Scholar (‘quidquid figuratur longum secundum modumorgani ante perfectam concordantiam, dicitur esse longum’). This seems to imply that the final note of a clausula can be a breve if the notation suggests it; Garlandia's second rule has been made into a more general principle, in effect restricting the principle of consonance, Garlandia's rule 1, entirely to opening simultaneities. Symptomatically, the St Emmeram Anonymous omits the first rule altogether from his text, undoubtedly through error, as he introduces the first of his remaining two rules with the words, ‘alia insequiturregula’ (ibid., p. 130, 1. 32). Compare Reckow, , Der Musiktraktat des Anonymus 4, ii, pp. 43–4Google Scholar, where it is argued that the anonymous writer deliberately omitted Garlandia's rule of consonance because he considered organum in speciali to be mensural in conception, and therefore not in need of the principle of consonance as Reckow interprets it.

⁷⁹ Reckow, , Der Musiktraktat des Anonymus 4, i, p. 86, 1. 22–89, 1. 2Google Scholar.

⁸⁰ Ibid., i, p. 89, ll. 3–4: ‘Quidquid contingit inter inceptionem supradictam et finem supradictum, pro mediatione habetur.’

⁸¹ Ibid., i, p. 86, ll. 14–19, p. 88, ll. 8–17. See also Roesner, ‘The Performance of Parisian Organum’, pp. 178–80.

⁸² Reckow, , Der Musiktraktat des Anonymus 4, p. 86, 1. 22– p. 87, 1. 9Google Scholar: ‘ Item omnis punctus duorum: primus, si fuerit in concordantia, longus; si fuerit in discordantia, brevis in quantum de se et non in quantum de paenultima praedicta. Item omnis punctus ultimus duorum: si fuerit concordans, longus; si fuerit discordans, minime in quantum de se etc.’, and so on.

⁸³ Ibid., i, p. 79, ll. 2–5: ’regula est: omnis paenultima ante pausationem, quae dicitur finis punctorum, longa est. Et si paenultima fuerit tonus in duplo supra tenorem ut in organo puro, optime erit concordans, quamvis tonus non sit concordantia.’ The passage goes on to compare the treatment of cadential dissonance ‘inter puros organistas et inter Lumbardos organizantes’, p. 79, 1. 5 – p. 80, 1. 6, the latter perhaps reflecting an earlier style such as that taught by Guido d'Arezzo.

⁸⁴ Compare the perceptive observation of Theodore Karp that by distinguishing between modus rectus and non rectus ‘Garlandia may have been seeking merely to draw a distinction between the strict confines of the core of modal theory and the liberties allowed in actual modal rhythms’; ‘Towards a Critical Edition of Notre Dame Organa Dupla’, The Musical Quarterly, 52 (1966), p. 358, no. 14Google Scholar. Although I have interpreted the theoretical evidence in a wholly different manner from William Waite, the results of the present study suggest a manner of transcription for sustained-tone organum that is not unlike the one presented in The Rhythm of Twelfth-Century Polyphony in its broad outlines, the major difference involving the treatment of structural points. On a more subjective level. I find many of Waite's readings unnecessarily fussy and ‘baroque’ in character. The notation of modus non rectus is inherently multiform in its rhythmic implications, and the problem of how to preserve this aspect of the musical style in modern transcription has yet to be adequately addressed.

⁸⁵ Rebecca Baltzer has pointed out the influence of Garlandia's mensural notation on the notation of the organa in Wolfenbüttel 1099; ‘Thirteenth-Century Illuminated Miniatures and the Date of the Florence Manuscript’, Journal of the American Musicological Society, 25 (1972), p. 17Google Scholar (she draws attention to discantus specifically, but these traits are equally evident in organum purum sections, as, for example, on fol. 47^r). The mensurally clarified notation in Berlin lat. 4° 523 is more striking still. In this manuscript it is precisely the material in modus non rectus that is notated mensurally, and not the discantus or the sustained-tone writing in modus rectus (which needs such clarification much less). The result is a tendency for the flexible rhythmic designs of the organum in speciali sections to be tightened into a kind of flow approaching that of the strict modal patterns, thereby neutralising the differences between the two kinds of modus. It is by no means clear whether Anonymous iv, writing two generations after Garlandia, intended his seventh chapter to be read in terms of a modally or a mensurally written organum source. His remarks concerning the ‘principle of consonance’ might suggest the former, but other aspects of his discussion — above all, his citation of a phrase of copula that seems to be identical with the redaction of this music in the mensurally written source, Montpellier H.196 (Reckow, , Der Musiktraktat des Anonymus 4, i, p. 84, ll. 14–19)Google Scholar — implies quite the opposite. So also does his long excursus on the development of mensural notation that follows immediately upon his famous account of Leoninus's magnus liber organi and Perotinus's ‘abbreviation’ of it (Ibid., i, p. 46, ll. 5–20). Anonymous iv notes with regard to music in cum littera notation that older generations had had to cope with copies of the repertory in non-mensural notation, and that the greatest part of their understanding of how to perform such modal redactions was communicated through precepts that were not related to any explicit representation in the notation (‘Maxima pars cognitionis antiquorum fuit in praedictis sine materiali significatione’; Ibid., i, p. 50, 11. 5–6), but that the introduction of mensural note-forms had greatlyexpedited the task of learning the music (see the whole passage, Ibid., i, p. 49, 1. 31 – p. 50, 1. 13). He then adds that ‘abbreviatio erat facta per signa materialia a tempore Perotini Magni e parum ante’ (i, p. 50, ll. 14–15), which I take to be a reference to the mensural redaction of the repertory. (On my interpretation of Perotinus's abbreviatio of Leoninus's liber organorum as an ‘edition’ or ‘redaction’, see Roesner, ‘The Problem of Chronology’, pp. 377–9.)

⁸⁶ See, for example, Smith, N. E., ‘Interrelationships among the Alleluias of the Magnus liber organi’, Journal of the American Musicological Society, 25 (1972), pp. 175–202, and Roesner, ‘The Problem of Chronology’, pp. 366–8CrossRefGoogle Scholar.

⁸⁷ See Roesner, ‘The Problem of Chronology’, pp. 385–98.