¹ Giovanni Spataro complained about this degeneration of the mensural system in a letter to Giovanni del Lago of 4 January 1529: ‘V. E. vede bene che a tempi nostri li signi ordinati da li antiqui sono tenuti in poco pretio et existimatione, et che solo usano questo signo ¢, et de le proportione solo uxano la sesqualtera.’ (‘You see well that in our times the signs established by the ancients are held in little esteem, and that they use only this sign ¢, and of the proportions, they use only sesquialtera.’) A Correspondence of Renaissance Musicians, ed. Blackburn, B. J., Lowinsky, E. E. and Miller, C. A. (Oxford, 1991), p. 336Google Scholar. Scholars often point out that signs of imperfect tempus could be used when the metre was actually triple, but this convention usually applied only to instrumental music.

² A few theorists of the early part of the century believed that even pieces in C should properly be measured in breves, but they admitted that this principle was not observed in practice. Lanfranco, G. M., Scintille di musica (Brescia, 1533; facsimile edn in Bibliotheca Musica Bononiensis, sez. II, 15, with preface by G. Massera (Bologna, 1970)), pt 2, p. 68Google Scholar, complains that many composers fail to observe the binary number of semibreves not only in C, but even in ¢, which is a much worse error. Spataro, in a letter of 30 January 1531 to Pietro Aron, acknowledges that he made an error in putting an odd number of semibreves in a motet in C, but says that the error is unimportant, because the signature calls for a semibreve tactus; if the piece had been in ¢, the error would have been serious. A Correspondence of Renaissance Musicians, ed. Blackburn, , Lowinsky, and Miller, , pp. 415–16Google Scholar.

³ There are exceptions, especially in the music of the second half of the century, that demonstrate the declining significance of the breve tempus at that time. For example, Casimiri, R., La polifonia vocale del secolo XVI e la sua trascrizione in figurazione musicale moderna (Rome, 1942), pp. 43–5Google Scholar, n. 94, lists pieces in ¢ by Palestrina that contain odd numbers of semibreves.

⁴ For summaries of sixteenth-century views on tactus, see Frobenius, W., ‘Tactus’ (1971). Handwörterbuch der musikalischen Terminologie, ed. Eggebrecht, H. H. (Wiesbaden, 1971–)Google Scholar, and Bank, J. A., Tactus, Tempo, and Notation in Mensural Music from the 13th to the 17th Century (Amsterdam, 1972), pp. 203–49Google Scholar. Praetorius, M., Syntagma musicum (Wolfenbüttel, 1614–1620; facsimile ed. Gurlitt, W., Documenta Musicologica, ser. 1, nos. 14, 15, 21 (Kassel, 1958–1959)), iii, pt 2, ch. 7, p. 49Google Scholar, says that a breve tactus is still used in some elegant chapels and schools in his time. According to the table in Bank, , Tactus, Tempo, and Notation, pp. 226–30Google Scholar, a minim tactus not representing augmentation is first mentioned as a possibility for unskilled singers in Sebastiani, C., Bellum musicale (Strasburg, 1563)Google Scholar, ch. 12, but Sebastiani says only that the tactus minor is for unskilled singers, and does not specify the value to which it applies. The earliest evidence for a minim tactus in C may therefore be Hoffmann, E., Musicae practicae praecepta (Wittenberg, 1572), ch. 10Google Scholar. Earlier references to half-tactus, such as that in Agricola, M., Musica figuralis deudsch (Wittenberg, 1532)Google Scholar, ch. 6, sigs. Giii^v–iiii^r, probably apply only to the semibreve in ¢. Agricola defines the half-tactus as a semibreve of ¢ or a minim of C, but all his examples of it apply to the semibreve.

⁵ Glarean, H., Dodecachordon (Basle, 1547; facsimile edn in Monuments of Music and Music Literature in Facsimile, ser. 2, 65 (New York, 1967)), bk 3, ch. 7, pp. 203–4Google Scholar (trans. C. A. Miller. Musicological Studies and Documents 6 (n.p., 1965), ii, p. 232), associates the choice of tactus with geography, saying that the breve tactus is still common in Germany, but that the French prefer the semibreve. He also points out that the semibreve tactus is easier for students. Ornithoparchus, A., Musice active micrologus (Leipzig, 1517)Google Scholar. bk 2, ch. 6, sig. Fiij^v (trans. J. Dowland (London, 1609), p. 46; facsimile edns of both in A Compendium of Musical Practice, ed. Reese, G. and Ledbetter, S. (New York, 1973), pp. 56 and 166)Google Scholar. similarly associates a semibreve tactus in ¢ with the unlearned. Zacconi, L., Prattica di musica (Venice, 1592, repr. 1596; facsimile edn in Bibliotheca Musica Bononiensis, sez. II, 1 (Bologna, 1983))Google Scholar, bk 1, ch. 36, fol. 25^r, points out the correlation between tactus and rhythmic character, saying that even though pieces in ¢ are usually sung with a semibreve tactus, they should not have that signature unless a breve tactus would also be comfortable for them.

⁶ See Bank, , Tactus, Tempo, and Notation, pp. 226–30Google Scholar, for a summary of the terminology applied to different types of tactus.

⁷ The meaning of the stroke through the mensuration signs ¢ and φ is the subject of several recent studies. Schroeder, E., ‘The Stroke Comes Full Circle: φ and ¢ in Writings on Music, ca. 1450–1540’, Musica Disciplina, 36 (1982), pp. 119–66Google Scholar, maintains that the stroke always meant diminution by half in ¢, but could indicate diminution by a third in φ until c. 1540. Berger, A. M. Busse, ‘The Myth of Diminutio per tertiam partem’, Journal of Musicology, 8 (1990), pp. 398–426CrossRefGoogle Scholar, and Mensuration and Proportion Signs: Origin and Evolution (Oxford, 1993), pp. 120–48Google Scholar, concludes that the theory of diminution by a third in φ was limited to a small number of German theorists and based on a misunderstanding of Johannes de Muris. Wegman, R. C., ‘What is “Acceleratio mensurae”?’, Music and Letters, 73 (1992), pp. 515–24CrossRefGoogle Scholar, argues that the stroke in both φ and ¢ indicated a semibreve tactus that was faster by an indeterminate amount than the tactus of the undiminished signs. The sign C became rare after c. 1520, but was revived in the note nere madrigal around 1540 and remained common in madrigals throughout the century. Dahlhaus, C., ‘Zur Entstehung des modernen Taktsystems im 17. Jahrhundert’, Archiv für Musikwissenschaft, 18 (1961), pp. 223–40 (esp. pp. 227–8)CrossRefGoogle Scholar, and Haar, J., ‘The Note nere Madrigal’, Journal of the American Musicological Society, 18 (1965), pp. 22–41 (esp. pp. 22–5)CrossRefGoogle Scholar, demonstrate that C requires a semibreve tactus that is slower than the semibreve of C, but not twice as slow, in that context.

⁸ The equivalence of the signs 3 and ³₂ is confirmed not only by their indiscriminate use by sixteenth-century composers (about which numerous theorists complained to no avail), but also by early seventeenth-century sources in which one of them appears in the vocal parts and the other in the basso continuo. For example, several pieces in da Viadana's, Lodovico GrossiCento concerti ecclesiastici (Venice, 1602)Google Scholar use the signature φ ³₂ in the vocal part simultaneously with 3 in the continuo as a sign of triple proportion of ¢. In Decantabat populus Israel, the barring of the continuo confirms the meaning of the proportion: it is in breves in both ¢ and 3, but the first perfect breve of the proportion is included in the same bar as the preceding semibreve when the proportion begins in the middle of a breve tempus, and must therefore have the same value as a semibreve of ¢. Modern edition in Viadana, Opere, ser. 1, i, ed. Gallico, C.. Monumenti Musicali Mantovani 1 (Kassel, 1964), p. 30Google Scholar.

⁹ Gaffurius, F.. Practica musicae (Milan, 1496; facsimile edn in Monuments of Music and Music Literature in Facsimile, ser. 2, 99 (New York, 1979))Google Scholar, bk 4, ch. 5, sigs. ggiij^viiij^v (trans. C. A. Miller. Musicological Studies and Documents 20 (n.p., 1968), pp. 180–1), maintains that a proportion cannot make notes perfect. Spataro, Giovanni devoted much of his Tractato di musica … nel quale si tracta de la perfectione da la sesqualtera producta in la musica mensurata exercitate (Venice, 1531; facsimile edn in Bibliotheca Musica Bononiensis. sez. II, 14, with preface by G. Vecchi (Bologna, 1970))Google Scholar to refuting the opinion of Gaffurius on this point. Theorists continued to disagree about the matter throughout the century.

¹⁰ For example, Heyden, S., De arte canendi (Nuremberg, 1540: facsimile edn in Monuments of Music and Music Literature in Facsimile, ser. 2, 139 (New York, 1969)), bk 2. ch. 5. pp. 86–7Google Scholar (trans. C. A. Miller, Musicological Studies and Documents 26 (n.p., 1972), p. 83). defines a proportion as ‘a measurement in which two different numbers are compared to notes or to notes and tactus’ (‘qua duo diversi numeri, aut Notularum inter se, aut Notularum & Tactuum, conferuntur’). Oridryus, J., Practicae musicae utriusque praecepta brevia (Düsseldorf, 1557)Google Scholar, ch. 13 (ed. in Federhofer-Königs, R., Johannes Oridryus und sein Musiktraktat, Beiträge zur Rheinischen Musikgeschichte 24 (Cologne, 1957), p. 152)Google Scholar. and Wilphlingseder, A., Erotemata musices practicae (Nuremberg, 1563). bk 2. ch. 11, p. 262Google Scholar. repeat Heyden's definition almost verbatim.

¹¹ The earliest source to use the term tactus proportionatus may be Wollick, N., Enchiridion musices (Paris, 1509)Google Scholar, bk 2, ch. 6 (sig. h vi^v of the 1512 edn). Wollick says that it applies to three semibreves in triple proportion (either in all voices or in one voice in a 3:1 ratio with others) and in sesquialtera. According to Frobenius, ‘Tactus’, p. 8, the first source to describe it unambiguously as an unequally divided tactus is Agricola, Musica figuralis deudsch, ch. 6.

¹² Sebald Heyden and his followers were the leading proponents of equal tactus for all types of mensurations and proportions. See below, note 41.

¹³ ‘Sono alcuni altri che inanzi pongono in principio del suo canto il segno seguente O, nel qual segno e diputato ciascuna semibreve passare per una misura, & con poca avertenza adducono la sesqualtera proportione con brevi & semibrevi, nel qual ordine & forma acadono tre effetti, dui contrarii, & uno difficile al pronuntiante over cantore. Per il primo havendo data la misura ne la semibreve, & volendo creare la sesqualtera, ne risulta tripla, perche prima passava per una battuta una semibreve, di poi ne passa tre. Per il secondo effetto contrario aviene, che se pur tu vuoi creare la sesqualtera proportione, a te e dibisogno mutarti da la prima misura quale era una semibreve per battuta, & entrare ne la misura qual si conviene a questo segno ¢ & e errore, per che tutte le proportioni drittamente si riferiscono a lo antecedente segno. Il terzo effetto di difficulta e che ben puoi creare la sesqualtera proportione ne le figure & forma medesima, senza rimuovere la misura del segno in questo modo, faccendo che ciascuna nota sia syncopata, laqual proportione resultera, che tutte le note resteranno dupplicate, per laqual cosa ne sara la giusta & vera sesqualtera, ma perche questo modo poco e usitato, avertirai quando sotto tal segno O tu pensarai formare una sesqualtera, fa le tue note di semibrevi & minime acompagnate, & non di brevi & semibrevi, nel qual processo verranno in battuta tre minime contra una semibreve, quale e sesqualtera, et cosi non incorrerai in tali errori da noi di sopra dimostrati. Per tanto ciascuna sesqualtera formata sotto la battuta di una breve, e di bisogno segnarsi con brevi & semibrevi vacue o piene, ma quella che si ritruova ne la battuta di semibrevi o minime, fa che la sua forma si mostri di semibrevi & minime, cosi a te sia manifesto de la semibreve sesqualterata ne la prolatione perfetta & imperfetta.’ Aron, P., Thoscanello de la musica (Venice, 1523; rev. edn with Aggiunta publishe as Toscanello in musica (Venice, 1529, 1539 and 1562): facsimile of 1539 edn, ed. Frey, G., Documenta Musicologica, ser. 1, 29 (Kassel, 1970))Google Scholar. bk 2. ch. 33, sig. Giij^r; trans. P. Bergquist, Colorado College Music Press Translations 4. 3 vols. (Colorado Springs, 1970), ii, pp. 47–8. My translation differs from Bergquist's in that I do not interpret the ‘ne’ in the expression ‘ne sara la giusta & vera sesqualtera’ as a negative. The wording of my translations often differs from that of the published translations cited, but my interpretations of the meanings are essentially the same unless otherwise noted.

¹⁴ This passage refers specifically to major sesquialtera of O, in which the syncopation is more complex than in major sesquialtera of C, because the two semibreves of O that are replaced by three in the proportion do not form a complete metrical unit. Aron evidently intends the same reasoning to apply to major sesquialtera of C, however, because he uses the sign ¢, not φ, in his second contradiction, and he states the general rule at the end of the passage in terms of the beat of the preceding mensuration, not the perfect or imperfect character of the tempus. The passage is discussed in Berger, , Mensuration and Proportion Signs, pp. 222–4Google Scholar.

¹⁵ ‘Aliud insuper non mediocre vitium universam fere musicorum manum temere sequi animadverti, cum ante sesquialteram ex his duobus alterum praeponant signum O C, idem namque sunt in mensura, sub his enim notulae signis militantes eandem imitantur mensuram, et singulae semibreves uno mensurantur ictu. Deinde ipsi sesquialtere proportionis breves ac semibreves dedicant notulas, inventes treis semibreves unico ictu contineri ac mensurari, quo nil ineptius est in sesquialtera proportione, id enim potius triplae congruit proportioni, ut solerti exploratori facile constabit.’ Vanneo, S., Recanetum de musica aurea (Rome, 1533; facsimile edn in Bibliotheca Musica Bononiensis, sez. II, 16, with preface by G. Vecchi (Bologna, 1969)), bk 2, ch. 30, p. 68Google Scholar; ed. and trans, of bk 2, chs. 20–37 (On Proportions) by A. Seay, Colorado College Music Press Texts/Translations 2 (Colorado Springs, 1979), pp. 32–3.

¹⁶ ‘Per fuggire la difficulta della Sincopa, & per dar tempo commodo alle note: & per non formare un'altra proportione in luogo della Sesqualtera: facciasi la detta Sesqualtera sotto a i segni del Tempo interi: o perfetti: o imperfetti che siano: proposti nel principio del Canto per fondamento delle proportioni: che seguono (chi non volesse cangiare per la via la misura: cosa che e da prattico semplice: & inconvenientissima) facciasi di Semibrevi: & di Minime. Ma essendo i detti segni traversati (percioche la misura e posta su la Breve) formasi la detta Sesqualtera di Brevi: & di Semibrevi.’ Scintille di musica, pt 2, p. 73.

¹⁷ ‘Per lo qual segno [¢] essi danno la misura sopra la semibreve, & questo inconveniente commettono per poter con piu loro facilita cantare, Et per tal modo incorrono in errore, & in non poca confusione, perche la misura che essi tengono nella semibreve verrà a dannare quello, che dal Compositore è inteso, & immaginato, conciosia cosa che dopo molte note nascera una sesqualtera habitudine, Per laqual cosa io vorrei un poco, che mi dicessero questi tali, che danno la misura sopra una semibreve per battuta, poscia che saranno giunti alla detta sesqualtera, che proportione sara quella? Certo non si puo dire, che essa habbia da essere sesqualtera. ma tripla si bene. Et questo averra perche essi passeranno tre semibrevi sopra la battuta della detta semibreve. La qual cosa è falsissima. … Et quanti ce ne sono anchora, che cantando tal segno per una semibreve per battuta giunti alla sesqualtera muteranno misura per accomodarsi a quella della sesqualtera.’ Aron, P., Lucidario in musica (Venice, 1545; facsimile edn in Monuments of Music and Music Literature in Facsimile, ser. 2, 68 (New York, 1978)Google Scholar, bk 3, ch. 6, fols. 20^r–v. The passage is discussed in Berger, , Mensuration and Proportion Signs, p. 224Google Scholar.

¹⁸ ‘Quae est Proportio Sesquialtera, Ea est in qua tres Minimae, aut Semibrevis & Minima uni Tactui accinuntur.’ Heyden, , De arte canendi, bk 2, ch. 5, p. 92Google Scholar; trans. Miller, p.86.

¹⁹ ‘Hemiolia Proportio quae est? Eadem est planè cum Sesquialtera? idem ipsum enim Graecis ⋯μιóλιος significat. Cur autem hanc ab illa distinctam voluerint Musici, nihil equidem firmi video. Nisi forte hoc nigror Notularum effecerit, quo solo ea fermè absque aliis signis Proportionalibus in Tempore, tres Semibreves: In Prolatione tres Minimas singulis Tactibus accommodandas significat.’ Ibid., p. 94; trans. Miller, p. 87.

²⁰ ‘Quae est Proportio Tripla? Ea est, in qua Brevis perfecta, aut tres Semibreves, unius integrae Semibrevis Tactui adaptantur.’ Ibid., pp. 89–90; trans. Miller, p. 85. In Petrucci, , Missarum Ghiselin (Venice, 1503)Google Scholar, the mensuration signs, beginning with the discantus, are: φ3, ¢³₂, ¢ and ¢; both tenor and bass are notated in black breves and semibreves. Trans. Miller, p. 135.

²¹ Modern edition of Heyden's example in De arte canendi, trans. Miller, , p. 87Google Scholar. The note values and mensuration signs in all my examples are the same as those in the sources, except that ties are used for notes that cross barlines. Barring is editorial unless otherwise noted. Glarean, , Dodecachordon, bk 3, ch. 11, p. 218 (trans. Miller, , ii, p. 291)Google Scholar, quoted this example and commented that the composer ‘has made sesquialtera and tripla equal under these signs, φ and O, which is understandable at any rate. However, because he has combined these two with the trochaic metre, and this in a twofold designation, namely, with the formula of prolatio in the tenor and of tempus in the bass, so we can indeed say that this perchance has been created through the freedom of a composer.’ (‘In quo Sesquialteram Triplamque aequavit his signis φ O. Quod utique intelligitur.Verum quod eas ambas Trochaicae rationi adplicuerit, Idque duplici pernotatione idelicet Prolationis formula in Tenore, Temporis autem in Basi, id vero est quod licentia factum Symphonetae forte dicere possemus.’) Ibid., p. 215; trans. Miller, p. 239. The same example, with Heyden's alteration of the original notation, was also quoted by Oridryus, , Practicae musicae utriusque praecepta brevia, ch. 13 (ed. Federhofer-Königs, , p. 155)Google Scholar, and Wilphlingseder, , Erotemata, bk 2, ch. 11, pp. 288–9Google Scholar.

²² Planchart, A., ‘Tempo and Proportions’, Performance Practice: Music before 1600, ed. Brown, H. M. and Sadie, S., The Norton/Grove Handbooks in Music (New York, 1989), p. 140Google Scholar. and Berger, , Mensuration and Proportion Signs, p. 224Google Scholar, mention the confusion between triple and sesquialtera proportions in sixteenth-century notation. Berger speculates, no doubt correctly, that the source of the confusion was the ambiguity of the tactus of ¢.

²³ ‘Tripla, Sesquialtera, Hemiola temporis & prolationis maxime in usu sunt apud Musicos, & idem de his est iudicium & eadem mensura, non tamen eodem modo pinguntur, ut patebit in exemplis sequentibus et tabula.’ Coclico, A. P., Compendium musices (Nuremberg, 1552; facsimile, ed. Bukofzer, M., Documenta Musicologica, ser. 1. 9 (Kassel, 1954))Google Scholar, pt 2, ‘De prolationibus usitatis’, sig. Gij^r; trans. A. Seay, Colorado College Music Press Translations 5 (Colorado Springs, 1973). p. 18.

²⁴ Modern edition in Coclico, Compendium, trans. Seay, Ex. 41. Seay transcribes the tenor mensuration sign as φ³₂. It is unclear in the source, but must be either ¢ or C³₂, because it is labelled ‘prolatio imperfect[a], sive sesquialtera’ and Coclico (sig. Fiij^v) defines prolation as perfect or imperfect on the basis of the quality of the tempus. The line through the sign is faint and may or may not be intentional. The clef sign in the tenor is between the mensuration sign and the ³₂. Seay transcribes the original clef of the bass incorrectly as f4.

²⁵ ‘Quinta est tripla, & sesquialtera aut Hemiola temporis & prolationis, ac in tres semibreves, vel minimas agit. .’ Ibid., ‘De tactu et mensura, diminutionis & augmentationis’, sig. Hiij^r; trans. Seay, p. 19.

²⁶ ‘Ad Sesquialteram Proportionem etiam Hemiola referatur, quae quando in una voce notatur ut Sesquialtera, quando simul in omnibus vocibus occurit, ad tactum proportionatum canitur.’ Dressler, G., Musicae practicae elementa (Magdeburg, 1571)Google Scholar, pt 3.ch. [7] (‘caput ultimum’), sig. N3^v. Dressler may have taken this idea from Agricola. who describes the difference between coloration in all voices and coloration in a single voice as follows: ‘And thus whenever all voices have three black semibreves at the same time, as in tripla, they are sung to the proportionate tactus, as the following example shows. But when the blackening of imperfect notes is not found simultaneously in all voices, the black notes must be sung in the manner demonstrated above for sesquialtera.’ (‘Und so werden alzeit/ wens alle stymmen zugleich haben drey schwartze Semibre. wie inn der Tripla/ auff den Proportien Tact gesungen/ wie das volgent Exem. ausweist. Wo aber die schwertzung der unvolkomen Noten/ nicht in alien stymmen zugleich erfunden/ so mussen die selbigen schwartzen Noten nach ausweisung/ wie oben von der Sesquialtera berürt/ gesungen werden.’) Musica figuralis deudsch, ch. 12, sigs. N^v–Nii^v.

²⁷ Listenius, N., Musica (Wittenberg, 1537; facsimile of Nuremberg, 1549 edn, ed. Schünemann, G., Veröffentlichungen der Musik-Bibliothek Paul Hirsch 8 (Berlin, 1927))Google Scholar, pt 2, ch. 12, sig. e8; trans. A. Seay, Colorado College Music Press Translations 6 (Colorado Springs, 1975), p. 42. Oridryus, , Practicae musicae utriusque praecepta brevia, ch. 13; ed. Federhofer-Königs, , pp. 154–5Google Scholar. Oridryus illustrates this point with Heyden's example (Example 1 above) in which three black minims in the tenor are equated with three black semibreves in the bass. Neither voice has a mensuration sign in Oridryus's version of the example.

²⁸ Faber, G.. Musices practicae erotematum libri II (Basle, 1553), bk 2, ch. 13, p. 227Google Scholar. Beurhaus, F., Erotematum musicae libri duo (Nuremberg, 1580; facsimile, ed. Thoene, W., Beiträge zur Rheinischen Musikgeschichte 47 (Cologne, 1961)), bk 1, ch. 13, p. 65Google Scholar.

²⁹ ‘Quae est [dimensio] binaria? Cum essentialem omnes notulae valorem retinent: seu cum semibrevis una, vel duae minimae in unam dimensionem veniunt. … Quae est mensura ternaria? Cum tres semibreves vel tres minimae uno tactu canuntur: & hac vulgo vocatur tripla.’ Raselius, A., Hexachordum (Nuremberg, 1589)Google Scholar. ch. 5. sig. E3^v.

³⁰ Modern edition of the original and the keyboard intabulation in Pierre Attaingnant Transcriptions of Chansons for Keyboard, ed. Seay, A., Corpus Mensurabilis Musicae 20 (n.p., 1961), pp. 150–1Google Scholar. Facsimile of the keyboard intabulation in Chansons und Tänze. ed. Bernoulli, E. (Munich, 1914), pp. 12–13Google Scholar. Modern edition of the voice and lute intabulation in Chansons au luth et airs de cours français du xvi^e siècle, ed. Mairy, A.. Thibault, G. and de la Laurencie, L., Publications de la Société Française de Musicologie (Paris, 1934), pp. 24–6Google Scholar. The barlines in Examples 4b and 4c are in the sources. The tie in the voice part of Example 4b at bars 4–5 is editorial.

³¹ All of the values are halved in the keyboard intabulation. The barring of the duple sections of the voice and lute intabulation differs from that of the vocal and keyboard versions because the intabulator omitted a semibreve of transitional material between the two statements of the opening section. The barring of the ternary passage does not follow the metrical groupings, which begin at the figure 3 and contain a semibreve plus minim (or three minims) each. The following errors have been corrected in the transcription of Example 4b, but not in the tablature notation: bar 15, note 1, lower note of the lute part is d, not f; bar 16, note 2 of the voice part is missing; bar 20, the first two rhythmic signs in the lute part are semiminims, not minims. The ‘copie exacte de l'original’ in the edition of this piece in Chansons au luth, p. 26, omits the c in bar 20, note 1, and includes an f with the ć in bar 17, note 3, that seems to have been erased in the source. The transcription of the piece Ibid., pp. 24–5. emends it much more drastically than I have.

³² Attaingnant, , Tres breve et familiere introduction pour entendre & apprendre par soy mesmes a jouer toutes chansons reduictes en la tabulature du lutz(Paris, 1529)Google Scholar, fols. xxxi^v–xxxii^r. Modern edition in Preludes, Chansons and Dances for Lute Published by Pierre Attaingnant, Paris(1529–1530), ed. Heartz, D. (Neuilly-sur-Seine, 1964), p. 27Google Scholar.

³³ A survey of twenty sources of German keyboard tablature from c. 1580 to c. 1620 indicates that this procedure was used in about two thirds of them. The repertory in these sources consists mostly of motets from c. 1560 to c. 1600, nearly all originally in ¢ with proportions notated as major sesquialtera or hemiolia. See Johnson, C., Vocal Compositions in German Organ Tablatures, 1550–1650: A Catalogue and Commentary (NewYork, 1989)Google Scholar, and Brown, H. M., Instrumental Music Printed Before 1600: A Bibliography (Cambridge, MA, 1965)CrossRefGoogle Scholar, for information on these sources. I am grateful to Professor Johnson for making his microfilms of them available to me.

³⁴ Modern edition of Wert's motet in his Opera omnia, ed. MacClintock, C. and Bernstein, M., Corpus Mensurabilis Musicae 24 (n.p., 1961–1977), XI, pp. 42–50Google Scholar. In the tablature notation, the vertical stroke represents a semibreve, the stroke with one flag a minim and the stroke with two flags a semiminim.

³⁵ This example is discussed in Shindle, W. R., ‘The Madrigals of Giovanni de Macque’ (Ph.D. dissertation, University of Indiana, 1971), i, pp. 89–91Google Scholar. I am grateful to Professor Shindle for making his microfilms of both editions of this piece available to me. The fourth note of the canto, bar 4, is ƒ# in the source of Example 6b. The text underlay in the alto, bars 4–5, is incorrect in the source of Example 6b and has been emended by comparison with Example 6a.

³⁶ Modern edition in Marenzio, , Sämtliche Werke, ed. Einstein, A., Publikationen älterer Musik 4.1, 6 (Leipzig, 1929–1931; repr. Hildesheim, 1967), i, pp. 16–18Google Scholar.

³⁷ Ein Schön Nutz unnd Gebreüchlich Orgel Tablaturbuch (Laugingen, 1583)Google Scholar and Thesaurus motetarum (Strasburg, 1589)Google Scholar.

³⁸ Modern edition of Rore's motet in his Opera omnia, ed. Meier, B.. Corpus Mensurabili Musicae 14 (np., 1959–1977). vi, pp. 164–8Google Scholar.

³⁹ The prevalence of the concept that three quarters of a white breve was one of the basic values of the black breve, rather than a variant of two thirds of a white breve, may he seen in Vincente Lusitano's incorrect etymology of the term ‘emiolus’. He says that the term means ‘one and a half’ and is applied to black notes because they are worth one and a half of the next smaller value in a binary context, (‘Ma nel numero binario, le figure maggiori nere perdeno la quarta parte, & chiamasi numero emiolus, perche la tal figura maggiore tiene tutta & mezza della minore che gli è aggiunta.’) Introdutione facilissima et novissima di canto fermo, figurato, contraponto semplice, et inconcerto (Rome, 1553: facsimile edn in Libreria Musicale Italiana Musurgiana 7 (Lucca, 1989)), sig. C (‘Della battuta’)Google Scholar. The term actually refers to one and a half notes taking the same amount of time as one note before the proportion. so that the value of each note is reduced to two thirds. The dotted rhythm in a which the larger value is three quarters of its white equivalent is a variant of the original meaning.

⁴⁰ ‘Per eam enim temeritatem variorum Tactuum, omnis ratio & natura Proportionum. quàm diversa signa inter sese habent, confusa, ac omnino deformata est. Quod quidem etiam nunc tanto aegrius ferimus, quanto minus opus fuerat plures, ac eas diversas Tactuum species excogitare. Cum enim tam multiplices Tactuum species ob hoc tantum excogitatas videamus, ut motum cantus subinde mutarent, nunc tardiorem, nunc concitatiorem, nunc properantissimum faciendo. Quaeso ergo, quid nam illos novatores, de Proportionibus, Augmentationibus, ac Diminutionibus intellexisse credamus? Certurn utique est, ex arte ipsa, quod illi per diversas species Tactus praestare voluerunt, idem veteres per integritatem, aut diminutionem Signorum, aut Proportiones, multo & rectius, & artificiosius praestitisse.’ Heyden, , De arte canendi. Dedication, sig. A3^r; trans. Miller, , p. 20Google Scholar. The wording implies that Heyden is objecting to changes of tempo within a piece, not to different tempos for different pieces. These tempo changes coincide with changes of signs and types of tactus, and must therefore include non-proportional interpretations of proportion signs.

⁴¹ Dahlhaus, C., ‘Zur Theorie des Tactus im 16. Jahrhundert’, Archiv für Musikwissenschaft, 17 (1960), pp. 22–39 (p. 32)CrossRefGoogle Scholar, claims that since Heyden used the term tactus proportionatus he did not reject the concept. Dahlhaus uses this argument to support his claim that the three kinds of tactus to which Heyden objects (De arte canendi, bk 1, ch. 5, p. 41: trans. Miller, , p. 53Google Scholar) are not the major, minor and proportionate tactus described by other theorists, but three different speeds of tactus. In fact, however, Heyden says explicitly that the tactus is always equally divided (bk 1, ch. 5, p. 40; trans. Miller. p. 53). In the dedication (sig. A3^v; trans. Miller, p. 20), he says that proportions cannot be sung properly with different types of tactus, and in bk 2, ch. 7, pp. 110–17 (trans. Miller, pp. 97–100), he offers various proofs that integral, diminished and proportionate signs are all sung to the same tactus, implying that these are the three types of measure that others associate with different types of tactus. Dahlhaus's reasoning is therefore not convincing.

⁴² Zacconi. Prattica di musica, bk 3, chs. 8–9, fols. 136^v–138^v; chs. 23–5, fols. 148^v–149^v: and chs. 31–4, fols. 151^v–152^v.

⁴³ Bank, , Tactus, Tempo, and Notation, p. 230Google Scholar, says that ‘the tactus proportionatus is always explicitly stated to be an independent tactus having a duration of its own, which really does not fit in very well with the theory of the one tactus’. I have not found such explicit descriptions of the independence of the tactus proportionatus from the binary tactus, but the failure of theorists to mention the issue seems to imply this conclusion.

⁴⁴ ‘Caeterurn in his quoque signis O3 C3 nostra aetas tactus diminutionem nimis licenter usurpavit, ut treis semibreveis uno tactu, magnifico quidem illo, & augustiore numerentur: Vulgus cantorum nunc triplam improprie vocat, quippe quae ad nullas unas notulas comparationem habeat, ut poscit tripla Ratio, sed in quatuor vocibus aequo valore incedit. Eam ego Trochaicam dicere malim. quanque lambum saepe in conclusionibus habet, & Tribrachyn, ut duobus his pedibus communem. … Sunt qui absque signo omnes denigrant notulas, quidam etiam denigrant cum signo, multi Hemioliam falso putant, cum Hemiolia vere sit sesquialtera. Triplae nomen ea diminutio inde haud dubie obtinuit, quod treis una mensura, hoc est uno tactu habeat semibreveis. Quidam hanc unico duntaxat ternarij Charactere praeposito innuunt. qui à Franchino recte reprehenduntur. Quidam super ternario circulum pingunt hoc modo ^o₃ significantes temporis perfecti rationem esse, sed tactus celeritate imminutam. quod numerus, ut aiunt, circulo additus diminutionem significet. Quidam etiam in sesquialtera ratione, quoties ea in minimis obvenit, semicirculum cum ternario C3 pingunt, qui non minus reprehendendisunt, quam qui triplam uno ternario pingere consuevere.’ Glarean, , Dodecachordon, bk 3, ch. 8, p. 206: trans. Miller, , ii, pp. 234–5Google Scholar.

⁴⁵ ‘Alij Trochaicam rationem simplici ternario indicant. Alij hoc signo ¢3. Iodocus ferme circulo integro, cui subijcit ternarium, quam rationem, & nos aliquando sequimur. Caeterum Triplam. Hemioliam, ac Trochaicam formam multi distinguere, imò discernere nequeunt. Cum Tripla ad Hemioliam, in celeritatis ratione sit dupla. At Trochaicae, & alia mensura, & alia canendi formula, longe à Tripla atque ab Hemiolia distincta.’ Ibid., ch. 11, p. 214; trans. Miller, ii, p. 239.

⁴⁶ Ibid., pp. 216–19 and 229–38; trans. Miller, ii, pp. 244–6 and 300–2.

⁴⁷ Ibid., pp. 206 and 215; trans. Miller, ii, pp. 234 and 246.

⁴⁸ ‘Hemiola, est quando Notae denigratae aequaliter in omnibus cantilenae partibus procedunt. Et quidem citius quam in Tripla proportione, idque ob colorem. Habet enim color plus agilitatis, quam albedo, quae in his Triplam proportionem, suo signo praefixo, efficit. Nonnunquam Hemiola simplicem contrapunctum refert, quod saltem admonendum duxi.’ Listenius, Musica, pt 2, ch. 12, sig. f^v; trans. Seay, p. 43.

⁴⁹ Modern edition of the example in Listenius, , Musica, trans. Seay, . p. 44Google Scholar. In the editions of 1543, 1548 and 1549, bar 3, note 2 of the discantus is a dotted minim, not a dotted semibreve. My correction of the bar is different from Seay's. The example, which is identical in the above three editions, contains several instances of unorthodox part-writing and dissonance treatment. I have not examined earlier editions. which might have a more correct version of it.

⁵⁰ ‘Hemiolia similis est Tripla vel sesquialtera. nisi quod celerius & subtilius canatur.’ Beurhaus, , Erotematum, bk 1, ch. 13, p. 67Google Scholar.

⁵¹ ‘Quid est Hemiolia proportio? Eadem planè est cum Tripla. nisi quod ea. propter nigredinem, plus agilitatis habeat, quam albedo. Was ist Hemiolia? Hemiolia ist gleich der Triplae proportioni/ allein das sie umb schwertze willen mehr behendigkeit hat in der Mensur/ dann die Tripla.’ Gumpelzhaimer, A., Compendium musicae, 6th edn (Augsburg, 1616)Google Scholar, ch. 9, fol. 16^r. (The first edition was published in 1591.)

⁵² ‘Quid est Hemiolia Proportio? Eadem plane est cùm Sesquialtera, nisi quòd ea. propter nigredinem, plus agilitatis habet, quam albedo. Nonnunquam etiam simplicem contrapunctum refert.’ Wilphlingseder, , Erotemata, bk 2. ch. 11. p. 287Google Scholar.

⁵³ ‘Heere is likewise another ensample wherein Tripla is in all the parts together. which if you pricke al in blacke notes, will make that proportion which the musitions falslie termed Hemiolia, when in deed it is nothing else but a round Tripla. For Hemiola doth signifie that which the Latines tearme Sesquipla or sesquialtra: but the good Munks finding it to go somwhat rounder then common tripla, gave it that name of Hemiolia for lacke of another.’ Morley, T., A Plaine and Easie Introduction to Practicall Musicke (London, 1597; facsimile edn Westmead, Farnborough, Hants, 1971), pt 1, p. 30; ed. Harman, R. A. (London, 1952; 2nd edn, 1963), p. 52Google Scholar. ‘If more blacke semibriefes or briefes bee togither, then is there some proportion, & most commonly either tripla or hemiolia, which is nothing but a rounde common tripla or sesquialtera.’ Ibid., annotations to p. 9, v. 18; ed. Harman, p. 115. (The word ‘round’ means ‘fast’ in these contexts.) Hell, H., ‘Zu Rhythmus und Notierung des “Vi ricorda” in Claudio Monteverdis Orfeo’, Analecta Musicologica, 15 (1975), pp. 87–157 (p. 123)Google Scholar, calls Morley an isolated and relatively early witness for a faster tempo for coloured notes, but Morley probably took the idea from Listenius and Beurhaus, both of whom he cites as authorities at the end of his book.

⁵⁴ ‘E se alcuno volesse salvare detta proportione, & dire che la battuta và alla ragione di breve, et che in la misura s'intende essa breve, & con tal ragione provare che detta proportione è ben detta sesqualtera; si risponderà che è necessario sentire cantare due contra tre, & non battere due contra tre.’ Vicentino, N., L'antica musica ridotta alla moderna prattica (Rome, 1555; facsimile, ed. Lowinsky, E. E., Documenta Musicologica, ser. 1, 17 (Kassel, 1959))Google Scholar, bk 4, ch. 31, fols.87^v–88^r.

⁵⁵ ‘Spesse volte i Compositori si obligano volontariamente senza esser astretti da nisciuno: perche nel comporre delle cantilene loro si servano nel principio come ho detto de i segni della Breve, & cosi dovendovi introdurre una Proportione si trovano esser astretti d'introdurvi la maggiore per far che i segni delle oppositioni circa la quantità delle figure si corispondino, & non fare ch'esse oppositioni sieno contrarie.’ Zacconi, Prattica di musica, bk 3, ch. 57, fol. 173^v. Zacconi recommends (Ibid.)that a composer who prefers to notate proportions in groups of semibreves in C, thinking the rhythm will be easier to read that way, should change the initial sign to ¢. Thus it is clear that a proportion notated in semibreves in ¢ has the same relationship to the preceding music as a proportion notated in minims in C, as in Examples 6a–b above.

⁵⁶ Tigrini, O., Il compendio della musica (Venice, 1588; facsimile edn in Monuments of Music and Music Literature in Facsimile, ser. 2, 25 (New York, 1966)). bk 4. ch. 22, p. 130Google Scholar. Adriano Banchieri uses the term proportioni d'equalità, but defines the tempos of these proportions in the traditional way, in his Cartella musicale (Venice, 1614), documents 4–7, pp. 29–32Google Scholar. The distinction between proportions of equality and proportions of inequality is also found in Wollick, Enchiridion musices, bk 5. ch. 6 (sig. h vi^v of the 1512 edn), where the two types are said to be performed to the same proportionate tactus.

⁵⁷ ‘Cura est & diligentia adhibenda, ut Isodimeris mensura, cantum plurimis coloratis notis non scatentem, non nimis protracta, nec quoque nimium properans demetiatur: Contextum verò coloratarum notarum protractione evidenti. Sic & Anisodimereos mensura mediocritatem teneat.’ Burmeister, J., Musica autoschediastike (Rostock, 1601)Google Scholar, accessio 3, sectio 2, sig. Aa4^r. On Burmeister's theories of mensuration and tactus. see Ruhnke, M., Joachim Burmeister: Ein Beitrag zur Musiklehre um 1600, Schriften des Landesinstituts für Musikforschung Kiel 5 (Kassel, 1955). pp. 76–84Google Scholar. Bank, , Tactus, Tempo, and Notation, p. 247Google Scholar, cites this passage as evidence that ternary passages had lost most of their proportional character by c. 1600, but non-proportional tempo relationships must have existed throughout the sixteenth century, with or without the approval of theorists.

⁵⁸ ‘Those [who call the unequal tactus “proportionate”] do not do so wrongly in my opinion, because there is a fixed proportion between the parts of the tactus, descent and ascent: they are in duple ratio with each other.’ (‘Quod illos meo judicio non injuria fecisse puto. Est enim inter Tactus partes depressionem & elevationem certa proportio. Habent enim ad se invicem ut analogia dupla.’) Ibid., sigs. Aa^v–Aa2^r.

⁵⁹ Praetorius, , Syntagma musicum, iii, pt 2. ch. 7, p. 52Google Scholar. Praetorius is probably referring to Adriano Banchieri's Cartella musicale, document 5, pp. 29–30. According to Aldrich, P.. Rhythm in Seventeenth-Century Italian Monody (New York, 1966), pp. 30–5Google Scholar, Antonio Brunelli (Regole utilissime per li scolari (Florence, 1606)Google Scholar) likewise advocates a literal interpretation of all types of sesquialtera and hemiolia proportions.

⁶⁰ ‘Ac licet quidam in ea sint opinione, etiam Sesquialteram & in Hemiola Notulas denigratas abolendas esse, cùm & hae & si quae aliae hujus generis sint, per unicam Triplam exprimi possint: Tamen incommodè fieri haud dixero, commodioris si distinctionis ergò in gratiam Canentium certis relinquantur Cantionum generibus: Tripla nempe in Motetis & Concertis; Sesquialtera verò in Madrigalibus, praesertim autem in Galliardis, Courrantis, Voltis & aliis id generis Cantionibus, in quibus celeriori Tactu necessariò opus est, retineatur: Propterea, quod pleraeque harum Cantionum tam celerem Tactum requirant, ut pro novitate rei, mihi de novis nominibus, non eum in modum antehac usurpatis, cogitandurn fuerit, atque adeò voce Sextupla vel Tactu Trochaico diminuto rem exprimere conatus sim.’ Praetorius, , Syntagma musicum, iii, pt 2, ch. 7, p. 53Google Scholar.

⁶¹ Praetorius says that madrigals and other pieces in C, which have many semiminims and fusas, move faster, while motets, which are signed ¢ and have many breves and semibreves, move more slowly. The semibreve tactus must be faster in ¢ than in C to mediate between the extremes of the note values, so that the slow tempo will not become tedious and the fast will not run away out of control, like the horses of the sun in the hands of Phaethon. (‘Jetztiger zeit aber werden diese beyde Signa meistentheils also observiret, dass das C fürnemlich in Madrigalien. das ¢ aber in Motetten gebraucht wird. Quia Madrigalia & aliae Cantiones, quae sub signo C, Semiminimis & Fusis abundant, celeriori progrediuntur motu; Motectae autem, quae sub signo ¢ Brevibus & Semibrevibus abundant, tardiori: Ideo hic celeriori, illic tardiori opus est Tactu, quò medium inter duo extrema servetur, ne tardior Progressus auditorum auribus pariat fastidium, aut celerior in Praecipitium ducat, veluti Solis equi Phaetontem abripuerunt, ubi currus nullas audivit habenas.’) Ibid., p. 50. He admits, however, that tempo cannot be chosen on the basis of the sign alone, but must also take account of the text and the character of the music. (‘Es kan aber ein jeder den Sachen selbsten nachdencken, und ex consideratione Textus & Harmoniae observiren, wo ein langsamer oder geschwinder Tact gehalten werden müsse.’) Ibid., p. 51.

⁶² Praetorius's views on tempo have been interpreted in various ways. Hiekel, H.-O., ‘Der Madrigal- und Motettentypus in der Mensurallehre des Michael Praetorius’, Archiv für Musikwissenschaft, 19–20 (1962–1963), pp. 40–55Google Scholar, maintains that Praetorius regards the breve of ¢ as equal to the semibreve of C and the semibreves of tripla in ¢ as equal to the minims of sesquialtera in Dahlhaus, C. C., ‘Zur Taktlehre des Michael Praetorius’, Die Musikforschung, 17 (1964), pp. 162–9Google Scholar, and P. Brainard, ‘Zur Deutung der Diminution in der Tactuslehre des Michael Praetorius’, Ibid., pp. 169–74, demonstrate why Hiekel's interpretation cannot be correct. Dahlhaus suggests (p. 168) that Praetorius may have intended the perfect breve of tripla to correspond to the semibreve of C, and the perfect semibreve of sesquialtera to the semibreve of ¢, even though this explanation contradicts Praetorius's earlier statements about the signs typical of motets and madrigals. In ‘Zur Entstehung des modernen Taktsystems’. pp. 232–3, Dahlhaus proposes a different explanation: tripla of ¢ relates to the breve tactus, and sesquialtera of C to the semibreve tactus. Neither of these explanations is convincing. Praetorius's description agrees with standard notational practice, and his view that the semibreve is the normal tactus of both C and ¢ is clear. Dahlhaus's doubts result from his assumption that the tactus of a proportion must be the same as that of the preceding sign, but what Praetorius says is exactly the opposite: the tempos of proportions, like those of binary signs, must be determined by the character of their music, and this will generally result in a slower tempo for proportions of C than for proportions of C. Praetorius is a late witness, but the music to which he refers in this chapter is mostly from the late sixteenth century and his comments make sense in relation to that repertory.

⁶³ Modern editions in Lassus, , Sämtliche Werke, ed. Haberl, F. X. and Sandberger, A. (Leipzig, 1894–1926), v, pp. 144–50 (Clare sanctorum), 60–3Google Scholar (Surgens Jesus) and 141–3.(Non vos me elegistis). The source has the word ‘laudes’ in place of ‘laudem’ in the lower four voices of Example 10b. The proportional sign in Example 10b is 3, rather than Ф³₂. in Le Roy and Ballard's 1571 edition of the book. In the modern edition. the rhythm of Example 10a, bar 17, bass, notes 3–4, is transcribed incorrectly as semibreve–minim.

⁶⁴ In Non vos me elegistis, there is a white breve simultaneous with three black semibreves in the penultimate measure of the proportion (not shown in Example 10d). This might he taken as evidence that three black semibreves are equivalent to a white breve before the proportion. but if the tempo of the proportion is different from that of the preceding section, the white breve could be understood to relate only to the tempo in effect at that point.

⁶⁵ All three motets are notated this way in the manuscript Regensburg, Fürstliche Thurn- und Taxissche Hofbibliothek, F. K. Musik no. 22. The same notation is found in, Non vos me elegistis in Rühling, J., Tabulaturbuch (Leipzig, 1583)Google Scholar, and Schmid, Bernhard (the father), Zwey Bücher: Einer Neuen Kunstlichen Tabulatur (Strasburg, 1577; RISM 1577 ¹²)Google Scholar, and in Surgens Jesus in Paix, Ein Schön Nutz unnd Gebreüchlich Orgel Tablaturbuch, Ruhling. Tabulaturbuch, and Regensburg, Fürstliche Thurn- und Taxissche Hofbibliothek, F. K. Musik no. 24. The proportion in Surgens Jesus is notated as major sesquialtera in Braunschweig, Stadtarchiv, MS G II 7:60; Munich, Bayerische Staatsbibliothek, Mus. MS 1640; and Passau, Staatliche Bibliothek, MS 115. See Johnson. Vocal Compositions, and Brown, Instrumental Music, for catalogues of these sources.

⁶⁶ Dahlhaus, ‘Zur Entstehung des modernen Taktsystems’, p. 223, characterises the difference between modern metres and Renaissance mensurations in this way.

⁶⁷ ‘Gleich wie sich die beide Ciffern 3 und 2 in Proportione sesquial. zu hauff haben/ also wird der Proporcien Tact wenn er langsam/ gegen dem gantzen/ odder gegen dem halben/ so er risch geschlagen wird/ geachtet-und abgemessen/ als ein Exempel. Der halbe Tact in diesem zeichen ¢ begreifft solcher ii. aber der Proporcien Tact alzeit der iii Darumb wird der Proporcien Tact/ soviel als eine Minima iii langsamer dann die andern beide gefüret/ Und dieweil er nach der art der sesquialtern/ gegen den andern Tacten geschatzt/ und sie anderthalb mal in ihm beschleust/ mag er billich sesquialteratus odder Proportionatus Tactus (wie die Musici schreiben) genant werden. Auch braucht man ihn nicht überal/ sondern allein in Prolatione perfecta/ wie im 4. Cap. berürt/ odder in Proportione Tripla/ Hemiola/ wenn sie alle stymmen zu gleich haben/ und so wird alzeit eine Semibre. nach der masse/ wie sonst eine Minima gesungen.’ Agricola, Musica figuralis deudsch, ch. 6. sig. Gv^r.

⁶⁸ Berger, , Mensuration and Proportion Signs, pp. 107–8Google Scholar, suggests that Agricola meant to say that the proportionate tactus is a minim faster, not a minim slower, than the other two tactus, but the passage makes sense and is internally consistent if the expression ‘nach der art der sesquialtern, gegen den andern Tacten geschatzt’ is understood to mean that the 3:2 proportion applies to the lengths of the tactus, not to the note values. Thus in a triple or hemiolia proportion of ¢, the tactus will include three semibreves, each equivalent to a minim of ¢, and will be 50% longer than the semibreve half-tactus of ¢. Berger transcribes the mensuration sign in this passage as φ, rather than ¢. The problem with Agricola's testimony is not so much in the ambiguity of this passage as in the contradictions between different descriptions of the proportionate tactus in the same treatise. For example, in the passage immediately preceding this one (sig. Giiii^v), he says that a semibreve in the proportionate tactus is almost as fast as, rather than equal to, a minim of ¢ (‘fast so risch/ als sonst eine Minima im halben Tact ¢’). Mendel, A., ‘Some Ambiguities of the Mensural System’, Studies in Music History: Essays for Oliver Strunk, ed. Powers, H. (Princeton, 1968), pp. 137–60 (pp. 140–8)Google Scholar, discusses additional contradictions among Agricola's statements about the tempo of the proportionate tactus.

⁶⁹ Ornithoparchus, , Musice active micrologus, bk 2, ch. 11, sig. Giij^v; trans. Dowland, , p. 57; A Compendium of Musical Practice, pp. 67 and 177Google Scholar.

⁷⁰ Gaffurius says nothing about coloration as duple proportion in bk 2, ch. 2, of the Practica musicae, but in bk 2, ch. 4, sig. aaiiij^r (trans. Miller, p. 76), he refers to Tinctoris's definition of black minims (called ‘semiminims’ by others) as ‘minims in duple proportion’. Tinctoris discusses the terminology in Proportionale musices, ch. 5, in Opera theoretica, ed. Seay, A., Corpus Scriptorum de Musica 22 (n.p., 1975–1978), nA, pp. 16–17Google Scholar: trans. A. Seay, Colorado College Music Press Translations 10 (Colorado Springs, 1979), p. 7. This usage may have been the source of later claims that coloration can indicate duple proportion.

⁷¹ Modern edition in A Compendium of Musical Practice, pp. 67 and 177. Picker, M., Henricus Isaac: A Guide to Research (New York, 1991), p. 172Google Scholar, is unable to identify the example in the surviving works of Isaac.

⁷² ‘Si vero plures Notulae denigratae fuerint, tum frequentius Proportionem Hemioliam, nonnumquam Duplam nigredo efficit, quod utrumque suis exemplis infra constabit.’ Heyden, , De arte canendi, bk 2, ch. 1, p. 62; trans. Miller, , p. 69Google Scholar.

⁷³ Hell, ‘Zu Rhythmus und Notierung’, pp. 97–105, uses similar arguments to demonstrate (convincingly, in my opinion) that the same proportion exists between the ritornello of ‘Vi ricorda, o boschi ombrosi’ and the preceding C in Monteverdi's Orfeo. He traces the origin of the relationship to certain canzonettas and dances of the later sixteenth century (Ibid., pp. 105–23). Machatius, F.-J., ‘über mensurale und spielmännische Reduktion’, Die Musikforschung, 8 (1955), pp. 139–51Google Scholar, calls this proportion ‘spielmännische Reduktion’ (‘players’ diminution'), in contrast to ‘mensurale Reduktion’ (diminution in which the tactus remains constant), because he believes that it originated in the kinds of dances and simple songs performed by popular instrumentalists. Dahlhaus, C., ‘Die Tactus- und Proportionenlehre des 15. bis 17. Jahrhunderts’, Hören, Messen und Rechnen in der frühen Neuzeit, ed. Zaminer, F., Geschichte der Musiktheorie 6 (Darmstadt, 1987), pp. 335–61 (p. 343)Google Scholar, citing the passage from Agricola quoted above (n. 67), asserts that this interpretations of sesquialtera was not limited to pieces of popular character, but coexisted in all genres with interpretations- in which the tactus remains constant.

⁷⁴ I am grateful to Michele Fromson for making her microfilm of this piece available to me. The rhythm of cantus 1 in bar 67 is semibreve–breve in the source. I have emended it to breve–semibreve by comparison with cantus 2, bar 69.

⁷⁵ I am grateful to Alexander Blachly for suggesting this possibility to me. His recordings of Lassus's Cantate Domino and Wert's O sacrum convivium with the Pomerium Musices on The Flemish Masters: Netherlanders in Italy in the 16th Century, I, Classic Masters CMCD-1007 are excellent examples of how it can work in practice. In the former the proportion is notated as minor hemiolia, with three black minims corresponding to a white semibreve in the same phrase (as in Example 10a), and in the latter it is notated as major sesquialtera under the sign φ³₂ (as in Example 5). In the performances of both, one third of the tactus of the proportion (a black minim or a white semibreve, respectively) is almost as slow as a minim of ¢, but the perceived relationship is between the complete tactus of the proportions and the semibreve of ¢, not between the smaller values.

⁷⁶ In a review of the performance of Taverner's Western Wynde mass by The Sixteen, directed by Harry Christophers, on Hyperion CDA 66507 (recorded in 1991). Earlier Music, 21 (1993), p. 138Google Scholar, David Mateer points out that the triple-time sections are performed in duple proportion to the duple-time sections (a semibreve of triple equals a minim of duple), and that this solution works well in practice despite its lack of theoretical support.

⁷⁷ Bowers, R., ‘Some Reflection upon Notation and Proportions in Monteverdi's Mass and Vespers of 1610’, Music and Letters, 73 (1992), pp. 347–98CrossRefGoogle Scholar, claims that all the proportions in Monteverdi's 1610 Vespers, including passages of major sesquialtera (in relation to C in the vocal part and ¢ in the basso continuo) in two sections of the Magnificat a7, should be interpreted literally. He cites several theorists, some of whose views he interprets in questionable ways, to support this opinion and calls those who suggest other possibilities ‘committed polemicists’ (p. 370, n. 48). Considering the common complaints of strict theorists about practitioners (composers as well as singers) who do not follow their precepts, however, the term ‘polemicists’ might better apply to them than to their more liberal colleagues. Bowers's interpretations of minor sesquialtera and triple proportion are plausible, but his interpretation of major sesquialtera (pp. 377–9) is not. The tactus of the sections in which the proportion occurs is clearly the semibreve, since the proportion sometimes begins in the middle of a breve tempus (a fact obscured by the irregular barring of the binary passages in Bowers's Example 21, p. 379), and the tradition of interpreting sesquialtera in relation to the tactus was nearly a century old by that time.