¹ ‘Ben voria che quella bella cosa di Tinttoris mi volevi dar al partir mio me la mandasti qui a Roma, et za me dicevati alora haver la resolucione qual facesti una volta per Zampiero, a chi Dio perdoni. Haveria molto a charo dicta cosa con qualche pocca dichiaratione, acciò apresso la resolucione io sapesse le rason perché, da la qual cosa molto ve ne saria obligato. Et voria veder de qui quello me saperiano dir alcuni valent ‘omeni vi sono.’ The letter forms part of the Spataro correspondence, Vatican City, Biblioteca Apostolica Vaticana, MS Vat. lat. 5318 [Vatican City Vat. lat. 5318], fol. 136. See Jeppesen, K., ‘Eine musiktheoretische Korrespondenz des früheren Cinquecento’, Acta Musicologica, 13 (1941), pp. 3–39CrossRefGoogle Scholar. An edition of the whole correspondence is being undertaken by Edward E. Lowinsky and Clement A. Miller. I wish to thank them warmly for allowing me to publish portions of some of the letters in their transcriptions in advance of publication. Moreover Professor Miller kindly read this paper and suggested an explanation for the puzzling word ‘dupliciter’ in some of the annotations. I am greatly indebted to Edward Lowinsky, who generously gave of his time and knowledge and whose criticism has considerably improved the final form of this paper.

² ‘la messa de Dixerunt discipuli cum le annotacion di quelli modi dove sarà bisognio … Haveria charo che facesti si possibil fosse in modo di uno comento sopra quello Difficiles di Tintoris acciò lo potesse bene intendere’ (letter of 20 December 1523, Vatican City Vat. lat. 5318, fol. 135^v). On the Mass, see below, p. 90 and note 119.

³ ‘et Tinctoris in tenore sui magistralis moteti. Difficiles alios delectat pangere cantus decenter consignavit’ (Gafurius, Franchinus, Practica musicae, trans. Miller, C. A., Musicological Studies and Documents 20 (n.p., 1968), p. 86Google Scholar). Young, Irwin (The Practica musicae of Franchinus Gafurius (Madison, 1969), p. 87)Google Scholar mistook the title of Tinctoris's motet for part of the succeeding sentence and entitled the motet Magistralis.

⁴ See Miller, C. A., ‘Early Gaffuriana: New Answers to Old Questions’, Musical Quarterly, 56 (1970), pp. 367–88, esp. pp. 374–80CrossRefGoogle Scholar.

⁵ ‘et ipsae proportiones virtutes proprias operari noscuntur ut diffusius patet in moteto nostro ad Joannem Tinctoris: Nunc eat & veteres et in quampluribus aliorum compositionibus’ (‘and the proportions themselves are known to effect their particular powers, as is more amply evident in our motet to Johannes Tinctoris, Nunc eat et veteres, and in many compositions of others’), Bologna, Civico Museo Bibliografico Musicale, MS a 69, fols. 2^v–3.

⁶ On the relations between Gafurius and Tinctoris, see Miller, ‘Early Gaffuriana’, p. 377.

⁷ The Segovia manuscript (Cathedral, MS s.s. [Segovia s.s.]) contains a small section in which are twelve proportional duos by Agricola, Obrecht, Adam, Roelkin, and Tinctoris (fols. cc–ccv^v). Of the six attributed to Tinctoris, two are also found in the Perugia manuscript.

⁸ A brief description may be found in Mazzatinti, G., Inventari dei manoscritti delle biblioteche d'Italia, v: Ravenna, Vigevano, Perugia (Forlì, 1895), p. 228Google Scholar. The table of contents given in Seay, A., ‘An “Ave maris Stella” by Johannes Stochem’, Revue Belge de Musicologie, 11 (1957), pp. 93–4CrossRefGoogle Scholar, is not entirely accurate. The revisions made by Herlinger, Jan in his description of the manuscript in ‘The Lucidarium of Marchetto of Padua: A Critical Edition, Translation, and Commentary’ (Ph.D. thesis, University of Chicago, 1978), pp. 100–2Google Scholar, have been incorporated here. I have not seen the manuscript itself but worked from a photocopy.

⁹ On this academy, see Atlas, A. W., ‘The Accademia degli Unisoni: A Music Academy in Renaissance Perugia’, A Musical Offering: Essays in Honor of Martin Bernstein, ed. Clinkscale, E. E. and Brook, C. (New York, 1977), pp. 5–23Google Scholar.

¹⁰ Medieval writers sometimes confused Jubal with his half-brother, attributing to him the invention of consonances and thus infusing the biblical passage with elements of Pythagoras's legendary discovery of the ratios of musical consonances in a blacksmith's workshop. The different traditions concerning the inventor of music have been traced by Beichner, P. E., The Medieval Representative of Music, Jubal or Tubalcain?, Texts and Studies in the History of Medieval Education 2 (Notre Dame, 1954)Google Scholar. For a broader historical inquiry see McKinnon, J. W., ‘Jubal vel Pythagoras, quis sit inventor musicae?’, Musical Quarterly, 64 (1978), pp. 1–28CrossRefGoogle Scholar.

¹¹ The unusual D♯ may be explained by the scribe's familiarity with Marchetto da Padova's Lucidarium, where this note is specified (see Herlinger, ‘The Lucidarium’, p. 353), or more likely the Calliopea legale of Johannes Hothby, whose ‘terzo ordine’ includes D♯ see de Coussemaker, E., Histoire de l'harmonie au moyen âge (Paris, 1852), p. 298Google Scholar.

¹² Seay, ‘An “Ave maris Stella” ’, p. 93, n. 3; Herlinger, ‘The Lucidarium’, p. 100, n. 3.

¹³ Incomplete and in a poor version. See Herlinger, ‘The Lucidarium’, pp. 100–2.

¹⁴ ibid., p. 100, n. 3.

¹⁵ Recently, Albert Seay has published an edition of this section of the manuscript: Anonymous (15th-Century), Quaestiones et solutiones, Colorado College Music Press Critical Texts 2 (Colorado Springs, 1977). At that time he was unaware of the identity of the part taken from Bonaventura, but in a note ‘To the Reader’ dated 26 November 1978 inserted in the publication, in which he thanks Professor Giuseppe Massera, he acknowledges the incorporation of a large section of Bonaventura's treatise. The table of contents of Perugia 1013 given on pp. i–iii of his edition is largely identical with that in his earlier article.

¹⁶ Modern edition in de Coussemaker, E., Scriptorum de musica medii aevi nova series, 4 vols. (Paris, 1864–1976), i, pp. 157–75Google Scholar (Scriptorum de musica is hereafter abbreviated as CS). See Gallo, F. A., ‘Tra Giovanni di Garlandia e Filippo da Vitry: note sulla tradizione di alcuni testi teorici’, Musica Disciplina, 23 (1969), pp. 13–20, esp. p. 18Google Scholar. In his edition of the Quaestiones, Seay points out the connection of the beginning with Garlandia's treatise.

¹⁷ For a facsimile, transcription, and explanation of this complex proportional duo, see Seay, ‘An “Ave maris stella” ’, pp. 96–108.

¹⁸ Modern edition by Seay, A. in ‘The Conditor alme siderum by Busnois’, Quadrivium, 12 (1971), pp. 225–34Google Scholar.

¹⁹ ‘Artis’ instead of ‘artem’ may be due to the absence of a following word, indicated by the ‘et cetera’.

²⁰ See, for example, the book prepared by Giovanniantonio Tagliente, the Venetian calligrapher, which provides samples of various types of chancellery hands, but also Venetian and Florentine mercantile hands, the Greek alphabet, and two kinds of Gothic, in addition to some highly fanciful personal styles (he was also the author of a book of patterns for embroidery). The 1525 edition of the Opera di Giovanniantonio Tagliente che insegna a scrivere de diverse qualità de lettere has been published in a facsimile, edited by J. M. Wells (Chicago, 1952). The Newberry Library possesses a manuscript (Wing MS fzw 141.46) of Gothic majuscule and minuscule alphabets, copied for the monastery of Santa Giustina in Padua, which is similar to the Perugia treatise; both series of letters are constructed geometrically. Perhaps Johannes Materanensis's interest in musical proportions was stimulated by his experience as a scribe – or vice versa. The mathematical foundation of calligraphy is stressed in one of the earliest printed treatises, the Theorica et pratica perspicacissimi Sigismundi de Fantis Ferrariensis in artem mathematice professoris de modo scribendi fabricandique omnes litterarum species, which came out in Venice in 1516. De Fantis undertook the book because lettering ‘anchora non è stata per qualunque modo dilucidata per essere li Praeceptori non bene in le Mathematice disciplinae adoctati’ (fol. +iii). I wish to thank James Wells, Associate Director of the Newberry Library, for acquainting me with the library's rich collection of manuscripts and early printed books on lettering, and for referring me to Meiss, M., Andrea Mantegna as Illuminator (New York, 1957)Google Scholar, which includes the informative chapter ‘The Alphabetical Treatises’.

²¹ The academy was founded in 1561; none of the members listed by Atlas in ‘The Accademia degli Unisoni’, however, has the initials ‘S.P.’.

²² On this manuscript, see Hamm, C., ‘The Manuscript San Pietro B 80’, Revue Belge de Musicologie, 14 (1960), pp. 40–55CrossRefGoogle Scholar. (Editor's note: see also C. Reynolds, ‘The Origins of San Pietro b 80’, p. 257–304 below.)

²³ Modern edition in Johanni[s] Tinctoris opera omnia, ed. Melin, W., Corpus Mensurabilis Musicae 18 (n.p., 1976), pp. 1–32Google Scholar. The opera are somewhat less than omnia, inasmuch as they do not include examples in the treatises excepting one duo that also appears in Segovia s.s. Unlike those of many other music treatises, a considerable number of examples in Tinctoris's Proportionate musices and Liber de arte contrapuncti are not only complete, but also musically significant compositions that should be presented among the composer's opera omnia, perhaps in a special appendix. (On this question, see also Strohm, R., ‘Die Missa super “Nos amis” von Johannes Tinctoris’, Die Musikforschung, 32 (1979), p. 40Google Scholar, n. 19.) Melin inexplicably omits the distichs heading the Mass in the Verona manuscript, the only known source: ‘Ferdinande sacer inter divos referende cantica tintoris suscipe parva tui’ (‘O Ferdinand, holy enough to be numbered among the gods, accept the small compositions of your Tinctoris’). These words, so characteristic of the Renaissance spirit of individualism and patronage, probably stem, directly or indirectly, from a presentation copy of Tinctoris's compositions (‘cantica’) made for King Ferdinand, now lost.

²⁴ ‘Sunt et alii et fere omnes qui maximo errore ducti se penitus expertes arithmeticae manifestantes, una tantum cyphra, videlicet, eius numeri qui ad alium refertur omnes quas praticant proportiones signant’ (Proportionate musices, Johannis Tinctoris opera theoretica, ed. Seay, A., Corpus Scriptorum de Musica 22 (n.p., 1975–1958), iia, p. 44Google Scholar. (Seay's edition of Tinctoris's theoretical works, which I have used throughout the paper, will be cited hereafter as Seay i, ii, and iia.) The criticism is echoed by Gafurius in his Practica musicae, trans. Miller, p. 156Google Scholar.

²⁵ Seay iia, p. 48.

²⁶ ‘Minima est nota valoris individui’ (‘a minim is a note of indivisible value’), states Tinctoris in his Diffinitorium (see Parrish, C., trans., Dictionary of Musical Terms by Johannes Tinctoris (New York and London, 1963), pp. 40–1Google Scholar). A century and a half earlier, Johannes de Muris had recognised and named a semiminim.

²⁷ The position that blackened minims are not semiminims forces Tinctoris into a lengthy discussion in his Proportionate on how to tell whether coloration means proportio dupla, sesquialtera, imperfection, or reductio (Seay iia, pp. 16–17 and 20–2).

²⁸ Charles, Hamm, in A Chronology of the Works of Guillaume Dufay Based on a Study of Mensural Practice (Princeton, 1964)Google Scholar, has put forward the thesis that ‘the beat was on the minim in pieces in major prolation dating from the first part of the century’ (p. 24). This view has been contested by Bank, J. A., Tactus, Tempo and Notation in Mensural Music from the 13th to the 17th Century (Amsterdam, 1972)Google Scholar, esp. chapter vii, ‘Mensura facit tactum’, in which he shows that the theorists quoted by Hamm speak of the tactus on the minim only when the part is to be read in augmentation.

²⁹ This example is concordant with Tinctoris's Virginis Mariae laudes in the counterpoint book (Seay ii, pp. 110–12). Contrary to his usual practice, Seay transcribed the beginning of the superius in integer valor, without so indicating.

³⁰ See the modern edition in Johanni[s] Tinctoris opera omnia, ed. Melin, pp. 74, 105 and 109Google Scholar.

³¹ Seay's transcription (iia, p. 54) of the first measure of the discantus should be emended to begin with a breve imperfected ad totum (six minims) and end with a semibreve G, thus eliminating the parallel ninths which are not sanctioned in Tinctoris's rules of counterpoint. Seay, who published his translation of the Proportionale in the same year that his first article on the Perugia manuscript appeared, failed to recognise any of the examples taken from Tinctoris's treatise. Of Difficiles alios he says no more than: ‘The Cantus appearing on f. 118^v has under its first notes the word “Tentoris” ’ (‘An “Ave maris Stella” ’, p. 95, n. 1).

³² On duos in fifteenth-century music, see Kämper, D., ‘Das Lehr- und Instrumentalduo um 1500 in Italien’, Die Musikforschung, 18 (1965), pp. 242–53Google Scholar; Kämper, D., Studien zur instrumentalen Ensemblemusik des 16, Jahrhunderts in Italien, Analecta Musicologica 10 (Cologne and Vienna, 1970), pp. 86–112Google Scholar; and Brown, H. M., ‘Eustachio's Historical Position’, Eustachio Romano, Musica duorum, Rome, 1521, ed. from the literary estate of David, Hans T. by H. M. Brown and E. E. Lowinsky, Monuments of Renaissance Music 6 (Chicago, 1975)Google Scholar, chapter vi. This edition includes a transcription of the duo on Le serviteur from the Perugia manuscript (pp. 38–40), particularly interesting because it runs through the series of proportions .

Recently Lawrence, F. Bernstein has made a survey, the results of which he published as ‘French Duos in the First Half of the Sixteenth Century’, Studies in Musicology in Honor of Otto E. Albrecht, ed. Hill, J. W. (Kassel and Clifton, N. J., 1980), pp. 43–87Google Scholar; in this article he discussed the duos of Perugia 1013. Independently he discovered a number of concordances with examples in Tinctoris's treatises. I wish to thank Professor Bernstein for placing the proofs of his article at my disposal before publication.

³³ Seay iia, pp. 47 and 57–8. Besseler dates this work c. 1435–40; see Guglielmi Dufay opera omnia, ed. Besseler, H., Corpus Mensurabilis Musicae 1, ii (n.p., 1960), p. vGoogle Scholar. This is the only Mass in which Dufay uses proportions other than sesquialtera, except for a brief passage in in the Missa L'homme armé. Charles Hamm has serious doubts about its authenticity, citing six examples of mensural usage that conflict with Dufay's normal practice; see A Chronology of the Works of Guillaume Dufay, pp. 105–10.

³⁴ Seay iia, p. 46, and ii, pp. 12, 143, 156. Seay noted (i, p. 25) that Tinctoris mentions or quotes six Masses found in Vatican City, Biblioteca Apostolica Vaticana, Archivio della Cappella Sistina, MS 14 [Cappella Sistina 14], suggesting that ‘Tinctoris may well have worked with this codex’. It also contains Faugues's Missa L'homme armé.

³⁵ It is not an example that Tinctoris would have viewed with favour, since it regards the notes under sesquialtera in the contratenor as in prolatio maior; see his criticism of Domarto in this respect in chapter 5 of Book iii of the Proportionale (Seay iia, p. 56; in sentence 16 on p. 56 the following words are missing after ‘Domarto’: ‘semibreves perfecit et breves imperfecit, secundus, scilicet Cousin, e converso’). Wolfenbüttel 287 and Dijon 517, which are quite similar, present better readings than the Perugia manuscript.

³⁶ Compositione di Meser Vincenzo Capirola, Lute-Book (circa 1517), ed. Gombosi, O. (Neuilly-sur-Seine, 1955), p. xxxviiiGoogle Scholar.

³⁷ See Seay, ‘The Conditor alme siderum’.

³⁸ Proportionale, Book iii, chapter 5 (Seay iia, p. 53).

³⁹ The readings do not differ substantially from the manuscripts of Tinctoris's treatises known at present. There is some variance, as might be expected, in ligatures and cadential ornamentation. In no. 20, concordant with Virginis Mariae laudes from the counterpoint treatise (Seay ii, pp. 110–12), white semiminims with double flags are used in place of black semiminims with single flags. An annotation to no. 61, concordant with an example in the Proportionale (Seay iia, p. 24), is helpful in correcting two passages in Seay's transcription that should have been read in sesquialtera: in bars 7 and 8 after the sign the three black semibreves are labelled ‘sesquialtera’. Therefore they are counted together with the following white semibreve to make a perfect tempus, and the preceding breve D must be perfect. The blackened breve and semibreve in bars 2–3 of the same section are also to be read in sesquialtera. J. A. Bank, in his transcription of this example (Tactus, Tempo and Notation, p. 164, Example 90), though purportedly based on Seay, falls into a different error; in bars 7–8 the black breve is transcribed as a semibreve. I do not understand the statement (p. 165) that ‘In the Tenor, Modus minor perfectus alternates with Modus maior perfectus and Modus minor imperfectus, indicated by means of special signs over the system, which Tinctoris thinks is the best way of notation.’ There is only one sign, of modus maior perfectus, minor perfectus at the beginning of the tenor, and blackened notes achieve imperfection of the modus. In none of his examples does Tinctoris change the modus within a composition.

⁴⁰ ‘Musicus est qui perpensa ratione beneficio speculationis canendi officium assumit’ (Tinctoris, Diffinitorium; Parrish, Dictionary, p. 44) – a definition that is largely taken over from Boethius: ‘Is vero est musicus, qui ratione perpensa canendi scientiam non servitio operis sed imperio speculationis adsumpsit’ (De institutione musica, ed. Friedlein, G. (Leipzig, 1867), p. 224)Google Scholar, but with the important difference that Boethius's musicus does not deign to sing.

⁴¹ For the literature, see Census-Catalogue of Manuscript Sources of Polyphonic Music 1400–1550, Renaissance Manuscript Studies 1, i (n.p., 1979), p. 68, to which should now be added Kirsch, W., Die Motetten des Andreas de Silva: Studien zur Geschichte der Motette im 16. Jahrhundert (Tutzing, 1977), pp. 24–7Google Scholar.

⁴² Gümpel, K.-W., ‘Das Enchiridion de principiis musice discipline des Guillermus de Podio’, Gesammelte Aufsätze zur Kulturgeschichte Spaniens, Spanische Forschungen der Görresgesellschaft, ser. i, 27 (Münister, Westphalia, 1973), pp. 359–96Google Scholar; see esp. pp. 359–62. I wish to thank Professor Gümpel for his kind answers to my questions.

⁴³ Census-Catalogue, i, p. 68. I have worked from a microfilm.

⁴⁴ Edited by Gümpel, ‘Das Enchiridion’, pp. 362–96.

⁴⁵ Modern edition by Anglés, H., ‘La notatión musical española de la segunda mitad del siglo XV. Un tratado desconocido de Guillermus … de Podio’, Anuario Musical, 2 (1947), pp. 161–173Google Scholar. Gümpel has recently edited a Catalan version of this treatise; see ‘Eine katalanische Version der Mensurallehre des Guillermus de Podio’, Orbis Musicae, 2 (1973–1974), pp. 41–52Google Scholar.

⁴⁶ On singing super librum, a practice discussed by Tinctoris, see Wright, C., ‘Performance Practices at the Cathedral of Cambrai 1475–1550’, Musical Quarterly, 64 (1978), pp. 295–328, esp. pp. 313–22CrossRefGoogle Scholar.

⁴⁷ ‘Improvised Ornamentation in the Fifteenth-Century Chanson’, Quadrivium, 12 (1971), pp. 235–58Google Scholar.

⁴⁸ ibid., pp. 239–40.

⁴⁹ Bologna a 71 was unknown to Bradshaw, Murray C., The Falsobordone: A Study in Renaissance and Baroque Music (n.p., 1978)Google Scholar. Although he believes falsobordone originated in Italy, most of the earliest sources are Spanish. Interestingly, Bologna a 71 has both Italian and Spanish elements. The falsobordone are in a style similar to Bradshaw's examples from the late fifteenth century.

⁵⁰ ibid., p. 24. Unlike other early sources, Bologna a 71 does not include a setting of Psalm 113, In exitu Israel.

⁵¹ De Silva's earliest datable work is his Gaude felix Florentia, written for the election of Leo x in 1513; see Lowinsky, E. E., ‘A Newly Discovered Sixteenth-Century Motet Manuscript at the Biblioteca Vallicelliana in Rome’, Journal of the American Musicological Society, 3 (1950), pp. 175–6CrossRefGoogle Scholar. Willaert was in Italy by July 1515, when he entered the service of Cardinal Ippolito d'Este in Rome; see Lockwood, L., ‘Josquin at Ferrara: New Documents and Letters’, Josquin des Prez: Proceedings of the International Josquin Festival-Conference, ed. Lowinsky, E. E. in collaboration with B. J. Blackburn (London, 1976), pp. 119–20Google Scholar. (An earlier date has been proposed by Valerie Weinhouse O'Donoghue; see note 55 below.)

⁵² Robert Stevenson has suggested that the manuscript was ‘intended for use among students in the Spanish college at Bologna’; see Spanish Music in the Age of Columbus (The Hague, 1960), p. 74Google Scholar. O'Donoghue, V. W., ‘A Music Manuscript from the Spanish College of Bologna: A Study of the Manuscript Bologna, Civico Museo Bibliografico Musicale, MS A 71 (olim 159)’ (M.M. thesis, University of Illinois at Urbana-Champaign, 1972)Google Scholar, accepts Stevenson's suggestion and points to the frequent underlining and marginal notes as evidence that the manuscript ‘was originally used for study purposes’ (p. 4). I wish to express my warmest thanks to Professor Donald Krummel of the University of Illinois, whose kind intervention made the thesis accessible to me.

⁵³ O'Donoghue is to be credited with the discovery of the concordances in the Perugia manuscript, although their source in Tinctoris's Liber de arte contrapuncti escaped her.

⁵⁴ For example, in Seay ii, p. 117, first measure of the upper voice at the top of the page, Bologna and Perugia give A as the third note – certainly not the correct reading; in the example on p. 120, bars 2–3, the change to sesquialtera occurs one measure earlier in both sources (with no change in harmony). On p. 126, third staff, bar 1, Perugia and Bologna agree with Bologna, Biblioteca Universitaria, MS 2573, in giving D as the fourth note of the lowest voice – a better reading since it removes the leap into a dissonance.

⁵⁵ O'Donoghue thinks it possible that the note is in the hand of Tinctoris; this is the basis for her suggestion that the manuscript may date before his death in 1511, though she concludes that 1510–15 is the most likely date (‘A Music Manuscript’, pp. 39–40). Since the handwriting of the note appears elsewhere in the manuscript, it seems more plausible to assume that the name Tinctoris was copied by the scribe together with the piece, a hypothesis considered conceivable by O'Donoghue (ibid., p. 37).

⁵⁶ Difficiles alios delectat pangere cantus seems to be a motto rather than the beginning of a text. I do not know its source.

⁵⁷ On this, see, the letters of Giovanni del Lago to Pietro Aaron and to Lorenzo Gazio (below, pp. 90 and 93–4).

⁵⁸ See the letter of Giovanni del Lago to Pietro Aaron of 27 August 1539, discussed below (p. 90), which shows the state of confusion in ordering the mensurations.

⁵⁹ And sometimes even the use of motifs. Compare the opening of the cantus of Difficiles alios with bars 1–3 of the tenor of Sancte Johannes Baptista (Seay ii, p. 125).

⁶⁰ I shall discuss the chronology of Tinctoris's treatises in another study.

⁶¹ Seay ii, p. 156. I use Lowinsky's, translation in ‘Renaissance Writings on Music Theory (1964)’, Renaissance News, 18 (1965), p. 366Google Scholar. In all his treatises, Tinctoris mentions no more than one of his own compositions, and that only because it serves as the catalyst for his treatise on alterations; see p. 82 below.

⁶² ‘De nona specie compositionis et valore notarum in ea existentium’: ‘Nona species compositionis fit ex utroque modo perfecto, tempore perfecto et prolatione minori. Itaque regulariter in ea valet maxima 3 longas, 9 breves, 27 semibreves et 54 minimas; longa 3 breves, 9 semibreves et 18 minimas; brevis 3 semibreves et 6 minimas; ac semibrevis 2 minimas.' (Seay i, p. 134.) The music example given by Seay lacks the three pauses covering three spaces each that are indicative of the major and minor perfect modes. There are also errors in pauses elsewhere in this treatise: in Example 30 there should be three pauses covering two spaces each; in Example 31 two pauses covering three spaces each, and in Example 35 two pauses covering three spaces each. In Example 29 a semibreve E is missing before the minim C. Del Lago, in a letter to Pietro Aaron of 27 August 1539, refers to ‘the glosses of the soprano, the prima pars of the tenor, and the secunda pars of the tenor’ of this composition as an example of how Tinctoris numbers the species of modes; see below, p. 90.

⁶³ ‘Tenor est cuiusque cantus compositi fundamentum relationis.’ (Parrish, Dictionary, pp. 64–5.)Google Scholar

⁶⁴ See Eggebrecht, H. H., Studien zur musikalischen Terminologie, Akademie der Wissenschaften und der Literatur: Abhandlungen des Geistes- und Sozialwissenschaftlichen Klasse 10 (Mainz, 1955), p. 879Google Scholar.

⁶⁵ de Muris, Johannes, Ars contrapuncti (CS iii, p. 60)Google Scholar: ‘Contrapunctus non est nisi punctum contra punctum vel notam contra notam ponere vel facere, et est fundamentum discantus. Et quia sicut quis non potest edificare, nisi prius faciat fundamentum, sic aliquis non potest discantare, nisi prius faciat contrapunctum.’ (Counterpoint is nothing but placing or making a point against a point or a note against a note and it is the foundation of discant. And just as no one can build unless he first makes a foundation, so no one can sing discant unless he first makes counterpoint.) In the chapter ‘De numerorum proportione’ in his Notitia artis musicae of 1321 (ed. Michels, U., Corpus Scriptorumde Musica 17 (n.p., 1972))Google Scholar, Johannes de Muris gives a table of the five species of proportions in which all the numbers are to be compared to the ‘fundamentum relationis’ which serves as the denominator (p. 54).

⁶⁶ Politics, viii, ch. 5, 7; see The Basic Works of Aristotle, ed. McKeon, R. (New York, 1941), pp. 1312, 1316Google Scholar.

⁶⁷ Strunk, O., Source Readings in Music History (New York, 1950), pp. 4–5Google Scholar.

⁶⁸ Glareanus, Henricus, Dodecachordon, trans. Miller, C., Musicological Studies and Documents 6, 2 vols. (n.p., 1965), i, p. 164Google Scholar. Glareanus, who prided himself on his knowledge of Greek, was apparently using a Latin translation of Plato or perhaps a Latin commentary on the De republica.

⁶⁹ ibid., i, p. 129.

⁷⁰ ibid., i, p. 130. The reference is to Gafurius, , De harmonia musicorum instrumentorum opus (Milan, 1518)Google Scholar, Book iv, chapters 2 and 5 (fols. 84, 85^v). Gafurius, however, is merely transmitting ancient reports about the effects of the modes (‘apud antiquos’ and ‘apud veteres’, as his chapter headings specify), and he does not connect these characterisations with music of his time.

⁷¹ Glareanus, , Dodecachordon, trans. Miller, i, p. 130Google Scholar.

⁷² ibid., i, p. 160.

⁷³ Required reading for scholars in Paris in 1304 were Aristotle's Politics and PseudoAristotle's Problems; see Carpenter, N. C., Music in the Medieval and Renaissance Universities (Norman, 1958), p. 50Google Scholar. These two books are mentioned frequently in Carpenter's study.

⁷⁴ ‘Namque Mixolydiam planctivam, Lydiam remissam, Phrygiam rigidam et Doriam mediam affirmat.’ (Seay i, p. 68.)

⁷⁵ Strunk, , Source Readings, p. 82Google Scholar. Tinctoris reports the story in his Complexus effectuum musices under the twelfth effect: ‘Musica voluntatem malam revocat’ (Seay ii, p. 172), in a version after Cicero that does not mention the Phrygian mode.

⁷⁶ On the distinction between the two musicians, who had been fused into one in Renaissance thought until Vincenzo Galilei discovered the difference in 1581, see Panofsky, E., Early Netherlandish Painting, 2 vols. (Cambridge, Mass., 1953), i, n. 2 to p. 197, pp. 435–6Google Scholar.

⁷⁷ See Weinmann, K.. Johannes Tinctoris (1445–1511) und sein unbekannter Traktat ‘De inventione et usu musicae’, ed. Fischer, W. (Tutzing, 1961), p. 39Google Scholar. The same anecdote is also to be found in Gafurius's Theorica musica, first printed in 1480 (fol. a2^v of the facsimile edition (Rome, 1934) of the 1492 edition). In her article ‘Spenser and Timotheus: A Musical Gloss on E.K.'s Gloss’, Publications of the Modern Language Association, Nan Carpenter has traced the Timotheos legend in humanistic writings of the Renaissance.

⁷⁸ ‘Auriumque item est admirabile quoddam artificiosumque iudicium, quo iudicatur et in vocis et in tibiarum nervorumque cantibus varietas sonorum intervalla distinctio et vocis genera permulta, canorum fuscum, leve asperum, grave acutum, flexibile durum, quae hominum solum auribus iudicantur.’ (Cicero, , De natura deorum, ii.lviii; cited from the translation by Rackham, H. (London and New York, 1933), pp. 262–5.)Google Scholar

⁷⁹ In music theory, remissus can also mean ‘lowered’; see Ramis, Bartolomeo, Musica practica, ed. Wolf, J. (Leipzig, 1901), p. 26Google Scholar: ‘vocis elevatio sive intensio et depressio sive remissio’.

⁸⁰ Mollis and durus of course also have another meaning in music theory, referring to the presence of Bb or B. Some of the adjectives for vocal quality are also listed by Isidore of Seville, who names the following: suave, perspicua, subtilis, pinguis, acuta, dura, aspera, caeca, vinnola (defined as ‘vox lenis, vox mollis atque flexibilis’) in his Etymologiarum sive originum libri xx, iii, ch. 20; see Gerbert, M., Scriptores ecclesiastici de musica sacra potissimum, 3 vols. (St Blasien, 1784), i, p. 22Google Scholar (Scriptores ecclesiastici is hereafter abbreviated as GS). An English translation of Isidore may be found in Strunk, , Source Readings, pp. 93–100Google Scholar. It seems to have gone unnoticed that practically the whole of chapter 5 of Book i of Aaron's, PietroThoscanello de la musica (Venice, 1523)Google Scholar, ‘Cognitione di voci, et suoni, et varii istromenti’, is translated from Isidore of Seville, not named, with some additions from other writers, who are named. It is a curious mixture of ancient and modern, for most of the instruments discussed are those of antiquity. Yet the characterisation of vocal quality must have seemed just as relevant to Aaron in 1523 as it did to Isidore in the seventh century.

⁸¹ Liber de arte contrapuncti (Seay ii, p. 12), and especially Proportionate (Seay iia, p. 10), where he speaks of ‘ars nova’. On Tinctoris's attitude toward the ‘new’, see Lowinsky, , ‘Music of the Renaissance as Viewed by Renaissance Musicians’, The Renaissance Image of Man and the World, ed. O'Kelly, B. (Columbus, 1966), pp. 129–77, esp. pp. 132–3;Google Scholar on his predecessors who praised or denounced the ‘moderni’, see ibid., p. 165, n. 5.

⁸² ‘Prima species compositionis fit ex utroque modo perfecto, tempore perfecto et maiori prolatione.’ (Seay i, p. 131.)

⁸³ ‘Quinta compositionis species fit ex utroque modo perfecto, tempore imperfecto et prolatione maiori.’ (Seay i, p. 133.)

⁸⁴ ‘But frequently unmeasured rests are written indicatively, when for example, they are placed at the beginning of a song before a circle or semicircle, the sign of tempus; then two rests of three tempora show modus maior perfectus, and another third rest shows modus minor perfectus, just as Tinctoris did in the tenor of his pedagogical motet, Difficiles alios delectat pangere cantus.’ (Gafurius, , Practica musicae, trans. Miller, p. 86.)Google Scholar The seeming contradiction between Gafurius's statement that the major perfect mode is shown by two long (longa) rests and Tinctoris's that it is shown by three such rests is resolved when we realise that in Tinctoris's system the major mode is never indicated by itself but always includes the minor mode, since the length of the maxima depends on the length of the long. If the minor mode is imperfect, then the three pauses occupy only two spaces each, showing that the long is comprised of two breves.

⁸⁵ ‘Si duae minimae solae in maiori prolatione ante semibrevem aut eius pausam inveniantur, ultima illarum alteratur.’ (Seay i, p. 179.)

⁸⁶ In the seventh general rule of his Liber imperfectionum notarum musicalium, Tinctoris shows how a long can be reduced to the value of eight minims (Seay i, pp. 149–50). Gafurius, in his Practica musicae, discusses another such example, ending with the remark that ‘a study of these varied kinds of imperfection produces little of value or practical application. In our opinion it should rather be avoided than encouraged’ (trans. Miller, p. 97). The Spanish theorist Guillermo de Podio vehemently denied that a ternary note could lose more than a third of its value; in his view, a note could be imperfected either as to the whole or as to its parts, but never as to both; see Stevenson, , Spanish Music in the Age of Columbus, pp. 78–9Google Scholar.

⁸⁷ This maxima contradicts Apel's, Willi statement that ‘the Mx itself was not admitted to be ternary; in other words, its maximum value was six, not nine, B’ (The Notation of Polyphonic Music 900–1600 (5th edn, Cambridge, Mass., 1953), p. 124)Google Scholar. In the Commentary, p. 440, he adds: ‘the equivalent of a maximodus perfectus is always represented by a group such as L L L (3 + 3 + 3), or Mx L (6 + 3), never by a single Mx (9)’. It is understandable that such a long note value is rarely encountered in compositions. Tinctoris has placed three maximae in his motet, the first two under major perfect mode, the third under major imperfect mode, but he has reduced them to less than half their full value through imperfection. The examples in major perfect mode in Tinctoris's Tractatus de regulari valore notarum, however, all begin with a perfect maxima (Seay i, pp. 127, 129, 132–6).

⁸⁸ This is not a case of simultaneous augmentation and imperfection, a practice Tinctoris deplores in his Liber imperfectionum notarum under the twelfth general rule (Seay i, pp. 153–4), giving as examples compositions by Domarto and Barbingant. Augmentation of an imperfect note by a dot is not the same as alteration.

⁸⁹ Perhaps the scribe was familiar with Johannes de Muris's Libellus cantus mensurabilis, in which one chapter begins ‘Maxima perfecta in toto et in partibus potest imperfici dupliciter’ (CS iii, p. 47). Dupliciter here means ‘in two ways’, as is evident from the continuation: ‘videlicet, quoad totum et quoad partes’. Johannes de Muris also uses dupliciter to describe another two ways of imperfecting the maxima: a parte ante and a parte post.

⁹⁰ The passage offered some difficulties toward the end. Alienanam has been emended to alienam. The unusual abbreviation difficili ^a was resolved as difficillima, following the suggestion of Professor Bernard Barbiche of the École des Chartes. The words ‘a nemine’ resisted all attempts at decipherment until the solution was found by Ségolène de DainvilleBarbiche. To Professor and Madame Barbiche I extend my warmest thanks for their expert assistance.

⁹¹ Practica musicae, trans. Miller, p. 132Google Scholar.

⁹² ibid. In the translation of the Loeb Classical Library: ‘For in the longer time there is opportunity for more sensation, whereas the brief and high note escapes attention because of its speed.’ (Aristotle: Problems, i, Books i–xxi, trans. Hett, W. S. (Cambridge, Mass., and London, 1936), pp. 391, 393.)Google Scholar

⁹³ ‘Why is it that the mistakes of singers are more noticeable in the low notes than in the high ones, if they sing out of tune? Errors in time are also more obvious in the low notes.’ (ibid., p. 391.) That the discussion involves the interval of an octave is made clear by two other problems in Book xix, Problem 18: ‘Why is the octave consonance alone used in singing?’ and Problem 17: ‘Why does not singing in fifths sound like the octave?’ (ibid., p. 389.)

⁹⁴ Book iii, De canto de órgano, chapter 50, fols. lix–lix^v.

⁹⁵ ‘Porqué en la composición de canto de órgano avemos de poner las consonancias mejores entre las bozes inferiores, y no podemos poner una dissonancia? Muchas causas pone Aristotiles para qué esto deva ser assí hecho. La Primera por ser la boz baxa más grave tiene más de tiempo, que la boz aguda: y en la boz que tiene más de tiempo, más se puede peccar, que en la que tiene poco tiempo. En el tiempo mayor la dissonancia grave por ser más tardía tiene mayor sonido: por la qual tardança es hecha más manifiesta la tal dissonancia. Empero el sonido veloz y agudo porque en menor tiempo da su sonido: está abscondida la tal dissonancia. De esta razón se sigue la segunda, que más hiere el oydo la dissonancia puesta en la parte grave, que si fuesse puesta en la aguda’ (Fol. lix^v.)

⁹⁶ ‘Si la boz grave dissonante por ser grave, haze más sensible a la aguda: luego la consonancia grave hará más sensible a la consonancia aguda.… Pues tan sensibles se hazen la quinta y octava con la consonancia grave: que no sentimos la dissonancia de la quarta.’ (ibid.)

⁹⁷ For a discussion of the position of the fourth in medieval theory, see Sachs, K.-J., Der Contrapunctus im 14. und 15. Jahrhundert: Untersuchungen zum Terminus, zur Lehre und zu den Quellen, Archiv für Musikwissenschaft, Beihefte 13 (Wiesbaden, 1974), esp. pp. 58–60, 62–5, 76–7, 124–9Google Scholar.

⁹⁸ Jacobi Leodiensis Speculum musicae, ed. Bragard, R., Corpus Scriptorum de Musica 3, vii (n.p., 1973), pp. 12–22Google Scholar.

⁹⁹ ‘Imperfectae discordantiae dicuntur, quando duo voces conjunguntur sic quod quodammodo se compati possunt secundum auditum, sed discordant’ (CS iv, p. 279). The treatise was erroneously attributed to Simon of Tunstede by Coussemaker; see Reaney, G., ‘Tunstede’, Die Musik in Geschichte und Gegenwart, ed. Blume, F., 16 vols. (Kassel, 1949–1979), xiii, cols. 979—81Google Scholar.

¹⁰⁰ For a lucid presentation of the changing view of consonance and dissonance in the Middle Ages and Renaissance, see Lowinsky, ‘Music of the Renaissance’, esp. pp. 136–47.

¹⁰¹ On the new appreciation for Aristoxenus and its significance for Renaissance theory, see ibid., pp. 137–8.

¹⁰² This phenomenon was first noticed and described by Charles Warren Fox; see his ‘Non-quartal Harmony in the Renaissance’, Musical Quarterly, 31 (1945), pp. 33–53Google Scholar.

¹⁰³ See Practica musicae, trans. Miller, p. 144Google Scholar.

¹⁰⁴ ‘Undecima species compositionis fit ex modo maiori imperfecto, modo minori perfecto, tempore perfecto et prolatione minori.’ (Seay i, p. 135.)

¹⁰⁵ ‘Quintadecima species compositionis fit ex modo maiori imperfecto, modo minori perfecto, tempore imperfecto et prolatione minori.’ (Seay i, pp. 136–7.)

¹⁰⁶ Notation of Polyphonic Music, pp. 148–50.

¹⁰⁷ it was this example that enabled Reinhard Strohm to identify Tinctoris's Mass, which was otherwise unknown. The three-voice Mass is found anonymously and without title in Strahov, Monastery Library, MS d.g.iv.47 (now in Prague, Památník Národního Písemnictví), fols. 114^v–116, 117^v–119. The Kyrie and Gloria also occur in Trent, Castello del Buon Consiglio, Biblioteca della Soprintendenza, MS 89, fols. 162^v–164 and the Credo in the Speciálník Codex (Hradec Králové, Krajske Muzeum, MS ii a 7), pp. 168–70 – all anonymous and without title; see Strohm, ‘Die Missa Super “Nos amis” von Johannes Tinctoris’, pp. 34–51, esp. p. 41.

¹⁰⁸ ‘Secunda generalis regula est quod non refert si duae notae solae inventae quarum ultima venit alteranda sint continuae aut syncopatae.’ (Seay i, pp. 174–5.)

¹⁰⁹ In this it differs from the example of 8:3 in Tinctoris's Proportionale (Seay iia, p. 35), where the note values are not changed; they must, however, be reduced by half to fit the tenor.

¹¹⁰ ‘Fugue is the likeness of the voice-parts in a composition as to the value, name, and shape of their notes and rests, and sometimes even to their degree on the staff.’ (Parrish, Dictionary, p. 33.)

¹¹¹ A good example of internal imitation, that is, imitation that does not begin in both voices after a pause but is prefaced in one or both voices by unrelated notes, is his chanson Hélas (Johanni[s] Tinctoris opera omnia, ed. Melin, pp. 130–1Google Scholar). See cantus and tenor, bars 4–8, 10–12, 15–18, 20, 21–2, and all three voices, bars 25–6.

¹¹² For a portrait of del Lago as drawn from his letters, see Harrán, D., ‘The Theorist Giovanni del Lago: A New View of the Man and his Writings’, Musica Disciplina, 27 (1973), pp. 107–51, esp. pp. 143–8Google Scholar.

¹¹³ Vatican City Vat. Lat. 5318, fols. 22–33^v.

¹¹⁴ ‘Et similmente da Tinctoris in uno suo canto a tre voci fatto sopra alcuni versi li quali principiano così: Defficiles [sic] alios delectat pangere cantus, nel tenore della prima parte, et similmente nella seconda parte del tenore, et nella parte suprema, et nel contratenore.’ (Vatican City Vat. lat. 5318, fols. 30^v–31.)

¹¹⁵ Paris, Bibliothèque Nationale, MS ital. 1110, fols. 68–77 (not dated), written during a period in which Spataro refused to answer personally any letter from del Lago.

¹¹⁶ ibid., fols. 73^v–74.

¹¹⁷ Vatican City Vat. lat. 5318, fols. 181–182.

¹¹⁸ Hothby's, ‘rota’ is not included in The Musical Works of John Hothby, ed. Seay, A., Corpus Mensurabilis Musicae 3 (n.p., 1964)Google Scholar, unless del Lago has in mind the proportional motet Orapro nobis (pp. 4–7), which, though not canonic, uses a variety of mensural signs, but not in a systematic manner. Nor have I found such a work in Hothby's treatises. (It might be merely a diagram in the shape of a circle, rota meaning ‘wheel’.) A brief explanation of Hothby's teachings on mensuration is included in his De cantu figurato (CS iii, p. 331), and Faenza, Biblioteca Comunale, MS 117, a collection of fifteenth-century treatises on musical theory, contains a table of mensural signs ‘secundum J. Hothbi’ (fol. 62^v).

¹¹⁹ The five-part Mass survives in Cappella Sistina 14, fols. 56 –65. Kiesewetter, R. G. included a transcription of the first Kyrie and the third Agnus Dei in his Geschichte der europäisch-abendländischen oder unsrer heutigen Musik (Leipzig, 1834), Appendix, pp. xii–xivGoogle Scholar. On Éloy d'Amerval, poet and musician, see Reese, G., Music in the Renaissance (rev. edn, New York, 1959), p. 263Google Scholar, and the literature cited there.

¹²⁰ ‘Perché Frate Gioanne Othobi tiene un'ordine, come appar in una sua rota, nella qual dimostra il valor delle notule. Et Eloi, in una soa messa composta sopra la antiphona Dixerunt discipuli ad beatum Martinum, tiene un'altro, diverso di quello che ha tenuto Othobi. Et Tintoris in uno suo canto a tre voci fatto sopra alcuni versi gli quali dicono cosí, Difficiles alios delectat pangere cantus, etc., similmente tiene un'altro modo quanto all'ordine molto diverso a quello che hanno tenuto gli duoi sopra nominati, cioè Othobi et Eloi, come appare nelle soe glosse fatte in dichiaratione di tal suo canto, cioè; nella glossa del soprano, et in quella del tenore della prima parte, et similmente in quella del tenore della seconda parte. Ma Eloi et Tintoris dicono esser sedici spetie di modi, et in questo solamente si concordano. Ma quanto all'ordine, cioè primo, secondo, terzo, etc., molto sono discrepanti l'un da I'altro.’ (Vatican City Vat. lat. 5318, fols. 181–181^v.)

¹²¹ For a survey of the various systems of indicating mensuration in the fifteenth and sixteenth centuries, see Wolf, J., Handbuch der Notationskunde, 2 vols., i (Leipzig, 1913), pp. 410–15Google Scholar.

¹²² Seay gives ‘modus minor imperfectus’, noting the variant reading ‘perfectus’ in only one source. The latter is correct, however, as the Tractatus de regulari valore notarum confirms: ‘Signum modi minoris perfecti est longalis pausa tria occupans spatia.’ (Seay i, p. 129.) The minor imperfect mode has two pauses occupying two spaces each. Gafurius, , in his Practica musicae, refers to the same Mass in much the same words, but without example (trans. Miller, p. 85)Google Scholar.

¹²³ ‘non solamente in quelli lochi che dice Messer Pre Zanetto, ma in molti altri epso tenore se trova falsissimo, per il che lauderia a Messer Adriano che quello che lui ha composto sopra esso tenore che per niente lo desse fora, perché certamente apud peritos più ge seria de vergogna che de honore. Voluntiera saperia chi è statto lo compositore. Se pur ve piacesse la resolution sua, giugando più presto a indivinar et interpretar la mente del compositor che per l'arte che sia in epso, i[o] ve la mando’. (Vatican City Vat. lat. 5318, fol. 175.)

¹²⁴ Letter to Lorenzo Gazio of 6 May 1535 (Vatican City Vat. lat. 5318, fols. 85–101^v).

¹²⁵ Particularly ironic, because Gazio was a friend of Gafurius, whom he calls ‘nostro amicissimo’ in a letter to del Lago of 9 September 1534 (Vatican City Vat lat. 5318, fol. 185^v).

¹²⁶ The composition has not survived, but it is mentioned by Pietro Aaron in his Trattato della natura et cognitione di tuttigli tuoni (Venice, 1525), fol. c2^v, as an example of the seventh mode (‘Multi sunt vocati pauci vero eletti del venerabele Messer Pre Zaneto Veneto’). Gazio was so upset by the thought that Willaert had composed something on del Lago's tenor that he himself went to Venice (undoubtedly before he had received del Lago's letter) and censured and heaped insult on the tenor; this we know from a letter Aaron wrote from Padua to del Lago on 12 May 1535 (Vatican City Vat. lat. 5318, fol. 171): ‘el nostra Don Lorenzo è stato dui giorni a Vinegia et andato da Messer Adriano et ha molto biasimato et sraachato el vostro tenore.’ Willaert's composition has not survived either.

¹²⁷ ‘Drawing together’ is used here to translate reducere, which means ‘to draw backwards, to lead back, to bring back’. In other contexts it may also be translated ‘to imperfect’ or ‘to be imperfected by’. In music theory it means to connect two or more notes so as to form a perfection and can refer to imperfection, alteration, or syncopation. In his Diffinitorium, Tinctoris defines reductio as ‘unius aut plurium notarum cum maioribus quas imperficiunt aut cum sociis annumeratio’ (‘reductio is the counting together of one or more notes with greater ones which they imperfect, or with their companions’). Carl Parrish's translation of aut cum sociis as ‘or their equivalents in value’, and the addition of a comma after maioribus lead him to the erroneous belief that ‘Reduction is simply the result of imperfection, in which a larger note in a perfect mensuration is reduced from its normal value of three to two values of the next smaller unit’ and that aut cum sociis ‘means that the smaller note which causes the imperfection might be either a single note or a group of notes of still smaller notes equivalent in value to it’; see Dictionary of Musical Terms, pp. 54–5, p. 91, n. 83. In this context, ‘reduction’ is a false cognate.

¹²⁸ ‘nel tenore della seconda parte del preditto suo canto, cioè Difficiles alios etc. fa alterare nel tempo perfetto la seconda semibreve inanzi la brieve per la reductione della prima semibreve alla seconda, come appar in eius glossa in dichiaratione di tale semibreve, la quale così dice: Ista semibrevis alteratur eo quod in tempore perfecto ante brevem ultima solarum inveniatur’. (Vatican City Vat. lat. 5318, fol. 92^v.) In his study of Vatican City Vat. lat. 5318 (cited in note 1) Jeppesen was the first scholar to draw attention to Tinctoris's motet. Citing the present letter and one other he remarks that both this motet and the Missa Elas ‘seem to have disappeared’ (p. 37, n. 42).

¹²⁹ ‘E tale modo, cioè tale imperfettione mediata, à stata usata da Tintoris nel suo prenominato concento, cioè Difficiles alios etc. nella prima parte del tenore. Sotto il segno di tempo imperfetto et prolatione perfetta reduce la sesta minima (la quale è puntata nella sumità della coda a parte posteriori) alla 2^a breve del predetto segno, la quale minima faimperfetta l'altra sua parte propinqua inclusa in essa breve quanto al suo tutto, la quale imperfettione mediata il predetto Tintoris la dimostra con il punto, qui[a] punctus significat transitum localem.’ (Vatican City Vat. lat. 5318, fol. 98^v.) The punctus transportationis or translationis is not mentioned by Tinctoris in his treatises, but Gafurius treats it in Book ii, chapter 12 of his Practica musicae (trans. Miller, pp. 104–5)Google Scholar.

¹³⁰ Vatican City Vat. lat. 5318, fols. 201–202.

¹³¹ ‘El secondo raodo, che ogni breve negra soto el tempo perfecto è diminuta de una tertia parte quale è una semibreve, et questo per essere essa breve formata de numero ternario, e così tute le semibreve negre apresso esse breve sono nela quantità come se fusseno vacue, ma solo stano in augumento del numero perfecto.’ (Vatican City Vat. lat. 5318, fol. 202; from Toscanello (1529 edn), fol. e3.) Spataro retains his own spelling when quoting from other theorists.

¹³² ‘Ma certamente circa tale vostra sententia asai dubito, et questo è solo perché ho trovato apresso a certi docti antiqui et moderni, come da Tintoris in uno suo mutetto multo arduo, del quale non me arecordo el nome, el quale Tintoris pone queste note ut hic: [example] overo così [example], nel quale predicto mutetto da esso Tintoris non è inteso che la breve plena et la sequente semibreve plena inseme coniuncte possano perficere et reintegrare uno tempo complecto de tre semibreve vacue, ma solo da lui son intese solamente havere el valore de due semibreve vacue, le quale con la precedente overo con la sequente semibreve vacua inseme coniuncte perficerano el valore de uno tempo perfecto, perché (in tale positione) quella breve plena non è facta plena perché sia convertita de perfecta in imperfecta, perché ancora che la fusse vacua, la non seria perfecta ma seria imperfecta de la sequente semibreve vacua. Pertanto tale pleno in tale breve dato più non potrà tendere ad imperficere, ma atenderà al sesqualterare, per la quale cosa advenirà che tale breve plena inseme con la sequente semibreve coniuncta non producerà una tempo perfecto, come da Vostra Excellentia è concluso. A me pare che Vostra Excellentia in questo passo doveva parlare con qualche exceptione. Imperò che quello che ha dicto Vostra Excellentia potrà stare ut hic, scilicet: [example], et altra declaratione non aduco perché scio bene che circa tale positione sapereti elegere el vero dal falso.’ (Vatican City Vat. lat. 5318, fol. 202.)

¹³³ Spataro's example shows the same principle as Tinctoris's usage, but it is so small that it makes little sense. A proper transcription would be but in practice it would have been performed , as is often the case when the figure is found in imperfect mensuration, and indeed, it is so specified by Aaron in the same chapter. In the following chapter, ‘On the black semibreve’, Aaron says that the figure is to be transcribed , ‘although some say that such figures ought to be in sesquialtera. But it hardly matters one way or the other because the value of these notes fills the space of a tempus; therefore take whichever pleases you, since it all comes to the same end.’ (Toscanello, fol. e3^v.)

¹³⁴ Vatican City Vat. lat. 5318, fol. 180, unconnected with any letter.

¹³⁵ Apel, Notation of Polyphonic Music, does not have a special category for this; he does, however, point out a brief example of it in Leonel Power's Anima mea (p. 134). On p. 158 he discusses the use of in combination with tempus perfectum. Tinctoris, in his treatise on proportions, gives an example of coloration indicating sesquialtera in perfect tempus in chapter 6 of Book i (Seay iia, p. 20).

¹³⁶ ‘Nona regula generalis est quod si completo numero aut nullis praecedentibus notis aliis, nota minor sola ante maiorem ab ea imperfectibilem inveniatur, ipsam imperficit.’ (Seay i, p. 150.)

¹³⁷ ‘Et si notae minores quae aliter iuxta praemissam regulam imperficerent aliquam maiorem impletae sunt, non ipsam imperficient, immo ad alias similiter impletas reducentur.’ (ibid., p. 151.) Here is an example that clearly illustrates ‘aut cum sociis’ in Tinctoris's definition of reductio (see note 127 above). The first black note in each group is counted together with its companions to form a perfect tempus. This meaning of coloration is not treated in Apel. Gafurius discusses it briefly on fol. bb7^v of his Practica musicae (trans. Miller, p. 103Google Scholar; unfortunately, the failure to show coloration in the transcription of the example makes Gafurius's point hard to understand), but he applies it to groups of three blackened notes of the same value.

In his Proportionate, Tinctoris gives an example showing four uses of the blackened minim (Seay iia, pp. 20–1). One of them, reductio, occurs twice in the tenor, bars 2–3 and 5. In the first instance, the three blackened minims (the first three notes in Seay's transcription should have crosses over them, indicating coloration) are counted together with the two blackened semibreves. Blackening the minims prevents imperfection of the breves between which they are placed. In the second instance, two blackened minims are counted together with two blackened semibreves and therefore do not imperfect the white semibreves between which they fall. This same example is found in the Perugia manuscript (no. 24), but the superius has been rewritten to remove both the passages in sesquialtera; apparently Johannes Materanensis, or whoever is responsible for the editing of the example, read Tinctoris's blackened minims as semiminims instead of minims in sesquialtera (the superius illustrates both usages of the blackened minim); he then had to adjust the superius to fit the tenor. Consequently, the piece– contrary to the anthology's purpose – has no proportions at all.

¹³⁸ As complicated a piece as Isaac's De radice Jesse goes no further than proportio quadrupla in the genus multiplex and sesquialtera in the genus superparticulare. On Isaac's virtuosity in using the mensural system of his time (whose proportions are mainly indicated by signs), see Gossett, P., ‘The Mensural System and the “Choralis Constantinus” ’, Studies in Renaissance and Baroque Music in Honor of Arthur Mendel, ed. Marshall, R. L. (Kassel and Hackensack, 1974), pp. 71–107Google Scholar. Another composer noted for his use of proportions is Jacob Obrecht, but they are mainly confined to augmentation and diminution of the tenor. An exception is his duo, Regina caeli, admirably explained by Hewitt, Helen in ‘A Study in Proportions’, Essays on Music in Honor of Archibald Thompson Davison (Cambridge, Mass., 1957), pp. 69–81Google Scholar. Obrecht employs an unusual system of indicating three proportional numbers in the first section in triple metre, which enables the singer to relate each proportion not only to the preceding proportion but also to the tenor.

¹³⁹ Ornithoparchus, trans. Dowland, J., A Compendium of Musical Practice, ed. Reese, G. and Ledbetter, S. (New York, 1973), p. 186Google Scholar.

¹⁴⁰ ibid.

¹⁴¹ Heyden, Sebald, De arte canendi, trans. Miller, C. A., Musicological Studies and Documents 26 (n.p., 1972), pp. 19, 18Google Scholar.

¹⁴² ibid., p. 90. I have changed the translation slightly at the end.

¹⁴³ Edward Roesner kindly reminded me of de Vitry's precedent. But de Vitry only refers to compositions, some of which can be established as his; Tinctoris places the music itself in his tracts and ties them to his text so intimately that the two become a unit.

¹⁴⁴ ‘Sed profecto frustra nisi quisquis in ipsa arte praeclarus evadere nitetur, diligenti cum assuetudine componat aut super librum canat. Nam, ut Cicero Ad Herennium ait, in omni disciplina informa est artis praeceptio sine summa assiduitate exercitationis.’ (Seay ii, p. 156.)

¹⁴⁵ For a translation of Trithemius's biographical notice on Tinctoris, see Reese, , Music in the Renaissance, p. 138Google Scholar.