¹ I warmly thank Professor Harold Powers for many formative and stimulating conversations while these thoughts were taking shape. Professor Edward Lowinsky graciously engaged in a lively correspondence in which he gave me the benefit of his experience and reactions to a more informal statement of my hypothesis. Many other colleagues, students and friends have helped and encouraged this enterprise by their comments and criticisms; I beg to defer the pleasant duty of thanking them by name until I have the opportunity to present a more extended and fully documented study. Two summer seminars under the auspices of the National Endowment for the Humanities provided a congenial workshop for performing from original notation and exploring practical ficta applications. Earlier versions of this paper were read at New York University in November 1982, subsequently at other institutions, and in 1983 at Oxford and at the Annual Meeting of the American Musicological Society in Louisville.

² One fairly constant distinction is invoked by Calvin Bower to demonstrate that ‘The translator has failed to distinguish between Guido's concept of qualitative pitch (vox) – sound defined by the intervals surrounding it – and discrete note (nota) – sound defined by a point on the system of a monochord and signified by a letter. Thus a subtle, but fundamental dualism of medieval musical thought has been obscured.’ Review in Journal of the American Musicological Society (hereafter JAMS), 35 (1982), p. 164Google Scholar.

³ Natasha Spender describes the faculty as a ‘sensory and aesthetic life-enhancer’ whose absence she finds analogous to colour blindness in an artist – an extreme statement of a common and wholly modern prejudice; the author finds no problem for her ‘absolutist’ acceptance of the modern phenomenon in the fact that ‘a listener with absolute pitch would now be disoriented to hear a C major work in the pitch of Mozart's day’; s.v. ‘Absolute pitch’, The New Grove Dictionary of Music and Musicians, ed. Sadie, S., 20 vols. (London, 1980)Google Scholar.

⁴ Carl Dahlhaus has addressed a number of such questions. The following quotations from his ‘Tonsystem und Kontrapunkt um 1500’, Jahrbuch des Staatlichen Instituts für Musikforschung preussischer Kulturbesitz 1969, ed. Droysen, D. (Berlin, 1970), pp. 7–17Google Scholar, are offered as samples rather than summaries of his important distinctions between counterpoint, tonal and tuning systems:

‘Ein System ist … ein Inbegriff von Relationen, nicht von bloßen Bestandteilen.’ ‘… Tonsysteme als Systeme von Tonrelationen [beruhen] auf Prinzipien, die nicht an einen bestimmten, immer gleichen Tonbestand gebunden zu sein brauchen. Form und Material sind nicht selten unabhängig voneinander.’ ‘Ein Tonsystem muß andererseits von der Stimmung oder Temperatur unterschieden werden, in der es erscheint oder sich verwirklicht. Eine Stimmung ist gleichsam die akustische Außenseite; und sie kann manchmal, wenn auch nicht immer, mit einer anderen vertauscht werden, ohne daß das Tonsystem, dessen äußere Darstellung sie ist, aufgehoben oder auch nur in seiner musikalischen Bedeutung modifiziert wäre.’

⁵ See Adkins, C. D., ‘The Theory and Practice of the Monochord,’ Ph.D. dissertation (Iowa, 1963Google Scholar), especially the table facing p. 94, which documents from a wide range of theorists both before and after Guido the use of letters as monochord labels with the semitones between different letters from our scale.

⁶ ‘Reperiuntur etiam tamen alii diversi modi cantandi ab istis et etiam inter se, quos scribere foret valde difficile et forte impossibile, eo quod tales diversimodi cantandi quodammodo infiniti sint, et diversis diversimode delectabiles, qua propter insurgit diversitas componentium, et quia intelectus noster infinita capere non potest, cum non sit infinite capacitas sed finite, eo quod aliter in hoc intelectui divino adequaretur, quod non est dicendum. Pro tanto huiusmodi modi a scriptura relinquendi sunt, nee adhuc scribi possent propter sui infinitatem … Scire autem ubi hec signa [of musica ficta] dulcius cadunt auri tue dimitto, quia de hoc regula dari non potest, cum hec loca quodammodo infinita sint’ (Prosdocimus, Contrapunctus, iv, v; ed. J. Herlinger, who very kindly made his work available to me in advance of publication). The above quotations embody the revisions of 1425 to the 1413 treatise. See also de Coussemaker, E., Scriptorum de musica medii aevi nova series (hereafter CS) (Paris, 1864–1876), iii, pp. 197–8Google Scholar.

⁷ Guido recommends use of the monochord in his prologue to the Antiphoner: ‘Duos enim colores ponimus, crocum scilicet & rubeum, per quos colores valde utilem tibi regulam trado, per quam aptissime cognosces de omni neuma & unaquaque voce, de quali tono sit, & de quali littera monochordi: si tamen, ut valde est opportunum, monochordum & tonorum formulas in frequenti habeas usu’ (Gerbert, M., Scriptores ecclesiastici de musica (hereafter GS), Saint Blaise, 1784, ii, 36aGoogle Scholar). But in the Epistola Michaeli he qualifies this advice: ‘Ad inveniendum igitur ignotum cantum, beatissime Frater! prima & vulgaris regula haec est, si litteras, quas quaelibet neuma habuerit, in monochordo sonaveris, atque ab ipso audiens tamquam ab homine magistro dicere poteris. Sed puerulis ista est regula, & bona quidem incipientibus, pessima autem perseverantibus. Vidi enim multos acutissimos philosophos, qui sed quia in hac sola regula confisi sunt, non dico musici, sed neque cantores umquam fieri, vel nostros psalmistas puerulos imitari potuerunt (GS ii, pp. 44b–45a). Translations of both passages in Strunk, O., Source Readings in Music History (New York, 1950), pp. 119, 123Google Scholar.

Ornithoparcus writes (in Dowland's English translation): ‘The Monochord was chiefly invented for this purpose, to be judge of Musical voices and intervals: as also to try whether the song be true or false: furthermore, to shew haire-braind false Musitians their errors, and the way of attaining the truth. Lastly, that children which desire to learne Musicke, may have an easie meanes to it, that it may intice beginners, direct those that be forward, and so make of unlearned learned’, A Compendium of Musical Practice, ed. Reese, G. and Ledbetter, S. (New York, 1973), i. 9Google Scholar.

⁸ Parvus tractatulus de modo monachordum dividendi; I again thank Professor Herlinger for access to the typescript of his new edition of this treatise. Ugolino's Tractatus monochordi (ed. A. Seay, Corpus Scriptorum de Musica (hereafter CSM), 7, iii, pp. 227–53) assigns yet a different set of letters to his recta and ficta divisions. See also Hughes, A., ‘Ugolino: the Monochord and musica ficta’, Musica Disciplina, 23 (1969), pp. 21–39Google Scholar and Lindley, M., ‘Pythagorean Intonation and the Rise of the Triad’, R.M.A. Research Chronicle, 16 (1980), pp. 4–61Google Scholar, for the systems of Prosdocimus and Ugolino, and n. 21 below.

⁹ Adkins (‘Monochord’) reviews the practical uses of the monochord in his chapter 7, but the pictorial evidence to which he refers does not weaken the general statement made here. It might be suggested that for purposes of theoretical demonstration it was symbolically important that the monochord remain essentially a monophonic instrument (despite later applications of the word to polychordal instruments). It precluded not only the checking of simultaneities, but also the efficient comparison of successive sounds.

¹⁰ They are not, normally, mutually accessible; the only way to travel between two nearby stations may be to ride back to a junction where their lines intersect. A note's presence in the system does not guarantee that it will be accessible from all points.

Dahlhaus has noted that the unqualified letter-names also include ‘altered’ pitches: ‘Zu Costeleys chromatischer Chanson’, Die Musikforschung, 16 (1963), pp. 253–65, n. 37Google Scholar and passim.

¹¹ The linking of monochord points by Prosdocimus and others with the corresponding ‘places’ of musica recta is a concession much less extreme than that of Johannes Boen's treatise of 1357 (ed. Frobenius, W., Johannes Boens Musica und seine Konsonanzlehre, Freiburger Schriften zur Musikwissenschaft, 1971)Google Scholar, which fixes, as would be necessary on a keyboard, what would result if the system were restrained from the fluctuations which may occur in a cappella vocal practice. Despite a few such attempts to equate actual points with notional recta sounds, it remains clear that, except for such purposes as monochord demonstration in principle, we are indeed dealing with the recta ‘scale’ as a set of relationships rather than as a pre-tuned system. Until late fifteenth-century keyboard-influenced attempts at reconciling the systems, Boen was virtually alone in attempting to expound the monochord and the gamut in a single operation, as distinct from using the gamut letters to label the monochord. He resorted to some unusual vocabulary in so doing, e.g. mansio (= lunar mansion?), and extorquere, for the removal of sounds from those proper places.

¹² See notes 8, 10 and 21. Only for the distinct purposes of tablature did letter-names thus indicate adjacency of keys by attaching genitive endings, as in fis: these endings often selected the ‘wrong’ enharmonic spelling of a note, showing that they were less tied to musical function than to keyboard designation.

¹³ ‘Clavis est littera localis per voces rectificata’ (clavis is a letter of a place [on the staff] adjusted to it by means of voces [contextual hexachord members]), Adam von Fulda, GS iii, p. 344. Du Cange gives rectificata = corrected (1332). ‘A Key is a thing compacted of a Letter and a Voyce;… A Key is the opening of a Song, because like as a Key opens a dore, so doth it the Song’ (Ornithoparcus, Compendium, I.3; see p. xxv for sources of these formulations in Guido and other earlier theorists).

¹⁴ Reckow, F., Handwörterbuch der musikalischen Terminologie (Wiesbaden, 1971–)Google Scholar, s.v. ‘Clavis’, has shown the likely derivation of the musical term from computus terminology, and his brief statement invites amplification of the parallels. The church calendar, with its fixed and moveable feasts, depends on three periodic cycles lacking a common measure: the seven days of the week (A–G); the lunar month (29.15 days); the solar year (365.25 days). The weekday sequence is repeated only every twenty-eight years (not seven, due to the taking up of irregularities in bissextile (leap) years); the lunar month and solar year coincide only every nineteen years (again, with some adjustment of irregularities). The moveable feast of Easter, together with feasts whose dates are dependent upon that of Easter, touches all these cycles: Easter is the first Sunday after the full moon that occurs on or after the vernal equinox (March 21). These calculations were performed with a number of aids, including forms of the wheel diagrams and hands shared with music theory. van Waesberghe, J. Smits, Musikgeschichte in Bildern, iii, Musikerziehung (Leipzig, 1969), pll. 57–8Google Scholar, has drawn attention to the existence of calendrical hands from as early as the so-called Guidonian hand (see pll. 55–84 on the musical hand in general), and there are some slightly later ones with even more significant musical analogy in that they link the seven-letter weekly cycle A–G with the permutations of the nineteen-year lunar cycle on which Easter depends and which gives rise to the so-called Golden Number – just as the musical hand links the seven-letter octave A–G with the permutating hexachord superstructures. The nineteen years of the lunar cycle, the nineteen places on the physical hand (knuckles and finger-tips), and the decision to confine the usable range of music to those nineteen positions on the Guidonian hand (excluding the later-added place for E la) present a striking analogy. Other calendrical hands show the A–G letters permutated with the so-called tabular of ‘fnugo’ letters, as does the musical hand with hexachords. See, for an example from a theorist also known to music history, van Wijk, W. E., Le nombre d'or, … massa compoti d'Alexandre de Villedieu (The Hague, 1936)Google Scholar. The sedes clavium were fixed dates, the earliest dates on which a feast could occur. The claves pasche are a series of nineteen numbers (11–39) which, when added to the sedes, provide a ready means of calculating the date of Easter. They are a convenient shorthand, a summary means of regulating the disparate cycles, just as musical claves regulate the disparate systems of proportional monochord tuning, the octave cycle, the functional hexachords. In both systems, the respective irregularities of leap years and commas have to be absorbed.

While it is not necessary to bring in the Pythagorean doctrine of the music of the spheres in order to establish a connection between the methods of calendrical and musical calculation and terminology, it is nonetheless worth recalling that the proportionate speeds of planetary revolution were the same set of duple and triple geometric proportions as underlie Pythagorean tuning. Haar (The New Grove Dictionary, s.v. ‘Music of the spheres’) has called it a kind of celestial monochord. It is hardly surprising that computists and musicians found common ways and terms for reconciling and illustrating those parts of their subject matter that were explained by geometric proportion, with the non-proportional structures that were to be superimposed on them.

¹⁵ E.g., his De natura et proprietate tonorum (CSM 22, i), chapter 2. Indeed, the places on the so-called Guidonian hand itself embody all those options and do not in themselves assist in making choices between the possible articulations, any more than the unadapted hand copes with musica ficta. Such adaptation is only rarely documented, for example by Ugolino; see note 8 above.

¹⁶ A myth perpetuated in The New Grove Dictionary, s.v. ‘Pitch’.

¹⁷ de Pareija, Bartolomeo Ramis, Musica Practica (Bologna, 1482)Google Scholar, [Prima pars] chapter 7, ‘Copulandi vocem cum instrumento modus subtilis’.

¹⁸ Micrologus, ed. van Waesberghe, J. Smits, CSM 4, chapter 8.10, p. 124Google Scholar.

¹⁹ Specifically allowed by Listenius, , Musica (Nuremberg, 1549)Google Scholar, chapter 5, discussed and rejected by Aron, , Lucidario in musica (Venice, 1545)Google Scholar, book i, chapters 8, 10.

²⁰ The few repertories that are not obviously for keyboard but for which score is characteristic (including organum and English discant) invite special consideration, but because they are not necessarily designed for or suited to use by one performer, they do not undermine the validity of the generalisation. The assumptions stated here and in what follows are shared with my ‘Resfacta and Cantare Super Librum’, JAMS, 36 (1983), especially pp. 376–8Google Scholar. That article also stresses that, for Tinctoris, the process of composition included not only operation of the rules of counterpoint, but the weighing of choices and priorities between them.

²¹ For examples of theoretical statements documenting the role of the ear in counterpoint see Crocker, R. L., ‘Discant, Counterpoint and Harmony’, JAMS, 15 (1962), p. 4Google Scholar; this article presents many important insights about the nature of medieval counterpoint and stresses the importance of trying to conceive it in contemporary terms.

To these references may be added a remarkable interpolation near the end of the revised version of Prosdocimus's Contrapunctus, ed. Herlinger, in which he says that the signs of musica ficta should be placed where they sound sweetly, that a choice between the discant and the tenor should be left to the ear, and that no rule can be given because the possibilities are infinite: ‘Scire autem ubi hec signa dulcius cadunt auri tuo dimitto, quia de hoc regula dari non potest, cum hec loca quodammodo infinita sint.’ If the ‘variatio’ sounds equally good in the tenor or the discant, it should be made in the discant. See below, section 9, on rules and priorities: this passage thus expresses a priority for applying ficta in a situation not otherwise discussed in this paper.

²² Typical theoretical statements expressing ideas of raising or lowering in terms of linear context rather than of individual pitch inflection include J. de Muris (CS iii, p. 73): ‘on la sol la (A G A) the sol should be raised and sung as fa mi fa’; Prosdocimus (ed. Herlinger): ‘♭ augments the ascent and diminishes it. The two signs do not augment or diminish [intervals] except by a major semitone’, Contrapunctus [v.4] Contrast the terminology of Johannes Boen (n. 11 above); see also the citation from the 1375 Paris anonymous and other relevant passages cited in Bent, , ‘Musica Recta and Musica Ficta’, Musica Disciplina, 26 (1972), p. 86Google Scholar and passim.

²³ These phrases are typical of contemporary ficta definitions, e.g. Tinctoris, Diffinitorium (1472): ‘ficta musica est cantus praeter regularem manus traditionem aeditus’; Prosdocimus, Contrapunctus [v]: ‘ficta musica est vocum fictio sive vocum positio in *loco ubi esse non videntur [revised version from *: aliquo loco manus musicalis ubi nullo modo reperiuntur], sicut ponere mi ubi non est mi, et fa ubi non est fa …’; Ornithoparcus (Compendium, i.10), ‘a Coniunct is this, to sing a Voyce in a Key which is not in it’.

To locate the notes on the monochord would involve a choice, but there is no standard monochord terminology that expresses or identifies any relationship between these notes and their neighbours, just as there are no functional names (such as F♯ and G♭) to distinguish the two possibilities. The two G♭s on the monochord, for example, are labelled by Prosdocimus N, R, by Ugolino O, P; the two F♯s X, 6 and S, 7 respectively. See also Dahlhaus ‘Zu Costeleys chromatischer Chanson’, p. 256: ‘daß das 16. Jahrhundert für den Ton, daß wir “heses” nennen, weder einen Namen noch ein Zeichen hatte’.

²⁴ L'Artusi overo delle imperfettioni della moderna musica (Venice, 1600; facsimile, Bologna, 1968)Google Scholar, Ragionamento Primo, ff. 21–21^v See Levitan, J. S., ‘Adrian Willaert's Famous Duo Quidnam ebrietas …’, Tijdschrift van de Vereniging voor Nederlandse Muziekgeschiedenis (hereafter TVNM), 15 (1938), pp. 166–233CrossRefGoogle Scholar, and Lowinsky, E. E., ‘Adrian Willaert's Chromatic “Duo” Re-examined’, TVNM, 18 (1956), pp. 1–36Google Scholar, where the piece is shown to have been a four-part composition. The two parts given here sufficed to demonstrate the problems that engaged theorists.

²⁵ Both principles are widely documented, e.g. Ornithoparcus (Compendium, i.10, p. 25 [145]): ‘Marking fa in b fa ♮ mi, or in any other place, if the Song from that shall make an immediate rising to a Fourth, a Fift, or an Eight, even there fa must necessarily be marked, to eschew a Tritone, a Semidiapente, or a Semidiapason, and inusuall, and forbidden Moodes …’ The example has leaps of those intervals; the principle is the same even where theorists differ in their insistence on what needs to be notated. See n. 27 below.

²⁶ The term modulation is used in contemporary theory only to describe how an interval is filled in melodically. It has no connotations of the kinds of change it has acquired in tonal theory, and must join the ranks of words and concepts that are out of place if applied to early music in a modern sense.

The cleffing is indeed so contrived that the two finals are at the same place on their respective staves.

²⁷ Tinctoris's famous statement discouraging as asinine the notation of unnecessary signs is given below, section 9, as is the passage from Aron which, rather than indiscriminately encouraging notated signs, requires them explicitly for cases that could not be anticipated by the singers. Prosdocimus [Contrapunctus, v.2] criticises composers for using ficta where it is not necessary, and makes it clear in the revision (see n. 6 above) that it is the unnecessary notating of signs to which he objects.

²⁸ His study of the Willaert composition is cited in n. 24 above. It is hard to single out for mention here anything less than Lowinsky's complete body of writings, so masterfully has he laid out a terrain that must continue to attract further investigation. The reader not already familiar with Lowinsky's writings is referred to the listing under his name in The New Grove Dictionary and, even better, to the first few pages of his ‘Secret Chromatic Art Reexamined’, Perspectives in Musicology, ed. Brook, B. S., Downes, E. O. D. and van Solkema, S. (New York, 1972), pp. 91–135Google Scholar, where he reviews not only his own contributions, starting with Secret Chromatic Art in the Netherlands Motet (New York, 1946), but also scholarly responses to it and relevant contributions by other scholars. Particularly germane in the present context are his studies of the Fortuna settings by Josquin, and Greiter, , and the study by Levy, K., ‘Costeley's Chromatic Chanson’, Annales musicologiques, 3 (1955), pp. 213–63Google Scholar (see also Dahlhaus, ‘Zu Costeleys chromatischer Chanson’).

²⁹ Marchettus lists them together with stems, dots, rests and ancillary markings in general in his ostentatiously Aristotelian Pomerium, ed. Vecchi, G., CSM 6, book i, part i. Heyden, De arte canendi (Nuremberg, 1540), p. 5Google Scholar, gives a different list of ‘accidentia necessary to the art of singing’: scala, clavis, tactus, nota, punctum, pausa, mensura, tonus.

³⁰ Letter to Aron, 1531: cited by Bergquist, P., ‘The Theoretical Writings of Pietro Aaron’, Ph.D. Dissertation, Columbia (1964), p. 440Google Scholar from Rome, Biblioteca Apostolica Vaticana, MS Vat. Lat. 5318, no. 86, f. 219^v.

³¹ Zarlino, , Le Institutioni Harmoniche (Venice, 1558), part iii, chapter 72Google Scholar (same chapter reference in edition of 1573).

³² Lockwood, L., ‘A Dispute on Accidentals in Sixteenth-Century Rome’, Analecta Musicologica, 2 (1965), pp. 24–40, especially pp. 28, 32Google Scholar.

³³ Aron, E.g., Toscanello in musica (Venice, 1529)Google Scholar book ii, chapter 40: ‘Che ne lo instrumento organico secondo il comune ordine, si ritrovano voci naturali di numero xxix, chiamati dal universale uso tasti bianchi: e accidentali di numero xviii, detti tasti negri, overo semituoni: per il qual ordine da noi sará diviso tasto per tasto: dimostrando ciascheduno intervallo del uno al altro cosi accidentali come naturali.’ Aron's arguments in general for the accidental status of B♭ are to be found in the Compendiolo (Milan, post-1545), chapter 10; the Libri tres de institutione harmonica (Bologna, 1516Google Scholar), book i, chapter 15; and in the Toscanello in musica book ii, chapter 5; also in the Aggiunta to that work.

³⁴ The relative sizes were reversed in Pythagorean (diatonic smaller than chromatic) and mean-tone (diatonic larger than chromatic) tunings. Dahlhaus has usefully separated consideration of the tuning system from the tonal system: ‘am Tonsystem … änderte der Wechsel der Stimmungen nichts’.

³⁵ Haar, J., ‘False Relations and Chromaticism in Sixteenth-Century Music‘, JAMS, 30 (1977), pp. 391–418Google Scholar.

³⁶ Zarlino, , Institutioni, part iii, especially chapters 76, 77Google Scholar.

³⁷ Vicentino, N., L'antica musica ridotta alla moderna prattica (Rome, 1555)Google Scholar, book iii, chapter 14 (ff. 46^v–47^v).

³⁸ Soprano, bars 4–5, 7–8; altus 18–19, 20–21. For a recent analysis of the prologue and references to earlier studies, see Berger, K., ‘Tonality and Atonality in the Prologue to Orlando di Lasso's Prophetiae Sibyllarum: Some Methodological Problems in Analysis of Sixteenth-Century Music’, The Musical Quarterly, 66 (1980), pp. 484–504CrossRefGoogle Scholar.

³⁹ Tinctoris, , Liber de arte contrapuncti, ii. xxxivGoogle Scholar: ‘Concordantiis perfectis que vel imperfecte vel superflue per semitonium chromaticum.’

⁴⁰ See Bent, , ‘Musica Recta and Musica Ficta’, p. 94Google Scholar.

⁴¹ Prosdocimus's well-known examples using tritones are given ibid., pp. 91–2 and in the yet unpublished editions of Herlinger. The late-medieval tolerance of the melodic tritone did of course constitute a departure from earlier abhorrence of it; the reinstatement of this rule in the late fifteenth century is only one of a number of ‘returns’ to earlier positions – another being the reversion of B♭ to accidental status. Tinctoris's statements on the use of the tritone are given in this section.

⁴² Some statements by modern scholars indicate that the distinctions between these terms are still not clearly understood:

(1) mi contra fa has sometimes been assumed to include relationships other than simultaneous vertical perfections (e.g. oblique false relations, melodic progressions), and

(2) implicit assumptions of enharmonic equivalence have led to confusion between the tritone and the diminished fifth. Clearly it is impossible to avoid, all the time, melodic and harmonic tritones and diminished fifths.

⁴³ Dated 1476; chapter 8. Ed. A. Seay, CSM 22, i, q.v. for music examples; translated A. Seay (Colorado Springs, 1976).

⁴⁴ Aaron, P., Toscanello in Musica (English translation, Bergquist, P., Colorado Springs, 1970)Google Scholar.

⁴⁵ (Edited in Josquin, Werken, ed. A. Smijers, Motets I.1.) Discussed by, amongst others, Carl Dahlhaus in ‘Tonsystem und Kontrapunkt um 1500’, pp. 15–16, with the rather different conclusion that ‘der Tonsatz abstrakt konzipiert ist und daß sich Josquin über die Unentschiedenheit, wie er zu realisieren sei, hinwegsetzte, da sie ihm gleichgültig war’. Dahlhaus thus posits compositional indifference to the actual resulting sounds, and that abstractly conceived counterpoint may have lacked either prior aural imagination of sounds or indeed any musically acceptable realisation. However, the size of an interval (as major or minor) may be determined by the musical context so clearly at crucial points in the contrapuntal fabric that the composer neither needed to specify it nor the contrapuntally experienced singer to be told what to do. Such choices must surely have been a matter of structural if not also aesthetic concern to the composer, even if the conventions of performance did not necessitate, nor the nature of the notation permit, its full prescription. Dahlhaus seems here to approach the notation from a more conventional view based on fixed pitches and alterations although elsewhere (‘Zu Costeleys chromatischer Chanson’) recognising a principle he felicitously names ‘relativ Fa-notation’. He there presents it as the special property of unusual pieces in which it is applied with extreme results, whereas I seek to bring that relative concept into play as a central and normal feature of renaissance notation.

⁴⁶ Hothby, , De arte contrapuncti, ed. Reaney, G., CSM 26, p. 90Google Scholar; Aron, , Libri tres de institutione harmonica (Bologna, 1516):Google Scholar this example is there explained only verbally. Similar passages by Vicentino and Lusitano are given by Ferand, E. T., ‘Improvised Vocal Counterpoint in the Late Renaissance and Early Baroque’, Annales musicologiques, 4 (1956), pp. 147–51Google Scholar. See also Monachus, G., De preceptis artis musicae, ed. Seay, A., CSM 11, p. 53Google Scholar.

⁴⁷ It may yet be demonstrated that the coincidence of the word nova with what we may anachronistically hear as a departure is significant, even though it does not fall within the kind of vocabulary supporting Lowinskian chromaticism. In urging that music must make sense independently of textual considerations that might have helped to shape it, I do not mean to underestimate considerations that cannot receive full treatment here.

⁴⁸ Munich, Bayerische Staatsbibliothek, Mus. MS 19 marks the uncontroversial bass B♭s shown in bars 1 and 5. Munich, Bayerische Staatsbibliothek Mus. MS 3154 (on which Thomas Noblitt based his dating of the piece in the 1470s) has a B♭ signature in the bass part until beyond this passage. London, Royal College of Music MS 1070 has a B♭ before the bass B in bar 3. (I am grateful to Lawrence Earp for extracting these from the computerised data of the Princeton Josquin project.)

⁴⁹ Manuscript accidentals in the altus: the late part-books Munich, Universitätsbibliothek, MS 8° 322–5 mark the B♭ shown in parentheses in bar 3. Observation of this ‘fa’-sign might have further consequences quite disruptive for the basic counterpoint of the other parts. A performer studying his part alone might have sung this ♭ (whether or not notated, and whether or not we call it ‘fa super la’). But on hearing the previously attacked B in the soprano (which also cannot be ‘changed’ without other consequences that are less readily defended than the version I propose), the altus is likely to sing B♮. The linear ‘rounding-off’ of this altus phrase with the ♭ was in any case an incompletely successful attempt to rescue what has to be admitted, here alone in the motet, as a less elegant line, subservient to the tight interlocking of the counterpoint between the other, primary, parts. I would therefore choose to override it, but without insisting that this passage would always have been solved in this way.

⁵⁰ For those who prefer to define diatonic in terms of segments that can be transposed to piano white notes, this can be done for the last limb of the sequence starting on B♭ if played a minor third lower. See also Example 7.

⁵¹ van Crevel, M., ‘Verwante Sequensmodulaties bij Obrecht, Josquin en Coclico’, TVNM, 16 (1941), pp. 119–21Google Scholar. Modern edition of Kyrie, ed. Smijers, A., Van Ockeghem tot Sweelinck (Amsterdam, 1939–1956), ii, 51Google Scholar.

⁵² Lowinsky, , ‘Secret Chromatic Art Re-examined’, n. 63aGoogle Scholar.

⁵³ See, for example, Bottrigari, E., Il Desiderio (Venice, 1594), p. 5Google Scholar: ‘Gli strumenti stabili, ma alterabili [as distinct from those ‘al tutto stabili’] sono tutti.quelli, che dapoi che sono accordati dal sonator diligente, si possono alterare con l'accrescere, & minuire in qualche parte, mediante il buon giudicio del sonatore toccando i loro tasti un poco più sù, un poco più giù’.

⁵⁴ Most modern writers presume just such a repertory of available pitches, aligning the gamut with the keyboard without recognising that vocal counterpoint and notation did not need to be so anchored. This is true of Berger's, Karol excellent study Theories of Chromatic and Enharmonic Music in Late 16th Century Italy (Ann Arbor, 1980)Google Scholar, from which a quotation will serve to illustrate where his view of tonal materials differs from mine: ‘Since steps [relatively defined pitches] are defined by means of intervals … it is possible to discuss the tonal system entirely in terms of intervals, that is, as a set of all intervals available to a composer (that is, the gamut) and its pre-compositional organization. Octave equivalence is basic to the sixteenth-century intervallic system; … the gamut consists of all intervals possible within the octave … Certainly more than twelve, and possibly even all twenty-one, different notes are used [notated] in practical sources. Although it can reasonably be assumed that the musicians of the Renaissance were able to notate all the steps and intervals they were using, it does not follow that all differently notated steps and intervals were indeed different’ (p. 98). For Prosdocimus on the infinity of sounds, see n. 21 above.

⁵⁵ Accidentien und Tonalität in den Musikdenkmälern. des 15. und 16. Jahrhunderts (Berlin, 1936).

⁵⁶ ‘Accidentals and Ornamentation in Sixteenth-Century Intabulations of Josquin's Motets’, Josquin des Prez, ed. Lowinsky, E. E. (London, 1976), pp. 475–522Google Scholar. ‘While the character and extent of disagreement on the practical application of the rules of musica ficta on the part of sixteenth-century intabulators differed from that of modern scholars, who do not even agree on the existence and applicability of the rules, there was nevertheless a considerable difference of judgement and taste among the former’ (p. 477 and passim).

⁵⁷ Gafurius, F., Practica musicae (Milan, 1496)Google Scholar, book iii, chapter 2: ‘Species seu elementa contrapuncti in instrumentorum fidibus atque vocali concentu gravium atque acutorum sonorum commixtionem qua harmonica consurgit melodia proportionabiliter consequantur necesse est.’ Burzius, N., Musices opusculum (Bologna, 1487)Google Scholar, Tractatus secundus, sig. e. iij, speaks of cantus; instrumental reference is not specific at this point.

⁵⁸ Vicentino, N., L'antica musica ridotta alla moderna prattica (Rome, 1555)Google Scholar; facsimile ed. E. E. Lowinsky (Kassel, etc., 1959, iii. xiv, ff. 46^v–47.

⁵⁹ van Benthem, J., ‘Fortuna in Focus: Concerning “Conflicting” Progressions in Josquin's Fortuna dun gran tempo’, TVNM, 30 (1980), pp. 1–50Google Scholar, argues against Lowinsky's reading of this piece (‘The Goddess Fortuna in Music, with a Special Study of Josquin's Fortuna d'un gran tempo,’ The Musical Quarterly, 29 (1943), pp. 45–77Google Scholar), on grounds which include the evidence of tablatures, the presence of ‘mi-fa’ false relations in other pieces by Josquin, and the unstated assumption that accidentals are a corruption of the text and should be kept to a minimum. That the result of Lowinsky's version is musically superior seems to me beyond question; the view of tonal materials here proposed helps to legitimate it against some arguments of its critics.

While no attempt has been made to assemble tablature evidence for application to the present examples, it is perhaps worth pointing out that the Kleber tablature arrangement of Josquin's Ave Maria avoids the linear contrapuntal approach of the vocal model but does correct the vertical fifth – with F♯! The passage in the L'homme armé Agnus containing the problem illustrated by Aron is avoided altogether by Kleber. For modern transcriptions of both pieces see Keyboard Intabulations of Music by Josquin des Prez, ed. Warburton, T. (Madison, 1980), pp. 32, 27Google Scholar.

⁶⁰ Pace formulations such as ‘used [musica ficta] as a “peccatum” … against the mode’; E♭ as a ‘violation of the fifth mode’: Meier, B., ‘The Musica Reservata of Adrianus Petit Cociico and its relationship to Josquin’, Musica Disciplina, 10 (1956), pp. 101, 103Google Scholar. See Brown, Howard Mayer (in Josquin des Prez, ed. Lowinsky, , p. 477)Google Scholar: ‘The idea that musicians of the time were guided by a desire to preserve the purity of the modes must be discarded once and for all. The profusion of accidentals incorporated into intabulations should lead those scholars who still advocate a policy of “utmost reserve” with respect to musica ficta to rethink their positions. Even so well-known a ‘radical’ in these matters as Edward Lowinsky would never gloss a reading as exuberantly as did some of the sixteenth-century lutenists.’

⁶¹ Powers, H. S., s.v. ‘Mode’, especially section iii, The New Grove Dictionary; ‘Modal Representation in Polyphonic Offertories’, Early Music History, 2 (1982), pp. 43–86CrossRefGoogle Scholar; ‘Tonal-Types and Modal Categories’, JAMS, 34 (1980), pp. 428–70Google Scholar.

⁶² Finck, H., Practica musica (Wittenberg, 1556)Google Scholar, sig. Rr iii^v.

⁶³ For statements reflecting the primacy that tonal organisation in its modern sense holds for much present-day scholarship, see Berger (Theories, p. 2): ‘There can be little doubt that the organization of a sixteenth-century work is primarily tonal, that it is the organization of various pitches in certain specific ways, whereas organization of other values (temporal, timbral, dynamic) is of secondary importance.’

⁶⁴ See, for example, Harvard Dictionary of Music, 2nd rev. edn (Cambridge, Mass., 1969)Google Scholar, s.v. ‘Musica ficta’: ‘In the music of the 10th to 16th centuries, the theory of the chromatic or, more properly, non-diatonic tones …’; ‘resulted from melodic modifications or from transpositions of the church modes’; ‘… disconcerting to find many long compositions completely lacking in any indication of accidentals’; ‘… the necessity for such emendations cannot be denied’; ‘Matters were carried much too far in many editions published between 1900 and 1930 … no doubt historically accurate view of adding as few as possible.’ And from The New Grove Dictionary: ‘The term used loosely to describe accidentals added to sources of early music, by either the performer or the modern editor. More correctly it is used for notes that lie outside the predominantly diatonic theoretical gamut of medieval plainchant, whether written into the source or not.’