1 As is widely documented, Guido took these syllables from his hymn to St John the Baptist ‘Ut queant laxis’, whose lyrics are about singing. This tune does not circulate outside of theoretical treatises until the nineteenth century. On the affinities, see Pesce, Dolores, The Affinities and Medieval Transposition (Bloomington and Indianapolis, 1987)Google Scholar.

2 In modern scholarship, the subtleties of this system are lost by casting this intervallic pattern under the label ‘hexachord’. Throughout this article, I have endeavoured to use technical terminology as it would have been understood by the readerships for the treatises in question. This will allow subtle nuances between conceptual layers of this system to be better understood.

3 One can imagine that this was most prevalent during the training of young musicians rather than with seasoned professional singers, as suggested by Gaforus. However, other theorists, such as Lanfranco, suggest that the syllables had a more fundamental position in the conceptual understanding of musical space: see Dean, Jeffrey, ‘Okeghem's Attitude Towards Modality: Three-mode and Eight-mode Typologies’, in Modality in the Music of the Fourteenth and Fifteenth Centuries/Modalität in der Musik des 14. und 15. Jahrhunderts, ed. Günther, Ursula, Finscher, Ludwig and Dean, Jeffrey, Musicological studies and documents 49 (Neuhausen-Stuttgart, 1996), 203–46Google Scholar, esp. 210–27. On the likelihood of a choirmaster signing individual parts, it should be noted that the position of choirmaster during the medieval period, and the activities and responsibilities of such a post, is something for which there is only scant documentary evidence. It seems likely that the Hand and solmisation system could have been used in performance to make quick corrections, but it is not entirely clear how this would have worked in practice, or whether it was especially widespread.

4 Such a theory has been advanced by Richard Crocker. Crocker was one of the strongest supporters of the idea of the sixth underpinning the diatonic conception of medieval and Renaissance music: see Crocker, Richard L., ‘Hermann's Major Sixth’, Journal of the American Musicological Society, 25/1 (1972), 19–37 CrossRefGoogle Scholar. This hypothesis was advanced from Handschin, Jacques, Der Toncharakter: eine Einführung in die Tonpsychologie (Zürich, 1948), 316–19Google Scholar. Crocker's theory, although apparently supported by some historical texts, does not take account of the fact that solmisation was probably just a pedagogical option, not a conceptual musical requirement. In all likelihood, the notion of overlapping hexachords was probably more an intellectual construction rather than a practical system.

5 See Mengozzi, Stefano, The Renaissance Reform of Medieval Music Theory: Guido of Arezzo between Myth and History (Cambridge, 2010)Google Scholar; idem, ‘The Heptachordal Basis of Hexachordal Theory: On the Semiotics of Musical Notation in the Middle Ages’, Plainsong and Medieval Music, 22/2 (2013), 169–95. See also idem, ‘Virtual Segments: The Hexachordal System in the Late Middle Ages’, Journal of Musicology, 23 (2006), 426–67, esp. 440–58; idem, ‘“Si quis manus non habeat”: Charting Non-Hexachordal Musical Practices in the Age of Solmisation’, Early Music History, 26 (2007), 181–218.

6 See, for example, Mengozzi, The Renaissance Reform, 45: ‘The question of how to integrate the six syllables into the existing diatonic order was not a high priority for eleventh- and twelfth-century theorists; indeed it may not have been a question at all.’ I am inclined to agree with this view, given that one has to question the extent to which this would have had a use in polyphonic music of the Renaissance, particularly as musical notation became more widely used as a means of distributing all manner of music.

7 The letters used to describe pitches were sometimes presented in capitalised or lower case forms depending upon their placement within the gamut, offering a degree of pitch specificity through the letter names alone.

8 The apparent simplicity and usefulness of this system was not universally accepted, with Johannes Gallicus arguing that solmisation syllables in such an arrangement added an unnecessary layer of ‘verbosity’. I return to this point later.

9 Paris, Bibliothèque nationale de France, lat. 11266, fol. 6v. See also n. 18.

10 The value of the so-called ‘Hollandrinus’ tradition of theoretical texts is only being realised fully in scholarship thanks to the recent survey of the texts making up this tradition; see, Traditio Iohannis Hollandrini, ed. Michael Berhard and Elzbieta Witkowska-Zaremba (Munich, 2010–15), 6 vols.

11 Bonnie Blackburn has characterised Tinctoris's treatises as surveying ‘fifteenth-century music theory with magisterial thoroughness’, and is not alone in asserting such a view; see Blackburn, Bonnie J., ‘Music Theory after 1450’, Music as Concept and Practice in the Late Middle Ages, ed. Strohm, Reinhard and Blackburn, Bonnie J. (Oxford, 2001)Google Scholar, 325.

12 The spread of these examples across his theoretical oeuvre is discussed in Adam Whittaker, ‘Musical Exemplarity in the Notational Treatises of Johannes Tinctoris (c.1435–1511)’, Ph.D. diss., Birmingham City University (2015), 89–99.

13 The two sources in question are: Valencia, Universitat de València, Biblioteca Històrica, 835 (V); Bologna, Biblioteca Universitaria, 2573 (Bu). These two manuscripts have been studied extensively in Christian Goursaud, ‘The Neapolitan Presentation Manuscripts of Tinctoris's Music Theory: Valencia 835 and Bologna 2573’, Ph.D. diss., Birmingham City University (2015). On the likely dating of V, see Ronald Woodley, ‘The Dating and Provenance of Valencia 835: A Suggested Revision’ (2013, rev. 2014), http://earlymusictheory.org/Tinctoris/Articles/DatingAndProvenanceOfValencia835/# (accessed 20 October 2014).

14 In his Proportionale musices and De arte contrapuncti, Tinctoris is well known for the inclusion of quite extended polyphonic miniatures to accompany his points. Nothing of this scale is seen in the Expositio manus as such examples would not have been appropriate for the theoretical material under discussion.

15 Tinctoris includes an image of the Hand, along with a number of other diagrams showing the range of the gamut. However, none of these show the overlapping hexachords, and by implication the mutations that needed to be effected to navigate the full range of the gamut, with the same clarity of Figure 1.

16 As already identified, such a figure can easily be traced as far back as the thirteenth century in the Ars musica attributed to Magister Lambertus. It also appears in a number of texts from the Hollandrinus tradition, most notably Trad. Holl. XIII (ed. Christian Meyer, vol. 4, 283–362), Trad. Holl. XV (ed. Elzbieta Witkowska-Zaremba, vol. 5, 1–84), and Trad. Holl. XX (ed. Christian Meyer, vol. 5, 365–420): see Traditio Iohannis Hollandrini, ed. Michael Berhard and Elzbieta Witkowska-Zaremba (Munich, 2010–15), 6 vols.

17 This list makes clear, by implication, which mutations are not possible.

18 A further example of such a practice can be seen in the ‘Ars musica’ attributed to Magister Lambertus in Paris, Bibliothéque nationale de France, lat. 11266, fol. 6v, which shows each of the points of mutation in a circular form with the ascending and descending motions marked out with syllabic annotations. Despite showing the mutations clearly, these examples do not make such a clear connection with staff-based musical notation perhaps, in itself, evidencing a shift in the conception of musical space.

19 One notable approach is taken by Prosdocimus de Beldomandis in his Plana musica, where the subject is discussed without any diagrammatic or musical examples. See Prosdocimus de Beldomandis, Plana musica, vi–viii, ed. and trans. Herlinger, Jan, Prosdocimo de’ Beldomandi's Plana musica and Musica speculativa (Urbana and Chicago, 2008), 46–55 Google Scholar.

20 There are a number of instances where solmisation syllables are entered directly onto the staff in some medieval central European elementary treatises and texts from the Hollandrinus tradition, particularly those dating from the early to mid-fifteenth century. Indeed, a number of examples on G-sol-re-ut, which include solmisation syllables placed at points of mutation with some specificity, can be seen on fol. 315r of Wrocław, Biblioteka Uniwersytecka IV Q 37. This text is an anonymous treatise used in one of the parochial schools in Wrocław that dates from sometime before 1478. See Hollandrini, Traditio XV in Traditio Iohannis Hollandrini, 5, ed. Bernhard, Michael and Witkowska-Zaremba, Elżbieta (Munich, 2014)Google Scholar, 2–83. I am grateful to the anonymous reviewer for drawing my attention to this example, and the breadth of the Hollandrinus tradition more generally.

21 Tinctoris, Expositio manus, 7.23, ed. and trans. Ronald Woodley, www.stoa.org/tinctoris/expositio_manus/expositio_manus.html (accessed 4 January 2016).

22 Such a close connection between the text and the musical examples holds throughout Tinctoris's whole oeuvre, evidencing his pedagogical orientation. This is particularly true of examples that illustrate each statement of the accompanying text in precise order, as seen numerous times in De arte contrapuncti, Proportionale musices and, slightly more problematically, De imperfectione notarum.

23 This type of example is almost always fairly short and was clearly conceived as a theoretical demonstration rather than as a standalone musical miniature. In this specific case, elements of the theoretical point are ‘projected’ into a slightly larger context, though the scope of the example is still quite limited. The instantiation model is explored more fully in Whittaker, ‘Musical Exemplarity’, 100–52.

24 These pitches, necessary to the example, situate the mutations in a ‘natural habitat’, rather than detracting from the key point.

25 Signa congruentiae, and their possible functions within examples, are further discussed in Whittaker, Musical Exemplarity in the Notational Treatises of Johannes Tinctoris (c. 1435–1511), 155–280.

26 Although it may be possible that re–la is missing before the upper octave, this does not appear in any of the surviving sources, some of which Tinctoris probably had a hand in the production of.

27 I am particularly grateful to Jeffrey Dean for his comments on this musical example.

28 On the manuscript source itself, see Derolez, Albert, The Library of Raphael de Marcatellis: Abbot of St. Bavon's, Ghent, 1437–1508 (Ghent, 1979), 227–34Google Scholar. This source includes the following of Tinctoris's treatises: Complexus effectuum musices, De natura et proprietate tonorum, De notis et pausis, De regulari valore notarum, De alterione notarum, De imperfectione notarum, Proportionale musices. An additional chapter is added to De punct which discusses ‘puncti acceptionis’ and does not survive in any other source. The additional chapter includes a polyphonic musical example which further strengthens the case for the attribution of this extra material to someone other than Tinctoris. This source does not, however, include the Expositio manus and De arte contrapuncti. In this source, the scribe misordered musical examples on one opening of De imperfectione notarum, resorting to numerical labels being added retrospectively to show which example should accompany a particular theoretical point.

29 Lebedev, Sergej, preface to Cuiusdam Cartusiensis monachi Tractatus de musica plana , ed. idem, Musica mediaevalis Europae occidentalis 3 (Tutzing, 2000), viii–xGoogle Scholar. The error stems from de Tervliet, Joseph Antoine Walwein, Catalogue des manuscrits de la Bibliothèque publique de la ville de Gand (Ghent, 1816), 31 Google Scholar; it was rejected by Coussemaker in Scriptorum de musica medii aevi novam seriem a Gerbertina alteram collegit nuncque primum edidit E. de Coussemaker (Paris, 1864–76), 2: xxv–xxvi, but uncritically accepted by Derolez, The Library of Raphael de Marcatellis, 233, and apparently on his authority by the Thesaurus Musicarum Latinarum, files http://boethius.music.indiana.edu/tml/15th/GENTSPE_MGRU70 and http://boethius.music.indiana.edu/tml/15th/GENTPRA_MGRU70 (including at the beginning some excerpts from the Quatuor principalia and omitting the last two parts). Lebedev appears not to have read Coussemaker's preface. Coussemaker evidently regarded all four parts as by the same author but in fact printed only the last two parts, Scriptores, 2: 434a–60b, followed by unrelated material from G. The notices of G in RISM B III/1: 65–9, and B III/6: 129–31 are understandably confused and misleading, since the boundaries of different texts between fol. 77vb (the end of Tinctoris's Complexus effectuum musices) and fol. 159vb (the scribe's colophon to the first section of his work) are indistinct, and the RISM entries fail to identify the correct origin of many excerpts. Lebedev, iii–iv, demonstrates the unity of these different texts as the work a single author.

30 Carthusian anonymous, Tractatus de musica plana, ed. Peter Slemon, online, www.chmtl.indiana.edu/tml/15th/GENTPRA_MGRU70.html (accessed 22 September 2014). Translation my own.

31 The sharp sign may have been added by a later hand, though this is far from clear from the available facsimile. I have not been able to conduct an autopsy of the original manuscript.

32 The text does not appear to have been conceived as a teaching text for beginners, given that it lacks the systematic comprehensiveness one would expect to find in such a text. Nevertheless, it is clear that a pedagogical logic clearly underpins its approach, though this is not necessarily the most didactically useful strategy.

33 The examples on folio 116 of G demonstrate these, though with less technical specificity than Tinctoris's examples.

34 The passage in question runs from fols. 117r to 119v in G.

35 The Carthusian anonymous discusses specific mutations in more detail later in the same chapter, though solmisation labels do not appear in any of the examples that accompany these points. Instead, he provides examples that include mutations without staff labels, except for a general caption, following the practice seen in countless other texts.

36 As outlined above, a number of texts of the Hollandrinus tradition include such labels, though these texts seem quite distant from the tradition with which Tinctoris is associated. I intend to explore the exemplification of mutation in these texts more fully in a future study, informed by recent studies of these texts.

37 For a more extended discussion of Marchetus's practice, see Whittaker, Musical Exemplarity, 68–82.

38 The passage in question is found in Tinctoris, De arte contrapuncti, 2.2, even if Tinctoris appears somewhat confused on this topic. The broader implications of Marchetus's writings on the chromatic semitone, and the ways in which Tinctoris may have misunderstood these, is discussed more fully in Woodley, Ronald, ‘Sharp Practice in the Later Middle Ages: Exploring the Chromatic Semitone and its Implications’, Music Theory Online, 12/2 (May 2006)Google Scholar. Jeffrey Dean also explores sharp tuning, arriving at somewhat different conclusions from Woodley, in ‘Okeghem's Attitude to Modality’.

39 All text references and translations to Marchetus's Lucidarium will be taken from Herlinger, Jan, The Lucidarium of Marchetto of Padua: A Critical Edition, Translation, and Commentary (Chicago, 1985)Google Scholar. I am grateful to Professor Herlinger for sending me a copy of his edition and for his continuing support of my investigation here. On the influence of the Lucidarium, see Herlinger, Jan, ‘L'influsso di Marchetto: prove manoscritti’, in La filologia musicale: istituzioni, storia, strumenti critica. 3: Antologia di contributi filologici, ed. Vela, Maria Caraci (Lucca, 2013), 201–28Google Scholar and its earlier version in Herlinger, , ‘Marchetto's Influence: The Manuscript Evidence’, in Music Theory and Its Sources: Antiquity and the Middle Ages, ed. Barbera, André (Notre Dame, IN, 1990), 235–58Google Scholar. Herlinger notes that, although the Pomerium has received a great deal of scholarly attention given its teachings on musical notation, it was the Lucidarium that was much more widely circulated. This wider circulation is further supported by the numerous references in other theoretical texts to the teachings of the Lucidarium.

40 Herlinger notes that ‘the organisation of this chapter parallels the discussion of mutation in the Introductio musice secundum Magistrum de Garlandia (Scriptores 1: 157–75, 160–62)’: see Marchetus of Padua, Lucidarium, 281, note ‘b’.

41 Marchetus of Padua, Lucidarium, 8.2.2–3, ed. and trans., Herlinger, 281.

42 Marchetus of Padua, Lucidarium, 8.2.5–7, 282–3.

43 The degree of variation between the sources of the Lucidarium might be partially accounted for by the fact that it continued to be copied for well over 150 years after its composition, increasing the likelihood of scribal revision and historical distance from the authorial original. Such variation might also be accounted for by the manner in which such exemplary content was indicated or notated in Marchetus's exemplar copies. Even more marked cases of variation in exemplary content can be seen in the surviving manuscripts of Boethius's De institutione musica, the most widely copied music theory text of the Middle Ages: see Elizabeth A. Mellon, ‘Inscribing Sound: Medieval Remakings of Boethius's De institutione musica’, Ph.D. diss., University of Pennsylvania (2011). However, it is important to be mindful of the significant differences between Marchetus's and Boethius's texts, and their place in music-intellectual culture.

44 None of the autographs seem to survive, making it impossible to deduce the makeup of a scribal exemplar with any certainty. However, the exemplar may have referred to a separate sheet or music volume, or perhaps only notated a partial example. Such an example could have been extended or condensed as the scribe saw fit.

45 Such variations are discussed in Whittaker, Musical Exemplarity, 20–88; Leitmeir, Christian Thomas, ‘Types and Transmission of Musical Examples in Franco's Ars cantus mensurabilis musicae ’, in Citation and Authority in Medieval and Renaissance Culture: Learning from the Learned, ed. Clark, Suzannah and Leach, Elizabeth Eva (Woodbridge, 2005), 29–44 Google Scholar.

46 This mutation would occur here if the rule, given by some theorists, that one must stay within a hexachord for as long as possible is followed. If not, then the first mutation could take place on the third note, and the second on the fourth note of the second section of the example. However, it seems safe to assume that the first case is most likely, with mutation only taking place when absolutely necessary. Although this is not stated explicitly, it is implied by this context.

47 Pg includes a number of passages from other theoretical works, and has been annotated extensively at points. This manuscript is explored more fully in Blackburn, Bonnie J., ‘A Lost Guide to Tinctoris's Teachings Recovered’, Early Music History, 1 (1981), 29–116 CrossRefGoogle Scholar.

48 For a facsimile and detailed catalogue record of the Einsiedeln manuscript, see www.e-codices.unifr.ch/en/searchresult/list/one/sbe/0689 (accessed 16 September 2016).

49 For a complete survey of the different versions of the examples to accompany the mutations on C fa-ut, see Herlinger, The Lucidarium of Marchetto of Padua, 282.

50 Gallicus, Ritus canendi, ed. and trans. in Mengozzi, The Renaissance Reform, 152.

51 The manuscript in question is London, British Library, Additional 22315, which was copied by Nicolaus Burtius, one of Gallicus's pupils. The indication of Gallicus's death in 1473 constitutes a terminus post quem for the copying date of the manuscript.

52 Mengozzi, The Renaissance Reform, 156. It is noteworthy that Gallicus's image of the Guidonian Hand also indicates pitch content using a letter accompanied by a single solmisation syllable. Thus, Gallicus clearly sought only to revise contemporary thinking on Guido's method, and not simply ‘throw it out’ as a less nuanced perspective might suggest.

53 There are a number of corrections and erasures to the letter/solmisation voice at the start of this example, showing that the scribe started in the wrong place. However, this is soon corrected. Red ink is used, maintaining a style used throughout the manuscript when musical examples are accompanied by annotations.

54 Indeed, Ramis de Pareia pioneered a heptachordal solmisation system, though he still retained solmisation labels for pitches rather than the letters A–G. His lack of access to print musical examples led him to describe the examples through text-based means, making their contents difficult to unpick. It is also worth pointing out that the use of solmisation syllables within theoretical examples is evidenced by some of the texts of the Hollandrinus tradition.

55 See, Agricola, Martin, Ein kurz deudsche Musica, fols. IXr–XVr (Wittenberg, 1528)Google Scholar, available at http://daten.digitale-sammlungen.de/~db/0010/bsb00104414/images/ (accessed 1 March 2016).

56 The extent to which Tinctoris's exposition might serve as a theoretical exposition of the possibilities of the system, rather than as a codification of its practical application is impossible to discern at this historical distance. However, it seems clear that something of a practical simplification did take place in the sixteenth century, and that this was considered sufficiently widespread for inclusion in theoretical texts.

57 The first edition was titled Musicae, id est, artis canendi libri duo (Nuremberg: Petreius, 1537); the second edition was the more developed De arte canendi (Nuremberg: Petreius, 1540). On Heyden, see Victor H. Mattfield, ‘Haiden’, Oxford Music Online: Grove Music Online (Oxford University Press), www.oxfordmusiconline.com/subscriber/article/grove/music/12175pg1 (accessed 13 January 2016).

58 On examples in Sebald Heyden's De arte canendi, see Judd, Cristle Collins, Reading Renaissance Music Theory: Hearing with the Eyes (Cambridge, 2002), esp. 90–108 Google Scholar.

59 It is noteworthy that the examples found in both editions of the treatise to make use of solmisation content in this way omit clefs for their notated passages. The arrangements of flats and the solmisation syllables would have been sufficient for most readers to ascertain which lines and spaces referred to specific pitches, especially in light of Prosdocimo's statement that round b is also a clef which, like the other clefs C and F, shows the position of fa. The implication of this is that the semitone could be established from the round b alone: see de Beldomandis, Prosdocimus, Plana musica I:x, ed. and trans. Herlinger, Jan, Prosdocimo de’ Beldomandi's Plana musica and Musica speculativa (Urbana and Chicago, 2008), 60–5Google Scholar.

60 Tinctoris, Expositio manus, 2: 75–8, ‘Hence there are some who call ut “on the line”, A re “in the space”, and so on alternately with the others. But it is the greatest error to speak in these terms, since ut is the line itself, and A re is the space itself, and so on alternately with the rest; they cannot, therefore, be said to be positioned “on” the line or “in” the space.’, ed. and trans. Ronald Woodley (online). This view is echoed in Ockeghem's music, as demonstrated in Jeffrey Dean, ‘Okeghem's Attitude Towards Modality’.