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The heptachordal basis of hexachordal theory: on the semiotics of musical notation in the Middle Ages

Published online by Cambridge University Press:  12 September 2013


By all appearances, the advent of a full-fledged hexachordal system in the late thirteenth century marked the culmination of a radical reconceptualisation of diatonic space that had begun over two centuries earlier with Guido of Arezzo's introduction of the syllables ut re mi fa sol la as an aid to sight-singing. An early witness of the new diatonic configuration is Johannes de Garlandia's Introductio musicae planae (c.1275), which presents the three proprietates per b quadro, per natura and per b molle as a fundamental articulation of the gamut that is necessary for conveying the intervallic distances between the diatonic pitches.

A contextual evaluation of the theory of the proprietates, however, indicates that the hexachordal parsing of the gamut was not meant to establish the major sixth as the regulative paradigm of diatonic space (i.e. as a normative scale), but was rather in line with the earlier understanding of the ut-la syllables as an optional aid for singing. The development of hexachordal theory was in fact an implicit confirmation of the heptachordal structure of the gamut that had been in place since around the year 1000 (and arguably earlier), when the cycle of seven A–G letters was first used to label the constituent pitches of the octave.

Research Article
Copyright © Cambridge University Press 2013 

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1 It is possible to overestimate the significance of the ‘Guidonian hand’ in musical instruction prior to the fifteenth century, as I have argued in The Renaissance Reform of Medieval Music Theory. The classic study of the hand as a mnemonic tool is Berger, Karol, ‘The Hand and the Art of Memory’, Musica disciplina, 35 (1981), 87120Google Scholar, now complemented by Berger, Anna Maria Busse, Medieval Music and the Art of Memory (Berkeley, 2005), 85110CrossRefGoogle Scholar. See also Weiss, Susan Forscher, ‘The Singing Hand’, in Writing on Hands: Memory and Knowledge in Early Modern Europe, ed. Sherman, Claire Richter and Lukehart, Peter M. (Carlisle, PA, 2000), 3545Google Scholar, and eadem, ‘Disce manum tuam si vis bene discere cantum: Symbols of Learning Music in Early Modern Europe’, Music in Art: International Journal for Music Iconography, 30 (2005), 3574Google Scholar. For a modern reproduction of numerous Guidonian hands found in medieval manuscripts, see van Waesberghe, Joseph Smits, Musikerziehung: Lehre und Theorie der Musik im Mittelalter. Musikgeschichte in Bildern, ed. Besseler, Heinrich and Bachmann, Werner, vol. III/3 (Leipzig, 1969), 133–45Google Scholar.

2 For a full transcription of Amerus's hand, see the edition of the Practica musice artis by Cesarino Ruini (n.p., 1977), 24. On the origin and meaning of the term properchant, see Blackburn, Bonnie J., ‘Properchant: English Theory at Home and Abroad, with an Excursus on Amerus/Aluredus and his Tradition’, in Quomodo cantabimus canticum? Studies in Honor of Edward H. Roesner, ed. Cannata, David B., Currie, Gabriela Ilnitchi, Mueller, Rena Charnin and Nádas, John L. (Middleton, WI, 2008), 8198Google Scholar.

3 The computus was the centuries-old method of calculating the date of Easter by mapping numbers on the left hand. On the similarities between it and the musical hand, see Berger, ‘The Hand and the Art of Memory’, 105–11. For Ciconia's reference (c.1410) to the compotus, see Ciconia, Johannes, Nova musica and De proportionibus, ed. and trans. Ellsworth, Oliver (Lincoln, 1993), 304–5Google Scholar. As I have shown elsewhere, in his treatises Ciconia indicates the musical pitches only by means of the seven litterae and of the Greek labels of the Greater Perfect System, from ‘proslambanomenos’ to ‘nete hyperboleon’. See my The Ciconian Hexachord’, in Johannes Ciconia: musicien de la transition, ed. Vendrix, Philippe (Turnhout, 2003), 181218Google Scholar.

4 Nam si quis manus non habeat, ergo cantum discere non potest? Id credere stultum est’, Gallicus, Johannes, Ritus canendi vetustissimus et novus, ed. Seay, Albert (Colorado Springs, 1981), 2:4Google Scholar.

5 Die sechs Töne setzen den siebenten voraus’, in Handschin, Jacques, Der Ton charakter: eine Einführung in die Tonpsychologie (Zurich, 1948; repr. Darmstadt, 1995), 328Google Scholar.

6 Crocker, Richard, ‘Hermann's Major Sixth’, Journal of the American Musicological Society, 25 (1972), 1937CrossRefGoogle Scholar, at 37.

7 Ibid., 37. In line with the portrayal of the hexachord as a central diatonic concept proposed by Crocker and others – a notion that may be traced back to the beginning of modern music historiography – a number of scholars have proposed hexachord-based analyses of numerous musical works by composers ranging from Guillaume de Machaut to J.S. Bach. I discuss some of these studies in The Renaissance Reform of Music Theory, 19–24.

8 In Dolores Pesce's recent edition of three treatises by Guido, the paragraph on the ut-la syllables corresponds to lines 112–45 of the Epistola. See Pesce, Dolores, Guido D'Arezzo's Regule rithmice, Prologus in Antiphonarium, and Epistola ad Michahelem. A Critical Text and Translation (Ottawa, 1999), 464–71Google Scholar.

9 For a compelling study of how such consolidation took place, see Atkinson, Charles, The Critical Nexus: Tone-System, Mode, and Notation in Early Medieval Music (Oxford, 2009)Google Scholar. However, Atkinson's monograph attributes a structural role of sorts to the hexachordal system, to the extent that it presents an extension of it – the theory of the coniunctae – as an essential step towards the conceptualisation and the notation of accidentals (ibid., 255–8).

10 See Affligemensis, Johannes, De musica cum tonario, ed. van Waesberghe, Joseph Smits (Rome, 1950), 50Google Scholar; Babb, Warren, ed. and trans., Hucbald, Guido, and John on Music: Three Medieval Treatises (New Haven, 1978), 104Google Scholar. John's comments on the reception of solmisation across Europe are discussed in Page, Christopher, ‘Towards: Music in the Rise of Europe’, The Musical Times, 136 (1995), 127–34, at 129CrossRefGoogle Scholar. For a more detailed reading of John's references to the Guidonian syllables, see The Renaissance Reform of Medieval Music Theory, 44–6.

11 For instance, Hucbald points to two notes an octave apart in the antiphon Undecim discipuli in Galilea, observing that they sound the same, even though one belongs to the vocal register of adult men, the other to that of children (‘Nam ultima sillaba, -runt, quem octavo loco a prima deponitur, si comparetur cum ipsa prima [ve-], eodem modo sonabit, quanvis haec voce virile illa puerile’, Chartier, Yves, L'oeuvre musicale d'Hucbald de Saint-Amand: les compositions et le traité de musique (Saint-Laurent, 1995), 150Google Scholar). For his part, the anonymous author of the Dialogus de musica uses capital letters to label the pitches of the lower octave (Γ–G), and small letters for the upper octave (a–g), insisting that the letters marking corresponding pitches in the two halves of the monochord be identical (et a medietate monochordi in antea eaedem sint litterae, quae sunt et in prima parte’, De Nardo, Lucia Ludovica, ed., Dialogus de musica (Udine, 2007), 80Google Scholar); the author later observes that ‘any melody will be sung in the same manner in the two parts’ (‘Unde fit ut omnis cantus similiter ut in pria, et in secunda parte cantetur. Sed primae partis voces ad voces secundae partis quasi viriles cum puerilibus vocibus concorditer cantant’, ibid., 88). On the recognition of octave equivalence before the Dialogus, see the editor's introduction, 32–4.

12 Thus, medieval theorists consistently refer to the seven litterae as the signa monochordi, and to the solmisation syllables as nomina vocum (or simply voces) superimposed to the signa. The distinction points to the syllables as ‘signs of signs’, thus further removed than the litterae from the sensible datum of the seven different soni identified on the monochord.

13 Such contrary evidence is provided, for instance, by tantalising references to the solmisation syllables in William of Conches, Glosae super Macrobium, and in Bernard Silvestris, Commentum in Martianum, discussed in Andrew J. Hicks, ‘Music, Myth, and Metaphysics: Harmony in Twelfth-Century Cosmology and Natural Philosophy’, Ph.D. diss., University of Toronto (2011), 83–4. My thanks to an anonymous reviewer for drawing my attention to this study.

14 Perhaps the clearest demonstration of the broad acceptance of the A–G letters as diatonic markers is the practice of integrating them to neumatic notation after c.1000 for the purpose of clarifying the pitches. Alma Colk Santosuosso has listed a total of almost sixty such practical sources from the ninth to the thirteenth century (not counting the music treatises) featuring alphabetic notations of various kinds and containing both chant and secular music (for a total of about three hundred pieces). See Santosuosso, Alma Colk, Letter Notation in the Middle Ages (Ottawa, 1999)Google Scholar. The use of solmisation syllables in medieval notation appears to be hardly documented at all, beyond the didactic examples of O Roma nobilis in Montecassino 318 and of course Ut queant laxis, which was usually transmitted as part of Guido's Epistola, but also independently from it (for an example, see item 5 in Table 2). For an overview of alphabetic notation in the Middle Ages, see also Phillips, Nancy, ‘Notationen und Notationslehren von Boethius bis zum 12. Jahrhundert’, in Die Lehre vom einstimmigen liturgischen Gesang, ed. Huglo, Michel, Atkinson, Charles, Meyer, Christian, Schlager, Karlheinz and Phillips, Nancy, Geschichte der Musiktheorie 4 (Darmstadt, 2000), 549623Google Scholar.

15 For a review of the arguments by several scholars who have proposed a ‘foundational’ interpretation of the hexachordal system, see chapter 1 of The Renaissance Reform of Medieval Music Theory, 19–29. See also footnote 31, below.

16 Thus, Christian Meyer has appropriately illustrated the emergence of the hexachordal system in a chapter titled ‘Die Formalisierungen in Zeitalter der Scholastik’, in Christian Meyer, ‘Die Tonartenlehre im Mittelalter’, in Die Lehre vom einstimmigen liturgischen Gesang, ed. Huglo et al., 197–215. See also Meyer, Christian, ed. Musica plana Johannis de Garlandia (Baden-Baden and Bouxwiller, 1998), 130Google Scholar.

17 For the text of Garlandia's treatise I have relied on Christian Meyer's edition (see footnote 16), which also includes the complete text of four additional ‘reportationes’ of Garlandia's musical teaching with relatively close textual links to the Introductio. Here and in the remainder of this article I will refer to Garlandia as the bona fide author of these texts. On the issue of the authorship of the Introductio and its relationship with the ‘reportationes’, see Meyer, ed., Musica plana Johannis de Garlandia, 129–31. The following section expands and partly modifies the discussion of Garlandia's account of solmisation that I presented in ‘Virtual Segments: The Hexachordal System in the Late Middle Ages’, The Journal of Musicology, 23 (2006), 426–67.

18 ‘Et ideo prima vox ut non ad alias voces, scilicet re, mi, fa, sol, la, reducitur, sed alie voces, scilicet re mi fa sol la ad ipsam primam vocem, scilicet ut, causa dignitatis referentur, quia a digniori inchoandum est, ut dicit philosophus, quia in omnibus rebus naturalibus tam integralibus quam etiam subjectivis constituitur fundamentum ad quod omnia referuntur et propter hoc dicimus quod ad semetipsum descendit et non ad alias, sed alie ad ipsum sicut dictum est.’ Ibid., 69. The ‘philosophus’ in the citation is perhaps Petrus Hispanus, the author of a Tractatus (also known as Summulae logicales, probably written between 1230 and 1245) that became one of the main texts of scholastic logic. References to the ‘integral whole’ (totum integrale) and its topical relationship to the ‘subjective part’ (pars subiectiva) are found in the fifth tract of the Tractatus (De locis). The term fundamentum, however, does not appear in that particular portion of the treatise. Here and elsewhere in this article, the translations from Latin are mine unless otherwise indicated.

19 ‘Voces attamen radicales omnium deductionum ex locis propriis et aliae quinque sequentes ex locis ipsarum vocum radicalium’ (Tinctoris, Johannes, Opera Theoretica, ed. Seay, Albert, Corpus Scriptorum de Musica 22, 3 vols. (Rome, 1975–78), 2:46Google Scholar; for the English translation, see Seay, Albert, ‘The Expositio Manus of Johannes Tinctoris’, Journal of Music Theory, 9 (1965), 194232CrossRefGoogle Scholar, at 214, although Seay's rendition of this particular sentence is somewhat misleading. The musical hand in Amerus's treatise refers to ut on C, F, and G as the radices (see Fig. 1).

20 As a matter of fact, in the Epistola Guido discusses the modi vocum of pairs of pitches as frequently as the modus vocis of individual ones. See, for instance, the section ‘De affinitatibus vocum per quatuor modos’ in lines 224–38, as well as lines 339–54 (Pesce, Guido of Arezzo's Regule rithmice, 490–5, 518–23). On one occasion, he focuses on the different diatonic positions of transposable portions of the gamut, as when he explains that ‘a.b.c. have two tones after them in descent [i.e. from a to f] and two tones after them in ascent [i.e. from c to e], whereas D.E.F. have only one tone in descent and three tones in ascent’ (lines 294–300, ibid., 508–9). The introduction of B flat (synemenon) will erase the difference (lines 301–8, ibid., 510–1). Compared analyses of the modi vocum of pitches a fourth or a fifth apart are also found in all the other treatises by Guido, a sure sign of the centrality of the theory of the diatonic affinities in this author. For a survey of the principal medieval theorists discussing the diatonic affinities, see Dolores Pesce, The Affinities and Medieval Transposition (Bloomington, 1987).

21 As flatly stated in the title of Chapter 9 of Guido's Micrologus, ‘affinity is perfect only at the octave’ (‘Item de similitudine vocum quarum diapason sola perfecta est’). See Guidonis Aretini Micrologus, ed. Joseph Smits van Waesberghe (n.p., 1955), 130; see also chapters 5–8, passim. The Epistola reiterates the same idea quite clearly in the section, ‘De diapason’, lines 217–23: ‘This diapason makes pitches agree so much that we do not call them similar, but the same’ (‘Hec diapason in tantum concordes facit voces ut non eas dicamus similes sed easdem’). See Pesce, Guido of Arezzo's Regule rithmice, 490–1.

22 For instance, D and A differ on the sixth step above, E and B on the fifth, F and C on the fourth, etc.

23 ‘Vides itaque, ut hec symphonia senis particulis suis a sex diversis incipiat vocibus? Si quis itaque uniuscuiusque particulae caput ita exercitatus noverit, ut confestim quamcumque particulam voluerit, indubitanter incipiat, easdem sex voces ubicumque viderit, secundum suas proprietates facile pronuntiare poterit.’ Pesce, Guido of Arezzo's Regule rithmice, 468–9. Here and elsewhere in this article, I quote Pesce's translation along with Guido's text.

24 Ibid., 27.

25 Ibid., 27, fn. 93.

26 The example is transcribed in ibid., 474–5.

27 Pesce, Dolores, ‘Guido d'Arezzo, Ut queant laxis, and Musical Understanding’, in Music Education in the Middle Ages and the Renaissance, ed. Murray, Russell. E. Jr., Susan Forscher Weiss and Cyrus, Cynthia J. (Bloomington, 2010), 2536Google Scholar, at 27.

28 Sunt autem in omni cantu tres proprietates, cum iste tres ad demonstracionem et evidentiam totius cantus sufficiant, scilicet .n. quadrum, natura vel proprius cantus quod idem est, et .b. molle. Ratio quare proprietates dicuntur talis est, quia regulariter proprie in notitiam vere canendi tonos et semitonos agendi legitime nos introducunt. Meyer, ed., Musica plana Johannis de Garlandia, 67.

29 The historical import of the emergence of a full-fledged hexachordal theory in the thirteenth century is not lost to Christian Meyer, who has observed: ‘Si l'utilisation des syllabes et du tableau de solmisation semble progressivement s'imposer dans l'enseignement de la musique au XIIe siècle, l'enseignement de Jean de Garlande révèle, dans les développements qui suivent, un effort de conceptualisation jusqu'alors inconnu’ (ibid., 118).

30 Ibid., 70.

31 Ibid., 93.

32 Meyer points out several textual links between the two treatises (ibid., 125); the list could be longer.

33 Et quamvis idem primus tonus per b mol secundum Guidonem naturaliter sit cantandus, est tamen ratio quaedam infalibilis et praecelsa cui non potest ab aliquo contradici: scilicet dyatesseron quae ostendit quod ibi est b mol. Et haec in omni tono sicut inferius continetur quae constat ex duobus tonis et semitonio, ef fit coniuncta et divisa. Quando ergo incipit in .f. coniuncta vel divisa de propinquo vel remoto unius ibi est b mol sine signo. Ibid., 89–90.

34 Perhaps the strongest articulation of this position is by Margaret Bent, when she stated that ‘it is only when yoked with hexachord syllables that the letters acquired unequivocal tone-semitone definition, even within the norms of musica recta’ (Bent, Margaret, ‘Diatonic ficta’, Early Music History, 4 (1984), 148CrossRefGoogle Scholar, at 10). Somewhat more ambiguously, David Cohen explains that ‘[the ut-la syllables] embody the intervallic relationships of each note [of the hexachord] to the others. The semitone, crucial for position finding, is always located between the two middle notes, mi and fa’ (Cohen, David, ‘Notes, Scales, and Modes in the Earlier Middle Ages’, in The Cambridge History of Western Music Theory, ed. Christensen, Thomas (Cambridge, 2002), 307–63Google Scholar, at 342). Similarly, Dolores Pesce has characterised the syllables as ‘signalling’ intervals and modes (Pesce, Dolores, ‘Theory and Notation’, in The Cambridge Companion to Medieval Music, ed. Everist, Mark (Cambridge, 2011), 276–90CrossRefGoogle Scholar, at 281). Given the consistent practice of indicating the diatonic intervals by means of only the litterae throughout the Middle Ages, a more accurate wording might be ‘in the system of solmisation, the recta semitones E–F, B–C, and A–B flat corresponded to the syllables mi–fa’.

35 The relevant passage from Virgil's Aeneid occurs in book VI, line 646. See The Aeneid of Virgil, trans. Charles J. Billson, 2 vols. (London, 1906), 1:292. Guido echoed Virgil in a section of the Regule titled ‘De vocum discriminibus’ beginning with ‘Semitonia et toni quia dissimiliter / septem sonis coaptantur, septem sunt discrimina / ut nullius vocis sonus idem sit in altera’ (Pesce, Guido of Arezzo's Regule rithmice, lines 126–9, pp. 354, 356). The musical meaning of vox as ‘inflected sound’ was mediated from grammar treatises (Laelius Donatus, Priscianus), and Aristotle's natural philosophy (De anima). On the reception of the term in the late Middle Ages, see Wittman, MichaelVox atque sonus: Studien zur Rezeption der Aristotelischen Schrift ‘De anima’ und ihre Bedeutung für die Musiktheorie (Pfaffenweiler, 1987)Google Scholar. My thanks to an anonymous reviewer for drawing my attention to this study.

36 ‘Deinde in ipso croco [i.e. the yellow line indicating letter C] est vox vel littera tercia in qua est tonus quintus vel sextus’, Prologus in antiphonarium, lines 96–7 (Pesce, Guido of Arezzo's Regule rithmice, 428–9; also line 108 at 432, and line 124 at 434).

37 On this hand the notae are positioned alternatively on the line and on the space from Γ-ut to d-la sol on the middle finger (a later hand added e-la to complete the seventh proprietas), according to the position of those same notae on the staff of musical notation. This hand also indicates the inception of the seven proprietates via the labels b quadrum, b molle, and naturale and the indication ‘incipit (¶) ut’. This particular representation of the gamut is somewhat confusing, in that the claves under the notae (consisting of one littera and one or more syllabae) are themselves positioned on the next line or space immediately below that of the notae. Yet the pedagogical intent is clear: having memorised the names of the notae and their position on either the line or the space, the novice will be ready to tackle the musical staff with its multiple lines. The hand in the Rio de Janeiro copy of Garlandia's Introductio (Biblioteca Nacional, Cofre 18, fol. 610v., possibly from southern France, c.1490) also places the notae alternatively on the line and on the space and attributes them to the deductions by means of abbreviations indicating the proprietates by B flat (flat sign), by B natural (bequadro sign), and by ‘nature’ (‘n’) positioned above each nota. This later hand also features the standard F and C clefs (signum) next to the corresponding notae. See the facsimile of this hand in Meyer, ed., Musica plana Johannis de Garlandia, 142. For other hands showing the lines and spaces of musical notation, see Smits van Waesberghe, Musikerziehung, 139–43.

38 Nevertheless the convention of indicating the notae on the musical hand will become the norm in later centuries. It is noteworthy that Johannes Gallicus, the most articulate critic of hexachordal solmisation in the fifteenth century, indicated only one nota for every ‘place’ in the hand in his Ritus canendi vetustissimus et novus (c.1460), though, to be sure, he never yoked more than one syllable for each littera. I refer to the copy of the treatise in London, BL Add. 22315, fol. 50r. For a modern reproduction of this musical hand, see Smits van Waesberghe, Musikerziehung, fig. 84.

39 For a more detailed discussion of these authors' views of six-syllables solmisation, see my monograph The Renaissance Reform, chapters 5 and 6.

40 References below refer to the facsimile of the 1508 edition of this text, Gregorius Reisch, Margarita philosophica nova, introd. Lucia Andreini, Analecta Cartusiana 179 (Salzburg, 2002).

41 Ibid., 235.

42 Ibid.

43 Ibid.

44 ‘Eo. Nam cum singulis clauibus singule ponuntur voces vna vel plures: vti patuit supra in monochordi descriptione. vnum posita claui cum nota in linea vel spacio: ipsam non vocem quamlibet, sed eam tantum que cum tali iungitur significare necesse est.’ Ibid.

45 Ibid.

46 See for instance the definition offered by Marchetto of Padua, among countless others: ‘Clavis est reseratio notarum in cantu quolibet signatarum, nam sicut per clavim reseratur ostium ad conservanda que intra ostium sunt, sic per clavim in cantu ipse reseratur cantus et conservantur notarum nomina et ab invicem esse distincta cognoscimus; et tales universaliter usitate sunt due, scilicet F grave et c acutum’ (‘The clef is the means for disclosing the notes written in any piece of music. Just as a key opens a door to preserve what is behind the door, so in melody does a clef disclose the melody and preserve the names of its notes; and it is through the clef that we recognize notes as distinct from one another. The clefs universally used are two, low F and high c’, Herlinger, Jan W., ed. and trans., The Lucidarium of Marchetto of Padua: A Critical Edition, Translation, and Commentary (Chicago, 1985), 538–9Google Scholar.

47 On the intellectual underpinnings of thirteenth-century Parisian theory, see Yudkin, Jeremy, ‘The influence of Aristotle on French University Music Texts’, in Music Theory and Its Sources: Antiquity and the Middle Ages, ed. Barbera, André (Notre Dame, 1990), 173–89Google Scholar; Tanay, Dorit, Noting Music, Marking Culture: The Intellectual Context of Rhythmic Notation, 1250–1400 (Holzgerlingen, 1999)Google Scholar.