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CHARTING BOETHIUS: MUSIC AND THEDIAGRAMMATIC TREE IN THE CAMBRIDGEUNIVERSITY LIBRARY’S DE INSTITUTIONEARITHMETICA, MS II.3.12

Daniel K. S. Walden

Early Music History / Volume 34 / October 2015, pp 207 - 228DOI: 10.1017/S0261127915000017, Published online: 23 September 2015

Link to this article: http://journals.cambridge.org/abstract_S0261127915000017

How to cite this article:Daniel K. S. Walden (2015). CHARTING BOETHIUS: MUSIC AND THEDIAGRAMMATIC TREE IN THE CAMBRIDGE UNIVERSITY LIBRARY’S DEINSTITUTIONE ARITHMETICA, MS II.3.12. Early Music History, 34, pp 207-228doi:10.1017/S0261127915000017

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Daniel K. S. Walden

Email: danielwalden@fas.harvard.edu

CHARTING BOETHIUS: MUSIC AND THEDIAGRAMMATIC TREE IN THE

CAMBRIDGE UNIVERSITY LIBRARY ’S DEINSTITUTIONE ARITHMETICA , MS II .3 .12

This article discusses a full-page schematic diagram contained in a twelfth-century manu-script of Boethius’ De institutione arithmetica and De institutione musica fromChrist Church Cathedral, Canterbury (Cambridge University Library MS Ii.3.12), whichhas not yet been the subject of any significant musicological study despite its remarkablescope and comprehensiveness. This diagrammatic tree, or arbor, maps the precepts of thefirst book of De institutione arithmetica into a unified whole, depicting the ways musicand arithmetic are interrelated as sub-branches of the quadrivium. I suggest that thisschematic diagram served not only as a conceptual and interpretative device for the scribeworking through Boethius’ complex theoretical material, but also as a mnemonic guide toassist the medieval pedagogue wishing to instruct students in the mathematics of musicaspeculativa. The diagram constitutes a fully developed theoretical exercise in its own right,while also demonstrating the roles Boethian philosophy and mathematics played in twelfth-century musical scholarship.

Scholars have often cited and reproduced images from CambridgeUniversity Library MS Ii.3.12, a twelfth-century manuscript fromChristChurch, Canterbury containing Boethius’ De institutione arithmetica(DIA) and De institutione musica (DIM), for its fantastical illustrated

I wish to thank Iain Fenlon, Sam Barrett, Teresa Webber, Calvin Bower, Susan Rankin,Thomas Forrest Kelly, Bonnie Blackburn, Elisabeth Giselbrecht, and the anonymousreaders for their helpful comments and suggestions. I am also grateful to the Universityof Cambridge Library for permission to reproduce images from the manuscript, and theGates Cambridge Trust and Richard F. French research fellowship from HarvardUniversity for their generous support. Parts of the article were presented at theconference ‘Revisiting the Legacy of Boethius in the Middle Ages’ at Harvard Universityin Mar. 2014.

The following abbreviations are used:CUL Cambridge, University LibraryDIA Boethius, De institutione arithmeticaDIM Boethius, De institutione musicaTCC Trinity College, Cambridge

Early Music History (2015) Volume 34. © Cambridge University Pressdoi:10.1017/S0261127915000017

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initials, detailed portraits of Greek music theorists and hundreds ofextravagant diagrams that painstakingly illustrate the arithmeticalprinciples and harmonic musical proportions discussed in the text.One of the most ambitious and complex of these diagrams coversalmost the entire fol. 22v, yet curiously this diagram, almost exclusivelyamong those in the manuscript, has been largely overlooked by con-temporary scholars. Folio 22v, discussed below, rewards further studyas an ambitious attempt by its scribe and artist to subject Boethius’ textto a complex schematics, synthesising all information, subject areasand particular vocabulary pertaining to arithmetic and music fromDIA I.1–31 into a single visual presentation. This twelfth-centurydiagram constitutes a map of the domains of the Boethian inquiryand their interrelations, presenting the precepts of DIA I as a unifiedwhole, with lines that establish connections between various areasof the text so as to illustrate how arithmetic and music serve as inter-related branches of the study of mathematics. This diagram servednot only as a conceptual and interpretative device for the scribeendeavouring to explicate Boethius’ complex theoretical material, butalso as a mnemonic guide that could assist the medieval pedagoguewho wished to instruct his students in the mathematics of musicaspeculativa. Finally, this diagram constitutes a fully developed theore-tical exercise in its own right, while also reflecting the role thatBoethian philosophy and mathematics played in twelfth-centurymusical scholarship.

I . H I S TOR I C A L CONT EXT

MS CUL Ii.3.12 serves as a particularly fine example of themeticulouslyillustrated medieval manuscripts produced by Christ Church, Canter-bury during the first third of the twelfth century.1 The manuscript is

1 For general overviews of themanuscript with accompanying citations, see A Catalogue of theManuscripts Preserved in the Library of the University of Cambridge, ed. C. Hardwick,J. E. B. Mayor and H. R. Luard, 6 vols. (Cambridge, 1856–67), iii, pp. 418–19(no. 1776); C. M. Kauffmann, Romanesque Manuscripts 1060–1190, Survey of ManuscriptsIllustrated in the British Isles, iii, ed. J. J. G. Alexander (London, 1975), p. 79 (no. 41);J. Alexander and M. Kauffmann, ‘Manuscripts’, in G. Zarnecki, J. Holt andT. Holland (eds.), English Romanesque Art, 1066–1200. Hayward Gallery, London, 5 April–8July 1989 (London, 1984), pp. 82–133, at p. 101 (no. 30); C. M. Bower, ‘De InstitutioneMusica: A Handlist of Manuscripts’, Scriptorium, 42 (1988), pp. 205–51, at p. 215 (no. 18);M. T. Gibson et al., Codices Boethiani: A Conspectus of Manuscripts of the Works of Boethius, 4vols. (London, 1995–2009), i, p. 42 (no. 6); T. Webber’s entry in P. Binski andS. Panayotova (eds.), The Cambridge Illuminations: Ten Centuries of Book Production in theMedieval West (London, 2005), p. 304 (no. 144); P. Binski and P. Zutshi, Western IllustratedManuscripts: A Catalogue of the Collection in Cambridge University Library (Cambridge, 2011),

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closely related to two other extant copies of DIM that were probablycopied in the same Christ Church scriptorium, albeit by differentscribes:2 Trinity College Cambridge MS R.15.223 and Wellington,Alexander Turnbull Library MS no. 16.4 These two manuscripts sharestrong textual similarities and even identical diagrams with MS CULIi.3.12, yet both situate DIM in a different context, bound with Guido ofArezzo’sMicrologus as a precursor to medieval studies of musical theoryand practice. MS CUL Ii.3.12 joins together the two surviving Boethiantexts on the quadrivial sciences as a single unit so as to position DIM asan outgrowth of the concepts surveyed in DIA,5 thereby presenting anintroduction to ancient arithmetic as a necessary precursor to anexamination of Boethian music theory and practice.6

pp. 22–3 (no. 21). There is some disagreement as to the exact dating of this manuscript,possibly because the text may have been transcribed and rubricated in the first quarter ofthe 12th c., while the detailed Romanesque historiated initials may date somewhat later toaround 1120–50. See M. Gullick and R. Pfaff, ‘The Dublin Pontifical TCD 98 [B.3.6]): StAnselm’s?’, Scriptorium, 55 (2001), pp. 284–94, at p. 91 and n. 17; also C. R. Dodwell, TheCanterbury School of Illumination 1066–1200 (Cambridge, 1954), p. 23. For more on itsprovenance, see also J. C. T. Oates, Cambridge University Library: A History (Cambridge,1996), p. 339.

2 M. M. Manion, V. F. Vines and C. de Hamel,Medieval and Renaissance Manuscripts in NewZealand Collections (London: Thames & Hudson, 1989), p. 123.

3 General overviews accompanied by citations in Bower, ‘De Institutione Musica’,pp. 214–15 [no. 17]; Gibson, Codices Boethiani, pp. 84–5 (no. 54); M. R. James, ‘TheJames Catalogue of Western Manuscripts. Manuscript Details. Shelfmark: R.15.22’, ed.by Librarians at Trinity College Cambridge, accessed per http://sites.trin.cam.ac.uk/james/viewpage.php?index=1175. The images are discussed further in Dodwell, TheCanterbury School, pp. 29, 32–3, 38–9, 64, 78–9, 121.

4 General overviews accompanied by citations in D. M. Taylor, The Oldest Manuscripts inNew Zealand (Wellington, 1955), pp. 63–71; Bower, ‘De Institutione Musica’, p. 242(no. 130); M. M. Manion et al., Medieval and Renaissance Manuscripts in New Zealand,pp. 122–4 (no. 140).

5 It is worth noting that some scholars have questioned whether Boethius had everconceived of his quadrivial texts as part of a single project or unified corpus, despite whatthe binding together in MS CUL Ii.3.12 might suggest (or, for that matter, what theproem of DIAmight indicate). See U. Pizzani, ‘The Influence of the De Institutione Musicaof Boethius up to Gerbert D’Aurillac: A Tentative Contribution’, in M. Masi (ed.),Boethius and the Liberal Arts (Berne, 1981), pp. 97–156, at p. 106: ‘notwithstanding thescheme articulated at the beginning of DIA the four treatises pertaining to thedisciplines of the quadrivium do not seem to have been conceived by Boethius as partof a single corpus, or at least not as far as form is concerned’. For more on thefragments on geometry and astronomy, see D. Pingree, ‘Boethius’ Geometry andAstronomy’, in M. T. Gibson (ed.), Boethius: His Life, Thought, and Influence (Oxford,1981), pp. 155–61.

6 T. Webber in Binski and Panayotova (eds.), The Cambridge Illuminations, p. 304 observesthat this juxtapositionmay not have been original, noting that the text of DIM begins on aseparate quire within a different ruling pattern; but even if this is correct, the two musthave been brought together within a single binding early in the twelfth century, on the

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At the time this manuscript was composed, Christ Church wasquickly establishing itself as home to one of the most productivescriptoria and vibrant literary communities in Europe. A fire in 1067had destroyed a significant portion of the pre-Norman library, pro-viding a powerful incentive to collect new manuscripts; the newlyappointed Norman archbishop Lanfranc (1070–89) spearheaded thecollection and reproduction of both sacred and secular works, amas-sing one of the most important collections of classical and medievaltexts on a wide variety of topics in the liberal arts.7Artists and scribes atChrist Church exchanged exemplars with scriptoria across NormanEngland, including nearby St Augustine’s of Canterbury, which haddirect links to mainland European sources through Mont-Saint-Michel.8 The creation of this new library took place during a timewhen Christ Church was opening up to new foreign influences fromFrance, the Low Countries and North Germany, which had a highlytransformative effect on its culture and pedagogical practices.9UnderLanfranc’s guidance, the library was filled with new Latin texts; stu-dents were well educated not only in Scripture and Christian doctrine,but also in Priscian’s grammar and the Latin classics.10Pupils were also

evidence that the frontispiece to DIM is located on the final page of the last quire of DIA.See also Binski, Western Illuminated Manuscripts, p. 23.

7 See M. R. James, The Ancient Libraries of Canterbury and Dover: The Catalogues of the Librariesof Christ Church Priory and St Augustine’s Abbey and of St. Martin’s Priory at Dover (Cambridge,1903), pp. xxix–xxxv; M. T. Gibson, ‘Normans and Angevins, 1070–1220’; and N. Ramsay,‘The Cathedral Archives and Library’, in P. Collinson, N. Ramsay andM. Sparks (eds.), AHistory of Canterbury Cathedral (Oxford, 1995), pp. 38–68 and 341–407, at pp. 48–55 and346–53. For a general overview of scribal practices and manuscript decoration in 12th-c.Canterbury, see R. Gameson, ‘English Manuscript Art in the Late Eleventh Century:Canterbury and its Context’ and T. Webber, ‘Script and Manuscript Production at ChristChurch, Canterbury, after the Norman Conquest’, in R. Eales and R. Sharpe (eds.),Canterbury and the Norman Conquest: Churches, Saints and Scholars, 1066–1109 (London,1995), pp. 95–144 and 145–58.

8 Gibson, ‘Normans and Angevins’, p. 52. Similarities between the historiated initials of MSCUL Ii.3.12, MS TCC R.15.22 and Durham Cathedral B.II.22 testify to the externalinfluence and reach of the Christ Church scriptorium’s ‘house style’: see A. Lawrence,‘The Influence of Canterbury on the Collection and Production of Manuscripts atDurham in the Anglo-Norman Period’, in A. Borg and A.Martindale (eds.),The VanishingPast: Studies of Medieval Art, Liturgy and Metrology Presented to Christopher Hohler (B. A. R.International Series III; Oxford, 1981), pp. 95–104, at pp. 98–100; also A. Lawrence-Mathers, Manuscripts in Northumbria in the Eleventh and Twelfth Centuries (Woodbridge,Suffolk, 2003), p. 54.

9 Cf. Gibson, ‘Normans and Angevins’, pp. 38–65; T. A. Heslop, ‘“Dunstanus Archiepiscopis”and Painting in Kent around 1120’, Burlington Magazine, 126:973 (Apr. 1984),pp. 195–204, at p. 204.

10 Gibson, ‘Normans and Angevins’, p. 45; Ramsay, ‘The Cathedral Archives and Library’,pp. 346–7.

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educated in music, with a combined instruction in both the music ofthe liturgy and the advanced mathematics of musica speculativa.11

The final five pages of the manuscript (fols. 135r–137r) contain abooklist of the secular and Classical manuscripts held in the ChristChurch library during the second half of the twelfth century (seeFigure 1).12 This booklist, composed later than the rest of the text andwritten in a different handwriting, contains 223 manuscripts in elevendifferent fields. Each manuscript is listed prefaced with a sign corre-sponding to a mark placed in the volume itself: the entry entitled‘Musica boecii’ is prefaced by the sign ‘EE’, which corresponds to a signon the first leaf of MS TCC R.15.22. 13 This list, which may have beenused to keep records of book borrowings,14 provides us with a glimpseinto the intellectual interests of Canterbury and an impression of thetypes of sources that may have been accessible to the literary community.The author has catalogued each duplicate copy of the text separately,indicating that there may have been a need to collect multiple copies forpedagogical purposes.15He also lists works in the disciplines of grammar,rhetoric, poetry, logic, astronomy, geometry, theology, medicine andlaw, as well as music theory and arithmetic. There are seven booksdealing with music and eight dealing with arithmetic; altogether, thereare eight volumes of Boethius, seven of which are copies of just one text,De consolatione philosophiae.16 Indeed, Boethius is one of the most highlyrepresented authors in terms of the number of copies of his works.

The statistics of the booklist are also reflective of contemporarygeneral surveys of extant Boethian manuscripts from medieval Europe.

11 N. C. Carpenter,Music in the Medieval and Renaissance Universities (New York, 1972), p. 20;R. Bowers, ‘The Liturgy of the Cathedral and its Music, c. 1075–1642’, in Collinson,Ramsay and Sparks (eds.), A History of the Canterbury Cathedral, pp. 341–407, atpp. 408–18.

12 See James, The Ancient Libraries, pp. xxxi–xxxiv and 3–12; Ramsay, ‘The CathedralArchives and Library’, pp. 350–1. These scholars argue that this booklist is incomplete orfragmentary; Ramsay suggests that the full collection of volumes would have numberedaround six to seven hundred. I am grateful to Teresa Webber for pointing out, however,that we should not assume that this list is partial without clearer evidence.

13 See James, The Ancient Libraries, pp. xxxii–xxxiii. None of these signs corresponds to anymarks on CUL Ii.3.12.

14 F. Wormald, ‘The Monastic Library’, in F. Wormald and C. E. Wright (eds.), The EnglishLibrary before 1700: Studies in its History (London, 1958), pp. 15–31, at p. 23.

15 James, The Ancient Libraries, p. xxxiv; Ramsay, ‘The Cathedral Archives and Library’,p. 351.

16 The only fields that outnumber music and arithmetic are grammar (26), dialectics (22),rhetoric (9) and astronomy (9). The authors with the highest number of copies of theirbooks are Martianus Capella (15) and Macrobius (11), while Horace, Virgil, Sallust andBoethius all come next with eight volumes each. Cf. James, The Ancient Libraries,pp. xxxiii–xxxiv.

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Figure 1 The booklist on fol. 135r of MS CUL Ii.3.12

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As many as a third of Boethian manuscripts produced between theninth and fifteenth centuries were copies of De consolatione philosophiae ;only about one in five contained versions of DIM and DIA.17 Very fewcopies of the quadrivial texts seem to have been in circulation in theninth to eleventh centuries,18 but the twelfth century shows a dramaticupsurge in production: as many as sixteen copies of DIA and fourteencopies of DIM, compared with only three and two copies respectively inthe century before. Indeed, over half of all extant medieval copies ofDIA and DIM come from the twelfth century alone, clearly suggestingthat this century was witness to a flourishing of Boethian musical andarithmetical scholarship. MS CUL Ii.3.12 was one of the earliest copiesproduced during this upsurge, demonstrating, perhaps, that Canter-bury was at the very vanguard of twelfth-century Boethian musical andarithmetical scholarship—making the rigour of the diagram on fol. 22v

all the more necessary as an introduction to new readers, and all themore striking for its complexity and insight.19

I I . I M AGE S AND D I AG R AM S

Although some scholars have disparaged the handwriting of MS CULIi.3.12 as ‘rather poor’20 or ‘mediocre’21 for exhibiting little concernfor scribal consistency, the exquisite illustrations, charts and diagramsfeatured in the manuscript inarguably suggest that the illustratorsobserved the very highest standards of decoration.22Folio 65v features anexquisite Romanesque historiated initial, depicting animal and humanforms in a bright palette of blue, green, red and violet (see Figure 2).

17 Gibson, Codices Boethiani, pp. 21–33.18 Cf. R. Gameson, The Manuscripts of Early Norman England (c. 1066–1130) (Oxford, 1999),

pp. 1–52.19 For more on the reception of DIA and DIM in Europe in general, see M. T. Gibson,

‘Boethius in the Carolingian Schools’, Transactions of the Royal Historical Society, V:32(1982), pp. 43–56; P. Kibre, ‘The Boethian De Institutione Arithmetica and the Quadriviumin the Thirteenth Century University Milieu at Paris’, M. Masi, ‘The Influence ofBoethius’ De Arithmetica on Late Medieval Mathematics’, and C. M. Bower, ‘The Role ofBoethius’ De Institutione Musica in the Speculative Tradition of Western MusicalThought’, in Masi (ed.), Boethius and the Liberal Arts, pp. 67–80, 81–96 and 157–74;A. White, ‘Boethius in the Medieval Quadrivium’, in Gibson (ed.), Boethius: His Life,Thought, and Influence, pp. 162–205; A. E. Moyer, ‘The Quadrivium and the Decline ofBoethian Influence’ and M. T. Rimple, ‘The Enduring Legacy of Boethian Harmony’, inN. H. Kaylor, Jr. and P. E. Phillips (eds.), A Companion to Boethius in the Middle Ages(Leiden, 2012), pp. 447–517.

20 Webber, ‘Script and Manuscript Production’, p. 153.21 Gibson, Codices Boethiani, p. 42.22 Dodwell, The Canterbury School of Illumination, pp. 23, 35, 37, 39, 64, 66, 74 and 119

provides an art-historical discussion of these images.

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Figure 2 The historiated initial on fol. 66v of MS CUL Ii.3.12

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Sandwiched between the texts of DIA and DIM on fol. 61v is aremarkable full-page illustration of the four most important musictheorists relevant to the text: Boethius, Pythagoras, Plato and Nico-machus (see Figure 3).23 The latter three theorists are ancient Greek,suggesting a strong continuity between ancient music theory, Boethiusand the medieval era.24An image on fol. 125v features what may be theearliest depiction in a manuscript of an English keyboard: a seriesof fifteen pipes divided into two sets, as well as an organ keyboardlabelled according to the a–p system of notation;25 if so, this imagerepresents an attempt by the illustrator to connect Boethian theory tocontemporary instrumental practice. The amount of coordinationthat must have been required between the scribe and the illustrator ofthe historiated initials, author portraits and organ image, as well as thecost of the inks used in the images, suggest that the creators of thismanuscript valued it highly.26

These illustrations provide a valuable set of clues for understandinghow Boethius was received, understood and interpreted in twelfth-century Canterbury. The text itself features only a few minor additions

23 Kauffmann, Romanesque Manuscripts, p. 79 argues that fol. 61v ‘may be described as thefirst fully Romanesque whole-page illustration produced at Canterbury’; Heslop,‘“Dunstanus Archiepiscopis”’, p. 204 also notes strong similarities between the artist’smanner of representing drapery and facial style and the work of Roger ofHelmharshausen, testifying to the influences of the Low Countries and North Germanyon 12th-c. Canterbury. This is not the only author portrait contained in the manuscript:on fol. 1r is a black-ink illustration of Boethius presenting his text to his adopted fatherand patron, the consul Quintus Aurelius Memmius Symmachus, dressed in rich robesand wearing a pointed hat topped by a cross. See also Gibson, Codices Boethiani, p. 42,arguing that this image seems to share a ‘common ancestry’ with the author illustrationfrom a 9th-c. presentation copy dedicated to Charles the Bald, MS Bamberg Staatsbibl.,Class. 9 HJ.IV.19. Gibson, ‘Boethius in the Carolingian Schools’, p. 50 identifies theBamberg Staatsbibliothekmanuscript as significant in demonstrating ‘the final assurancethat by the mid-ninth century the De Arithmetica had court recognition’.

24 Cf. E. C. Teviotdale, ‘Music and Pictures in the Middle Ages’, in T. Knighton andD. Fallows (eds.), Companion to Medieval and Renaissance Music (Berkeley, 1992),pp. 179–88, at pp. 186–8.

25 C. Page, ‘The Earliest English Keyboard’, Early Music, 7 (1979), pp. 308–14, with areproduction of the image on p. 309, ill. 1; a more sceptical approach to thisinterpretation is found in P. Williams, The Organ in Western Culture, 750–1250(Cambridge, 1993), pp. 183–4. A practically identical version of this same image appearsin MS TCC R.15.22 on fol. 90r. See also A. C. Browne, ‘The a–p System of LetterNotation’,Musica Disciplina, 35 (1981), pp. 5–54 for more on the history of the origins ofthe a–p system of notation and its transmission through Boethius, with references to itspossible use in 12th-c. Canterbury.

26 See Gibson, ‘Boethius in the Carolingian Schools’, p. 52. It is also worth noting that thehistoriated initials are always outlined in great detail, but towards the end of the text theyare not filled in with colour inks; perhaps too much expense had already been paidtowards the diagrams?

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to the text of Boethius, in the forms of glosses, maniculae andmarginalia, and therefore the clearest demonstration of interpretativeactivity can be found in the elaborate graphic presentation. Mary

Figure 3 The portrait of Boethius, Pythagoras, Plato and Nicomachus on fol. 61v

of MS CUL Ii.3.12

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Carruthers proposed that images of this sort in medieval manuscriptsconstitute ‘pictorial writing’: ‘the visual presentation of a text was con-sidered, at least by the learned, to be part of its meaning, not limited tothe illustration of its themes or subjects but necessary to its properreading, its ability-to-be significant and memorable’.27 Diagrams wereespecially important as interpretative, pedagogical and mnemonictools: ‘they can serve as fixes formemory storage, and as cues to start therecollective process. The one function is pedagogical, in which thediagram serves as an informational schematic, the other is meditationaland compositional.’28 These diagrams, however elaborate, shouldexhibit clarity, brevity and be able to be taken in by a single gaze; thusthe graphic arrangements of words and illustrations on the page willhelp make complex concepts more readily understandable and thusmore memorable.29 Such diagrams often aided medieval musicians inthe memorisation of modal and rhythmic theory useful for musicalpractice and composition.30 The focus in the diagram on fol. 22v, how-ever, is not on musical practice but on the mathematics of speculativemusical theory, demonstrating how arithmetical and music theoreticalconcepts in Boethius’ writings unfold in parallel but are also integrated,while displaying the numerical relationships and proportions thatunderlie musical intervals, scales andmodes and making them easier toconceptualise and grasp than they are in the text itself.

I I I . T H E A R B O R OF FO L . 2 2 V

Boethius frequently refers to the presence of explanatory diagrams,indicating that the text was elaborated with images from the verybeginning; part of DIA I.11 and the entire DIA I.12 are dedicated toa description of a diagram outlining the relationships between oddtimes even numbers, which appears in MS CUL Ii.3.12 on fol. 10r

27 M. Carruthers, The Book of Memory, 2nd edn (Cambridge, 2008), pp. 274 and 278.28 Ibid., p. 332. Also cited in E. A. Mellon, ‘Inscribing Sound: Medieval Remakings

of Boethius’s De Institutione Musica’ (Ph.D. diss., University of Pennsylvania, 2011),p. 168.

29 M. Carruthers and J. M. Ziolkowski, ‘General Introduction’, in Carruthers and Ziolkowski(eds.), The Medieval Craft of Memory: An Anthology of Texts and Pictures (Philadelphia, 2002),pp. 1–31; Mellon, ‘Inscribing Sound’, esp. pp. 159–252; P. Jones, ‘The MedievalEncyclopaedia: Science and Practice’, in Binski and Panayotova (eds.), The CambridgeIlluminations, p. 298; M. Bernhard and C. M. Bower, Glossa maior in institutionem musicamBoethii, 4 vols. (Munich, 1993–2011), i, p. xli. See also Manion, Medieval and RenaissanceManuscripts in New Zealand, p. 124, who observes that the decoration on a diagramdepicting mathematical proportions applied to music with an elephant, ‘symbol ofmemory, reinforces the mnemonic quality of the diagram’.

30 A. M. B. Berger, Medieval Music and the Art of Memory (Berkeley, 2005).

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(see Figure 4). Diagrams of this complexity are common in medievalmanuscripts from a variety of different centuries, locations and scribalpractices; often they were transferred between codices just as carefullyas the principal text itself.31Nevertheless, the arbor of fol. 22v is uniquein its ambition, its comprehensiveness and the extraordinary level ofdetail it applies in its summary of the first book of DIA; nothing com-parable appears even in the closely related manuscripts MS TCCR.15.22 or MS Turnbull no. 16.32 (See Figure 5). It is not a marginalgloss by a later reader. Situated just before the opening of the finalchapter of Book I, DIA I.32, it was clearly composed by the scribe andillustrator who completed the rest of the manuscript, written in thesame hand and with the same colour scheme as the remaining textand images. The large amount of space reserved on the page indicatesthat the scribe had realised that readers might be confused by thechallenging text up to this point, and determined that an elaboratediagram was necessary in order to orient the reader by recapitulatingand clarifying the previous material in the book.33

While Boethius does not always progress in a clear order throughhis argument, the illustrator has attempted to project a unifying logicby charting the logical flow between subdisciplines. The transcriptionof fol. 22v provided in Appendix I aims to demonstrate the conceptualframework that the creators of the Cambridge manuscript found inDIA, preserving the form and dimensions of the original illustrationwhile reproducing them in translation. As this transcription demon-strates, fol. 22v represents the material of the first book of DIA in theform of a diagrammatic tree, or arbor, in which larger, more generalconcepts are divided into subsections and aremade eventually to focuson specific examples.34 By scanning the chart from top to bottom, thereader is able to conceptualise how each subject flows logically into thenext. Appendix II shows how the illustrator of the chart has reorga-nised different sections of the text in order to present the conceptual

31 Bernhard and Bower, Glossa maior, i, p. xlvii.32 Bernhard and Bower, ibid., ii, pp. 7–8 identify similar diagrams in manuscripts of DIM,

but these do not approach the same level of specificity. For a point of reference ondiagrams on DIA, see also M. Bernhard, ‘Glossen zur Arithmetik des Boethius’, in M.Bernhard and S. Krämer (eds.), Scire litteras: Forschungen zum mittelalterlichen Geistesleben(Munich: Bayerische Akademie der Wissenschaften, Abhandlungen der philosophisch-historischen Klasse, Neue Folge 99 (1988)), pp. 23–34.

33 Scribes often ‘doubled as page designers’, plotting out the amount of space left for theilluminations and diagrams that would be filled in later by the artists: S. Panayotova andT. Webber, ‘Making an Illuminated Manuscript’, in Binski and Panayotova (eds.), TheCambridge Illuminations, pp. 23–36, at pp. 31–2.

34 See Bernhard and Bower, Glossa maior, i, p. xlvi; Mellon, ‘Inscribing Sound’, p. 146.

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Figure 4 The diagram described by Boethius in DIA I.11–12, depicting odd timeseven numbers, on fol. 10r of MS CUL Ii.3.12

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Figure 5 The arbor summarising the various branches of essentia as described inthe first book of DIA I, on fol. 22v of MS CUL Ii.3.12

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organisation in a more clearly logical progression than the Boethiantext. Examples and specifics in many chapters that appeared to theillustrator to digress from the subjects of interest have been excludedfrom the chart entirely. Indeed, the analysis invested by the scribe inthe complex editing and reorganisation of this chart amounts to athorough reinterpretation of the entire first book.35

Appendix II also reveals how the diagram divides into three regions.Region I, which includes lines 1–5 of the chart, represents all thematerial of DIA I.1 introduces two types of essences sensible by per-ception: continuous essence, which is composed of single elements(e.g. a ‘tree or a stone’), and disjoined essence, which is composed of acollection of separate and distinct elements (e.g. ‘a populace, achorus’).36 The numerical qualities of continuous essence are mea-sured by magnitudes, which may be movable or immovable. The studyof movable magnitude is subsumed by the field of astronomy, whichquantifies and studies themovement of stars in harmonic intervals; thestudy of immovable magnitude falls within the domain of geometry,which studies the dimensions of fixed shapes such as circles orsquares.37 The numerical qualities of disjoined essence, on the otherhand, are determined by multitudes, which may exist independentlyon their own (‘per se’), or in proportion to other multitudes (‘relata’).The study of independent multitudes constitutes the field ofarithmetic, while the study of proportional multitudes is part of thefield of music theory. ‘Numbers’, Boethius explains in DIA I.2, are the‘principal exemplar in the mind of the creator’ and are thus essentialfor understanding the natural world.38 These four quadrivial fields

35 Fol. 22v is not the only example of scribal reorganisation at work in MS CUL Ii.3.12: seeC. M. Bower, ‘Introduction’, in Fundamentals of Music, trans. C. M. Bower and ed. C. V.Palisca (New Haven, 1989), pp. xli–xliv, at p. xxiii; also Mellon, ‘Inscribing Sound’, pp.298–99. These authors note that the chapter arrangement of DIM in MS CUL Ii.3.12 hasbeen labelled differently than usual, although the text stays the same: the proemium ismarked as prologus, the table of contents as I.1, with the beginning of the treatise at I.2.Mellon observes that this seemingly subtle reorganisation has the significant effect ofreshaping DIM so that it more closely resembles medieval music theory treatises such asGuido of Arezzo’s Micrologus.

36 Boethius, Boethian Number Theory: A Translation of the De Institutione Arithmetica, trans. withintrod. and notes by M. Masi (Amsterdam, 1983), p. 72: ‘There are two kinds of essence.One is continuous, joined together in its parts and not distributed in separate parts, as atree, a stone, and all the bodies of this world which are properly called magnitudes. Theother essence is of itself disjoined and determined by its parts as though reduced to asingle collective union, such as a flock, a populace, a chorus, a heap of things, thingswhose parts are terminated by their own extremities and are discrete from the extremityof some other. The proper name for these is a multitude.’

37 Ibid., p. 75.38 Ibid., p. 76.

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which complete Region I ‘allow the mind to ascend from the study ofthings perceived by the senses to the more certain matters of theintellect . . . Through number, the scholar moves from the unity of thedivine mind to the multiplicity – and quantitative order – of createdthings.’39

Region II of fol. 22v, which includes the left-hand side of lines 6–10,sets forth the basic principles of arithmetic. In DIA I.3–6, Boethiusintroduces the distinction between even and odd numbers. Directlyfollowing a brief discussion of the nature of unity in DIA I.7 – a passageexcluded from the arbor – Boethius explains in DIA I.8 that evennumbers are subdivided into three different categories: pariter par(even times even), elaborated in greater detail in DIA I.9; pariter inpar(even times odd), elaborated in DIA I.10; and inpariter par (odd timeseven), elaborated in DIA I.11–12. Odd numbers are also divided intothree different subdivisions, introduced in DIA I.13: prime andincomposite, elaborated in DIA I.14 and I.17; secondary and compo-site, elaborated in DIA I.15 and I.17; and those numbers in the middleof the two categories, elaborated in DIA I.16–18.40He returns to evennumbers inDIA I.19 in order to explain a second way in which they canbe subdivided into yet another set of three categories: perfect,superabundant and deficient.41 Boethius relates these three types ofodd numbers to ‘virtues and vices’ in DIA I.20, noting that perfect

39 A. E. Moyer, ‘The Quadrivium and the Decline of Boethian Influence’, in Kaylor andPhillips (eds.), A Companion to Boethius in the Middle Ages, pp. 479–517, at pp. 481–2.

40 Boethius, Boethian Number Theory, pp. 89–92: Prime and incomposite numbers areindivisible: they have ‘no other factor but that one which is a denominator for the totalquantity of that number so its fraction is nothing other than unity’, such as 3, 5, 7, 11, 13,et al. Secondary and composite numbers are odd numbers that are not prime: they are‘formed by the same properties as an odd number . . . [It is] composed of other numbersand has parts named both in relation to itself and in relation to other terms’, such as 9, 15,21, 25, 27, et al. The third category is of odd numbers that are not prime, yet when theyare compared proportionally to another such number they have no commondenominators, such as 9 and 25. They are thus in the ‘middle’, for they are ‘compositeand secondary . . . but when compared to each other they become primary andincomposite because no othermeasure will fit both, except unity, which is a denominatorfor both’.

41 See Boethius, Boethian Number Theory, pp. 96–7. Superabundant numbers are those inwhich ‘the sum of their parts factored out of the total body are found to be larger thanthat sum’, i.e., those whose factors added together exceed the number itself, e.g. 12 and24. Deficient numbers are those whose parts, ‘when put together in the same way, areexceeded by the multitude of the whole term’, i.e., those whose factors added togetherare less than the number itself, e.g., 8 and 14. Perfect numbers are those which areneither superabundant nor deficient, but ‘hold the middle place between the extremeslike one who seeks virtue’ because ‘the sum of their parts is not more than the total nordoes it suffer from a lack in comparison with the total’, i.e., numbers whose factors addedtogether equal the number, e.g., 6 and 28.

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numbers occur only ‘rarely’ yet superabundant or deficient numbersappear to be ‘many and infinite’, and ‘arranged randomly andillogically’.42

Region III describes the types of proportions that comprise musicaltheory. Boethius’ proportions are divided into two types, enumeratedin DIA I.21–2: equal (a:a, e.g. 10:10) and unequal (a:b). Equal pro-portions are indivisible by nature, for ‘just as a friend is a friend to afriend, or a neighbor is a neighbor to a neighbor, so is equal said to beequal to equal’.43He explains in DIA I.22 that unequal proportions, onthe other hand, are divided into major and minor inequalities, both ofwhich are subdivided into five different types. The five types of majorinequalities are: (1) multiplex (a× b:a), elaborated in DIA I.23 andagain in I.26–7; (2) superparticular (a + 1:a), elaborated in DIA I.24–5;(3) superpartient (a + b:a), elaborated in DIA I.28; (4) multiplexsuperparticular ((b× a + 1/c):a) elaborated in DIA I.29–30; and(5) multiplex superpartient (b× a + (a-c):a), elaborated in DIA I.31.44

The five minor inequalities are the submultiplex, subsuperparticular,and so forth. According to Boethius, through the edifying study of suchcategories of arithmetic, one can learn better to distinguish goodfrom evil.45

Region III is of vital importance, as all Pythagorean proportions thatgovern the basic building blocks ofmusic – the octave at 1:2, the fifth at2:3, the fourth at 3:4, and so on – can be identified as one of these typesof major or minor inequalities. Boethius writes in Book I:

The logical force of numbers is also prior to music, and this can especially bedemonstrated because not only are numbers prior by their nature, since theyconsist of themselves and are thus prior to those things which must be referred toanother in order to be, but also musical modulation itself is denoted by the namesof numbers . . . The names diatessaron, diapente and diapason are derived fromthe names of antecedent numerical terms. The proportion of their sounds isfound only in these particular relationships and not in other number relationships. . . [T]he rest of this work will demonstrate [this idea] without any doubt.46

Music, for Boethius, emerges from this intricate mathematicalworld. The quadrivial sciences are interdependent; arithmetic is priorto the remaining three sciences of music, astronomy and geometryand thus the logical place to begin the quadrivial studies: ‘[S]ince,

42 Ibid., p. 98.43 Ibid., p. 100.44 Cf. Moyer, ‘The Quadrivium’, p. 482.45 Boethius, Boethian Number Theory, p. 114; Moyer, ‘The Quadrivium’, p. 482.46 Boethius, Boethian Number Theory, pp. 74–5.

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as it is obvious, the force of arithmetic is prior, we may take up thebeginning of our exposition’.47

The artist has produced a cleaner representation of the logicalunfolding of music in Region III than of mathematics in Region II.Region II is the only section of the chart in which the lines cross overeach other. The line extended from per se to arithmetica crosses over theline connecting immobilis and geometria; the lines connecting impar toits subdivisions interfere with the lines connecting numerus par topariter inpar and impariter par. The illustrator of fol. 22v appears to haveproduced Region II last, cramming its material into the small spaceleft after illustrating Regions I and III. Even though arithmetic is thetitular subject of DIA – and, according to Boethius, logically prior tomusic – the artist who composed this schematic map seems to havedevoted his attention first to illustrating the synopsis of music inRegion III.

For this artist, it appears, music is of critical importance, as aperceptible demonstration of the qualities of abstract numericalrelationships which appear in all fields of the quadrivium.Music theory could also provide a sensory route for students tounderstand mathematics and the cosmic harmonies: fol. 76v depictshow the proportions classified in fol. 22v can be applied to celestialharmonies, illustrating the orbits of the moon, sun and planets andtheir corresponding musical intervals (see Figure 6).48 Folio 22v thusreveals in detail the necessary frameworks for understandingmore abstract applications of music in geometry and astronomy, aswell as helping to make perceptible and comprehensible to studentsthe basic components of mathematics and the harmonies ofthe spheres.

By presenting Book I of DIA as an arbor, the illustrator employsvisual aids to explicate the circuitous and confusing subject matter inthe same manner as a medieval teacher, who would strive to presentconcepts by means of a system of organisation that assigns each con-cept to a logical place, making the overall study coherent andthus easier to commit to memory. And as any teacher knows fromplanning a syllabus, the work of distilling complex information intoa comprehensible set of schematics constitutes a theoretical act:for as Leslie Blasius explains, ‘the organization of diverse topics isno longer a preliminary to theorizing but rather a mode of theorizing

47 Ibid., p. 75.48 A practically identical version of this same diagram appears on fol. 24v of the closely

related MS TCC R.15.22.

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Figure 6 The cosmic harmonies on fol. 76v of MS CUL Ii.3.12

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in itself’.49 Boethius sums it up succinctly in his own description ofDe Arithmetica, which, he explains, is itself a distillation of Nicomachus’Arithmetike eisagoge:

I do not restrict myself slavishly to traditions of others, but with a well-formed ruleof translation, having wandered a bit freely, I set upon a different path, not thesame footsteps. Those things which were discussed in a rather diffuse manner byNicomachus concerning numbers, I have put together with moderate brevity;those things which demanded a greater care of understanding, but are gonethrough quickly, I clarified with a small additional explanation and I have evenused formulae and diagrams for greater clarity of matters. The careful reader willeasily recognize that this involved for us many nights of labor.50

Simply put, the medieval scholars have treated Boethius’ text just asBoethius treated Nicomachus, creating a diagram that shows twelfth-century musical theory at its most complex and ambitious. But aboveall, the diagram is visually stunning, seamlessly integrating word andimage to delight the eye while leading the intellect towards a betterunderstanding of the ways in which music and arithmetic are inter-related as branches of the mathematical quadrivium.

Harvard University

49 L. Blasius, ‘Mapping the Terrain’, in T. Christensen (ed.), The Cambridge History of WesternMusic Theory (Cambridge, 2002), p. 30.

50 Boethius, Boethian Number Theory, p. 67.

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A P P END I X I

Transcription of MS CUL Ii.3.12, fol. 22v

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A P P END I X I I

Organisation of Topics from the First Book of DIA, in fol. 22v

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