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Published online by Cambridge University Press: 23 September 2015
This article discusses a full-page schematic diagram contained in a twelfth-century manuscript of Boethius’ De institutione arithmetica and De institutione musica from Christ Church Cathedral, Canterbury (Cambridge University Library MS Ii.3.12), which has not yet been the subject of any significant musicological study despite its remarkable scope and comprehensiveness. This diagrammatic tree, or arbor, maps the precepts of the first book of De institutione arithmetica into a unified whole, depicting the ways music and arithmetic are interrelated as sub-branches of the quadrivium. I suggest that this schematic diagram served not only as a conceptual and interpretative device for the scribe working through Boethius’ complex theoretical material, but also as a mnemonic guide to assist the medieval pedagogue wishing to instruct students in the mathematics of musica speculativa. 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.
I wish to thank Iain Fenlon, Sam Barrett, Teresa Webber, Calvin Bower, Susan Rankin, Thomas Forrest Kelly, Bonnie Blackburn, Elisabeth Giselbrecht, and the anonymous readers for their helpful comments and suggestions. I am also grateful to the University of Cambridge Library for permission to reproduce images from the manuscript, and the Gates Cambridge Trust and Richard F. French research fellowship from Harvard University for their generous support. Parts of the article were presented at the conference ‘Revisiting the Legacy of Boethius in the Middle Ages’ at Harvard University in Mar. 2014.
The following abbreviations are used:
CUL Cambridge, University Library
DIA Boethius, De institutione arithmetica
DIM Boethius, De institutione musica
TCC Trinity College, Cambridge
1 For general overviews of the manuscript with accompanying citations, see A Catalogue of the Manuscripts 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); Kauffmann, C. M., Romanesque Manuscripts 1060–1190, Survey of Manuscripts Illustrated in the British Isles, iii, ed. J. J. G. Alexander (London, 1975), p. 79Google Scholar (no. 41); Alexander, J. and Kauffmann, M., ‘Manuscripts’, in G. Zarnecki, J. Holt and T. Holland (eds.), English Romanesque Art, 1066–1200. Hayward Gallery, London, 5 April–8 July 1989 (London, 1984), pp. 82–133Google Scholar, at p. 101 (no. 30); Bower, C. M., ‘De Institutione Musica: A Handlist of Manuscripts’, Scriptorium, 42 (1988), pp. 205–251CrossRefGoogle Scholar, at p. 215 (no. 18); Gibson, M. T. et al., Codices Boethiani: A Conspectus of Manuscripts of the Works of Boethius, 4 vols. (London, 1995–2009), i, p. 42Google Scholar (no. 6); Webber’s, T. entry in P. Binski and S. Panayotova (eds.), The Cambridge Illuminations: Ten Centuries of Book Production in the Medieval West (London, 2005), p. 304Google Scholar (no. 144); Binski, P. and Zutshi, P., Western Illustrated Manuscripts: A Catalogue of the Collection in Cambridge University Library (Cambridge, 2011), pp. 22–23CrossRefGoogle Scholar (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 of the 12th c., while the detailed Romanesque historiated initials may date somewhat later to around 1120–50. See Gullick, M. and Pfaff, R., ‘The Dublin Pontifical TCD 98 [B.3.6]): St Anselm’s?’, Scriptorium, 55 (2001), pp. 284–294Google Scholar, at p. 91 and n. 17; also Dodwell, C. R., The Canterbury School of Illumination 1066–1200 (Cambridge, 1954), p. 23Google Scholar. For more on its provenance, see also Oates, J. C. T., Cambridge University Library: A History (Cambridge, 1996), p. 339Google Scholar.
2 Manion, M. M., Vines, V. F. and de Hamel, C., Medieval and Renaissance Manuscripts in New Zealand Collections (London: Thames & Hudson, 1989), p. 123Google Scholar.
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, ‘The James 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, The Canterbury School, pp. 29, 32–3, 38–9, 64, 78–9, 121.
4 General overviews accompanied by citations in Taylor, D. M., The Oldest Manuscripts in New Zealand (Wellington, 1955), pp. 63–71Google Scholar; Bower, , ‘De Institutione Musica’, p. 242Google Scholar (no. 130); Manion, M. M. et al., Medieval and Renaissance Manuscripts in New Zealand, pp. 122–124Google Scholar (no. 140).
5 It is worth noting that some scholars have questioned whether Boethius had ever conceived of his quadrivial texts as part of a single project or unified corpus, despite what the binding together in MS CUL Ii.3.12 might suggest (or, for that matter, what the proem of DIA might indicate). See Pizzani, U., ‘The Influence of the De Institutione Musica of Boethius up to Gerbert D’Aurillac: A Tentative Contribution’, in M. Masi (ed.), Boethius and the Liberal Arts (Berne, 1981), pp. 97–156Google Scholar, at p. 106: ‘notwithstanding the scheme articulated at the beginning of DIA the four treatises pertaining to the disciplines of the quadrivium do not seem to have been conceived by Boethius as part of a single corpus, or at least not as far as form is concerned’. For more on the fragments on geometry and astronomy, see Pingree, D., ‘Boethius’ Geometry and Astronomy’, in M. T. Gibson (ed.), Boethius: His Life, Thought, and Influence (Oxford, 1981), pp. 155–161Google Scholar.
6 T. Webber in Binski and Panayotova (eds.), The Cambridge Illuminations, p. 304 observes that this juxtaposition may not have been original, noting that the text of DIM begins on a separate quire within a different ruling pattern; but even if this is correct, the two must have been brought together within a single binding early in the twelfth century, on the 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 James, M. R., The Ancient Libraries of Canterbury and Dover: The Catalogues of the Libraries of Christ Church Priory and St Augustine’s Abbey and of St. Martin’s Priory at Dover (Cambridge, 1903), pp. xxix–xxxvGoogle Scholar; Gibson, M. T., ‘Normans and Angevins, 1070–1220’; and N. Ramsay, ‘The Cathedral Archives and Library’, in P. Collinson, N. Ramsay and M. Sparks (eds.), A History of Canterbury Cathedral (Oxford, 1995), pp. 38–68Google Scholar and 341–407, at pp. 48–55 and 346–53. 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 Webber, T., ‘Script and Manuscript Production at Christ Church, 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. 95–144Google Scholar and 145–58.
8 Gibson, ‘Normans and Angevins’, p. 52Google Scholar. Similarities between the historiated initials of MS CUL Ii.3.12, MS TCC R.15.22 and Durham Cathedral B.II.22 testify to the external influence and reach of the Christ Church scriptorium’s ‘house style’: see Lawrence, A., ‘The Influence of Canterbury on the Collection and Production of Manuscripts at Durham in the Anglo-Norman Period’, in A. Borg and A. Martindale (eds.), The Vanishing Past: Studies of Medieval Art, Liturgy and Metrology Presented to Christopher Hohler (B. A. R. International Series III; Oxford, 1981), pp. 95–104Google Scholar, at pp. 98–100; also Lawrence-Mathers, A., Manuscripts in Northumbria in the Eleventh and Twelfth Centuries (Woodbridge, Suffolk, 2003), p. 54Google Scholar.
9 Gibson, Cf., ‘Normans and Angevins’, pp. 38–65Google Scholar; 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. 45Google Scholar; Ramsay, , ‘The Cathedral Archives and Library’, pp. 346–347Google Scholar.
11 Carpenter, N. C., Music in the Medieval and Renaissance Universities (New York, 1972), p. 20Google Scholar; 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, at pp. 408–18.
12 See James, , The Ancient Libraries, pp. xxxi–xxxiv and 3–12Google Scholar; Ramsay, , ‘The Cathedral Archives and Library’, pp. 350–351Google Scholar. These scholars argue that this booklist is incomplete or fragmentary; Ramsay suggests that the full collection of volumes would have numbered around 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–xxxiiiGoogle Scholar. None of these signs corresponds to any marks on CUL Ii.3.12.
14 Wormald, F., ‘The Monastic Library’, in F. Wormald and C. E. Wright (eds.), The English Library before 1700: Studies in its History (London, 1958), pp. 15–31Google Scholar, at p. 23.
15 James, , The Ancient Libraries, p. xxxivGoogle Scholar; Ramsay, , ‘The Cathedral Archives and Library’, p. 351Google Scholar.
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 their books are Martianus Capella (15) and Macrobius (11), while Horace, Virgil, Sallust and Boethius all come next with eight volumes each. James, Cf., The Ancient Libraries, pp. xxxiii–xxxivGoogle Scholar.
17 Gibson, , Codices Boethiani, pp. 21–33Google Scholar.
18 Gameson, Cf. R., The Manuscripts of Early Norman England (c. 1066–1130) (Oxford, 1999), pp. 1–52Google Scholar.
19 For more on the reception of DIA and DIM in Europe in general, see Gibson, M. T., ‘Boethius in the Carolingian Schools’, Transactions of the Royal Historical Society, V:32 (1982), pp. 43–56CrossRefGoogle Scholar; P. Kibre, ‘The Boethian De Institutione Arithmetica and the Quadrivium in the Thirteenth Century University Milieu at Paris’, M. Masi, ‘The Influence of Boethius’ De Arithmetica on Late Medieval Mathematics’, and C. M. Bower, ‘The Role of Boethius’ De Institutione Musica in the Speculative Tradition of Western Musical Thought’, 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 of Boethian Influence’ and Rimple, M. T., ‘The Enduring Legacy of Boethian Harmony’, in N. H. Kaylor, Jr. and P. E. Phillips (eds.), A Companion to Boethius in the Middle Ages (Leiden, 2012), pp. 447–517CrossRefGoogle Scholar.
20 Webber, , ‘Script and Manuscript Production’, p. 153Google Scholar.
21 Gibson, , Codices Boethiani, p. 42Google Scholar.
22 Dodwell, , The Canterbury School of Illumination, pp. 23Google Scholar, 35, 37, 39, 64, 66, 74 and 119 provides an art-historical discussion of these images.
23 Kauffmann, , Romanesque Manuscripts, p. 79Google Scholar argues that fol. 61v ‘may be described as the first fully Romanesque whole-page illustration produced at Canterbury’; Heslop, ‘“Dunstanus Archiepiscopis”’, p. 204 also notes strong similarities between the artist’s manner of representing drapery and facial style and the work of Roger of Helmharshausen, testifying to the influences of the Low Countries and North Germany on 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 father and patron, the consul Quintus Aurelius Memmius Symmachus, dressed in rich robes and 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 illustration from 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 the Bamberg Staatsbibliothek manuscript as significant in demonstrating ‘the final assurance that by the mid-ninth century the De Arithmetica had court recognition’.
24 Teviotdale, Cf. E. C., ‘Music and Pictures in the Middle Ages’, in T. Knighton and D. Fallows (eds.), Companion to Medieval and Renaissance Music (Berkeley, 1992), pp. 179–188Google Scholar, at pp. 186–8.
25 Page, C., ‘The Earliest English Keyboard’, Early Music, 7 (1979), pp. 308–314CrossRefGoogle Scholar, with a reproduction of the image on p. 309, ill. 1; a more sceptical approach to this interpretation 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 appears in MS TCC R.15.22 on fol. 90r. See also Browne, A. C., ‘The a–p System of Letter Notation’, Musica Disciplina, 35 (1981), pp. 5–54Google Scholar for more on the history of the origins of the a–p system of notation and its transmission through Boethius, with references to its possible use in 12th-c. Canterbury.
26 See Gibson, , ‘Boethius in the Carolingian Schools’, p. 52Google Scholar. It is also worth noting that the historiated initials are always outlined in great detail, but towards the end of the text they are not filled in with colour inks; perhaps too much expense had already been paid towards the diagrams?
27 Carruthers, M., The Book of Memory, 2nd edn (Cambridge, 2008), pp. 274CrossRefGoogle Scholar and 278.
28 Ibid., p. 332. Also cited in Mellon, E. A., ‘Inscribing Sound: Medieval Remakings of Boethius’s De Institutione Musica’ (Ph.D. diss., University of Pennsylvania, 2011), p. 168Google Scholar.
29 Carruthers, M. and Ziolkowski, J. M., ‘General Introduction’, in Carruthers and Ziolkowski (eds.), The Medieval Craft of Memory: An Anthology of Texts and Pictures (Philadelphia, 2002), pp. 1–31Google Scholar; Mellon, ‘Inscribing Sound’, esp. pp. 159–252; P. Jones, ‘The Medieval Encyclopaedia: Science and Practice’, in Binski and Panayotova (eds.), The Cambridge Illuminations, p. 298; Bernhard, M. and Bower, C. M., Glossa maior in institutionem musicam Boethii, 4 vols. (Munich, 1993–2011), i, p. xliGoogle Scholar. See also Manion, , Medieval and Renaissance Manuscripts in New Zealand, p. 124Google Scholar, who observes that the decoration on a diagram depicting mathematical proportions applied to music with an elephant, ‘symbol of memory, reinforces the mnemonic quality of the diagram’.
30 Berger, A. M. B., Medieval Music and the Art of Memory (Berkeley, 2005)Google Scholar.
31 Bower, Bernhard and, Glossa maior, i, p. xlviiGoogle Scholar.
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 on diagrams on DIA, see also Bernhard, M., ‘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. 23–34Google Scholar.
33 Scribes often ‘doubled as page designers’, plotting out the amount of space left for the illuminations and diagrams that would be filled in later by the artists: S. Panayotova and T. Webber, ‘Making an Illuminated Manuscript’, in Binski and Panayotova (eds.), The Cambridge Illuminations, pp. 23–36, at pp. 31–2.
34 See Bower, Bernhard and, Glossa maior, i, p. xlviGoogle Scholar; Mellon, ‘Inscribing Sound’, p. 146.
35 Fol. 22v is not the only example of scribal reorganisation at work in MS CUL Ii.3.12: see Bower, C. M., ‘Introduction’, in Fundamentals of Music, trans. C. M. Bower and ed. C. V. Palisca (New Haven, 1989), pp. xli–xlivGoogle Scholar, 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 has been labelled differently than usual, although the text stays the same: the proemium is marked 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 of reshaping DIM so that it more closely resembles medieval music theory treatises such as Guido of Arezzo’s Micrologus.
36 Boethius, , Boethian Number Theory: A Translation of the De Institutione Arithmetica, trans. with introd. and notes by M. Masi (Amsterdam, 1983), p. 72Google Scholar: ‘There are two kinds of essence. One is continuous, joined together in its parts and not distributed in separate parts, as a tree, a stone, and all the bodies of this world which are properly called magnitudes. The other essence is of itself disjoined and determined by its parts as though reduced to a single collective union, such as a flock, a populace, a chorus, a heap of things, things whose parts are terminated by their own extremities and are discrete from the extremity of some other. The proper name for these is a multitude.’
37 Ibid., p. 75.
38 Ibid., p. 76.
39 A. E. Moyer, ‘The Quadrivium and the Decline of Boethian Influence’, in Kaylor and Phillips (eds.), A Companion to Boethius in the Middle Ages, pp. 479–517, at pp. 481–2.
40 Boethius, , Boethian Number Theory, pp. 89–92Google Scholar: Prime and incomposite numbers are indivisible: they have ‘no other factor but that one which is a denominator for the total quantity 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 numbers and 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 they are compared proportionally to another such number they have no common denominators, such as 9 and 25. They are thus in the ‘middle’, for they are ‘composite and secondary . . . but when compared to each other they become primary and incomposite because no other measure will fit both, except unity, which is a denominator for both’.
41 See Boethius, , Boethian Number Theory, pp. 96–97Google Scholar. Superabundant numbers are those in which ‘the sum of their parts factored out of the total body are found to be larger than that sum’, i.e., those whose factors added together exceed the number itself, e.g. 12 and 24. Deficient numbers are those whose parts, ‘when put together in the same way, are exceeded by the multitude of the whole term’, i.e., those whose factors added together are less than the number itself, e.g., 8 and 14. Perfect numbers are those which are neither superabundant nor deficient, but ‘hold the middle place between the extremes like one who seeks virtue’ because ‘the sum of their parts is not more than the total nor does it suffer from a lack in comparison with the total’, i.e., numbers whose factors added together equal the number, e.g., 6 and 28.
42 Ibid., p. 98.
43 Ibid., p. 100.
44 Moyer, Cf., ‘The Quadrivium’, p. 482Google Scholar.
45 Boethius, , Boethian Number Theory, p. 114Google Scholar; Moyer, , ‘The Quadrivium’, p. 482Google Scholar.
46 Boethius, , Boethian Number Theory, pp. 74–75Google Scholar.
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.
49 Blasius, L., ‘Mapping the Terrain’, in T. Christensen (ed.), The Cambridge History of Western Music Theory (Cambridge, 2002), p. 30Google Scholar.
50 Boethius, , Boethian Number Theory, p. 67Google Scholar.