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ʿUMAR AL-KHAYYĀM’S CONTRIBUTION TO THE ARABIC MATHEMATICAL THEORY OF MUSIC

Published online by Cambridge University Press:  26 August 2010

MICHELE BARONTINI
Affiliation:
Dipartimento di matematica, Università di PisaLargo Pontecorvo 5, 56127 Pisa, Italy Email: mpeyglus@free.fr, tonietti@dm.unipi.it
TITO M. TONIETTI
Affiliation:
Dipartimento di matematica, Università di PisaLargo Pontecorvo 5, 56127 Pisa, Italy Email: mpeyglus@free.fr, tonietti@dm.unipi.it

Abstract

We here present the Arabic text, with an English translation, of certain pages dedicated by al-Khayyām to the mathematical theory of music. Our edition is based on a manuscript extant in a library in Manisa (Turkey), and corrects the mistakes found in another transcription. Lastly, we compare the theory of al-Khayyām with other Arabic theories of Music, and with those coming from other traditions.

Résumé

Nous présentons ici le texte arabe, avec traduction anglaise, de certaines pages consacrées par al-Khayyām à la théorie mathématique de la musique. Notre édition se fonde sur un manuscrit conservé dans une bibliothèque de Manisa (Turquie) et corrige les erreurs d'une autre transcription. Nous comparons enfin la théorie d'al-Khayyām avec d'autres théories arabes de la musique, et avec certaines théories issues d'autres traditions.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2010

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References

1 Humai, Jamal al-Din, Khayyam-Nama (Teheran, 1967), pp. 340344Google Scholar . The Iranian scholar acknowledged the following sources. “The present epistle is to be found in a collection of hand-copied treatises also containing other mathematical works by Khayyam, in file 509, on pages 97–9, in the Central Library of the University of Teheran, in the section composed of books transferred from Turkey to Teheran through the agency of Fadil Karami Agha. The date marked on the back of the original is VII-VIII century Egira [XIV–XV century A.D.]. We have simply re-copied, in turn, the text already copied at the University Library (though we are not sure of its correctness, especially as regards the numbers), without omitting anything; we have also transcribed a few paragraphs of hand-written comments in the margin, which complete the text and demonstrate that at least the copy that is used here is a complete whole. It would have been much better if we had been able to dispose of another copy of the same text, in order to compare them and correct them against each other!” The manuscript found in the Central University Library in Teheran was quoted by Youschkevitch, A. P. & Rosenfeld, B. A., entry ‘Al-Khayyāmī’, Dictionary of Scientific Biography (New York, 1981), vol. V, pp. 323334Google Scholar , p. 332, as “509, fols. 97–99”. A copy of this manuscript is said to be present today also in the Biruni Library at Tashkent; private communication by Amnon Shiloah.

2 Shiloah, Amnon, The Theory of Music in Arabic Writings (München, 1979), pp. 296297Google Scholar .

3 We use square brackets [ ] to indicate our additions which may help the reader to understand the text more easily. In the text, the asterisks * sign those Arabic words, and numbers which either do not coincide in the Manisa, and Teheran manuscripts, or are corrected by us in the translation.

4 This was a recovery of the Greek genera. The 4:3 tetrachord contemplated three of them. The diatonic one was formed by two tones and one semitone; the chromatic one included a minor third and two semitones. The enharmonic tetrachord consisted of a major third and two micro-intervals, similar to quarters of a tone. Sachs, Curt, The Rise of Music in the Ancient World. East and West (New York, 1943), pp. 206207Google Scholar .

5 The ratios of the four numbers are those indicated. For example, 56 can be obtained from 49 by dividing by 7 and multiplying by 8; 64 can be obtained from 56 in the same way. The three ratios united together, 8:7, 8:7, 49:48, give 4:3, which is thus divided into three parts. If the players of musical instruments were to follow a similar theory, they would tune them by placing the notes that divide the interval of the fourth on the basis of these ratios. For example, the re and the mi in the interval of the fourth do-fa, in accordance with modern Italian notation. In order to prepare the comment on the following section, we will compare the species, as al-Khayyām lists them, with those that are found in the works of al-Fārābī and Ibn Sīnā (Avicenne). This first way of dividing the fourth is also found in Ibn Sīnā, extrait du Kitāb al-Shifāʾ [The Book of Healing], ‘Section des sciences éducatives, chapitre XII, La musique’, in Rodolphe d'Erlanger (ed.), La musique arabe, 4 vols. (Paris, 1930 and 1935), vol. II, p. 146. It is also described by al-Fārābī, Kitāb al-Mūsīqā al-kabīr [Grand traité de la musique], in La musique arabe, vol. I–II, I, p. 109. Tito M. Tonietti, Eppur si ode (to be published), ‘Capitolo 4’.

6 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 109. Ibn Sīnā, al-Shifāʾ, p. 149.

7 This goes back to the Greek Pythagorean tradition. Barker, A., Greek Musical Writings (Cambridge, 1984 and 1989)Google Scholar . Tonietti, Tito M., ‘The mathematical contributions of Francesco Maurolico to the theory of music of the 16th century (The problems of a manuscript)’, Centaurus, 48 (2006): 149200CrossRefGoogle Scholar . Tonietti, Eppur si ode, ‘Capitolo 1’.

8 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 109.

9 If we desire to maintain the same proportions as in the text, the number 62, which is found both in the Manisa manuscript and in Humai's edition, should be corrected to 63, because this is the result of nine-eighths of 56. If, on the contrary, we desire to maintain 62 in the sequence, then the proportions should be changed in the text. Al-Fārābī, Kitāb al-Mūsīqā, I, p. 111. At this point, in his edition, Humai states: “I say this without being completely sure of the accuracy of the calculations for this subject and the following one. And Allah knows more about this.” This sentence authorises us to suspect that the editor of the printed edition did not know a lot about the Greek and Arabic theories of music.

10 The numbers of the sequence do not correspond to the proportions in the text; 168 should be corrected to 160, and 165 to 135. In Humai's edition, there is a sign of a horizontal parenthesis with a dot, placed next to 18, below 1/5. It probably derives from the zero 0 with a line over it in the Manisa manuscript. Zero was often written like an omicron with a line over it in manuscripts. Al-Fārābī, Kitāb al-Mūsīqā, I, p. 111.

11 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 111. On the contrary, Humai's edition gives the partly mistaken numbers 225 198 18 165LA. This last symbol, LA, lām alif is probably due to the written form for zero in the Manisa manuscript.

12 The error in the Manisa manuscript, 60 instead of 70, was aggravated in Humai's edition by the numbers 85 65 63 6?, again probably due to the different notation for zero. Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 112 and 114. Ibn Sīnā, al-Shifāʾ, p. 147.

13 353 should be corrected to 352. Instead of the 0 of 320, Humai's edition has the Arabic letters for LA.

14 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 112. For this interval, equivalent to the whole Pythagorean tone 9:8, cf. also Cowl, Carl, ‘The Risāla fī ḫubr taʾlīf al-alḥān of al-Kindī', The Consort, 23 (1966): 129166Google Scholar .

15 This lacuna was indicated by the copyist in the margin of the Manisa manuscript, but it was ignored in Humai's edition. Anyway, the sequence should be completed by “one whole unit plus eleven one hundred and seventeenths” [128:117], which, together with the other proportions, gives 4:3. But then, it is necessary to start from 13:12, otherwise the whole units would not be correctly divisible. Consequently, the sequence of numbers becomes 156, 144, 128, 117. Ibn Sīnā wrote: “468 (13/12) 432 (9/8) 384 351” (al-Shifāʾ, p. 150). It is interesting that the ratio 128:117 is also missing in the edition for Ibn Sīnā of Erlanger between the last two numbers, and that the numbers in the sequence are exactly three times our numbers. If this ratio were to be missing also in the most ancient Arabic editions, might this indicate a (perfectly plausible) direct contact between this text of al-Khayyām and that of Ibn Sīnā?

16 Ibn Sīnā, al-Shifāʾ, p. 148. With the ratios in a different order, 8:7, 13:12, 14:13, the species is to be found also in al-Fārābī, Kitāb al-Mūsīqā, I, p. 114.

17 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 104 and 105.

18 Humai's edition has 35 instead of 30. Did copyists often confuse 5 with 0? In reality, there are various versions of zero designed by the Manisa manuscript copyists, which go from a kind of Greek gamma to a sort of Arabic He, or even a little circle like the modern-day 5. There is no doubt, on the contrary, about the five: a sort of upside-down B in the most ancient forms of writing. Al-Fārābī, Kitāb al-Mūsīqā, I, p. 107. Ibn Sīnā, al-Shifāʾ, p. 153.

19 The third, in fact, we did not find in al-Fārābī, Kitāb al-Mūsīqā.

20 458 should be completed to read 48 45. Humai's edition has wa-juzʾun min khamsatin wa-ʿishrīn kullin [and one part out of twenty-five of a whole unit]. On the contrary, the Manisa manuscript reads wa-juzʾun min khamsata ʿashara min kullin [and one part out of fifteen of a whole unit], but it also contained two misleading dots under the letter ra of ʿashar, thus indicating a dual and creating confusion. Al-Fārābī, Kitāb al-Mūsīqā, I, p. 113. Ibn Sīnā, al-Shifāʾ, p. 153.

21 In the sequence, the 1 of 14 is missing, and the ratios are calculated in the opposite order. Al-Fārābī, Kitāb al-Mūsīqā, I, p. 104. Ibn Sīnā, al-Shifāʾ, p. 152.

22 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 107. Ibn Sīnā, al-Shifāʾ, p. 152.

23 The ratios that link that four numbers in the proportion 4:3, are, on the contrary, in the order 10:9, 36:35 and 7:6. Ibn Sīnā, al-Shifāʾ, p. 152.

24 Humai's edition continued to write the latters LA instead of 0 in 40. We have already seen that this derives from the version of zero similar to an Arabic lām-alif. Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 104–5. Ibn Sīnā, al-Shifāʾ, p. 154.

25 The second 0 of 100 is missing in Humai's edition. In this sequence, we did not find the species either in al-Fārābī, Kitāb al-Mūsīqā, or in Ibn Sīnā, al-Shifāʾ. But Ibn Sīnā, on p. 154, gave a different order to the ratios, obtaining the numbers: “80(40/39)78(26/25)75(5/4)60”.

26 The 0 of 108 is missing in the manuscript; furthermore, the sequence of numbers fits in with a different order of ratios: 5:4, 28:27, 36:35.

27 The third species, that is to say the enharmonic one, composed in the order 5:4, 28:27, 36:35, is to be found in al-Fārābī, Kitāb al-Mūsīqā, I, p. 113. In the form “36(36/35) 35(5/4) 28(28/27) 27”, it is also found in Ibn Sīnā, al-Shifāʾ, p. 154.

28 The number 47 does not fit in with the ratio 46:45, and should be substituted in the sequence by 46. Al-Fārābī, Kitāb al-Mūsīqā, I, p. 107. The mistakes in the numbers written in Arabic figures, which are present in the sequences found both in the Manisa text and in the one in Teheran, are probably due to copyists and the editor. However, there are considerably fewer in the Manisa manuscript than in Humai's edition. By contrast, the ratios expressed in words in the Manisa manuscript do not contain any.

29 Rashed, Roshdi & Vahabzadeh, Bijan, Al-Khayyām Mathématicien (Paris, 1999), pp. 376377Google Scholar . The passage is presented in the translation of Michele Barontini and Ron Packham.

30 ‘The “putting together” and “taking away” in the passage, which would mean, for musicians, adding and subtracting the intervals between notes, when transferred to the ratios between numbers, become multiplying and dividing, because according to Greek theory, the numbers of music formed a geometrical sequence. On ratio and proportionality in al-Khayyām, see also Vahabzadeh, Bijan, ‘Al-Khayyām's conception of ratio and proportionality’, Arabic Sciences and Philosophy, 7 (1997): 247263CrossRefGoogle Scholar .

31 Humai, Khayyam-Nama, p. 340.

32 The main sources regarding the life of al-Khayyām are: Samarqand, Nizami-I-Arudi, Chahar Maqala [Four Discourses], ed. Browne, Edward G. (Cambridge, 1921), pp. 7174 and 134140Google Scholar ; Zhukovsky, V. A., ‘Omar Khayyam and the wandering quatrains’, Journal of the Royal Asiatic Society, 30 (1898): 349366Google Scholar . For the Quatrains, see the edition of Forughi (Teheran, 1942); Arberry, Arthur J., The Rubaʾīʿāt of Omar Khayyam: the Chester Beatty MS (London, 1949)Google Scholar . Arberry, Arthur J., Omar Khayyam: a New Version (London, 1952)Google Scholar . Those that appreciate music, or the tar, in the most ancient manuscript (Cambridge's 604 H./1207 A.D.), are to be found under the numbers 39, 70, 80, 85, 138, 147, 148, 169, 174, 181, 214, 230. In Arabic music in general, it may be useful to consult: Farmer, Henry George, ‘The music of Islam’, in New Oxford History of Music, 11 vols. (1957), vol. I, pp. 421477Google Scholar . Shiloah, Music in Arabic Writings; Shiloah, Amnon, Music in the World of Islam (Detroit, 1995)Google Scholar ; Chaik-Moussa, Abdallah, ‘Considérations sur la littérature d’Adab, présence et effets de la voix et autres problèmes connexes’, Al-Qantara, XVII, 1 (2006): 2562CrossRefGoogle Scholar ; Amnon Shiloah, ‘La scienza della musica negli scritti arabi’, in La civiltà islamica, ed. Roshdi Rashed, vol. III of Storia della scienza, 10 vols. (Roma, 2002), pp. 525–38. Shiloah, Amnon, Music and its Virtues in Islamic and Judaic Writings (Abingdon UK, 2007)Google Scholar .

33 Farmer, Henry George, ‘Greek theorists of music in Arabic translation’, ISIS, XIII (1930): 325333CrossRefGoogle Scholar . Roshdi Rashed, ‘Dal greco all’arabo: trasmissione e traduzione', in La civiltà islamica, pp. 31–49. Amnon Shiloah, ‘La scienza della musica negli scritti arabi’. Tonietti, Eppur si ode, ‘Capitolo 4’.

34 Ibn Sīnā, al-Shifāʾ, pp. 129 and 148.

35 Aristoxenus, , Aristoxeni Elementa Harmonica, ed. Da Rios, Rosetta (Roma, 1954)Google Scholar . Ptolemaeus, Claudius, Harmonicorum libri tres, ed. Wallis, John (Oxonii, 1682), pp. 3950 and 6162Google Scholar . Barker, Greek Musical Writings. Tonietti, Eppur si ode, ‘Capitolo 1’.

36 It was Marin Mersenne who finally succeeded, as is described in his Harmonie universelle (Paris, 1636), at least in particular cases, based on the inspiration of the pendulum. Tonietti, Eppur si ode, ‘Capitolo 7’.

37 Ibn Sīnā, al-Shifāʾ, p. 111.

38 Bausani, Alessandro, L'enciclopedia dei fratelli della purità (Napoli, 1978), p. 196Google Scholar . Diogenes Laertius attributed a comparison between sounds and the waves of the sea to Zeno (third century BC), who was a Stoic. But it is also written of these philosophers that they “conjectavere sonum esse corpus, ictum utpote aerem” [“hypothesised that sounds were substance, beats, that is to say, air”] (Charles Burnett, ‘Sound and its perception in the Middle Ages’, in Charles Burnett, Michael Fend, and Penelope Gouk [eds.], The Second Sense [London, 1991], pp. 43–69, p. 56; ‘Introduction’, p. 3). It is not at all clear, however, how they succeeded in reconciling waves with beats. Seeing that the substance of sound was their famous pneuma, might be “beats” be the strokes of the pneuma against the ear? Furthermore, as far as we know, the main Greek traditions that influenced Arabic musical culture were those of Pythagoras-Ptolemy or Aristotle. Boethius used a similar image at a much later date. “deinde [saxum] maioribus orbibus undarum globos spargit […] Ita igitur cum aër pulsus fecerit sonum, pellit alium proximum et quodammodo rotundum fluctum aeris ciet” [“therefore [the stone thrown into water] spreads the spheres of the ripples in widening circles […] In the same way, when air creates the sound of the impulse, it would propel the air next to it, and in a certain sense will give rise to the spherical air waves”]. (Sev. Boethius, De Institutione Musica Libri Quinque, ed. Godofredus Friedlein [Lipsiae, 1867], p. 200). Tonietti, Eppur si ode, ‘Capitolo 5’. The debate renewed at Marin Mersenne's times, who quoted Aristotle's De Anima lib. II [ch. 4]. One of the main opponent to the wave theory was Isaac Beeckman. Mersenne, MarinCorrespondance (Paris, 19451988), I, pp. 341342Google Scholar , 348–9; II, pp. 282, 293–4. Cohen, H.F., Quantifying Music (Dordrecht, 1984), p. 275CrossRefGoogle Scholar . Francis Bacon supported wave theory, even though in an ambiguous way. Gouk, Penelope, ‘Music in Francis Bacon's natural philosophy’, in Francis Bacon, ed. Fattori, Marta (Roma, 1984), pp. 139149Google Scholar . Tonietti, Eppur si ode, ‘Capitolo 7’.

39 Ibn Sīnā, al-Shifāʾ, p. 116.

40 Ibn Sīnā, Danesh-Nama, trad. Achena, Mohammad & Massé, Henri, Avicenne. Le Livre de Science, 2 vols. (Paris, 1986), vol. II, p. 223Google Scholar .

41 Boetius, Severinus, De Institutione Arithmetica Libri Duo - De Institutione Musica Libri Quinque, ed. Friedlein, Godofredus (Lipsiae, 1867)Google Scholar . Tonietti, ‘The mathematical contributions of Maurolico’, pp. 153–6.

42 Burnett, Charles, ‘Teoria e pratica musicali arabe in Sicilia e nell’Italia meridionale in età normanna e sveva', Nuove Effemeridi, III n. 11 (1990): 7989Google Scholar .

43 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 172. Cf. Shiloah, Music in the World of Islam, p. 112, and ‘La scienza della musica’, p. 535.

44 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 76.

45 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 33 and 51–2.

46 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 55–9.

47 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 60–1. Emphasis in the translation of Erlanger. Cf. Collangettes, M., ‘Étude sur la musique arabe’, Journal Asiatique, (Novembre-Décembre 1904): 365422Google Scholar ; (Juillet-Août 1906): 149–90.

48 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 61 and 66.

49 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 94–100.

50 Amnon Shiloah, Music in the World of Islam. Tauzin, Aline, ‘Femme, musique et Islam. De l’interdit à la scène', CLIO, histoire, femmes et sociétés, 25 (2007): 143163Google Scholar .

51 Ptolemaeus, Harmonicorum, p. 159.

52 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 158–60.

53 Quite rightly, in his French translation, Erlanger added, as a title to the paragraph: “Construction d'un instrument pour la vérification expérimentale de la théorie” (p. 158).

54 Tonietti, Eppur si ode, ‘Capitolo 4’.

55 Thérèse-Anne Druart, ‘Scienza e filosofia’, in La civiltà islamica, pp. 72–6, p. 74.

56 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 161–2.

57 Instrument with two or three strings and a long neck; it was a kind of pandora.

58 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 218–19, 227, 241, 229–30, 242–62. For the ṭunbūr of Baghdad, Henry Farmer wrote explicitly of quarter tones. Farmer, ‘The music of Islam’, pp. 447, 456, 463.

59 Bausani, L'enciclopedia dei fratelli della purità.

60 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 28. Ibn Sīnā, al-Shifāʾ, p. 106.

61 Ptolemaeus, Harmonicorum, pp. 33–4, 157–8. Tonietti, Eppur si ode, ‘Capitolo 1’.

62 Al-Fārābī, Kitāb al-Mūsīqā, I, pp. 263–8.

63 1 fen equals approximately one third of a cm.

64 Tonietti, Tito M., ‘The mathematics of music during the 16th century: The cases of Francesco Maurolico, Simon Stevin, Cheng Dawei, and Zhu Zaiyu’, Ziran kexueshi yanjiu [Studies in the History of Natural Sciences], 22, 3 (2003): 223244Google Scholar . Tonietti, Tito M., Le matematiche del Tao (Roma, 2006)Google Scholar . Tonietti, Eppur si ode, ‘Capitolo 2’.

65 Galilei, Vincenzio, Dialogo della musica antica et moderna (Firenze, 1581; repr. Roma, 1934Google Scholar ). Galilei, Vincenzio, Discorso intorno all'opera di messer Gioseffo Zarlino … (Firenze, 1589)Google Scholar . Palisca, Claude V., ‘Scientific empiricism in musical thought’, in Rhys, H.H. (ed.), Seventeenth Century Science, and the Arts (Princeton, 1961)Google Scholar . Tonietti, ‘The mathematics of music’. Tonietti, Eppur si ode, ‘Capitolo 5’. In the 17th century, Vincenzio Galilei was to be followed, at least partially, by Marin Mersenne, who succeeded in adding his numbers and his experience to the work of the Florentine; Mersenne, Harmonie Universelle.

66 For example, Gerhard Endress, ‘Scienza e filosofia nel tardo ellenismo’, in La civiltà islamica, pp. 29–31. Pascal Crozet, ‘Aritmetica’, in La civiltà islamica, pp. 498–506. Casari, Mario & Speziale, Fabrizio, ‘La scienza islamica in India’, in Pingree, David & Torella, Raffaele (eds.), La scienza indiana (Roma, 2001), pp. 908928Google Scholar .

67 Bharata, , Gitalamkara, ed. Daniélou, Alain & Bhatt, N.R. (Pondichéry, 1959), p 139Google Scholar .

68 ṣadja [born from six] is the first note.

69 madhyama [middle] is the fourth note.

70 Natya Śastra, ed. a board of scholars (Delhi, 1996), chap. 28, 23 and 27–28 (ed. 1996, pp. 388–9); chap. 19, 38–40 (ed. 1996, p. 268); chap. 30, 11–13 (ed. 1996, p. 415); chap. 33, 29–35 (ed. 1996, p. 486).

71 Sanyal, A.N., Ragas and raginis (Calcutta, 1959), p. 4Google Scholar . Perinu, Roberto, La musica indiana (San Giuliano Milanese-Milano, 1981), pp. 40, 2425, 2930, 44Google Scholar .

72 Al-Fārābī, Kitāb al-Mūsīqā, I, p. 57.

73 Bharata, Gitalamkara, pp. 138–9.

74 I do not agree with Alain Daniélou when he writes, in his edition and comments on the Gitalamkara: “une parenté certaine avec les théories musicales de la Grèce”, p. v.

75 The Śulbasutras, ed. Sen, S.N. & Bag, A.K. (New Delhi, 1983)Google Scholar . Tonietti, Eppur si ode, ‘Capitolo 3’.

76 Datta, B. & Singh, A.N., History of Hindu Mathematics, 2 vols. (Bombay, 1935 [1962]), vol. I, p. 7Google Scholar .

77 Tito M. Tonietti, ‘Toward a cross-cultural history of mathematics. Between the Chinese and Arabic mathematical theory of music: the puzzle of the Indian case’, a talk given at the conference “Advances in Mathematics: Historical Developments & Engineering Applications”, Pant Nagar (Uttarakhand, India), (December 19–22 2007). Tonietti, Eppur si ode, ‘Capitolo 3’. Giacomo Benedetti & Tito M. Tonietti, ‘Sulle antiche teorie indiane della musica. Un problema a confronto con altre culture’, Rivista di Studi Sudasiatici (2010), to appear.

78 Rashed & Vahabzadeh, Al-Khayyām mathématicien, p. 378.

79 Farmer, Henry George, ‘“Ghosts” An excursus on Arabic musical bibliographies’, ISIS, XXXVI (1946): 123130CrossRefGoogle Scholar .

80 Cf. Youschkevitch & Rosenfeld, ‘Al-Khayyami’, p. 326. However, they attributed 22 species to the Persian mathematician and poet, though the text only contains 21. Nor did they notice that in actual fact, Al-Khayyām had proposed only two new species, and not three, as he had declared.

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