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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  and
TITO M. TONIETTI
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.

Information

Type
Research Article
Information
Arabic Sciences and Philosophy , Volume 20 , Issue 2 , September 2010 , pp. 255 - 280
Copyright
Copyright © Cambridge University Press 2010

I. THE TEXT: “AL-QAWL ʿALĀ AJNĀS ALLADHĪ BI-AL-ARBAʿA” [DISCUSSION ON SOME OF THE GENERA CONTAINED IN A FOURTH]

The text was established by comparing the manuscript housed in the Il Halk Kutuphanesi at Manisa (Manisa Public Library, Turkey) with the version edited by Jamal al-Din Humai in the appendix to Khayyam-Nama [The book of Khayyam].Footnote 1 We would like to thank Amnon Shiloah, who had already brought attention to the manuscript with his description of it,Footnote 2 for kindly placing his copy at our disposal. We would also like to thank Soghra Nobakhtian and Jafar Zafarani for sending us the appendix to the book of Humai. The editor of the printed version has the merit of calling attention to the pages of musical theory attributed to ʿUmar al-Khayyām. The differences between our edition of the manuscript and the printed version, with its relative mistakes, are pointed out in the notes to the English translation. This new edition is the result of collaboration between Michele Barontini, a musician, and a scholar of Arabic humanities, and Tito M. Tonietti, a historian of science, and a scholar of the relationships between science and music. The English translation is by Ron Packham.

“The ratio of the sample-unit plus one third [4:3]Footnote 3 can be further subdivided into three ratios, which correspond to the three abʿād [intervals] limited to four sounds. For this reason, the ratios of the sample-unit plus one third was called a fourth [a tetrachord]. These three abʿād [intervals] either do not include an interval whose ratio is greater than the sum of the others, or else they include an interval whose ratio is equal to twice the remaining two. The first of the ajnās [among the species of the fourth] was called qawī or ṭanīn [strong, diatonic], the second mulawwan [coloured, chromatic] or muʿtadil [moderate, median], and the third rikhw [weak, enharmonic] or taʾlīf [compound].Footnote 4

The first of the strong species is the first, which doubles the interval; it is one whole unit plus one seventh [8:7], one whole unit plus one seventh [8:7], and one whole unit plus one of forty-eight parts of a whole [49:48]. It corresponds to the numbers 64, 56, 49, 48.Footnote 5 This would be an extremely strong, valid species, if it were not for this interval, that is to say, one forty-eighth [49:48], which is a value very distant [from practice].

The second of the strong species is the second which doubles [the interval]; it consists of one whole unit plus one eighth of a unit [9:8], with one whole unit plus one eighth of a unit [9:8], and one whole unit plus thirteen out of two hundred and forty-three parts [256:243], corresponding to the numbers 324, 288, 256, 243.Footnote 6 This is a species that is very close to normal practice, and as a result, it is the only one that is [commonly] used in most countries.Footnote 7

The third species is the third that doubles [the interval]; it consists of a whole unit plus one ninth [10:9], one whole unit plus one ninth [10:9] and one whole unit plus six parts out of seventy-five [81:75 = 27:25]. The numbers are 100, 90, 81, 75. It was discovered by al-Fārābī and, as far as I can judge, it was not among the most common.Footnote 8

The fourth is the strong conjunct of the first species; it is made up of one whole unit plus one seventh of a unit [8:7], one whole unit plus one eighth of a unit [9:8], and one whole unit plus one twenty-seventh of a unit [28:27]. The numbers are 72 62[63] 56 54.Footnote 9 This is a very good calculation.

The fifth is the strong conjunct of the second species, and is made up of one whole unit plus one eighth of a unit [9:8], one whole unit plus one ninth of unit [10:9], and one whole unit plus one fifteenth of a unit [16:15]; its numbers are 180 168[160] 144 165[135].Footnote 10 This species is the best of all, in my opinion.

Among the conjuncts of the third species, the sixth is made up of one whole unit plus one ninth [10:9], one whole unit plus one tenth [11:10], and one whole unit plus one eleventh [12:11]. Its numbers are 220* 198 180* 165.Footnote 11 This, too, is a good species.

The seventh, the first disjunct, is made up of one whole unit plus one seventh [8:7], one whole unit plus one ninth [10:9], and one whole unit plus one twentieth [21:20]. Its numbers are 80* 60*[70] 63 60*.Footnote 12 This, too, is good and appropriate.

The eighth species, the second of the strong disjunct ones, is made up of one whole unit plus one eighth [9:8], one whole unit plus one tenth [11:10], and one whole unit plus twenty-three two hundred and ninety-sevenths [320:297]. Its numbers are 396 353[352] 320 297.Footnote 13 This species was discovered by al-Fārābī, but it is not appropriate, in spite of the fact that he identified the so-called ṭanīnī interval in it.Footnote 14

Another species was discovered by the Imam of Saviours, Ibn Sīnā, and is made up of one whole unit plus one eighth [9:8], one whole unit plus one twelfth [13:12], [lacuna],Footnote 15 and he sustained that it is made up of one whole unit plus one seventh [8:7], one whole unit plus one thirteenth [14:13], and one whole unit plus one twelfth [13:12], the numbers of which are 16, 14, 13, 12. In my opinion, this species is far from being usable, on account of the small difference between its two similar intervals. Other examples can be found among these disjunct [species]; I have limited myself to these few disjuncts, because [by proceeding in this way, we find species which] are not used, and because they are far from any permitted iʾtilāf [consonance].Footnote 16

The first of the coloured species is one whole unit plus one fifth [6:5], one whole unit plus one nineteenth [20:19], and one whole unit plus one eighteenth [19:18]. The relative numbers are 24, 20, 19, 18.Footnote 17

The second of the coloured species is made up of one whole unit plus one fifth [6:5], one whole unit plus one fourteenth [15:14], and one whole unit plus one twenty-seventh [28:27]. The numbers are 36, 30, 28, 27.Footnote 18

The third of this species is one whole unit plus one fifth [6:5], one whole unit plus one thirty-ninth [40:39], and one whole unit plus one twelfth [13:12]. Its numbers are 48, 40, 39, 36. I believe that this was not found by al-Fārābī, and that the second and the third are far from practical use, in spite of the fact that [their ratios] are acceptable.Footnote 19

The fourth of the species is one whole unit plus one fifth [6:5], one whole unit plus one twenty-fourth [25:24], and one part out of fifteen together with one whole unit [16:15]. The numbers are 60, 50, 8 45 [48, 45].Footnote 20 This species is close to common use.

The fifth is one whole unit plus one sixth [7:6], one whole unit plus one fourteenth [15:14], and one whole unit plus one fifteenth [16:15]. The numbers are 16 15 4[14] 12.Footnote 21 This species is good, in spite of the fact that we calculated the largest part at the end of the group, as we preferred to lighten it, but this does not create too much disturbance.

The sixth is one whole unit plus one sixth [7:6], one whole unit plus one eleventh [12:11], and one whole unit plus one twenty-first [22:21]. The numbers are 28 24 22 21, and this is good, too.Footnote 22

The seventh is one whole unit plus one sixth [7:6], one whole unit plus one ninth [10:9], and one whole unit plus one thirty-fifth [36:35]. The numbers are 40 36 35 30.Footnote 23 Here, too, we have placed the largest interval at the end, but this set is far removed from practical use.

As regards the first of the compound [enharmonic] species, it is one whole unit plus one quarter [5:4], one whole unit plus one thirty-first [32:31], and one whole unit plus one thirtieth [31:30]. The numbers are 40 32 31 30.Footnote 24

The second one is one whole unit plus one quarter [5:4], one whole unit plus one thirty-ninth [40:39], and one whole unit plus one twenty-fifth [26:25]. The numbers are 100 80 78 75.Footnote 25 This is a good proportion.

The third one is one whole unit plus one quarter [5:4], one whole unit plus one thirty-fifth [36:35], and one whole unit plus one twenty-seventh [28:27]. The numbers are 140 112 1[0]8 105.Footnote 26 These two species are not quoted in the books of the ancient masters [Kutub al-Qudamāʾ], despite their validity. I am unable to suggest any reason for their absence, other than some form of negligence [in copying the texts].Footnote 27

The fourth species is one whole unit plus one quarter [5:4], one whole unit plus one twenty-third [24:23], and one whole unit plus one forty-fifth [46:45]. The numbers are 60 48 47[46] 45.Footnote 28 This was discovered [by the ancient masters], but this species is not as consonant as the second and the third. Other examples of these species could be given, but they would not be usual.”

II. AL-KHAYYĀM'S TEXT IN THE LIGHT OF ARABIC THEORIES OF MUSIC

As regards the compositions of ratios, another passage about music is to be found in al-Khayyām's “Third Book” of his Commentary on the difficulties of certain postulates of Euclid's work.Footnote 29

As regards the taʾlīf al-nisba [composition of the ratio], mentioned in the science of music, this is something different. It is a question of tarkīb [putting together] and nuqṣān [taking away]. The term taʾlīf [composition] is used in this case by a sort of tacit agreement and not by a real consensus of opinion. Euclid dealt with the taʾlīf [composition] of the ratio in a well-known passage of his “Eighth Book”, using it in a shakl [proposition] which he could have omitted in his book, and ignoring the meaning mentioned by us [in this work]. The tarkīb al-nisba [putting the ratio together], on which certain parts of music are based, is numerical. Euclid spoke of this at length in his eighth book. As regards nuqṣān al-nisba [taking the ratio away], when it is analysed and one reflects over it, it is nothing other than a ṣanfun min al-tarkīb [a category derived from tarkīb]. The way to understand these things is the same, for the person who possesses an acute spirit of observation and good intuition. We have already partly recalled this subject in the Sharḥ al-Mushkil min Kitāb al-Mūsīqā [Explanation of problematic passages in the ‘Book on Music’].Footnote 30

Unfortunately, the Sharḥ al-Mushkil min Kitāb al-Mūsīqā is no longer to be found. We are entitled to believe that the “Discussion on the Genera Contained in a Fourth” presented above was a part of it.Footnote 31 At present, we can only hypothesise about the subject of the whole work, pending (hopefully) its discovery. They might be justified here both by everything else we know about the writings and the life of ʿUmar al-Khayyām, and by the works about music by other scholars, to which he made reference. It is highly likely that the Book on Music mentioned by him was the treatise by al-Fārābī with the same title, from which the Persian mathematician had taken some of the tetrachords listed in the Manisa manuscript.

Al-Khayyām judged the validity of choices of how to divide the fourth, partly by how these notes were used by the musicians of his period and his culture: not only in theory, but also in musical practice. Unfortunately, information about al-Khayyām is particularly scarce and riddled with legends. In the opinion of scholars, his famous Quatrains gained him more popularity than certainties, arousing all kinds of biographical and religious controversies on his account. We may add to these another note of our own: as there were some quatrains that celebrated the tar, that is to say, the Persian lute with the long neck, we may wonder how much experience al-Khayyām had of this.Footnote 32

The rise of Islam among the Omayyads in Syria and the Abbassids at Baghdad was successfully consolidated by means of several famous translations from Greek. The countless scientific texts gathered in many places also included the musical theories of Euclid, Aristoxenus and Ptolemy, to mention only the best-known.Footnote 33 The most famous of the Arabic natural and theoretical philosophers of music, such as al-Fārābī, Ibn Sīnā, Ṣafī al-Dīn, regularly started from the ancient Greeks, who were sometimes quoted by name,Footnote 34 adding their own varied and different divisions of the tetrachord, that is to say, of the fourth.

The greater variety of musical genera calculated by the Arabs, compared with the orthodox Greek Pythagorean-Ptolemaic traditions, probably depended on a considerable more complex tradition, which had accumulated pre-Islamic music, Persian, Egyptian and Turkish music, with all the relative variants of the dervishes and the Sufis. This music was conceived for a rich variety of instruments, each of which required its own different manner of tuning and playing technique. How could the few Greek canonical divisions into genera have been tolerated?

It is striking, in this connection, that as far as we can see from known texts, there were no followers of Aristoxenus among the Arabic scholars. But this Greek musician, who was a disciple of Aristotle, had been criticised by Ptolemy, because he was said to have insisted on dividing the tone and the octave into parts that were all equal, without any numerical ratios, but rather using the ear.Footnote 35

What were the major innovations that Ibn Sīnā and al-Fārābī may be said to have introduced with respect to the Greek theories of Pythagoras and Ptolemy? For these “ancients”, as they were generally called in Arabic texts, sounds were thought to arrive on the ear as “ictus” [beats] in rapid succession. In this way, they could theoretically be counted, and consequently included without difficulty in the discrete theory of Pythagorean arithmetic, thus participating in the famous Quadrivium. But in actual fact, good Pythagoreans could neither think of truly counting the beats, in order not to allow their ideal theories to degenerate among real movements, nor, as far as we know, did they ever try.Footnote 36 On the contrary, for Ibn Sīnā, sounds took the form of waves in the air. Among the various causes of pitch described by Ibn Sīnā, one was described as “une forte compression des couches d'air par le mouvement ondulatoire qui transporte le son.”Footnote 37 The continuous wave model was reproposed in the Asbāb ḥudūth al-ḥurūf [Causes which produce the sounds of the tongue], where the term used for sounds was tamawwuj al-hawāʾ [air waves].Footnote 38

And yet, even if he was undoubtedly influenced by the Aristotelian aspect of Neoplatonism, but in a different way compared with Aristoxenus, also for him, it was only the ratios between whole numbers that could produce consonances. He therefore excluded, on the grounds of dissonance, for example, the kind produced “par une portion d'une corde dont la tension est indéterminée, et l'autre par cette corde dans toute sa longueur, si ces deux longueurs de corde sont entre elles dans le rapport du côté d'un carré à sa diagonale.”Footnote 39 And in actual fact, this ratio was incommensurable for the Greeks, and could not be represented by means of numbers; at that times a surd, nowadays we would call it an irrational.

With Ibn Sīnā, the Dār-al-Islām [Abode of Islam] could take advantage of a valid variant on the predominant Greek theory. He demonstrated a sufficient understanding of the subject when he wrote that for those numbers, placed in geometrical sequences, the differences between adjacent terms were not constant, as in the case of the arithmetical ones.Footnote 40 By contrast, the general musical model in the corresponding field in Europe, for several centuries, was Severinus Boetius, who confused geometrical sequences with arithmetical ones in at least one delicate case.Footnote 41 Unfortunately, even if they had translated the Arabic texts, the Europeans of the period would generally have avoided translating into Latin the parts devoted to music.Footnote 42 Boetius probably appeared to them to be a sufficiently authoritative surrogate of the Greek sources, which were still missing.

Al-Fārābī divided the Greek tetrachord, that is to say, the fourth 4:3, in a greater variety of ways: these were still numerical, but their inspiration was even Persian. These include: the mujannab furs [close to the Persian index] 162:149, the mujannab Zalzal [close to the Zalzal index] 54:49, the wusṭā furs [Persian middle finger] 81:68 and the wusṭā Zalzal [Zalzal middle finger] 54:44.Footnote 43 It should be noted that what the Greeks called “epimorios”, or superparticular ratios of the kind 1 + (1/n), that is to say, n + 1:n, were not appropriate for the Persian notes.

Thus the ears of the Persians and of al-Fārābī judged other intervals to be consonant, different from those of the Greeks, who generally limited themselves to the superparticular kinds or to multiples. Consequently, they were forced to exclude also the interval of the fourth plus the octave, because it was represented by the ratio 8:3. And yet it was so close to the fourth that even Ptolemy, following his ear, decided to admit it among the consonances. Following their famous Alexandrian, also al-Fārābī seems to have a bad judgement of Aristoxenus.

Nous avons montré par ce qui précède comment il nous faut envisager les notes et les intervalles pour en déterminer la valeur et la figurer à l'aide de nombres entiers; et aussi à quel point de vue il faut les concevoir pour les figurer à l'aide de nombres fractionnaires. Ce sont là deux manières de voir bien différentes; l'une est celle des Pythagoriciens, l'autre celle des Aristoxéniens. Il est facile d'en conclure quelle est la méthode à suivre en musique.Footnote 44

It should be understood that here, the Arabic-Persian theoretician was attributing the whole numbers to the Aristoxenians and the fractional numbers to the Pythagoreans. Because, once we have determined (with the ear) the equable semitone or the equable quarter of a tone, any interval will on this basis contain a whole number of them. Whereas the Pythagorean ratios continually produced fractional numbers.

Again taking his inspiration from Ptolemy, al-Fārābī was trying to reconcile attitudes that were difficult to follow together in his period and with the mathematical instruments that he had chosen: his ear and Greek numerical theories. For example:

On ne peut donc connaître les principes fondamentaux de la théorie musicale qu'à l'aide de la sensation, de l'expérience. […] Il nous importe seulement de savoir que la pratique musicale est antérieure de beaucoup à la théorie.

He criticised or praised other Arabs, unfortunately without naming them.

Parmi ceux qui ont voulu fixer dans des livres le nombre de toutes les notes reproduisibles à l'octave, il nous faut compter les mathématiciens grecs de l'antiquité et les théoriciens de l'empire arabe proche de notre temps. Parmi ces derniers, certains ont suivi la voie des mathématiciens grecs, les autres n'en ont tenu aucun compte; habiles praticiens, rompus à la musique, ils se sont uniquement fiés à leur oreille. […] Ils sont donc plus près de la vérité que ceux des théoriciens de notre époque qui ont voulu suivre la voie des mathématiciens grecs de l'antiquité. Ils les ont suivis, mais ils ne possédaient pas leur savoir ni l'expérience et le sens esthétique des praticiens plus récents.Footnote 45

At a certain point, he must have abandoned ratios, in order to consider the octave as a single quantity to be divided into 144 parts: “nous attribuons à l'octave la valeur 144 (considérant l'octave comme une quantité)”. After giving it this value, it was subdivided into a fourth and a fifth of 60/144 and 84/144, respectively. The fourth was then distinguished in various ways, depending on the genus: 24/144, 24/144, 12/144, that is to say, tone, tone, semitone [the diatonic genus]; 36/144, 12/144, 12/144, tone+semitone, semitone, semitone [chromatic]; 48/144, 6/144, 6/144, ditone, quarter tone, quarter tone [enharmonic]. It is clear that he now divided the octave into 12 semitones or 24 quarter tones which were all equal. Was he trying to translate Aristoxenus into numbers? Then he believed that he had to give up ratios. But why was he trying to do so? Because the music that entered his ears, on the lute, used notes that could not be inserted into orthodox Greek theory: the Persian notes, and those of Zalzal that we have seen above.Footnote 46

The context of the musical culture of the Arabs must have made itself felt also when al-Fārābī attributed the particular effects to each of the genera. The first genera were called “forts” because they are considered to create strong impressions, whereas the others are “doux”. Among the latter, “Les genres trop doux produisent une impression faible, superficielle, comparable à celle d'un dessin simplement ébauché; […] Certains peuples, dans l'antiquité, qualifiaient les genres doux de féminins, rappelant la douceur de la femme; par opposition, les genres forts étaient appelés masculins.”Footnote 47

But this treatment of the theory should have seemed superficial (feminine?) to the Persian scholar who had emigrated to Baghdad, and so he immediately continued “d'une façon plus approfondie”. In the end, he, too, would choose to trust the Greek musical theory based on proportions.

Notre oreille à elle seule ne nous permet pas de définir tous les états d'une note; la théorie de son côté ne nous fournit pas les moyens de reconnaître si une note est naturelle ou non. Il nous faut donc avoir, en même temps, recours à la théorie et à la pratique musicales. […] En un mot, la théorie et la pratique musicales se complètent l'une l'autre, et leur ensemble constitue la science musicale.Footnote 48

Al-Fārābī split the Pythagorean tone 9:8 into two parts, 18:17 and 17:16, or into four parts, 36:35, 35:34, 34:33, and 33:32, which were clearly not equal.Footnote 49 It is clear that he did not want to use the geometrical mean of 9 and 8, that is to say, $9:\sqrt{9 \times 8} = \sqrt{9 \times 8} : 8$. He could certainly have calculated it, like everybody else, and he undoubtedly knew that the incommensurable $\sqrt{2}$ was hidden in there. He therefore continued to follow Greek orthodoxy, which, as is known, from Euclid to Ptolemy had excluded its use. Why? If he had not done so any longer, he would have found the most effective way of all to reconcile hearing with the theory and the practice of Persian (and, as we know, non-Persian) musicians. But the new mathematical model would have raised two problems for him: he would have abandoned Greek mathematical orthodoxy that did not consider $\sqrt{2}$ to be a number, and believed that all ratios deriving from it were unbearably dissonant in music. Furthermore, he would have given too much space to “doux” genera, which would undoubtedly have worsened the judgements of the most orthodox ʿulamāʾ regarding music.Footnote 50 These are all hypotheses that we can advance to explain the lack of a passage which would otherwise have been relatively easy and natural.

And yet, nevertheless, this Persian theoretician was to offer us something very different from the tradition found in Greek books. “Le but de toute théorie est de nous faire atteindre la vérité. Or, la vérité est une croyance conforme à la réalité objective.” The means to attain it consisted in constructing a suitable musical instrument with fifteen strings. The notes could be verified on this: “leur existence doit donc pouvoir nous être confirmée par la sensation.” He judged it necessary to “décrire un instrument imaginé par un ancien [Ptolemy?Footnote 51] et conçu d'une façon générale. Après l'avoir construit et doté des corps susceptibles de produire des notes selon leur rang et avec les qualités que nous leur avons attribuées, on lui fera rendre ces notes, conformément à ce qui a été exposé.” He was even worried about any possible “bourdonnement” that might disturb the test.Footnote 52

Al-Fārābī confronted us with a veritable experimental apparatus in order to verify whether his mathematical model for music was in harmony with the practice of auditory sensations. It should thus be compared with the philosophy of scientific experimentation proposed by Francis Bacon in Europe a few centuries later.Footnote 53 And we may note the greater delicacy of the experiment, which is suitable to penetrate the secrets of nature, proposed by the Arab-Persian, compared with the well-known, aggressive dissecare naturam [cutting nature to pieces] of the English.Footnote 54 Where would a scholar as fond of classifications as al-Fārābī, and famous for those of the sciences,Footnote 55 have placed music?

[…] pour les mathématiciens, la musique est une partie des mathématiques. Faut-il donc alors rechercher le but de l'existence de chacune des choses qui appartiennent à la musique? La philosophie mathématique ne s'enquiert pas, selon l'avis de tous, du pourquoi des choses qui lui appartiennent. Elle ne nous donne, dans ses définitions, qu'une seule des quatre espèces de causes, elle nous dit ce qu'est la chose définie. Quant aux autres espèces de causes, et en particulier aux causes finales et à celle qui répond à la question pourquoi est la chose, on ne s'en occupe pas en mathématique.

However, here he had given his “Éléments de l'Art Musical”, establishing its principles. Anyone who masters them “saura remonter, […], vers les causes des principes que la musique a tirés de l'expérience et de la sensation. Il sera capable aussi de distinguer ce qu'il y a de juste et de faux dans les dires des divers théoriciens”.Footnote 56

Our Arab-Persian writer also tells us how they used a different method to tune the ṭunbūr Footnote 57 at Baghdad or in Khurasan. The former was “le plus en faveur dans la contrée où nous écrivons cet ouvrage”. For this, he proposed a distribution of frets and tunings, following the principles so far expounded, but different from the one that was in use at the time. The instrument must have been derived from a pre-Islamic tradition, because its frets were called jāhiliyya [infidels]. They were generally placed on the neck at the same distance. Our theoretical musician judged it “un instrument incomplet dans sa première disposition, et nous avons montré comment on peut le compléter et le parfaire.” This ṭunbūr was considered to be suitable for “doux, relâchés” genera; its system of tuning was defined as “féminin”. In the traditional arrangement of the frets, it was capable of playing microintervals. A chord of the length of 40 was taken, and 7/8 of this was cut off, obtaining 35. The other frets were distributed at equal distances: 36, 37, 38, 39. In this way, intervals quite distant from the classic Pythagorean ones were generated: 40 to 39 (as a fraction 1.0256), which was fairly close to the tempered quarter of a tone (1.0293); 40 to 38 (1.0526), which is close to the semitone (1.0594); 40 to 36 = 10 to 9 (1.1111), which is less comparable to the tempered tone (1.1224), whereas the Pythagorean tone corresponds to 9 to 8 (1.125). The ṭunbūr of Khurasan, on the contrary, was considered more suitable for the ‘masculine’, “strong” genera. Its tuning was left in the Pythagorean manner, and its notes enjoyed a correspondence with those of the lute.Footnote 58

The theory of music expounded in the Fourth and Thirty-first of the Fifty-two Rasāʾil Ikhwān al-Ṣafāʾ [Epistles of the Brethren of Purity] was explicitly full of elements taken from the Pythagorean sects, with even a direct reference to a mythical Pythagoras.Footnote 59 Both Ibn Sīnā and al-Fārābī criticised it, rejecting its Pythagorean-Platonic music of the heavenly spheres.Footnote 60 In the few available pages presented above, al-Khayyām completely ignored it.

III. COMPARISONS WITH GREEK AND OTHER THEORIES OF MUSIC

In the pages dedicated by Ptolemy to musical harmonies, it can be seen how far Greek theory was from any interest in tuning pipes for wind instruments. He avoided listening to the sounds of the “αύλων, tibia” [flute] or those obtained by hanging weights on the strings.

Nam, in tibiis & fistulis; praeterquam quod sit admodum difficile omnem irregularitatem inibi cavere: et am termini, ad quos sunt exigenda longitudines, latitudinem quandam admittunt indefinitam: atque (in universum) Instrumentorum inflatilium pleraque, inordinatum aliquid adjunctum habent; & praeter ipsas spiritus injectiones. [For in flutes and pipes, besides the great difficulty in avoiding all irregolarities, the terms, whose length we must evaluate, admit a certain indefinite breadth; and (in general) the vast majority of wind instruments also have something unruly, as well as the breathing introductions.]Footnote 61

On the contrary, al-Fārābī made some attempts also with flutes.

L'acuité et la gravité dépendent aussi du diamètre du conduit tubulaire que l'air traverse. En effet, plus ce conduit est étroit, plus les molécules de l'air qui le traversent sont serrées; et plus les chocs qui se produisent entre elles et contre les parois, sont violents. Le son naît alors plus aigu. Au contraire, plus ce conduit est large […] est alors plus grave.

Initially, he considered the two factors that determine the pitch of a sound to be independent. Pipes of the same diameter would emit notes in proportion to their length. Furthermore, for him, pipes of the same length would emit notes in proportion to their diameter. But then he also wrote: “Toutes ces causes peuvent s'ajouter les unes aux autres et concourir à déterminer le degré des notes.” Thus, in the end, he prudently concluded, echoing Ptolemy: “On ne peut guère fixer la place des notes sur des instruments de ce genre en se guidant sur l'échelle des autres [stringed] instruments”.Footnote 62 This is the only part of the Kitāb al-Mūsīqā where the ratios between the magnitudes involved (length and circumference) were not fixed with the precision of numbers, in order to obtain the values of the heights for the notes. For a musician who used his own ears, the orthodox Greek theory based on the monochord could not be extended directly to the pipes of wind instruments, and in any case, al-Fārābī did not do so. Thus, the only ancient mathematical theory, to my knowledge, for sounds generated by pipes, that took into adequate account the effect of the diameter on the tone is that of the Chinese.

The Chinese mathematical theory of music was based on lülü [standard pipes], for which, starting from the 1st century A.D., precise dimensions were given in Qian Han Shu [Former Han Books], so that they would emit the required notes. As they were solid, the lülü had not only a length, but also a diameter. Even though other Chinese theories existed, which maintained the same diameter, varying only the length, in the pages of the Han Shu also the diameter was changed to generate three different notes. It should be noted that the diameter followed the same ratio as the length. These literati had invented a mathematical rule that satisfied the ears. If the note Huangzhong [yellow bell] is 90 fen Footnote 63 long, and 9 fen open, in order to get the Linzhong [bell of the forest], it is not sufficient to shorten only the length by 2 : 3, that is, to 60 fen; otherwise, the pitch would be less high than required by the ears for this note (the Western fifth). Good tuning requires that also the diameter should be shortened by the same 2 : 3, to 6 fen. The same rule generated the Dazu [big bunch], with an increase from 60 to 80, applying the ratio 4 : 3, both in length and in diameter.

It was extremely important for Confucius, and for the relative imperial rituality, that music was based on pipes, rather than on strings, as it was for the Arabs and the Greeks, because it was through them that the energy-giving vital breath called qi was to pass. This was a central factor for the classic Chinese cosmogony and their conception of the world, where its role was to regulate every phenomenon, and not only the natural ones. For historians, it is interesting that, to my knowledge, no other culture ever invented such a way of tuning instruments based on pipes.Footnote 64

Furthermore, no other culture ever attempted a mathematical procedure that worked so well, even for pipes. In Europe, only a scholar who departed sufficiently from the prevailing Pythagorean tradition, e.g. from Boetius, reappraising Aristoxenus, would have dealt in the same way with wind instruments. Thus, more than 1,000 years were to pass until Vincenzio Galilei, in the 16th century, explored the question of pipes. He considered them to be solids, and therefore it was necessary to measure their volume, and not only their length. As a result, the selected ratios were different from those of Pythagoras, because they required the cubes of the numbers. But this lute player and composer, the father of Galileo Galilei, had to battle against the European environment of his epoch, which was still largely Pythagorean, and therefore hostile to other musical theories.Footnote 65

Relationships between the Arabs and the Indians were intense, and are well documented, as is well known.Footnote 66 However, their ideas regarding music appeared to be so different that there were practically no exchanges in this field. It is sufficient to recall here that in the most ancient Indian texts, music certainly did occupy a pre-eminent position, but its theory appears to have ignored ganita [mathematical calculation]. The rules for intoning the gita [songs], were handed down directly from the brahmin to his disciple, and were therefore based, above all, on the ear.

Bharata wrote in the Gitalamkara [Rhetoric of singing] (perhaps a few centuries B.C.) about three sthana [locations, registers].

Syllables can be [pronounced] in three sthana: from the chest, from the throat, from the head. People learned in the Scriptures say they are named after the three libations [the drink-offerings to the Gods, the soma, in the morning, at noon, and in the evening].

Unfortunately, at this central point, the text is mutilated, and annotators completed it by following books of phonetics or music much later.Footnote 67

The Natya Śastra [Science of drama] (first few centuries A.D.) explained how to tune stringed instruments.

These notes become low or high, according to the adjustment of the strings, and the diversity of the danda [neck, measuring 120 cm] of the viṇa, and of the sense organs. […] The difference that occurs in pa [pancama, the fifth note], when it is raised or lowered by one śruti [hearable], and when the consequential marda [slackness] or ayatatva [tension] of strings occurs, will indicate a typical śruti. We shall explain the system of these śruti. Two viṇas should be made ready, with the dandas, strings, of a similar measure, and with a similar adjustment of the latter in the ṣadja grama [gamut on ṣadja].Footnote 68 Then one of these should be tuned in the madhyama grama [gamut on madhyama],Footnote 69 by lowering pa by one śruti. The same viṇa, by adding one to pa, will be tuned in the ṣadja grama. This is the meaning of decreasing a śruti. […] The notes of the vamśa [flute] should be perfected, and accomplished with the help of the viṇa and of the human throat […] A unison of the human throat, the viṇa, and the flute is specially praised.

Sound is airy [i.e. air-dependent], and it is considered to be of two kinds, one equipped with svaras [notes], and the other with abhidhana [name-words with meaning] […] Seven svaras have been proclaimed in the viṇa as well as in human vocal cords. The same are produced in atodyas [instruments] as well. Notes coming out of the human body are transmitted to the wooden viṇa, then to the puṣkara [drums], and ultimately to the ghana [solid instruments] […].

There are three sthana [locations, registers]: 1) the uras [chest], 2) the kaṇtha [throat], and 3) the śiras [head].Footnote 70

Here, to tune the viṇa, one took into account the length of the strings, adjusting their tension. The musician let his “sense organs” guide him, and he added or took away these “hearable” microintervals called śruti. The sthana was divided into 22 śruti. I must caution readers not to equate the sthana simply with the Greek octave. The relative sets of seven svara [notes] were called a saptaka [septenary], and not an aṣtaka [octonary]. In India, the notes are not arranged along a scale, but around a grama [village].Footnote 71 Al-Fārābī alluded to the origins, in the three parts of the human body, of the Indian sthana in order to criticise them.Footnote 72

Śarngadeva (between the 9th and 13th centuries A.D.) wrote the Samgita Ratnakara [Mine of gems for singing]. The book is best preserved in the comment of Simhabhupala (13th century A.D.). A passage from it is pertinent here. “In practice, there are three parts: in the heart [the octave] considered mandra [grave, low], in the throat the madhya [middle] one, in the head the tara [acute, high] one, each doubly ascending.”Footnote 73

I have not been able to find any previous text with a precise statement about an octave described as “the double”, that is to say, in a proportion of 2 to 1. Did a mathematical octave appear only at this time, in the 13th century, in India? Anyway, it is only when one should divide the octave into 22 śruti, that these could become quarter-tones.

At this time, in the North of India, was there not already, perhaps, a strong presence of an Arabic culture with its Greek-inspired mathematical theory of music? Right from its origins, the Indian theory of music did not share the numbers and proportions with that of the Greeks. The former could have been worked out only by means of the ear, the latter (in its orthodox form) derived its numbers from the heaven of ideas.Footnote 74 In Greece, only Aristoxenus followed the judgments of his ears, and divided the octave also into (equal) quarter-tones. Had Indian scholars known him, perhaps they would have appreciated this unorthodox Greek side of the story. But it is improbable that Arabic scholars brought him to India, as we saw above. The fact that Brahmins and scholars preferred to trust the accuracy of their ears, rather than numbers, should not become a shortcoming, or (worse) a defect. It is a matter of difference. Anyway, it is a puzzle. Why did the famous inventors of our universal digits not use numbers in their theory of music?

The Śatapatha Brahmana [Brahmana of a Hundred Paths] gave the exact geometrical measurements to build the female altar vedi [earth], and the male altar agni [fire]. Here, exactness was crucial, otherwise the rite would prove to be wrong, and ineffective. Therefore Brahmins developed rules to protect their precise liturgy. At the beginning, these were handed down from master to disciple orally, and later they were written down in books. The Śulbasutra [Aphorisms of the string] (controversal dating, perhaps 3rd century B.C.) gave rules to be precise in the construction of altars. In these pages, we find the oldest examples of Indian geometry. There, sometimes, we can even read proofs of the relative rules. For example, to compute the area of a trapezoidal Mahavedi, the calculation was justified by a geometric proof.Footnote 75

The Sama Veda [Knowledge of melodies] is sung. The ritual mantra [instruments to think, formulas] must be pronounced in the traditional way, and intoned exactly as prescribed. The well-known Panini's Astadhyayi [Collection in eight sections] (5th-4th century B.C.) gave rules of phonetics and grammar. But how to sing would seem to be a subject to learn directly from the master. That is to say, the shape of the altars in Vedic rituals was fixed by geometry, and a few numbers. But numerical ratios were not used in the art of singing mantra, to make the musical performance precise. Hence the famous phrase, “As the crest on the head of peacocks, as the gems on the hoods of snakes, so is ganita at the top of sciences known as the Vedanga [Limbs of the Veda]”,Footnote 76 should be completed with: “Like peacocks, in India, so mathematical sciences seem to be birds that did not sing.”Footnote 77 Pending the decision of Indian historians to dig up the thousand texts still buried in their archives, here we cannot go any farther.

IV. BACK TO AL-KHAYYĀM

Let us now return to al-Khayyām, enriched by the idea that the context, which was the background of the extant pages of his theory, was thus complex, and rich in problems and cultural varieties. But until a more complete version of the book is found, we cannot know in what different way the Persian mathematician, astronomer and poet would have dealt with them. We can only suspect that whereas al-Fārābī may have timidly drawn back, conforming to the Euclidean-Ptolemaic and religious orthodoxy in the matter of music, perhaps al-Khayyām would have dared to depart somewhat more from it. We know that he, more than anyone else, ventured beyond the prohibitions that Greek scholars had imposed regarding the numerical representation of incommensurable magnitudes. Consequently, in his Ṣharh mā ashkala min muṣādarāt Kitāb Uqlīdis [Commentary on the difficulties of certain postulates of Euclid's work], he considerably expanded the current notion of number.

Et l'on ne devra pas concevoir la grandeur G en tant qu'elle est une ligne, une surface, un solide ou un temps. Il faudra au contraire la concevoir en tant qu'elle est abstraite dans l'intellect de ces caractères adjoints, et en tant qu'elle est rattachée au nombre: non en tant que nombre absolu véritable, car le rapport entre A et B pourrait ne pas être numérique, de sorte que l'on ne pourra pas trouver deux nombres selon leur rapport. […] Il faudra donc que l'on sache […] que la grandeur G doit être considérée comme un nombre, comme nous l'avons mentionné, quelque grandeur qu'elle soit.Footnote 78

But as regards the question of how far ʿUmar al-Khayyām was prepared to use his surds in his theories of music, thus completely abandoning Euclidean-Ptolemaic orthodoxy, we can only advance hypotheses at the moment. We thus stop here, in the hope that al-Khayyām's book on music will not prove to be a “Farmer's ghost”.Footnote 79 Without doubt, however, his extant books reveal a sufficiently creative personality, not afraid of proposing even radical innovations to the tradition of Greek mathematics. Why should he not have done so, following his ear and his tastes, also in the music with which he very probably freely accompanied his Quatrains?

ACKNOWLEDGEMENTS

Our thanks are due to Chiara Letta and Roberto Lorenzi, of the Library of Physics, Information Sciences and Mathematics of the University of Pisa, for their skill and promptness in obtaining some of the texts quoted, and to Fausto Soriani for changing TeX into Word. We would like to express our thanks to Ron Packham for conferring an impeccable English style on the manuscript. We are also grateful to the referees, who have forced us to improve the text, thanks to their criticism.

This work is dedicated to the mathematician Franco Giannessi, the mathematician and pianist Giovanni Gronchi, and our friend, the composer and pianist Paolo Marzocchi.

APPENDIX

The Species of the Fourth by al-Khayyām

Diatonic

  1. 1. 64 -(8:7)- 56 -(8:7)- 49 -(49:48)- 48

  2. 2. 324 -(9:8)- 288 -(9:8)- 256 -(256:243)- 243

  3. 3. 100 -(10:9)- 90 -(10:9)- 81 -(81:75)- 75

  4. 4. 72 -(8:7)- 63 -(9:8)- 56 -(28:27)- 54

  5. 5. 180 -(9:8)- 160 -(10:9)- 144 -(16:15)- 135

  6. 6. 220 -(10:9)- 198 -(11:10)- 180 -(12:11)- 165

  7. 7. 80 -(8:7)- 70 -(10:9)- 63 -(21:20)- 60

  8. 8. 396 -(9:8)- 352 -(11:10)- 320 -(320:297)- 297

  9. 9. 156 -(13:12)- 144 -(9:8)- 128 -(128:117)- 117

  10. 10. 16 -(8:7)- 14 -(14:13)- 13 -(13:12)- 12

Chromatic

  1. 11. 24 -(6:5)- 20 -(20:19)- 19 -(19:18)- 18

  2. 12. 36 -(6:5)- 30 -(15:14)- 28 -(28:27)- 27

  3. 13. 48 -(6:5)- 40 -(40:39)- 39 -(13:12)- 36 *

  4. 14. 60 -(6:5)- 50 -(25:24)- 48 -(16:15)- 45

  5. 15. 16 -(16:15)- 15 -(15:14)- 14 -(7:6)- 12

  6. 16. 28 -(7:6)- 24 -(12:11)- 22 -(22:21)- 21

  7. 17. 40 -(10:9)- 36 -(36:35)- 35 -(7:6)- 30

Enharmonic

  1. 18. 40 -(5:4)- 32 -(32:31)- 31 -(31:30)- 30

  2. 19. 100 -(5:4)- 80 -(40:39)- 78 -(26:25)- 75 *

  3. 20. 140 -(5:4)- 112 -(28:27)- 108 -(36:35)- 105

  4. 21. 60 -(5:4)- 48 -(24:23)- 46 -(46:45)- 45

The * is used to indicate the species that we have not found, either in al-Fārābī's Kitāb al-Mūsīqā, or in Ibn Sīnā's al-Shifāʾ in the same order of ratios.Footnote 80

References

1 Humai, Jamal al-Din, Khayyam-Nama (Teheran, 1967), pp. 340–344Google 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. 323–334Google 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. 296–297Google 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. 206–207Google 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): 149–200CrossRefGoogle 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): 129–166Google 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. 376–377Google 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): 247–263CrossRefGoogle 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. 71–74 and 134–140Google Scholar ; Zhukovsky, V. A., ‘Omar Khayyam and the wandering quatrains’, Journal of the Royal Asiatic Society, 30 (1898): 349–366Google 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. 421–477Google 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): 25–62CrossRefGoogle 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): 325–333CrossRefGoogle 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. 39–50 and 61–62Google 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, 1945–1988), I, pp. 341–342Google 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. 139–149Google 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): 79–89Google 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): 365–422Google 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): 143–163Google 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): 223–244Google 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. 908–928Google 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, 24–25, 29–30, 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): 123–130CrossRefGoogle 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.