Musical Mathematics

on the art and science of acoustic instruments


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Part IV: Arabian, Persian, and Turkish Music


Section 11.52


          With respect to fretted lutes, the term hamila signifies a stationary bridge. However, on stringed instruments without frets, hamila also refers to a moveable bridge situated between two immovable nuts. In his treatise entitled Kitab al-kafi fi’l-musiqi (Book of sufficiency on music),[1] Al-Husain ibn Zaila (d. 1048) observes


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          And of those (instruments) possessed of strings without frets to determine the places (pitch) of the notes, but whose difference between the places (pitch) is in the length or shortness of the string itself, as in the sanj and the shahrud, or in the length or shortness of the string and the similarity of the bridges (hamilat) and the supports (amida) as in the anqa.[2] (Bold italics mine. Text in parentheses in Farmer’s translation.)


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Furthermore, in a passage quoted later from a treatise entitled Risala fi’l-musiqi (Treatise on music),[3] Abu l-Salt describes the placement of a hamila under a canon string. These definitions and references are of paramount importance because they lead to irrefutable theoretical and musical connections between the Arabian canon and the Arabian lute. For example, significant passages in the treatises of Al-Farabi (d. c. 950), Ibn Sina (980–1037), and Abu l-Salt (d. 1134) indicate that Arabian music theorists and musicians considered a dastan (Persian, lit. fret) on the fingerboard of an ‘ud as having the same mathematical and musical function as a hamila (Arabic, lit. carrier) on the soundboard of a qanun. Here the Arabicized word qanun does not refer to the modern zither, built in the form of a trapezoid and equipped with strings of different lengths, but rather to the ancient Greek kanon, described at length in Ptolemy’s Harmonics.[4] Before we discuss these passages in full detail, let us first acquaint ourselves with an Arabian version of this instrument as described by Al-Jurjani (d. 1413):


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On the construction of an instrument to test the rules of music.


     Having shown everything that relates to the elements of melody: the notes, the small intervals, the genera . . . all that remains is for us to explain the large elements, those whose role, in a melody, is comparable to that of verses in a poem. . . .


     You know now, in theory, everything that relates to the ratios of notes. If you want to realize [the theory] through experience and sensation, you need only to play an instrument; I am going to tell you how to build it. It will allow you to easily distinguish consonance from dissonance, whether it concerns notes, intervals, genera, or systems.


     Here are the requirements for building the instrument.


     Take a wood frame with four sides . . . at right angles, in the shape of a parallelepiped. It may be either rectangular or square. Choose one of the sides of this frame for the base [right side] of the instrument, and the one that is opposite it, for the top [left side]; if it is a rectangle, these will be the two short sides. These two sides must, however, be long enough to accommodate a line of fifteen tuning pegs. Cover the face [soundboard] of the instrument with a flat surface made of thin wood, and its back [bottom], with a surface of curved wood.


     Fasten a semi-cylindrical rod, made of ivory or hardwood, and of the thickness of at least one finger, along the edge of the face, on the [left side]. On the other edge of the face, on the [right side], attach a type of [string holder], similar to that of the lute; or else place fifteen hitch pins at the [right side] of the instrument, [along] the side of the face, in the same way that one attaches the hitch pin[s] of the tunbur. Place each one of these hitch pins opposite one of the fifteen tuning pegs that are aligned [along] the [left side].


     Next, equip the instrument with strings. Coming from the hitch pins, or the holes made in the string holder, these strings go towards the tuning pegs, after passing over the semi-cylindrical rod through notches cut opposite the tuning pegs on one side, and opposite the string holder holes or hitch pins on the other.


     Tension the strings equally so that they all produce identical notes.


     Then make a ruler having the length of the distance from the [left side] to the [right side], or a little more; and determine on it a distance equal to the vibrating section of the [open] strings. Divide the edge of this ruler according to the various divisions . . . of a particular genus . . . ; and inscribe on each division the sign of the note to which it corresponds.


     Next, make fourteen bridges of ivory or of hardwood. The base of these bridges must be flat so that they can be positioned at a right angle to the face of the instrument and stay perfectly fixed there. The top of each of these bridges, i.e., their surface over which the string passes, must be rounded and must include one notch, so that its contact with the string approaches the ideal contact of a point with a straight line. The top of the bridges must be slightly higher than the semi-cylindrical [rod].


     Then determine on the strings the points that correspond to each of the divisions on the ruler. Move the bridges to place them in line with each of these points, so as to make each of the strings produce one of the notes of the desired system. Once this is done, when the strings are set in motion, they will make audible the system that we have organized . . . ; all that we know in theory will then be made perceptible to the ear, and we will thus be able to realize, through sensation, the consonance of intervals, and their dissonance . . .


     We can, without difficulty, call this a perfect instrument; in fact, it serves the theory as completely as the practice of the Art of Music.[5] (Text in brackets mine. Bold italics in brackets my correction of the “cordier” translation error in La Musique Arabe.)


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          Since no illustrations of ancient Arabian canons have survived, Figures 11.46(a) and 11.46(b) depict my interpretations of these two different canon descriptions. Many years of canon building and playing have sensitized me to recognize key passages in Al-Jurjani’s text. In my opinion, the following quotations attest to Al-Jurjani’s direct experiences with such instruments. I find his observations (1) that the contact of a bridge “. . . with the string approaches the ideal contact of a point with a straight line . . . ”, and (2) that “. . . the top of the bridges must be slightly higher than the semi-cylindrical [rod] . . .” especially noteworthy. Observation 1 has two consequences: a triangular-shaped bridge with a sharp point (i) gives an accurate pitch at the exact measured location along a string’s length; in contrast, a wide or blunt point has the effect of shortening the string, which in turn causes an inadvertent increase in the string’s frequency. (ii) A sharp point also produces accurate length ratios on both sides of the bridge. (See Chapter 3, Figures 15 and 17, and Chapter 12, Section 3.) Observation 2 is a caveat against using a bridge that is too high; such a bridge stretches a string in an upward direction, which effectively increases the string’s tension, and thereby causes an unintentional increase in frequency.





          Now, let us consider a less detailed but equally significant description of a canon by Abu l-Salt, who lived approximately 300 years before Al-Jurjani. Abu l-Salt was born in Andalusia in 1067, and spent most of his life in Egypt and Tunis. His treatise on music has survived in a Hebrew translation; the Arabic original is lost. Hanoch Avenary, who translated this treatise into English,[6] states that it was a part of a much larger opus entitled Sefer ba-Haspaqah (Book of Sufficiency), a work of encyclopedic proportions. In any event, the Hebrew text gives the following title on the first page: “Fourth Discipline of the Second Section: The Science of Music.” Apparently, the “fourth discipline” refers to the quadrivium, or to the four branches of mathematics: arithmetic, geometry, astronomy, and music. As such, the quadrivium was taught in all leading Medieval and Renaissance universities throughout Europe. In Chapter 4, Paragraph 1, Abu l-Salt states


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     When you want to determine the above-mentioned intervals by means of the [measuring-] cord, take for that purpose two strings of the same substance, equal with regard to all the qualities of length, thickness and smoothness, and assemble them on any instrument. Divide the surface of the instrument in parallel to one of them according to the ratio of any System and Genus you wish, and inscribe marks at the single dividing points; these will be on the straight line parallel to the string, and will determine the place of the frets* when they are touched. Next put your finger on the place of any note, and touch the other [string] by putting your finger wherever you wish: thus you will hear the notes of the intervals accurately . . . You can hear them [the notes of an interval] also from one string alone which is divided lengthwise according to the demanded ratios. Touch it as an open string, then put the finger on the place of the required note, and touch a second time. However, this transition of the hand formerly mentioned is difficult and troublesome on this instrument. The most advantageous and perfect [method] is to take 15 strings, equal in every respect, and to stretch them on a rectangular instrument. Then take a measuring stick [al-mistara] as long as one of them, and divide it according to the intervals of that System which you want to investigate by means of the [measuring-] cord. Then fix it alongside the string next to the first string from which the first note shall be sounded. At the place of the second of the fifteen notes put a bridge [hamila] made of a body with a broad seat and a sharp edge, and let the second string ride on it in a notch [fi tahziz] at the upper end. Fix the measuring stick near the third string as well. Let the third note be above it, and in this manner go on till the last string. This is the instrument on which all the mentioned notes can be heard, and it is the most perfect of all the given instruments and contains their entire compass.


     *On p. 53 the translators note, “In the original text, the term fret or stopping place is always rendered by the Persian-Arabic designation al-dastan.”[7] (Bold italics mine.)


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          In this quote, Abu l-Salt establishes the mechanical and conceptual interchangeability of the dastan and the hamila. Both components serve exactly the same function, namely, to provide a mechanical means for stopping a string at a theoretically infinite number of measured locations along its vibrating length. Since the hamila of the kanon is much older than the dastan of the ‘ud, the intriguing possibility exists that the inventors-developers of the fretted ‘ud consciously conceived of this instrument as a highly practical and portable kanon. In any event, when viewed from this perspective, the evolutionary process from kanon to ‘ud is both profound and unique because it stems from a desire to bestow on the ‘ud the tuning accuracy and tuning flexibility for which all kanons are known.



[1] Farmer, H.G. (1965). The Sources of Arabian Music, p. 38. E.J. Brill, Leiden, Netherlands.



[2] Farmer, H.G. (1978). Studies in Oriental Musical Instruments, First and Second Series. First Series, p. 8. Longwood Press Ltd., Tortola, British Virgin Islands.



[3] The Sources of Arabian Music, p. 41.



[4] Barker, A., Translator (1989). Greek Musical Writings, Volume 2. Cambridge University Press, Cambridge, England.



[5] Forster Translation: in La Musique Arabe, Volume 3, pp. 360–362.


D’Erlanger, R., Bakkouch, ‘A.‘A., and Al-Sanusi, M., Translators (Vol. 1, 1930; Vol. 2, 1935; Vol. 3, 1938; Vol. 4, 1939; Vol. 5, 1949; Vol. 6, 1959). La Musique Arabe. Librairie Orientaliste Paul Geuthner, Paris, France.


Volume 1: Kitab al-musiqi al-kabir, Books 1 and 2, by Al-Farabi.


Volume 2: Kitab al-musiqi al-kabir, Book 3, by Al-Farabi.

Volume 2: Kitab al-shifa’, by Ibn Sina.


Volume 3: Risalat al-Sharafiya fi’l-nisab al-ta’lifiya, by Safi Al-Din.

Volume 3: Sharh Maulana Mubarak Shah [on the Kitab al-adwar], by Al-Jurjani.


Volume 4: Risala fiilm al-musiqi, by Al-Shirwani.

Volume 4: Risalat al-fathiya fi’l-musiqi, by Al-Ladhiqi.


Volume 5: Attempt at codification of the customary rules of modern Arabic music:

Volume 5: General scale[s] of sounds.

Volume 5: Modal system.


Volume 6: Attempt at codification of the customary rules of modern Arabic music:

Volume 6: The rhythmic system.

Volume 6: Various forms of artistic composition.


     In The Sources of Arabian Music, Farmer credits Al-Sanusi as translator of all six treatises contained in Volumes 1–4.


Avenary, H., Translator (1974). The Hebrew version of Abu l-Salt’s treatise on music. Yuval III, pp. 7–82.



[7] Ibid., p. 52.