on the art and science of acoustic instruments
Also available from the publisher at Chronicle Books, San Francisco.
2000–2013 Cristiano M.L. Forster
All rights reserved.
CHAPTER 11: WORLD TUNINGS
Part IV: Arabian, Persian, and Turkish Music
North American musicians who do not read Arabic, French, or German have very limited opportunities to study ancient Arabian music and tuning theory from original sources. Of the treatises on music written by Al-Farabi (d. c. 950), Ibn Sina (980–1037), Safi Al-Din (d. 1294), Al-Jurjani (d. 1413), Al-Ladhiqi (d. 1494), and Al-Shirwani (d. 1626), not a single work has ever been translated into English. Furthermore, due to intractable religious, linguistic, and intellectual prejudices against Islam, Christian-dominated institutions throughout Europe — such as Catholic and Protestant churches, schools and universities, and the craft guilds — managed by 1600 to completely eradicate the Arabian influence from the written history of European music. For example, the works of Michael Praetorius (1571–1621) and Marin Mersenne (1588–1648) offer no information on the origins of one of the most important instruments of their time: the lute. First in Arabian history (from approximately 700) and later in European history (to approximately 1700), the fretted lute served for a thousand years as an instrument of scientific exploration and musical expression. Henry George Farmer (1882–1965), an eminent historian of Arabian music, gives this etymology of the lute (from the Arabic al-‘ud, the lute; lit. flexible stick or branch):
* * ↓ * *
Western Europe owes both the instrument and its name to the Arabic al-‘ud, as we see in the Portuguese alaud, the Spanish laud, the German Laute, the Dutch Luit, the Danish Lut, the Italian liuto, the English lute, and the French luth.
* * ↑ * *
Are we to naïvely accept the highly improbable possibility that while Europeans inherited the lute from the Arabs, European musicians learned absolutely nothing about tuning from Arabian musicians? Consider the following fact: by the end of the 13th century, Arabian literature included not only a voluminous and highly sophisticated collection of works on the art and science of music, but on the precise mathematics of lute tunings as well.
Between approximately a.d. 750 and 1250, many nations in the West experienced the religious, scientific, and artistic influences of what I call the Arabian Renaissance. After the life and death of the prophet Mohammed (c. 570 – d. 632), a stunning series of military campaigns brought Spain, Sicily, North Africa, Egypt, Syria, al-‘Iraq, Persia (modern Iran), Farghanah (Central Asia), Tukharistan (modern Afghanistan), and Western India (modern Pakistan) under Moslem control. Coincidentally, most of these territories were conquered by Alexander the Great (356–323 b.c.) a thousand years earlier. To administer their newly conquered empire, Moslem rulers created two great cultural centers. In 762, Baghdad became the capital of the empire in the east, and subsequent to the invasion of 711 into Spain, in 756, Cordova became the capital of the empire in the west. The former was destroyed by Mongols in 1258, and the latter, reconquered by Christians in 1236. The Reconquista (Reconquest) of Spain continued until the final defeat of the Moslems at Granada in 1492.
Reminiscent of the building of Alexandria by Alexander the Great, Baghdad and Cordova boasted running water, paved and lighted streets, world-renowned architectural monuments, international markets, universities, hospitals, and above all, libraries that contained hundreds of thousands of volumes. If it were not for these libraries, and the care Arabian translators and scholars bestowed on ancient texts, the works of Homer, Hippocrates, Plato, Aristotle, Euclid, Archimedes, Nicomachus, and Ptolemy, to name only a few, would probably not have survived. The task of translating these volumes began in Baghdad in approximately 750, and later became centralized at the famous Bayt al-Hikmah (House of Wisdom) in 830. By the end of the 10th century, most of the translations were completed. This phenomenal achievement raises the inevitable question, “Is the Italian Renaissance indebted to the Arabian Renaissance?” To contemplate the profound interdependence of these two civilizations, consider this biographical account from Philip K. Hitti’s exhaustive work entitled History of the Arabs:
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Al-Kindi . . . flourished in Baghdad. His pure Arabian descent earned him the title “the philosopher of the Arabs,” and indeed he was the first and last example of an Aristotelian student in the Eastern caliphate who sprang from Arabian stock. Eclectic in his system, Al-Kindi endeavored in Neo-Platonic fashion to combine the views of Plato and Aristotle and regarded the Neo-Pythagorean mathematics as the basis of all science. Al-Kindi was more than a philosopher. He was astrologer, alchemist, optician and music theorist. No less than two hundred and sixty-five works are ascribed to him, but most of them unhappily have been lost. His principal work on geometrical and physiological optics, based on the Optics of Euclid in Theon’s recension, was widely used in both East and West until superseded by the greater work of ibn-al-Haytham [d. c. 1039]. In its Latin translation, De aspectibus, it influenced Roger Bacon [c. 1214 – d. 1292]. Al-Kindi’s three or four treatises on the theory of music are the earliest extant works in Arabic showing the influence of Greek writers on that subject. In one of these treatises Al-Kindi describes rhythm ( iqa‘ ) as a constituent part of Arabic music. Measured song, or mensural music, must therefore have been known to the Moslems centuries before it was introduced into Christian Europe. Of Al-Kindi’s writings more have survived in Latin translations, including those of Gerard of Cremona [d. 1187], than in the Arabic original. (Dates in brackets mine.)
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The oldest extant source on Arabian music is a work entitled Risala fi hubr ta’lif al-alhan (On the composition of melodies), by Ishaq Al-Kindi (d. c. 874). Because this text only survived as a fragmented 17th-century transcription of a 13th-century copy, many pages are missing; this explains why the text begins in mid-sentence. Fortunately, the fragments provide enough information to impart Al-Kindi’s ‘ud tuning, which bestows the following six incipient contributions on the history of music:
●1. Outside China, this is the first mathematical description of a 12-tone chromatic scale. Although Al-Kindi’s scale also consists of a spiral of “fifths,” it differs from the Chinese model in that the tonic, ratio 1/1, simultaneously functions as the origin of two different spirals: one ascends four “fifths,” or four 3/2’s, and the other descends seven “fifths,” or seven 3/2’s. (See Section 11.47, Table 11.22.)
● 2. Al-Kindi’s 12-tone scale is the first tuning that uses identical note names to identify the tones of the lower and upper “octave.” In his text, Al-Kindi specifically states that the musical qualities of tones separated by an “octave” are identical.
● 3. This is the first mathematically verifiable scale that accounts for the comma of Pythagoras. In his ‘ud tuning, Al-Kindi distinguishes between the apotome [C#], ratio 2187/2048, and the limma [Db], ratio 256/243.
● 4. This is the first mathematically verifiable example of a Greek tetrachord on an actual musical instrument.
● 5. On the Bamm, or the lowest sounding string, Al-Kindi defines four ancient length ratios — 9/8, 32/27, 81/64, 4/3 — which appear in all subsequent ‘ud tunings through the 17th century.
● 6. Al-Kindi gives the first mathematical description of a fifth string, which I call Zir 2, for the purpose of taking his 12-tone scale to its logical conclusion, namely, to sound the “double-octave,” ratio 4/1, above the open Bamm.
Although an English translation of this work exists, for technical reasons I translated the next excerpts from a German translation by Robert Lachmann and Mahmud el-Hefni. Note carefully that these paragraphs deal primarily with discussions on tuning, and that I excluded several sentences that include missing text, that make no sense, or that violate mathematical logic. Even though the copyist complains, “The model copied was written at the end of Sawwal in 621 [a.d. 1224] in Damascus from a defective, unreliable copy,” I seriously doubt that he fully comprehended, and was therefore unable to correctly interpret, muddled or spurious passages of the 13th-century copy.
Due to Arial diacritical font limitations, in the following translation and discussion,
Ĥ represents H with a dot below, and Ţ represents T with a dot below.
See Figure 11.41, and Table 22.
* * ↓ * *
(1.) . . . and K to A is a whole plus an eighth of a whole [ancient length ratio: 1 + 1/8, or 9:8] of it, and we have already explained, that the difference between a fifth [3:2] and a fourth [4:3] is a whole plus an eighth of a whole. From this, the distance of the W, which is the open string of the Mathlath, from the A, which is the first fret of the Mathna, is the interval of a fifth. And the octave is composed of a fifth and a fourth. From this, the distance of the A of the Bamm from the A of the Mathna is the octave. From this, the relationship of the A of the Bamm to the A of the Mathna is the relationship of the doubling 2 : 1. Consequently, according to the previous statement, the A of the Mathna is of the quality as the A of the Bamm.
(2.) From this example, the tones that succeed one another on the basis of similarity, succeed one another on the basis of quality. Therefore, the B of the Mathna is of the same quality as the B of the Bamm; the utilization of the B of the Bamm with respect to the frets is already clarified. Likewise, the G of the Mathna is equal to the G of the Bamm, and the D of the Mathna is equal to the D of the Bamm and the D of the Zir [Zir 1]. Likewise the W of the Zir is equal to the W of the Mathlath, and the Z of the Zir is equal to the Z of the Mathlath, which is not used; and the Ĥ of the Zir is equal to the Ĥ of the Mathlath, and the Ţ of the Zir is equal to the Ţ of the Mathlath and to the Ţ of the lower Zir [Zir 2], and the I of the lower Zir is equal to I of the Mathlath, and the K of the lower Zir is equal to the K of the Mathlath and to the K of the Mathna, and the L of the lower Zir is equal to the unused L of the Mathna; and the A of the lower Zir is equal to the A of the Mathna, and the B of the lower Zir is equal to the B of the Mathna, and the G of the lower Zir is equal to the G of the Mathna, for compelling reasons which we indicated before . . .
. . . (4.) In order now to give an example in numbers, we assign the number 16 to the A of the Bamm. Then the W of the Bamm equals 12; because A is a whole plus a third of a whole [1 + 1/3, or 4:3] from W. And the W of the Bamm equals the W of the Mathlath. Then the K of the Mathlath equals 9; because the W of the Mathlath is a whole plus a third of a whole from the K of the Mathlath . . .
. . . (2.) Of the double octave, there are two kinds. One is called the conjunct [system]; it is the one in which the A of the Mathna participates at the end of the first and in the beginning of the second octave. In the disjunct system, the beginning of the first octave is the A of the Bamm, and its end the A of the Mathna, and the beginning of the second, the G of the Mathna, and its end, the G of the second Zir [Zir 2]; these two systems are disjointed by the distance from A up to G of the Mathna, which forms the interval of the whole tone, i.e., the relationship of a whole plus an eighth of a whole. These indicated tones limit the disjunct system, and that which is below it.
(3.) After the locations of the tones and the used tones, we must discuss the tones in the system of the double octave. We account for the number of their locations. They amount to 20; because on each string there are 4 [stopped] tones in the range of the fourth [tetrachord], and there are 5 strings; in addition to this, comes the tone G of the second Zir, so that through it, the whole range of the octave completes itself, if it is used instead of the A of the second Zir. And with respect to the number of used tone-locations, so W of the Bamm and of the Mathlath are identical, and K of the Mathlath and of the Mathna are identical, and D (of the Mathna and) of the first Zir [Zir 1] are identical, and Ţ of the first and of the second Zir are identical . . .
(4.) . . . Of the total number of tones used in the genera, 10 are firm and do not change their locations . . . The unchangeable [tones] lie at the two ends of the frets [that is, at the two ends of the vertical fret pattern, which includes 5 tones of Fret 1: G, Ĥ, A, W, K at the top end, and 5 tones of Fret 4: W, K, D, Ţ, B at the bottom end]; the changeable [tones] lie in between. The utilization of the modes change the ones lying between the ends [of the fret pattern]; because the first mode of the diatonic genus is used differently than the second and third . . .
(5.) . . . Since in the diatonic genus the first and second mode is used most often, its beginning is at the first fret; in contrast, the beginning of the third mode is at the open string . . . of the Bamm . . .
(9.) Regarding the number of tones, and their quality in the system of the double octave, there are 7 tones from which all the melodies arise. Since the A of the Mathna has the same quality as the A of the Bamm, the octave consists of 7 tones, because its two ends, with respect to quality, are one and the same tone. (Bold italics, and text and ratios in brackets mine. Text in parentheses in Lachmann’s and El-Hefni’s translation.)
* * ↑ * *
Figure 11.41 is a fret diagram of Al-Kindi’s ‘ud tuning. The only note not mentioned in the fragment is H, ratio 81/64 [E], of the Bamm, and H, ratio 81/32 = 81/64 [E'] of the Zir 1. However, since the text explains that Al-Kindi named the twelve notes of the lower and upper “octaves” after the first twelve letters of the old Arabic alphabet: A, B, G, D, H, W, Z, Ĥ, Ţ, I, K, L, the presence of H is self-evident. A brief examination of the treatise reveals the following tuning. Chapter 1, Paragraph 4 assigns 16 units to A [C], ratio 1/1, of the open Bamm, and defines W [F] — Fret 4 — of the Bamm as ancient length ratio 16:12, or 4/3. Al-Kindi then tunes W [F] — open Mathlath — in unison to the latter fret; therefore, K [Bb] — Fret 4 — of the Mathlath must be ratio 4/3 × 4/3 = 16/9. Chapter 2, Paragraph 3 defines K [Bb] — open Mathna — tuned in unison to the latter fret, which means D [Eb'] — Fret 4 — of the Mathna must be ratio 16/9 × 4/3 = 64/27 = 32/27. He then defines D [Eb'] — open Zir 1 — tuned in unison to the latter fret, which means Ţ [Ab'] — Fret 4 — of the Zir 1 must be ratio 32/27 × 4/3 = 128/81. At the end of the paragraph, he defines Ţ [Ab'] — open Zir 2 — tuned in unison to the latter fret, which means B [Db"] — Fret 4 — of the Zir 2 must be ratio 128/81 × 4/3 = 512/243 = 256/243. Although the ‘ud (pl. ‘idan, a‘wad) is not specifically mentioned in the fragment, the open strings of short-necked lutes were traditionally tuned in ascending “fourths.”
Chapter 1, Paragraph 2 states that D [Eb'] of the Mathna is equal to D [Eb] — Fret 2 — of the Bamm, which means that the latter sounds an “octave” below the former. Since we know that all open strings are tuned in “fourths,” the Fret 2 location determines the ratios of Ţ [Ab], B [Db'], Z [Gb'], and L [Cb'] above D [Eb]. Al-Kindi also states that B [Db'] of the Mathna has the same quality as B [Db] — Anterior Fret 1 — of the Bamm, which means that the latter sounds an “octave” below the former fret. (The special function of Anterior Frets 1 and 2 is discussed below.) The Anterior Fret 1 location determines the ratios of Z [Gb] and L [Cb] above B [Db].
Chapter 1, Paragraph 1 states that A [C] of the Bamm and A [C'] — Fret 1 — of the Mathna constitute an “octave,” which means that the latter sounds a 2/1 above the former; the Fret 1 location determines the ratios of G [D] and Ĥ [G] below A [C'], and of W [F'] and K [Bb'] above A [C'].
Chapter 2, Paragraph 2 states that the span from A [C'] — Fret 1 — of the Mathna to G [D'] — Fret 3 — of the Mathna is interval ratio 9/8. Therefore, the Fret 3 location determines the ratios of H [E] and I [A] below G [D'], and of Ĥ [G'] and A [C"] above G [D'].
Chapter 1, Paragraph 2 states that I [A'] — Anterior Fret 2 — of the Zir 2 is equal to I [A] of the Mathlath, which means that the former sounds an “octave” above the latter; the Anterior Fret 2 location determines the ratio of H [E'] below I [A'].
Finally, Chapter 2, Paragraph 2 states that G [D'] of the Mathna and G [D"] of the Zir 2 constitute an “octave,” which means that the latter sounds an “octave” above the former, and a “double-octave” above G [D] of the Bamm. The G [D"] of the Zir 2 does not indicate the location of a fret. Musicians played this tone through a “shift” of the hand, which places the index finger at the location of L [Cb'], and the ring finger at the location of G [D"]. (See Ibn Sina’s explanation in Section 11.59.) Many modern scholars insist that the fifth string constitutes a theoretical addition, included solely to complete the “double-octave” between the open Bamm and Fret 3 of the Zir 2. However, the renowned poet and musician Ziryab, who lived in the southern region of Spain called Andalusia, already added a fifth string to his ‘ud sometime between 822 and 852. So, while Ziryab is credited with this invention in the West, Al-Kindi was first to implement it in the East.
The greatest difficulty with respect to fret locations occurs in Chapter 1, Paragraph 2. Here Al-Kindi states that Z [Gb] of the Mathlath and L [Cb] of the Mathna are not used. However, he also reminds the reader that “. . . the utilization of the B [Db] of the Bamm with respect to the frets is already clarified.” This statement refers to a discussion that did not survive in the fragment. Since a careful analysis of the treatise reveals that tones B [Db], Z [Gb], and L [Cb] reside on the same fret, the extant text does not give a consistent description of the function of this fret. To resolve this apparent contradiction, first consider Sequence A, which gives the ratios of an ascending sequence of seven 3/2’s, and then Sequence B, which gives the ratios of a descending sequence of seven 3/2’s. (See Chapter 10, Figure 16.)
↑1/1 [C], 3/2 [G], 9/8 [D], 27/16 [A], 81/64 [E], 243/128 [B], 729/512 [F#], 2187/2048 [C#] (A)
↓1/1 [C], 4/3 [F], 16/9 [Bb], 32/27 [Eb], 128/81 [Ab], 256/243 [Db], 1024/729 [Gb], 4096/2187 [Cb] (B)
Now, imagine L and Z occur on Anterior Fret 2, which determines the H [E'] of the Zir 1 and the I [A'] of the Zir 2. Under such circumstances, L would sound a “fourth” below H [E' = 81/32], which yields ratio 81/32 ÷ 4/3 = 243/128, [B]; and Z would sound a “fourth” below L, which yields ratio 243/128 ÷ 4/3 = 729/512 [F#]. Since Sequences A and B indicate that the interval between the former L [B] and the L of the Zir 2 [Cb] is not an “octave,” and since the interval between the latter Z [F#] and the Z of the Zir 1 [Gb] is also not an “octave,” Al-Kindi’s earlier statement with respect to L and Z makes perfect sense. He advises against these two tones because they produce “octaves” that are a comma of Pythagoras flat, or “octaves” that are a comma of Pythagoras too narrow. The interval between B and Cb' is 243/128 ÷ 4096/2187 = 531441/524288 and the interval between F# and Gb' is 729/512 ÷ 1024/729 = 531441/524288. Next, suppose B of the Bamm sounds a “fourth” below Z [F#]; such an interval yields ratio 729/512 ÷ 4/3 = 2187/2048 [C#], also known as the apotome. (See Chapter 10, Sections 15 and 18.) With respect to the B [Db'] of the Mathna, this tone also produces an “octave” that is a comma of Pythagoras flat, which means the interval between C# and Db' is 2187/2048 ÷ 256/243 = 531441/524288. However, remember that Al-Kindi explicitly states, “. . . the B [Db'] of the Mathna is of the same quality as the B [Db] of the Bamm . . . ” Therefore, to sound an exact “octave,” we must move the B [C#] of the Bamm from Anterior Fret 2 to Anterior Fret 1. If we position the latter fret a comma of Pythagoras above the former fret, such a location produces ratio 2187/2048 ÷ 531441/524288 = 256/243 [Db], also known as the limma. (See Chapter 10, Sections 10, 15, 18.) Suppose we also move Z and L from Anterior Fret 2 to Anterior Fret 1. Now B [Db], Z [Gb], and L [Cb] of the Bamm, Mathlath, and Mathna, respectively, produce three exact “octaves” with B [Db'], Z [Gb'], and L [Cb'] of the Mathna, Zir 1, and Zir 2, respectively. This construction resolves the apparent contradiction and leaves Al-Kindi’s original scale intact; that is, the frets of the ‘ud produce two identical 12-tone scales that span the interval of a “double-octave.” Finally, observe that Anterior Fret 1 provides only three tones, and Anterior Fret 2, only two tones of Al-Kindi’s “double-octave” tuning.
With respect to the tonic, Table 22 shows that Al-Kindi’s ‘ud tuning consists of an ascending spiral of four “fifths” above 1/1, and of a descending spiral of seven “fifths” below 1/1. The ascending progression represents the ubiquitous pentatonic scale known the world over as ratios: 1/1, 9/8, 81/64, 3/2, 27/16, [2/1]; in Al-Kindi’s notation, A, G, H, Ĥ, I, [A'].
We may attribute portions of the descending sequence to Greek tetrachordal theory. Arabian music theorists frequently refer to the most venerated Greek tetrachord — the diatonic genus of Philolaus (fl. c. 420 b.c.) — in their descriptions of four cardinal frets on the ‘ud. These frets represent ratios: 9/8, 32/27, 81/64, 4/3. In Chapter 2, Paragraphs 4 and 5, Al-Kindi describes three different modes of the diatonic genus. Because the extant text does not specify an exact order of intervals, let us assume Al-Kindi’s first mode constitutes Philolaus’ original tetrachord. If we now systematically rotate the limma from its initial position as first interval ratio, to the next position as second interval ratio, and to the final position as third interval ratio, the pattern below emerges.
The Greeks called these the Dorian, Phrygian, and Lydian harmoniai. Al-Kindi states in Chapter 2, Paragraph 5, that the first two modes begin on the first fret, and the third mode, on the mutlaq (lit. free; hence, fig. open string). His descriptions refer to the ancient Arabic majra (lit. course or path; hence, fig. mode) of the wusta (lit. middle finger), and the majra of the binsir (lit. ring finger). One distinguished these two majari by a “semitone” interval between the sabbaba (lit. index finger) and middle finger, and by a “whole tone” interval between the index finger and ring finger. Figure 11.41 confirms Mode 1 in sabbaba fi majra al-wusta because this sequence begins with the index finger and proceeds to the middle finger; Mode 2, in sabbaba fi majra al-binsir because it begins with the index finger and proceeds to the ring finger; and, Mode 3, in mutlaq fi majra al-binsir because it begins with the open string, followed by the index finger and ring finger. In Western terms, sabbaba fi majra al-binsir sounds a tetrachord with a “minor third”: 1/1, 9/8, 32/27, 4/3, and mutlaq fi majra al-binsir, a tetrachord with a “major third”: 1/1, 9/8, 81/64, 4/3. Because ancient Arabian music theory decreed that 32/27 [Eb] and 81/64 [E] are ‘incompatible’ (see Section 11.50), musicians abstained from playing these two tones in the same mode. This simple rule foreshadowed what would eventually become the distinction between the minor and major tonalities of Western music.
 Farmer, H.G. (1954). “‘Ud.” In Grove’s Dictionary of Music and Musicians, Volume 8, 5th ed., E. Blom, Editor, p. 631. St. Martin’s Press, Inc., New York, 1970.
 Lewis, B., Editor (1976). Islam and the Arab World, p. 225. This volume consists of an anthology of many excellent articles. For a history of the Arabian influence in Andalusia, consult “Moorish Spain,” by Emilio García Gómez, and for a survey of Arabian music, “The Dimension of Sound,” by A. Shiloah. Alfred A. Knopf, New York.
On p. 225, Gómez quotes the famous Spanish philosopher José Ortega y Gasset (1883–1955), “I do not understand how something which lasted eight centuries can be called a reconquest.”
Hitti, P.K. (1937). History of the
Arabs, pp. 370–371. Macmillan and Co. Ltd., London, England, 1956.
 Hitti, P.K. (1937). History of the Arabs, pp. 370–371. Macmillan and Co. Ltd., London, England, 1956.
 Cowl, C., Translator (1966). Al-Kindi’s essay on the composition of melodies. The Consort, No. 23, pp. 129–159.
 Lachmann, R. and El-Hefni, M., Translators (1931). Risala fi hubr ta’lif al-alhan [Über die Komposition der Melodien], by Al-Kindi. Fr. Kistner & C.F.W. Siegel, Leipzig, Germany.
 For example, in the Lachmann/El-Hefni and Cowl translations, Chapter 2, Paragraph 5, Sentences 8 and 9 contradict the first four and a half lines of Chapter 2, Paragraph 7. Since the former passage makes perfect sense, and the latter passage is confused and has missing text, I excluded the latter from the discussion.
 The Consort, 1966, p. 149.
 Forster Translation: in Risala fi hubr ta’lif al-alhan [Über die Komposition der Melodien], pp. 21–25.
 (a) Farmer, H.G. (1978). Studies in Oriental Musical Instruments, First and Second Series. Second Series, p. 47, 90. Longwood Press Ltd., Tortola, British Virgin Islands.
This volume consists primarily of articles by Henry George Farmer first published in The Journal of the Royal Asiatic Society of Great Britain and Ireland. In this book, the division into First and Second Series means that the first set of articles is numbered pp. 3–107, and the second set is numbered pp. 3–98.
(b) Ribera, J. (1929). Music in Ancient Arabia and Spain, pp. 100–107. This work was translated and abridged from the Spanish by Eleanor Hague and Marion Leffingwell. Stanford University Press, Stanford University, California.
 In Figure 11.41, the physical distance between Anterior Frets 1 and 2 indicates the aural interval of the comma of Pythagoras, which equals 23.5 ¢.
 See Chapter 10, Section 10.
 See Chapter 10, Section 6.