---------------------------------------------------- The Zalzalian Diatonic Tetrachords of Ibn Sina Placing his `Oud tuning in a larger context by Margo Schulter ---------------------------------------------------- As a philosopher, physician, and musician, Ibn Sina has enjoyed the admiration of the Islamic world for a millennium, being likewise honored in Latin Europe (under the name of Avicenna). His fascination with fine shades of intonation, and in his interest in small commas and near-equivalences between superparticular and more complex forms of Zalzalian or neutral second steps, evince the same spirit of artistic refinement and intellectual analysis which may be found in the qanun tunings of Julien Jalal Ed-Dine Weiss (here JJW). In attempting this brief overview and survey of Ibn Sina's admirable and often remarkable observations, I am deeply indebted both to Stefan Pohlit, who has invaluably documented JJW's qanun tunings and masterfully nuanced techniques in interpreting traditional maqam music as well as in composing new music and providing materials for other musicians and composers to explore, and of course to JJW himself, both of whom have shown me friendship, generosity, and grace. This paper is therefore dedicated to them both, with due notice that while they are responsible for much of whatever virtue it may have, the fault for any errors or infelicities lies solely with me. --------------- 1. Introduction --------------- In his _Kitab al-Shifa_, Ibn Sina (980-1037) presents a fascinating survey of what I will here term "Zalzalian diatonic" tetrachords, those involving a tone at 8:7 or 9:8 plus two neutral second steps. Having surveying these tetrachords, including two earlier discussed by al-Farabi (c. 870-950), he offers some observations about the intonational practices of his musical contemporaries. These passages (d'Erlanger 2001-II: 148-150) are critical in understanding his later remarks on the placement of the wusta Zalzal (wZ) or neutral third fret on the `oud (ibid. at 235). For Ibn Sina, as for al-Farabi, the superparticular ratios are of special importance, a preference which follows the Greek tradition of Ptolemy, as Ibn Sina himself emphasizes in a remarkable passage describing the division of the 4:3 fourth into 16:14:13:12, or a large tone at 8:7 plus Zalzalian or neutral second steps at 14:13 and 13:12 or 231-128-139 cents (d'Erlanger, ibid. at 148). The "very noble jins" has all superparticular steps, and Ibn Sina regards as musically equivalent the two permutations with 8:7 as the lowest step. With a 9:8 tone as the large step of a Zalzalian diatonic tetrachord, the situation becomes inevitably more complex, because a 4:3 fourth less 9:8 leaves a minor third at 32:27 to be divided between the two neutral second steps. In Ibn Sina's tetrachords, one of these steps is a superparticular neutral second at 11:10, 12:11, 13:12, or 14:13 (respectively 165, 151, 139, or 128 cents); while the other is a more complex "remainder," which he sees as analogous to the Greek _limma_ (remainder) in the diatonic at 9:8-9:8-256:253 (204-204-90 cents). Just as the ear accepts the 256:243 semitone in this usual diatonic (d'Erlanger, ibid. at 148-149), so it may accept the complex neutral second which necessarily arises when a 32:27 minor third is divided into two neutral steps, since no division is available where both neutral steps are superparticular. As he repeatedly observes, however, the nonsuperparticular or "remainder" step may be taken as closely approximating a superparticular ratio. His explorations of these approximations survey some interesting small commas, although he does not focus on the precise ratios of these commas (ibid. at 149-150). Before delving into Ibn Sina's remarks about these Zalzalian diatonic tetrachords, we may find it helpful to list the forms he surveys and adopts in this main discussion, along with the related tetrachord in his `oud tuning (ibid. at 235) at 9:8-13:12-128:117 (204-139-156 cents), the basic for his Mustaqim modality with wZ at 39/32 (342 cents). In addition to the tetrachords which he makes a part of his system, there are two he cites from al-Farabi (without attribution) but does not include in his own preferred set. -------------------------------------------------- Table 1: Ibn Sina's Zalzalian Diatonic Tetrachords Including two from al-Farabi not adopted (B1, B2) -------------------------------------------------- A. Tetrachords with 8:7 tone ("most noble jins" of Ibn Sina) A1. 8:7-13:12-14:13 or 104:91:84:78 (231c-139c-128c) Zalzalian 3rd at 26/21 (370 cents) A2. 8:7-14:13-13:12 or 16:14:13:12 (231c-128c-139c) Zalzalian 3rd at 16/13 (359 cents) B. Tetrachords with 9:8 tone B1. 9:8-11:10-320:297 or 396:352:320:297 (204c-165c-129c) Zalzalian 3rd at 99/80 (369 cents) -- al-Farabi B2 9:8-12:11-88:81 or 108:96:88:81 (204c-151c-143c) Zalzalian 3rd at 27/22 (355 cents) -- al-Farabi `oud B3a 13:12-9:8-128:117 or 468:432:384:351 (139c-204c-156c) Zalzalian 3rd at 39/32 (342 cents) -- Ibn Sina B3b 9:8-13:12-128:117 or 468:416:384:351 (204c-139c-156c) Zalzalian 3rd at 39/32 (342 cents) -- Ibn Sina `oud B4 9:8-14:13-208:189 or 252:224:208:189 (204c-128c-166c) Zalzalian 3rd at 63/52 (332 cents) Ibn Sina's two permutations with an 8:7 tone feature larger Zalzalian or neutral thirds at 26/21 (369.7 cents) and 16/13 (359.5 cents), while his preferred tetrachords with 9:8, both in his listing of ajnas and in his `oud tuning, have smaller neutral thirds at 39/32 (342.5 cents), his wZ, and 63/52 (332.2 cents). However, he does not adopt the two Zalzalian diatonic tetrachords of al-Farabi which he mentions, with larger neutral thirds at 99/80 (368.9 cents) and 27/22 (354.5 cents), the latter al-Farabi's notable placement of wZ. It is easy to note one contrast because the chosen tetrachords of al-Farabi and Ibn Sina with a 9:8 tone. While al-Farabi prefers to place the larger Zalzalian second step before the smaller, Ibn Sina prefers the converse arrangement. Thus their respective wZ positions at 27/22 (355 cents) and 39/32 (342 cents). Clearly Ibn Sina is interested in fine shadings of neutral steps, which he tells us (ibid. at 150) his contemporaries often confuse, for example using indifferently either 14:13 (128 cents) or 13:12 (139 cents). This discussion of intonational practices may give clues not only to his own musical outlook, but as to the local tastes of his time and place. His report that the placement of wZ varied (ibid. at 150, 235) is much in accord with the observations of Safi al-Din al-Urmawi (c. 1216-1294) some two centuries later. Ibn Sina's recognition and use of all four superparticular neutral or Zalzalian seconds is a development still very influential in the precise and extremely sophisticated just qanun tunings of Julien Jalal Ed-Dine Weiss, an Ars Intonationis Subtilissima or "most subtle intonational art" for the 21st century. More modestly, the MET-24 temperament supports these same four shadings of neutral intervals, albeit imprecisely, by tempering out the small commas which Ibn Sina notes and the qanun tunings faithfully observe. From this perspective, let us consider Ibn Sina's treatment of the individual Zalzalian diatonic tetrachords he surveys. ---------------------------------------------------- 2. Tetrachords with an 8:7 tone: A "most noble jins" ---------------------------------------------------- in presenting Zalzalian diatonic tetrachords, Ibn Sina begins with two "truly consonant genera" which he describes as "very noble." When, to the interval 8:7, one joins 13:12, the complementary interval will be 14:13. (d'Erlanger, op. cit., at 148). He then demonstrates in mathematical terms how this division, with all superparticular steps, may be elegantly derived on the monochord, using an approach which I here illustrate using a length of 16 for the whole string to show his aliquot arithmetic divisions. In setting this genre, most noble, one has effected the last partition by a halving of intervals [i.e. string-length ratios]. The interval of the double octave, partitioned in half, has, indeed, yielded the octave. (Ibid. at 148) [Illustration added -- M.S.] 2:1 2:1 |------------------------------------------|-----------------|.... 16 8 4 After this first partition, Ibn Sina proceeds by a series of arithmetic means. This last interval [i.e. the octave], partitioned by a means, has given the fifth and fourth. (Ibid. at 148) [Illustration added -- M.S.] 4:3 3:2 |-----------------------|-------------------| 16 12 8 This partition of the 2:1 octave into the 4:3 fourth and 3:2 fifth is, of course, one of the basic divisions of the Pythagoreans, here realized as 16:12:8 (16:12 or 4:3 fourth plus 12:8 or 3:2 fifth). Partitioned in the same fashion, the interval of the fourth has engendered the intervals 8:7 and 7:6; the last in turn has given 13:12 and 14:13. (Ibid. at 148) [Illustration added -- M.S.] 8:7 7:6 |-----------|-----------|-------------------| 16 14 12 8 8:7 14:13 13:12 |-----------|-----|-----|-------------------| 16 14 13 12 8 The partition of 16:12 or 4:3 into 8:7:6 (16:14:12) is the division of Archytas, as used by Ptolemy. Here Ibn Sina has carried the process a step further, arriving at a remarkable result: the Zalzalian division of 7:6 into neutral second steps of 14:13 and 13:12. Ptolemy prefers the genre at which we proceed to arrive above all others. The numeric expression is as follows: [Illustration added by M.S., cf. Cris Forster (2010, at ) 231 139 128 8:7 13:12 14:13 |-------------|--------|-------| 104 91 84 78 1/1 8/7 26/21 4/3 0 231 370 498 Although I know of no such tetrachord in Ptolemy, Ibn Sina very reasonably asserts that this division superbly realizes Ptolemy's method and ethos of seeking elegant superparticular ratios arranged in a musically pleasing way! This style of division might be described as "Zalzalian-Archytan," since it relies both upon the Archytan division of 4:3 into 8:7 and 7:6, and upon Zalzal's division of a minor third into two neutral second steps (here 13:12 and 14:13). From a mathematical perspective, the Zalzalian division of the 7:6 small minor third, here 91:78, into lengths of 91:84:78 is harmonic, since the differences of adjacent terms have a ratio of 7:6, also the ratio of the extreme terms, or the interval being partitioned. [Illustration added -- M.S.] 7 6 |--------------|-----------| 91 84 78 |--------------------------| 7:6 From a musical perspective, this division produces a tetrachord of 1/1-8/7-26/21-4/3 or 0-231-370-498 cents, and steps of 8:7-13:12-14:13 (231-139-128 cents). It has a bright Zalzalian or neutral third at 26/21 or 370 cents, typical of a modern Syrian Rast in the practice of Aleppo, and of some historical and modern Ottoman practices, as illustrated for example by a baglama from Istanbul as reported by Cameron Bobro (who measured a neutral third fret at 370 cents). The important distinction between Ibn Sina's tetrachord and these practices, of course, is the higher position of the tone at 8/7 rather than a usual Rast where it is at or near 9/8 (204 cents). Ibn Sina then proceeds to the other permutation of this tetrachord with the 8:7 tone as the first or lowest interval, the form which he has already presented in his Ptolemaic demonstration of multiple aliquot divisions (ibid. at 148): When, on the other hand, one joins to the interval of 8:7, the interval of 14:13, the complement will be 13:12 and the genre obtained will be identical to the previous one: 231 128 139 8:7 14:13 13:12 |-------------|-------|--------| 16 14 13 12 1/1 8/7 16/13 4/3 0 231 359 498 Here the large Zalzalian third is at 16/13 or 359 cents, typical of a Rast third in the Syrian practice of Damascus. Ibn Sina evidently finds the effect musically equivalent to the 8:7-13:12-14:13 form, since he described the two tetrachords as "identical." Both could be described as septimal variations on what is in modern terms a bright Rast. To some Syrian tastes, for example, both permutations with their respective neutral thirds at 26/21 and 16/13 might be congenial. Interestingly, Bozkurt and colleagues (2009, at 46) find that this genus "is resemblant of quotidian Arabic rendition of the cadence region of _maqam Segah_" -- that is, the lower Rast tetrachord rast-dukah-sikah-jaharkah, as often intoned in making a cadence on the step sikah, the final of this maqam. In such an Arab practice, this tetrachord with rast-sikah at around 16/13 would have a normal tuning in theory, and likely in practice, at around 1/1-9/8-16/13-4/3 or 0-204-359-498 cents, with a quite large neutral second step between dukah and sikah (9/8-16/13) at 128:117 or 156 cents. However, in cadencing on sikah, there is evidently a desire for a higher leading tone and a smaller neutral step. Thus dukah, as reported by Bozkurt and his colleagues, may be raised by about a comma, from 9/8 to Ibn Sina's 8/7, resulting in a small cadential neutral second at 14:13 (128 cents). This small adjustment might be related to the concept in Arab theory of a _dint_, the raising of a leading tone in approaching the final or another principal note of a given maqam (Marcus 1989, 612-616). While a _dint_ inflection, at least in recent modern theory, is generally by a semitone, here it is more subtly by a comma. The 8:7-13:12-14:13 and 8:7-14:13-13:12 divisions of Ibn Sina are of special interest for two reasons. First, they are the only divisions he adopts which have a large neutral third step at 26/21 or 16/13 -- in contrast to the lower neutral thirds he prefers when a tetrachord has a tone at 9/8. Secondly, these are the only Zalzalian diatonic tetrachords where all three steps are superparticular: the 8:7 tone leaves a 7:6 minor third to be divided between the two neutral second steps, neatly partitioned into 13:12 and 14:13 (or vice versa). When the tone is at 9:8, this leaves a 32:27 or regular Pythagorean minor third (294 cents), whose division into two neutral seconds must inevitably result in a nonsuperparticular interval, introducing the complication of various small commas into the tuning equation. Ibn Sina approaches these complications with great insight, and at the same time may be reflecting some musical tastes of his time and region. ---------------------------------------------------------------- 3. Tetrachords with a 9:8 tone: remainders, tastes, and shadings ---------------------------------------------------------------- For Ibn Sina, the appearance of "dissonant" or nonsuperparticular steps, often with strikingly complex ratios, is a phenomenon worthy of exploration. Why is it that these ratios, despite their deviation from the Ptolemaic ideal of epimoric or superparticular steps, nevertheless are often acceptable to the ear? His discussion of this problem is an apt prelude to his survey of the Zalzalian diatonic tetrachords with a tone at 9:8, where it is impossible to obtain all superparticular steps (at least as long as one keeps the fourth at a just 4/3). Here Ibn Sina's method is to present a series of tetrachords combining a 9:8 tone with another superparticular step ranging downward in size: thus 9:8, 10:9, 11:10, 12:11, 13:12, and 14:13. In his classification of intervals, 14:13 is noteworthy as the smallest superparticular step which, when doubled, is equal to the greater part of a 4/3 fourth -- 196:169, or 256.6 cents (leaving a lesser part of 169:147, or 241.5 cents). --------------------------------------------------------------- 3.1. The Ditonic Diatonic (9:8-9:8-256:253) and the "remainder" --------------------------------------------------------------- Following this method, he begins with the familiar Pythagorean or ditonic diatonic at 9:8-9:8-256:256 (204-204-90 cents), and offers some observations providing a framework for his survey of the Zalzalian tetrachords that follow (d'Erlanger, op. cit. at 148-149). As for genera having for a basis the interval 9:8, the first among them results from a repetition of this interval in the middle of the tetrachord; it is described as diatonic (or ditonic). This genus is composed of a tone followed by a tone, and of a _remainder_ (limma), which wrongly is called a "demitone," and which is not consonant. The dissonance of this remainder interval is however mitigated by the richness of sonority of the two intervals of the tone, and also by the selfsame nature of these two intervals. They belong, indeed, to the series of intervals whose denominators are a number twice even [or equal?]. In other words, the simplicity of the genus as a whole, "which is compromised of two intervals of 9:8," may make the more complex step of 256:243 (90 cents) acceptable. He explains, ibid. at 149: The ear has little by little habituated itself to the remainder interval. It may happen, perhaps, that this is not a genus of which the consonance of the complementary interval should be doubtful, and so the ear accepts it since it accepts the diatonic. We have sufficiently informed the reader to permit an understanding of why [this genus] has been pleasant to adopt. The numeric expression for this genus of fourth, which is comprised of two intervals of 9:8, is the following: [Illustration from d'Erlanger, ratios and cents for notes added] 204 204 90 9:8 9:8 256:243 324 288 256 243 1/1 9/8 81/64 4/3 0 204 408 498 Thus is established the principle that if a tetrachord includes two pleasing superparticular steps, felicitously arranged, then a more complex or "remainder" interval will not offend the ear. He concludes his discussion of the Pythagorean Diatonic, or Ditonic Diatonic of Ptolemy, by addressing a problem raised also by Ptolemy: a rational ratio which may approximate a true "demitone," or half of a 9:8 tone. The ratio of the remainder interval is then 256:243. If we seek a number which with 256 gives us the ratio for the demitone, we find 241 [256:241, 104.5 cents], or yet 240 [256:240 or 16:15, 111.7 cents], makes up a larger half of a tone. These numbers are both smaller than that which, with 256, constitutes the ratio of the remainder interval. The remainder interval is thus smaller than a demitone. (Ibid. at 149) Ptolemy, faced with the same problem, estimated the demitone at around 258/243 or 86/81 (103.7 cents), as compared to the actual and irrational value at 101.955 cents. --------------------------------------------------------------- 3.2. An interlude: 10:9-9:8-16:15 (Ptolemy's Syntonic Diatonic) --------------------------------------------------------------- Ibn Sina now moves to a tetrachord where a 9:8 tone is complemented by a small tone at 10:9 (182.4 cents), here interestingly placed so that the 10:9 step is lowest, ibid. at 149: When the interval of a tone [9:8] is followed by that coming immediately after it [in the series of superparticular ratios], which would be 10:9, the complement in the tetrachord will have a ratio of 16:15. The intervals so obtained are truly consonant. The numeric expression of this genus is the following: [Illustration from d'Erlanger, ratios for notes added] 182 204 112 10:9 9:8 16:15 20 18 16 15 1/1 10/9 5/4 4/3 0 182 386 498 In modern Arab terms, at least, this is not a Zalzalian tetrachord; but for the Systematist theorist Qutb al-Din al-Shirazi (1236-1311), the permutation 9:8-10:9-16:15 does define the Rast genus, for which he gives the monochord ratios 180:160:144:135 (204-182-112 cents), see Wright (1978, ). Modern Turkish theory likewise favors this tuning of Maqam Rast, and follows Safi al-Din (who used the Pythagorean diminished third or double limma at 65536:59049 or 180.45 cents, and the apotome at 2187:2048 or 113.7 cents, as equivalents for the simpler 10:9 and 16:15) and Qutb al-Din in considering 10:9 and 16:15 as _mujannab_ or "middle" steps, with sizes somewhere between those of the 9:8 tone and 256:243 limma. ------------------------------------------- 3.3. Two Zalzalian tetrachords of al-Farabi ------------------------------------------- Without mentioning al-Farabi by name, Ibn Sina next presents his predecessor's two Zalzalian tetrachords which in modern terms are forms of Rast, the second better known because it appears also in al-Farabi's famous `oud tuning with wZ at 27/22 (354.5 cents). He first, however, considers the tetrachord with a higher neutral third at 99/80 (368.9 cents), ibid. at 149. When the interval of a tone is followed by an interval of 11:10, the intervals of the genre obtained will not be consonant. The ratio of the complementary interval will be that of 320:297, very close to 14:13 [at 129.1 and 128.3 cents respectively]. We have already shown what one ought to think of such a ratio. [Illustrations added -- M.S.] 204 165 129 9:8 11:10 320:297 396 352 320 297 1/1 9/8 99/80 4/3 0 204 369 498 Ibn Sina observes that "the intervals of the genre obtained will not be consonant" -- evidently referring to the complex remainder interval of 320:297 or 129.1 cents -- and adds that "[we] have already shown what one ought to think of such a ratio." One possible reading is that, although Ibn Sina does not himself adopt this tetrachord, neither does he necessarily consider it unacceptable -- given his previous discussion of 9:8-9:8-256:243, where he explains that a complex remainder interval need not prevent a genus from pleasing the ear. Of special interest is Ibn Sina's observation that al-Farabi's remainder interval of 320:297 is "very close to 14:13," a difference in fact equal to the small comma of 2080:2079 (0.833 cents). This is the amount by which the 26/21 neutral third (369.7 cents) in Ibn Sina's jins of 8:7-13:12-14:13 is larger than al-Farabi's 99/80 (368.9 cents). Note that a tetrachord of 9:8-11:10-14:13 (204-165-128 cents), with all superparticular steps, would produce a fourth slightly narrow of 4/3 by this same small comma, at 693/520 (497.2 cents). However, to obtain a division with a tone plus a large neutral step where all intervals are superparticular and the fourth is at a just 4/3, Ibn Sina's 8:7 tone is necessary rather than al-Farabi's 9:8. Ibn Sina now presents al-Farabi's famous Zalzalian tetrachord still closely approximated in many Arab tunings of Rast, ibid. at 149-150: When the interval joined to the tone is of the ratio 12:11, the ratio of the complementary interval will be 88:81, which approximates 13:12 [respectively 143.5 and 138.6 cents]. We have already shown a similar case. [Illustrations added -- M.S.] 204 151 143 9:8 12:11 88:81 108 96 88 81 1/1 9/8 27/22 4/3 0 204 355 498 This is al-Farabi's jins with the famous wusta Zalzal or wZ at 27/22. Ibn Sina notes that the "complementary" or remainder interval at 88:81 (143.5 cents) "approximates 13:12" (138.6 cents); the difference is 352:351, or 4.9 cents. This is the amount by which 27/22 (354.5 cents) is smaller than Ibn Sina's 16/13 (359.5 cents) in his 8:7-14:13-13:12 jins. It would be possible to devise a tetrachord where the two neutral steps are at 12:11 and 13:12, but this would require compromising or altering one of the basic definitive intervals of medieval Near Eastern theory: either the 9:8 tone or the 4/3 fourth. One possibility is 44:39-12:11-13:12 (209-151-139 cents), or 176:156:143:132, where the fourth is pure, but the 44:39 tone (208.8 cents) is larger by 352:351 than 9:8. Another is 9:8-12:11-13:12 (204-151-139 cents), where all steps are superparticular but the fourth at 117/88 (493.1 cents) is narrow by the same comma. ------------------------------------- 3.4. Ibn Sina's Zalzalian tetrachords ------------------------------------- Having presented al-Farabi's two Zalzalian tetrachords, not specifically recommending them but arguably implying that their "complementary" or nonsuperparticular intervals may be acceptable, he turns to the jins which, in another permutation, will form the basis for the placement at wZ at 39/32 (342.5 cents) in his `oud tuning. See ibid. at 150: When the interval joined to the tone has for its ratio 13:12, the ratio which completes the fourth will not be consonant, but its ratio [128:117] much resembles 12:11 [respectively 155.6 and 150.6 cents]. This genus is in favor; we give its numeric expression: [Illustration from d'Erlanger, ratios for notes added] 139 204 156 13:12 9:8 128:117 468 432 384 351 1/1 13/12 39/32 4/3 0 139 342 498 In this tetrachord, unlike the others, the lowest interval is the superparticular neutral step at 13:12 (138.6 cents), followed by the 9:8 tone, and finally by the complementary interval at 128:117 (155.6 cents). Ibn Sina points out that this last interval "will not be consonant" in itself, but notes that "[t]his genus is in favor" -- statements by no means contradictory, as he has explained in discussing the Ditonic Diatonic with its 256:243 step. In observing that 128:117 "much resembles 12:11" (150.6 cents), he again calls our attention to the 352:351 comma earlier noted in the discussion of al-Farabi's second jins with 12:11 and 88:81. In each jins, the comma is the difference between the 32:27 minor third (at 294.1 cents) to be divided between the two neutral steps, and the slightly smaller 13:11 (289.2 cents) which would result from having both neutral steps superparticular at 13:12 and 12:11. In al-Farabi's jins, 12:11 is just, with 88:81 thus larger than 13:12 by the comma; in Ibn Sina's jins, 13:12 is just while 128:117 is a comma larger than the 12:11 ratio which it "much resembles." The 13:12-9:8-128:117 arrangement of this jins might serve as an ideal form of the tetrachord leading up to the final in the Persian Avaz-e Bayat-e Esfahan, where the intonation as described by Hormoz Farhat might be around 6-9-7 commas (or, in Farhat's pragmatic approach, something like 135-205-160 cents). See Farhat (199 , ). This permutation, like that used in his `oud tuning at 9:8-13:12-128:117, has a 39/32 neutral third at 342 cents, with the large neutral second at 128:117 or 156 cents as the highest step of the tetrachord. Then follows his second Zalzalian diatonic jins with 9:8, ibid. at 150. If the interval joined to the tone is the smallest of the emmeles [intervals apt for melody], the remaining interval will have for its ratio 208:189. The genus is numbered: [Illustration from d'Erlanger, ratios for notes added] 204 128 166 9:8 14:13 208:189 252 224 208 189 1/1 9/8 63/52 4/3 0 204 332 498 The complementary interval has here a ratio which approximates, but rather distantly, 10:9 [respectively 165.8 and 182.4 cents]; it has had {? little significance in music ? -- I am not sure of correct translation}. In Ibn Sina's system, as already mentioned, 14:13 (128.3 cents) is the smallest of the emmeles in the "middle" category where twice the interval is equal to the greater part of a fourth. From a modern perspective, it may be considered the smallest of the superparticular neutral or Zalzalian seconds, although in later medieval theory and modern Turkish theory, 15:14 (119.4 cents) and 16:15 (111.7 cents) are also regarded as "middle" intervals, with 16:15 (or its Pythagorean near-equivalent 2187:2048) as the upper step of Rast. In Persian, Turkish, and certain flavors of Arab music, steps of around 14:13 (say 125-130 cents) seem rather common. As in Ibn Sina's last tetrachord, the smaller neutral step precedes the larger, here 14:13 and 208:189 (165.8 cents). The small neutral third is at 63/52 (332.2 cents), which might be described as a low Mustaqim, Ibn Sina's name for the mode featuring 9:8-13:12-128:117 (204-139-156 cents), literally in Arabic the "right, correct, usual" mode -- equivalent to the later Persian term Rast. Possibly it might be found in modern Persian practice as a tuning of Dastgah-e Afshari, or Gushe-ye Shekaste of Dastgah-e Mahur, with a very low neutral third; I much use it in my own peripheral practice. In discussing the "complementary interval" of this tetrachord, the large neutral second at 208:189 (166 cents), Ibn Sina presents us with a surprise. Earlier, in discussing al-Farabi's 9:8-11:10-320:297 (204-151-129 cents), he observed that 320:297 is "very close" to 14:13 (128 cents) -- the difference being the 2080:2079 comma (0.8 cents). We might likewise expect him to compare 208:189 to al-Farabi's almost identical large neutral second at 11:10 (165.0 cents), narrower by this same comma. However, he finds instead that 208:189 "approximates, but rather distantly, 10:9" -- a difference of 165.8 vs. 182.4 cents, or 105/104 (16.567 cents). Possibly this comparison reflects the kind of response I got from one Turkish musician, who suggested to me that a tempered tetrachord such as 207-127-163 cents (334-cent neutral third) might represent, not so much a low Mustaqim, as a high Nihavend, a maqam for which he generally favored a third around 6/5 (9:8-16:15-10:9), but with the very small neutral third a pleasant variation. Ibn Sina's statement thus may represent, not a slip of the pen (10:9 where 11:10 was meant), but a kind of categorical perception that a very small neutral third may serve as an extra-large minor or "supraminor" third, and thus the upper step completing the fourth as in effect a very small tone, somewhat like 10:9. -------------------------------------------------- 4. Ibn Sina's comments on musicians and intonation -------------------------------------------------- Having surveyed his two "most noble" Zalzalian ajnas with steps of 8:7-13:12-14:13 (with the upper steps arranged in either order), the tetrachords of al-Farabi where 9:8 is followed by a larger neutral second at 11:10 or 12:11, and his own ajnas where 9:8 is preceded by 13:12 (a genus "in favor") or followed by 14:13, he turns to the intonational practices of some of his contemporaries, ibid. at 150. The musicians of our day confuse the complementary intervals, the intervals of relaxation, and the smallest intervals in the series of large emmeles [ranging from 5:4 to 14:13]; they play the one for the other, without perceiving the differences by which they are constituted. Thus, they use indifferently a tone augmented by the interval of 13:12 or 14:13. These remarks about the frequent imprecision of intonation around a millennium ago include a possible hint as to Persian tastes in the earlier 11th century: "they use indifferenctly a tone augmented by the interval of 13:12 or 14:13." Ibn Sina does not mention the larger superparticular steps favored in the Zalzalian diatonic tetrachords of al-Farabi at 11:10 and 12:11, the latter, of course, the basis for his mode of Zalzal (9:8-12:11-88:81) and his `oud tuning with wZ at 27/22. The discussion of 13:12 and 14:13, incidentally, brings into play another comma: their difference of 169:168 (10.274 cents). His next comment moves from the confusion of 13:12 and 14:13 to the tuning of the `oud, ibid. at 150: When they set the place of the middle finger of Zalzal, some indeed fix the fret higher, and others lower, and certain people [set it] halfway between the index finger [9/8] and ring finger [4/3], as will be seen below. This description tells us, first, that tastes vary in placing wZ, so that there is not a single generally accepted algorithm, although "certain people" favor the aliquot division of 72:64:59:54 or 9:8-64:59:54 (204-141-153 cents) which places wZ at 72/59 or 344.7 cents, just a tad higher than Ibn Sina's 39/32 (342.5 cents) -- a comma of 768:767 (2.256 cents). Taken together, these comments tell us that people indifferently use "a tone augmented by the interval of 13:12 or 14:13" -- that is, a smaller neutral third of 39/32 or 63/52, as in his own two Zalzalian diatonic tetrachords with 9:8; and that some place wZ higher or lower, but with 72/59 or 345 cents as one practice, where a tone presumably around 9:8 would be followed by a step (64:59, 140.8 cents) very slightly larger than 13:12. Thus suggests the hypothesis that Ibn Sina -- who is well familiar with al-Farabi's tetrachords joining 9:8 with 11:10 or 12:11, as well as his own "very noble" Zalzalian ajnas where 8:7 is followed by a high neutral third at 26/21 or 16/13 -- has made an artistic choice in favor of a lower neutral third, and more specifically a lower wZ, when 9:8 is the tone rather than 8:7. The fact that his contemporaries vary in the placement of wZ, but evidently often lean to a lower position such as 72/59 -- and that their reported confusion involves 14:13 and 13:12, rather than al-Farabi's 11:10 and 12:11 -- suggests that Ibn Sina and his Persian contemporaries may have a different preference than either al-Farabi or the many modern Arab musicians likewise favoring a higher wZ somewhere between around 27/22 (355 cents) and 99/80 (369 cents), as is illustrated by the range of Syrian practices. One factor influencing al-Farabi's placement of wZ at 27/22 may be the tuning of two other frets that makes it possible to find this interval by an aliquot division: wZ 355 27/22 |------------------|----|----| 81 68 66 64 1/1 81/68 81/64 0 303 408 wF ditone Here wZ may be placed very conveniently midway between the wusta Fars or "Persian third" at 81/68 (302.9 cents) and the Pythagorean major third or ditone at 81/64 (407.8 cents) -- giving it a location at 27/22 or 354.5 cents. The absence of wF in the `oud tuning described by Ibn Sina might make this placement less obvious or convenient -- but he is fully aware of the 9:8-12:11-88:81 tetrachord of al-Farabi, and nevertheless expresses a preference for a smaller neutral second step of 13:12, and thus a lower position for wZ at 39/32. Since he mentions how some of his contemporaries use 14:13 and 13:12 interchangeably -- the former not obviously so close to the 64:59 step after 9:8 that results from the 72:64:59:54 procedure -- it appears that there is considerable variation in intonation, but generally tending to smaller neutral second intervals and neutral third steps in tetrachords with 9:8. The closing portion of this discussion on intonation in practice might tend to confirm this general impression, ibid. at 150: They confuse also the intervals of the remainder (limma) and that which separates the two frets of the middle finger, a quartertone [Ibn Sina's favored scheme places these frets at 32/27 and 39/32, a difference of 1053:1024 or 48.3 cents], and they play indifferently the one for the other. It is not, however, impossible to meet artists whose ears are quite refined in distinguishing these differences. If we assume that Ibn Sina's contemporaries likewise placed the minor third fret at around 32/27 (294 cents), then a "quartertone" up to wZ would imply a placement at somewhere around 39/32 (1053:1024, 48.3 cents); or 72/59 (243:236, 50.6 cents); or possible 11/9 (33:32, 53.3 cents). In contrast, a higher placement such as 16/13 (27:26, 65.3 cents) or 26/21 (117:112, 75.6 cents) would produce more of a thirdtone, comparable to the 28:27 of Archytas (63.0 cents) and indeed somewhat larger. This may be another clue that Ibn Sina and his contemporaries generally favor a lower wZ, and smaller neutral steps following 9:8, rather than al-Farabi's 11:10 or 12:11. ----------------------------------------- 5. Ibn Sina's placement of wZ on the `oud ----------------------------------------- Having established a context by surveying Ibn Sina's approach to Zalzalian diatonic tetrachords and his description of contemporary intonational practices -- often marked by the kind of imprecision or confusion we see alleged by European theorists when dealing with other aspects of practice -- let us now consider his statement about the placement of wZ, ibid. at 235: The moderns have fixed another fret for the medius, about halfway between the index and the auricular. Some place it lower, others higher, obtaining therefore diverse genera of the fourth. But in our day they do not distinguish these differences. As in the general discussion on contemporary practice, Ibn Sina notes the popularity of a position "about halfway between the index [9/8] and the auricular [4/3]," with some placing it lower, others higher. The aliquot division was evidently popular, because Safi al-Din both reports it as a much-favored choice some two centuries later, and makes the resulting Rast tetrachord of 9:8-64:59:59:54 one of his principal ajnas -- almost identical to Ibn Sina's Mustaqim. The greater part do this so that the interval between the index and medius is 13/12. The approximate interval between the medius and the auricular will thus be 12/11, and the real interval 128/117, which permits the formation of all of the genres/genera we have cited. (Ibid. at 235) Here Ibn Sina, who himself has found a jins of 13:12-9:8-128:117 "in favor," reports that the permutation 9:8-13:12-128:117 is the basis for for the practice of "[t]he greater part." He then repeats his point made in the tetrachord discussion that 128:117 "much resembles" the simpler superparticular step of 12:11 -- the latter and tidier ratio being the "approximate interval," and 128:117 "the real interval" found between the wZ at 39/32 and the 4/3 fourth. While his language about the aliquot division indicates that some musicians indeed use 72/59 as the placement of wZ, I do not see tha language about an "approximate interval" of 12:11 between wZ and the 4/3 fourth as indicating a practical tuning with neutral steps at 13:12 followed by 12:11 -- as excited as I might be by such a practical use in the 11th century of 13:11, the basis of my favorite maqam tunings! It may be helpful to compare the 72/59 aliquot tuning later reported as prevalent by Safi al-Din, Ibn Sina's 39/32 tuning he reports is favored by the "greater part" of musicians, and an interpretation of a tuning literally implementing neutral second steps of 13:12 and 12:11 by tuning down from the 4/3 fret -- something I intuit would be unlikely in practice, except perhaps by indirection. Aliquot tuning -- wZ midway between 9/8 and 4/3 (72/59, 344.7 cents) 204 141 153 9:8 64:59 59:54 0 204 345 498 1/1 9/8 72/59 4/3 72 64 59 54 Tuning of "greater part" -- wZ at 9:8 + 13:12 (39/32, 342.5 cents) 204 139 156 9:8 13:12 128:117 1/1 9/8 39/32 4/3 0 204 342 498 468 416 384 351 Hypothetical tuning with upper 13:12 + 12:11 (11/9, 347.4 cents) 204 139 151 44:39 13:12 12:11 0 209 347 498 1/1 44/39 11/9 4/3 44 39 36 33 The hypothetical 44:39-13:12-12:11 tuning actually has simplest string ratios at 44:39:36:33. However, it requires making the traditional tone at 9/8 larger by 352:351 -- the comma which Ibn Sina addressed, although without specifying its ratio, in his discussion of tetrachords -- so as to arrive at 44:39 (208.8 cents), and an upper minor third at 13:11 rather than 32:27, which neatly divides 13:12:11 (139-151 cents). Such a tuning might happen by serendipity or creative indirection either on an 11th-century `oud or on a modern Persian tar, setar, or santur -- in the modern setting, at least, we know through actual measurements that tones a bit larger than 9/8, minor thirds at around 13/11 rather than the slightly larger 32/27, and also fourths slightly smaller than 4/3 are all within the usual range of variation. However, Ibn Sina himself seems simply to be following his bent for often noting how a complex "complementary interval" or remainder such as 128:117 may "much resemble" a nearby superparticular ratio such as 13:12. The idea of actually placing the usual 9/8 fret a few cents higher at 44/39 does not seem to be part of his practical agenda, although, given the variations of intonational practice he reports, it might well happened at various points during the centuries when both he and Safi al-Din report tunings close to 44:39:36:33 much in favor. ------------------------------------------- 6. Summary and some tetrachord permutations ------------------------------------------- Ibn Sina presents to us a rich variety of Zalzalian diatonic tetrachords, two "very noble" ones using 8:7-13:12-14:13 (upper steps in either order), derived from the mathematically elegant and musically beautiful neo-Ptolemaic division 16:14:13:12. These tetrachords may be described in later terms as forms of "septimal Rast" with large or bright neutral thirds at 26/21 (370 cents) or 16/13 (359 cents). He then, after briefly addressing the Ditonic Ditonic at 9:8-9:8-256:243 as an illustration of how a "dissonant" or nonsuperparticular step such as 256:243 can nevertheless be part of a pleasing jins, and noting 10:9-9:8-16:15 with all superparticular ratios, turns to Zalzalian diatonic tetrachords with a 9:8 tone. He first describes al-Farabi's 9:8-11:10-320:297 and the better-known 9:8-12:11-88:81 with its neutral third at 27/22, not finding these tunings in favor, but possibly implying that, as with the Ditonic Diatonic, the complex "remainder" intervals need not make these ajnas unpleasant. Then, he addresses a jins "in favor" with 9:8 -- 13:12-9:8-128:117, a fine example of what might today might be called the "Old Esfahan" in Persian music with a rather low neutral second, here 39/32 or 342 cents. The permutation of this at 9:8-13:12-128:117, likewise with the smaller neutral step preceding the larger, serves as the basis for his `oud tuning, which he reports the "greater part" of musicians use. He also describes a jins with the smallest "large emmele," also in modern terms the smallest superparticular neutral third, at 14:13 -- the jins 9:8-14:13-208:189, with a small neutral third at 63/52 or 332 cents. Ibn Sina is very interested in resemblances between complex nonsuperparticular intervals which inevitably arise when dividing a 32:27 minor third (4:3 less 9:8) into two neutral steps, and the small commas involved -- thus al-Farabi's 320:297 as near 14:13, and 88:81 as near 13:12, as well as Ibn Sina's 128:117 as near 12:11. The first resemblance involves the 2080:2079 comma at 0.8 cents, while the other two bring into play the 352:351 comma at 4.9 cents. The discussion is especially fascinating because it presents the idea of coupling 9:8 with each of the superparticular neutral seconds: thus four shadings at 11:10, 12:11, 13:12, and 14:13 (165, 151, 139, and 128 cents), producing neutral thirds at 99/80, 27/22, 39/32, and 63/52 (369, 355, 342, and 332 cents). Turning to the real-world intonational practices of his contemporaries, he describes how these musicians often confuse or use indiscriminately the ratios of 13:12 and 14:13, more specifically when a tone is "augmented" by one of these intervals, evidently resulting in a neutral or Zalzalian third at 39/32 or 63/52 if a 9:8 tone is meant. Given his "most noble" jins with an 8:7 tone followed by 13:12 and 14:13 in either order, however, this observation could also refer to the interchangeable and sometimes indiscriminate use of the two forms of this jins also, with a Zalzalian third at 26/21 (370 cents) or 16/13 (359 cents). What Ibn Sina does mention in this discussion of actual practice is joining a tone directly to a larger neutral step at 11:10 (165 cents) or 12:11 (151 cents), as happens with 9:8 in al-Farabi's two Zalzalian diatonic ajnas. Rather, in his own tetrachords, Ibn Sina always joins a tone (at 8:7 or 9:8) to a neutral step at 13:12 or 14:13, with any large neutral step (as in 9:8-13:12-128:117 or 9:8-14:13-208:189) relegated to the "complementary interval" completing the fourth. His comments in this discussion on practice describe how different people tune wZ on the `oud lower or higher, but often it is at or in the vicinity of 72/59 (345 cents), or midway between 9/8 and 4/3 (72:64:59:54), a placement later reported as much favored by Safi al-Din, who adopts the 9:8-64:59-59:54 tetrachord as one of his principal ajnas. A reference to the 32/27 and wZ frets as a "quartertone" apart (1053:1024 or 48 cents with Ibn Sina's 39/32 placement (342 cents), or 243:236 or 51 cents with the 72/59 placement he reports some people use) tends to confirm an impression that a relative low placement of wZ generally prevails. If it were around 16/13 or 26/21, say, as in the two version of his jins with 8:7, then the 32/27 and wZ frets would be some kind of "thirdtone" apart, rather than the "quartertone" that Ibn Sina describes. His remarks focusing specifically on the placement of wZ again note that a placement at or near 72/59 is common, with "the greater part" of musicians choosing 39/32 (9:8 + 13:12). As in his discussion of the ajnas, where he addressed 13:12-9:8-128:117, he notes that the complementary and complex interval of 128:117 (156 cents) is an "approximate" 12:11 (151 cents) -- a difference we know as the 352:351 comma at 4.9 cents. Thus in his 9:8-13:12-128:117 jins (204-139-156 cents) on the `oud, the basis of his Mustaqim modality, he says that wZ at 39/32 will be 13:12 from the 9/8 fret, an an "approximate" 12:11 but a "real" 128:117 from the 4/3 fret. Ibn Sina, therefore, does not himself appear to be proposing or documenting a tetrachord literally at 44:39:36:33 or 44:39-13:12-12:11 (209-139-151 cents), with a 44/39 fret higher than the usual 9/8 by a 352:351 comma, leaving an upper 13/11 minor third (289 cents) smaller than 32/27 (294 cents) by this same comma, and permitting a neat aliquot division of this third into 13:12:11. Nor does he appear to be suggesting some kind of temperament where 352:351 is distributed between two or more intervals, e.g. 207-139-150 cents (tone slightly wider than 9:8 but smaller than 44:39, fourth slightly narrow, and 13:12 and 12:11 very close to just). However, his discussion of how 128:117 resembles 12:11, and likewise with al-Farabi's 88:81 and the superparticular ratio of 13:12, certainly sets the stage for divisions such as 44:39:36:33 and their tempered variations. ------------------ In surveying the Zalzalian diatonic tetrachords which Ibn Sina addresses, including the two from al-Farabi, we may find it helpful to look at all six permutations of each tetrachord (as later recommended by Safi al-Din), and to do so in a way that identifies what Ibn Sina might call "resemblances" between forms differing only by the small commas which he notes (specifically 2080:2079 and 352:351). Although not listed in his discussion of tetrachords, the aliquot division he reports as sometimes favored on the `oud, 72:64:59:54, is also of interest. This tetrachord, with its neutral third at 72/59, brings into play the additional commas of 768:767 (2.256 cents) and 649:648 (2.670 cents) -- the amounts by which 72/59 (344.7 cents) is larger than Ibn Sina's wZ at 39/32 (342.5 cents), but smaller than 11/9 (347.4 cents), the neutral third in a permutation of al-Farabi's famous 9:8-12:11-88:81 to 9:8-88:81-12:11 (204-143-151 cents). For each permutation, I give a description both in terms of the later Systematist classification of Safi al-Din and his followers, with T as a tone (e.g. 9:8 or 8:7), and J as some kind of neutral or "middle" second step; and in terms of maqam/dastgah categories. For Ibn Sina's two forms of the jins with steps of 8:7-13:12-14:13, I make a comparison with al-Farabi's two Zalzalian diatonic tetrachords in order to note the comparable sizes of neutral thirds, but produced in al-Farabi's approach by joining a 9:8 tone to a large neutral step at 11:10 or 12:11, i.e. 9:8-11:10-320:297 or 9:8-12:11-88:81. The table compares ajnas of Ibn Sina (IS) and al-Farabi (AF). The arrangement generally follows the Systematist classification of tetrachords, as presented by Safi al-Din, whose seven types of tetrachords include three types relevant to this survey IV, V, VI). with T (tanini) as a tone; B (bakiye) as a limma; and J (mujannab) as a neutral or Zalzalian second. I. T T B (current Arab `Ajam or Persian Mahur, e.g. 9:8-9:8-256:243) II. T B T (current Arab Nahawand or Persian Nava, e.g 9:8-256:243-9:8) III. B T T (current Arab or Turkish Kurdi, e.g. 256:243-9:8-9:8) IV. T J J (current Arab or Turkish Rast, e.g. 9:8-12:11-88:81) V. J J T (current Arab Bayyati or Persian Shur, e.g. 13:12-128:117-9:8) VI. J T J (Buzurg, current Persian Segah or Esfahan, e.g. 13:12-8:7-14:13) VII. J J J B (Systematist Isfahan, e.g. modern 13:12-12:11-14:13-22:21) Each complete set of permutations takes the order of the two forms of IV, then V, and then VI. Note that ajnas 1, 4, and 5 are Ibn Sina's own forms; ajnas 2 and 3 are those he addressed from al-Farabi; and jins 6 is the `oud tuning of 72:64:59:54 which he mentions, and Safi al-Din later adopts as one of his principal ajnas. 1. Ibn Sina's "most noble" jins (8:7, 13:12, 14:13) (Thirds of 1.1 and 2.1 differ by 2080:2079 or 0.833 cents; Thirds of 1.2 and 3.1 differ by 352:351 or 4.925 cents) IS gives 1.1, 1.2 AF gives 2.1, 3.1 1.1. Systematist TJJ 2.1 Systematist TJJ "Septimal Rast, 26/21" "High Rast" (26/21 third, 369.7c) (99/80 third, 368.9c) 104 91 84 78 396 352 320 297 1/1 8/7 26/21 4/3 1/1 9/8 99/80 4/3 0 231 370 498 0 204 369 498 8:7 13:12 14:13 9:8 11:10 320:297 231 139 128 204 165 129 1.2. Systematist TJJ 3.1 Systematist TJJ "Septimal Rast, 16/13" "Medium-high Rast" (16/13 third, 359.5c) (27/22 third, 354.5c) 16 14 13 12 108 96 88 81 1/1 8/7 16/13 4/3 1/1 9/8 27/22 4/3 0 231 359 498 0 204 355 498 8:7 14:13 13:12 9:8 12:11 88:81 231 128 139 204 151 143 1.3. Systematist JJT No comparison made "Higher septimal Shur, Bayyati, or Ushshak" 364 336 312 273 1/1 13/12 7/6 4/3 0 128 267 498 14:13 13:12 8:7 128 139 231 1.4. Systematist JJT "Lower septimal Shur, No comparison made Bayyati, or Ushshak" 28 26 24 21 1/1 14/13 7/6 4/3 0 128 267 498 14:13 13:12 8:7 128 139 231 1.5. Systematist JTJ No comparison made "Higher Buzurg, Avaz-e Bayat-e Esfahan, or Byzantine Soft Chromatic, 8-13-7 steps of 68 in system of Chrysanthos of Madytos" 52 48 42 39 1/1 13/12 26/21 4/3 0 139 370 498 13:12 8:7 14:13 139 232 128 1.6. Systematist JTJ No comparison made "Higher Buzurg, Avaz-e Bayat-e Esfahan, or Byzantine Soft Chromatic, 8-13-7 steps of 68 in system of Chrysanthos of Madytos" 112 104 91 84 1/1 14/13 16/13 4/3 0 139 370 498 13:12 8:7 14:13 139 232 128 ------------------------------------------------ 4. Ibn Sina's jins "in favor" (9:8, 13:12, 128:117) 3. Al-Farabi's 2nd cited jins (9:8, 12:11, 88:81) (Differences of 352:351 or 4.925 cents) IS gives 4.1, 4.5 AF gives 3.1 4.1. Systematist TJJ 3.2. Systematist TJJ "Higher Mustaqim, "Higher Mustaqim or Arab Dastgah-e Afshari, Rast Jadid, 9-6-7 commas Gushe-ye Shekaste" (cf. Rast, 9-7-6 commas)" 468 416 384 351 396 352 324 297 1/1 9/8 39/32 4/3 1/1 9/8 11/9 4/3 0 204 342 498 0 204 347 498 9:8 13:12 128:117 9:8 88:81 12:11 204 139 156 204 143 151 4.2. Systematist TJJ 3.1. Arab Rast, Byzantine "Medium-high Arab Rast, Diatonic (Chrysanthos) Low Turkish Rast" 9-7-6 commas 144 128 117 108 108 96 88 81 1/1 9/8 16/13 4/3 1/1 9/8 27/22 4/3 0 204 359 498 0 204 355 498 9:8 128:117 13:12 9:8 12:11 88:81 204 156 139 204 151 143 4.3. Systematist JJT 3.4. Systematist JJT "Moderate Shur, Arab "Moderate Arab Bayyati, Bayyati, Turkish Ushshak" 6-7-9 commas" 416 384 351 312 352 324 297 264 1/1 13/12 32/27 4/3 1/1 88/81 32/27 4/3 0 139 294 498 0 143 294 498 13:12 128:117 9:8 88:81 12:11 9:8 139 156 204 143 151 204 4.4. Systematist JJT 3.3. Systematist JJT "Moderate Arab Huseyni, "Moderate Arab Huseyni, Low Turkish Huseyni, 7-6-9 commas" 128 117 108 96 96 88 81 72 1/1 128/117 32/27 4/3 1/1 12/11 32/27 4/3 0 156 294 498 0 151 294 498 128:117 13:12 9:8 12:11 88:81 9:8 156 139 204 151 143 204 4.5. Systematist JTJ 3.6. Systematist JTJ "Persian Old Esfahan" "Arab `Iraq, in theory, 6-9-7 commas" 468 432 384 351 88 81 72 66 1/1 13/12 39/32 4/3 1/1 88/81 11/9 4/3 0 139 342 498 0 143 347 498 13:12 9:8 128:117 88:81 9:8 12:11 139 204 156 143 204 151 4.6. Systematist JTJ 4.5. Systematist JTJ "Medium Persian Esfahan" "Medium Persian Esfahan, 7-9-6 commas" 128 117 104 96 108 99 88 81 1/1 128/117 16/13 4/3 1/1 12/11 27/22 4/3 0 156 359 498 0 151 355 498 128:117 9:8 13:12 12:11 9:8 88:81 156 204 139 151 204 143 ------------------------------------------------ 5. Ibn Sina's jins (9:8, 14:13, 208:189) 2. Al-Farabi's 1st cited jins (9:8, 11:10, 320:297) (Differences of 2080:2079 or 0.833 cents) IS gives 5.1 AF gives 2.1 5.1. Systematist TJJ 2.2. Systematist TJJ "Low Mustaqim, Afshari, Same as 5.1 or Shekaste; possibly High Turkish Nihavend" 252 224 208 189 360 320 297 270 1/1 9/8 63/52 4/3 1/1 9/8 40/33 4/3 0 204 332 498 0 204 333 498 9:8 14:13 208:189 9:8 320:297 11:10 204 128 166 204 129 165 5.2. Systematist TJJ 2.1. Systematist TJJ "High Syrian Rast, or Same as 5.2 Medium Ottoman Rast" 468 416 378 351 396 352 320 297 1/1 9/8 26/21 4/3 1/1 9/8 99/80 4/3 0 204 370 498 0 204 369 498 9:8 208:189 14:13 9:8 11:10 320:297 204 166 128 204 165 129 5.3. Systematist JJT 2.4. Systematist JJT "Low Shur, Lebanese Folk Same as 5.3 Bayyati, or Turkish Ushshak" 448 416 378 336 320 297 270 240 1/1 14/13 32/27 4/3 1/1 320/297 32/27 4/3 0 128 294 498 0 129 294 498 14:13 208:189 9:8 320:297 11:10 9:8 128 166 204 129 165 204 5.4. Systematist JJT 2.3. Systematist JJT "Turkish Huseyni or Same as 5.4 High Arab Huseyni" 416 378 351 312 352 320 297 264 1/1 208/189 32/27 4/3 1/1 11/10 32/27 4/3 0 166 294 498 0 165 294 498 208:189 14:13 9:8 11:10 320:297 9:8 166 128 204 165 129 204 5.5. Systematist JTJ 2.6. Systematist JTJ Low Arab `Iraq Same as 5.5 (as theoretical jins), High Turkish Segah, Low Persian Old Esfahan 252 234 208 189 320 297 264 240 1/1 14/13 63/52 4/3 1/1 320/297 40/33 4/3 0 128 332 498 0 129 333 498 14:13 9:8 208:189 320:297 9:8 11:10 128 204 166 129 204 165 5.6. Systematist JTJ 2.5. Systematist JTJ Possible High Same as 5.6 Persian Segah 208 189 168 156 396 360 320 297 1/1 208/189 26/21 4/3 1/1 11/10 99/80 4/3 0 166 370 498 0 165 369 498 208:189 9:8 14:13 11:10 9:8 320:297 166 204 128 165 204 129 ------------------------------------------------ 5. Ibn Sina's jins "in favor" (9:8, 13:12, 128:117) 6. Ibn Sina's jins "some" use (9:8, 64:59, 59:54) 2. Al-Farabi's Zalzal jins (9:8, 12:11, 88:81) (Differences: 5-6, 768:767 or 2.256 cents; 6-2, 649:648 or 2.670 cents; 5-2, 352:351 or 4.925 cents) 4.1. Systematist TJJ 6.1 Systematist TJJ 3.2. Systematist TJJ "Higher Mustaqim, Same as 4.1 "Higher Mustaqim or Arab Dastgah-e Afshari, Rast Jadid, 9-6-7 commas Gushe-ye Shekaste" (cf. Rast, 9-7-6 commas)" 468 416 384 351 72 64 59 54 396 352 324 297 1/1 9/8 39/32 4/3 1/1 9/8 72/59 4/3 1/1 9/8 11/9 4/3 0 204 342 498 0 204 345 498 0 204 347 498 9:8 13:12 128:117 9:8 64:59 59:54 9:8 88:81 12:11 204 139 156 204 141 153 204 143 151 4.2. Systematist TJJ 6.2. Systematist TJJ 3.1. Arab Rast, Byzantine "Medium-high Arab Rast, Same as 4.2 Diatonic (Chrysanthos) Low Turkish Rast" 9-7-6 commas 144 128 117 108 2124 1888 1724 1593 108 96 88 81 1/1 9/8 16/13 4/3 1/1 9/8 59/54 4/3 1/1 9/8 27/22 4/3 0 204 359 498 0 204 357 498 0 204 355 498 9:8 128:117 13:12 9:8 59:54 64:59 9:8 12:11 88:81 204 156 139 204 153 141 204 151 143 4.3. Systematist JJT 6.3 Systematist JJT 3.4. Systematist JJT "Moderate Shur, Arab Same as 4.3 "Moderate Arab Bayyati, Bayyati, Turkish 6-7-9 commas" Ushshak" 416 384 351 312 64 59 54 48 352 324 297 264 1/1 13/12 32/27 4/3 1/1 64/59 32/27 4/3 1/1 88/81 32/27 4/3 0 139 294 498 0 141 294 498 0 143 294 498 13:12 128:117 9:8 64:59 59:54 9:8 88:81 12:11 9:8 139 156 204 141 153 204 143 151 204 4.4. Systematist JJT 6.4. Systematist JJT 3.3. Systematist JJT "Moderate Arab Huseyni, Same as 4.4 "Moderate Arab Huseyni, Low Turkish Huseyni, 7-6-9 commas" 128 117 108 96 1888 1724 1593 1416 96 88 81 72 1/1 128/117 32/27 4/3 1/1 59/54 32/27 4/3 1/1 12/11 32/27 4/3 0 156 294 498 0 153 294 498 0 151 294 498 128:117 13:12 9:8 59:54 64:59 9:8 12:11 88:81 9:8 156 139 204 153 141 204 151 143 204 4.5. Systematist JTJ 6.5. Systematist JTJ 3.6. Systematist JTJ "Persian Old Esfahan" Same as 4.5 "Arab `Iraq, in theory, 6-9-7 commas" 468 432 384 351 576 531 472 432 88 81 72 66 1/1 13/12 39/32 4/3 1/1 64/59 72/59 4/3 1/1 88/81 11/9 4/3 0 139 342 498 0 141 345 498 0 143 347 498 13:12 9:8 128:117 64:59 9:8 59:54 88:81 9:8 12:11 139 204 156 141 204 153 143 204 151 4.6. Systematist JTJ 6.6. Systematist JTJ 4.5. Systematist JTJ "Medium Persian Esfahan" Same as 4.6 "Medium Persian Esfahan, 7-9-6 commas" 128 117 104 96 236 216 192 177 108 99 88 81 1/1 128/117 16/13 4/3 1/1 59/54 59/48 4/3 1/1 12/11 27/22 4/3 0 156 359 498 0 153 357 498 0 151 355 498 128:117 9:8 13:12 59:54 9:8 64:59 12:11 9:8 88:81 156 204 139 153 204 141 151 204 143 Rough preliminary draft -- not complete Margo Schulter June 7, 2013