The question of whether or how neutral thirds -- also known as Zalzalian thirds (see below) or middle thirds -- are used in Near Eastern music is a complex one, with various views expressed by practical musicians and theorists beginning with al-Farabi (c. 870-950). First, in practice, the range of these neutral or middle thirds -- somewhere between a large minor third at 6/5 (316 cents) and a small major third at 5/4 (386 cents) -- is quite wide, with the spectrum of possible sizes nicely surveyed by the great philosopher, physician, and music theorist Ibn Sina (c. 980-1037). And this amazing variety continues today in the many local and regional practices of Arab, Turkish, Kurdish, and Iranian music. Here I will show how each of the answers proposed in the question may have some grain of truth, but with qualifications that suggest a more complex reality over the last 1200 years or a bit more of Near Eastern music in practice and theory. To sum up in advance: (1) Theoretical writings of the 10th-15th centuries show a wide range of middle thirds in use, mostly based on a diverse set of melodic steps using superparticular ratios of the pattern (n+1:n), e.g. the 9:8 tone and middle second steps such as 11:10, 12:11, 13:12, and 14:13; as well as more complex ratios such as 88:81 or 128:117 that inevitably occur in these tunings also. While 11:9 at 347 cents may sometimes be found (e.g. when adjacent steps of 9:8 at 204 cents and 88:81 at 143 cents occur), these middle thirds cover a wide spectrum from around 63:52 (332 cents) to 26:21 (370 cents). (2) The method of deriving middle intervals from very long chains of pure or virtually pure fifths -- of which a chain of 52 plus a pure octave make a musical circle for all practical purposes -- would fit the "53-comma" theory of modern Turkey and Syria. There are indications that some Turkish performers on flexible-pitch instruments may produce sizes for certain intervals which are almost identical to those of the 53-comma system. However, the comma system seems more of a general guide to interval categories and regions than an explanation of how intervals are tuned in practice. (3) The concept of "an intonational inflection not linked to the harmonic series or the framework of Pythagorean intervals" may reflect the fact that intonational expressiveness and small nuances are valued for their own sake in Near Eastern traditions, and that the tuning of the Persian tar, setar, or santur is very much to taste. This does not mean that some linkage to other intonational parameters of the music, such as the tuning of regular fifths and fourths, does not exist; but that indeed the performer has great freedom and discretion, with a given theoretical step realized in practice by a "cluster" of pitches at different points in a performance. ------------------------------------------------------- 1. Classic Near Eastern Theory: The 10th-15th centuries ------------------------------------------------------- Beginning with al-Farabi and Ibn Sina, Near Eastern writers of the 10th-15th centuries sought to document and measure the tunings in use, for example in fretting the `oud (an instrument of which the European lute is an offshoot both in name and in design). One starting point was the famous `oudist Mansur Zalzal of Baghdad (?-791). Zalzal was renowned for introducing a middle finger or _wusta_ fret at a middle third above the open string, known in his honor as a Zalzalian third, and with its fret called the _wusta Zalzal_, or middle finger fret of Zalzal. However, whatever Zalzal's own preferences may have been in 8th-century Baghdad, later musicians had a variety of placements for this fret. Near Eastern music theory of the 9th-15th centuries represented a creative synthesis between the Classic Greek tradition and the indigenous practices of the Arab, Persian, and other peoples of the region. Especially influential were Pythagoras and his followers, who emphasized the basic concords of the octave (2:1), fifth (3:2), and fourth (4:3); and also Ptolemy, who favored a wide range of intervals and ratios which had the property of being superparticular. A superparticular ratio follows the pattern of (n+1:n): e.g. 3:2, 4:3, 5:4, 6:5, 7:6, 8:7, etc. These favored ratios may be quite small, for example 28:27 (63 cents) as a semitone or thirdtone much esteemed by Archytas for his melodic modes (also cited by Ptolemy); or Ptolemy's 22:21 step (81 cents). Also, like Classic Greek theory, Near Eastern theory often focused on tetrachords or divisions of the 4/3 fourth using four notes or three intervals. These tetrachords could be combined to build scales over an octave or more; and pentachords, or divisions of the fifth (typically using five notes or four intervals) sometimes come up both in 10th-15th century theory and in modern approaches to Near Eastern theory. For al-Farabi, a tetrachord which became part of his `oud tunings was formed by the steps 9:8-12:11-88:81 (204-151-143 cents), or 1/1-9/8-27/22-4/3 (0-204-355-498 cents). His Zalzal fret was thus placed at 27/22, or 355 cents, the sum of the two lower superparticular steps in this tetrachord: the usual 9:8 tone, plus the 12:11 middle or Zalzalian second step. Note that while 11/9 (347 cents) would be the simplest ratio for a neutral or Zalzalian third, al-Farabi's 27/22 (355 cents) is a bit higher, a small difference of 243:242 or 7.139 cents. In melodic terms, note that in his tetrachord of 9:8-12:11-88:81 or 204-151-143, the larger Zalzalian step at 151 cents comes before the smaller one at 143 cents. This preference is also found in current versions of Arab and Turkish Rast, which feature a similar tetrachord with a tone, followed by a larger and smaller Zalzalian step. Interestingly, al-Farabi's tuning of what would later be known as Rast in the simple "textbook" form of an octave scale -- Near Eastern modes are far more intricate, and involve many types of inflections and modulations to related modalities! -- does include some 11:9 thirds, but not in relation to the main resting note or _qarar_, which we'll here call the final. This "Mode of Zalzal" is built from two of al-Farabi's tetrachord plus an upper 9:8 tone to complete the octave. Here, following modern Arab conventions, we'll take the 1/1 step or final as C. Lower tetrachord Upper tetrachord 9:8 |----------------------|--------------------|.......| C D Ed F G Ad Bb C 1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1 0 204 355 498 702 853 996 1200 9:8 12:11 88:81 9:8 12:11 88:81 9:8 204 151 143 204 151 143 204 Here the symbols Ed and Ad use ASCII "d" like a modern Arab half-flat, to show that a note is lowered by a small interval which might be close to some variety of "quartertone," but with a range of shadings possible. For example, the same composition might be performed with notated Ed a bit lower in Egypt and a bit higher in Syria. As Sami Abu Shumays observes, the fine shade of tuning preferred in a given locality or region becomes a kind of mark of the distinctive musical "dialect" for that group of musicians. We have clues that this may have also been the situation in the 11th-14th century Near East. Thirds at 11:9 or 347 cents may be found in al-Farabi's tuning at 27/22-3/2 (Ed-G), and 18/11-2/1 (Ad-C). In approaching Near Eastern music, it's very important to recognize that the focus is above all on pure melody, not on complex stable or unstable vertical sonorities of a kind which by around 1200 were a central feature of European composition (e.g. Perotin). While Near Eastern theorists had a very sophisticated concept of relative degrees of consonance and dissonance, their interest especially concerned melodic grace and subtlety, with the many shadings of Zalzalian steps and intervals a vital aspect of this focus. In the early 11th century, Ibn Sina described a local practice that preferred to place the fret of Zalzal a bit lower, so that the smaller Zalzalian step comes before the larger one. Like al-Farabi, he started the tetrachord of Zalzal used in his suggested `oud tuning with a 9:8 tone, here followed by a 13:12 Zalzalian step at 139 cents (a superparticular ratio, like al-Farabi's 12:11 at 151 cents), and then a larger Zalzalian step at 128:117 (156 cents) to complete the 4/3 fourth. His Mode of Zalzal, as it might be called, is like this: Lower tetrachord Upper tetrachord 9:8 |-----------------------|---------------------|........| C D Ed F G Ad Bb C 1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1 0 204 342 498 702 841 996 1200 9:8 13:12 128:117 9:8 13:12 128:117 9:8 204 139 156 204 139 156 204 Here the Zalzal fret is at 39/32 or 342 cents, a bit lower than 11/9 at 347 cents. We may also notice that while al-Farabi's Zalzalian steps of 151-143 cents are not too far from equal, Ibn Sina's 139-156 cents differ by 17 cents, and provide more melodic contrast. In the 11th century or today, the subtle differences in degrees of contrast between large and small Zalzalian steps is one dimension of melodic creativity in Near Eastern music. While al-Farabi and Ibn Sina illustrate differences in taste as to exactly how high the third of Zalzal should be -- and Ibn Sina notes that some people place the fret higher, and others lower -- their tetrachords show a common element that helps us understand how the theories of the Pythagoreans and Ptolemy both influenced these writers. The type of mode we are considering was called Mustaqim by Ibn Sina, Arabic for "right, correct, standard, usual." By the 13th century, the Persian name Rast had caught on for this type of mode -- also meaning "right, correct, standard, usual." The name Rast has become the norm, although Mustaqim can be useful for describing a variety of Rast where the smaller Zalzalian step comes before the larger, as in Ibn Sina's tuning. In Rast, as tuned by al-Farabi or Ibn Sina, the lower step is the standard 9:8 tone of Pythagoras and his followers. This tone is equal to two pure 3:2 fifths (each 702 cents) less a 1200-cent octave at 2/1, or 204 cents; and also to the difference between the 3:2 fifth (702 cents) and the 4:3 fourth (498 cents). Since both the pure 4:3 fourth and the 9:8 tone are givens for this type of tetrachord, the two Zalzalian steps completing the tetrachord must add up to the difference between 4:3 and 9:8, a usual Pythagorean minor third at 32:27 or 294 cents. In different tunings of al-Farabi and Ibn Sina, including the two most famous given above, the lower 9:8 tone is followed by a superparticular Zalzalian step at 11:10 (165 cents), 12:11 (151 cents), 13:12 (139 cents), or 14:13 (128 cents). However, as it turns out, it is impossible to divide a 32:27 minor third precisely into two superparticular ratios each within the range from 11:10 to 14:13. Thus, as Ibn Sina discusses, there must be a less tidy "remainder" interval to complete the 4:3 fourth which will not itself be superparticular, but which the ear will accept as part of the total experience of the tetrachord with its other superparticular or "consonant" steps. Surveying the different ways that a 9:8 tone can be joined to a superparticular Zalzalian step, Ibn Sina shows the range of Zalzalian or middle third sizes known by the early 11th century. Our examples above fall in the middle of the range, with 9:8 joined to 12:11 or 13:12; here we add al-Farabi's tetrachord using an 11:10 step (quoted by Ibn Sina), and another tetrachord of Ibn Sina using a 14:13 step: (a) Al-Farabi, 9:8-11:10-320:297 (204-165-129 cents) Zalzalian or neutral third at 99:80 (369 cents) Lower tetrachord Upper tetrachord 9:8 |----------------------|--------------------|.......| C D Ed F G Ad Bb C 1/1 9/8 99/80 4/3 3/2 33/20 16/9 2/1 0 204 369 498 702 867 996 1200 9:8 11:10 320:297 9:8 11:10 320:297 9:8 204 165 129 204 165 129 204 (b) Al-Farabi, 9:8-12:11-88:81 (204-151-143 cents) Zalzalian or neutral third at 27:22 (355 cents) Lower tetrachord Upper tetrachord 9:8 |----------------------|--------------------|.......| C D Ed F G Ad Bb C 1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1 0 204 355 498 702 853 996 1200 9:8 12:11 88:81 9:8 12:11 88:81 9:8 204 151 143 204 151 143 204 (c) Ibn Sina, 9:8-13:12-128:117 (204-139-156 cents) Zalzalian or neutral third at 39:32 (342 cents) Lower tetrachord Upper tetrachord 9:8 |-----------------------|---------------------|........| C D Ed F G Ad Bb C 1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1 0 204 342 498 702 841 996 1200 9:8 13:12 128:117 9:8 13:12 128:117 9:8 204 139 156 204 139 156 204 (d) Ibn Sina, 9:8-14:13-208:189 (204-128-166 cents) Zalzalian or neutral third at 63:52 (332 cents) Lower tetrachord Upper tetrachord 9:8 |-----------------------|---------------------|........| C D Ed F G Ad Bb C 1/1 9/8 63/52 4/3 3/2 21/13 16/9 2/1 0 204 332 498 702 830 996 1200 9:8 14:13 208:189 9:8 14:13 208:189 9:8 204 128 166 204 128 166 204 From this range of tunings and melodic colors, we can learn, as Ibn Sina was already aware a millennium ago, there is no "One True Size" for a Near Eastern Zalzalian third, or the melodic steps making up a Rast mode. A small Zalzalian third like Ibn Sina's 63/52 at 332 cents may have a "supraminor" quality, and a large Zalzalian third like al-Farabi's 99/80 at 369 cents, or Ibn Sina's similar 26/21 at 370 cents, may give a "submajor" impression. Sizes such as al-Farabi's `oud fret at 27/22 (355 cents), or Ibn Sina's at 39/32 (342 cents), may give a range of intermediate impressions. Also, while al-Farabi's famous 9:8-12:11-88:81 tuning has near-equal Zalzalian steps of 151-143 cents, Ibn Sina's 9:8-13:12-128:117 with 139-156 cents shows greater contrast. In al-Farabi's 9:8-11:10-320:297 with 165-129 cents, or Ibn Sina's 9:8-14:13-208:189 with 128-166 cents, the difference is one of over 35 cents. It's also worthy of notice that ease of measuring frets can sometimes lead to quite mathematically sophisticated and complex solutions -- which, however, are very simple to implement in practice! Thus Ibn Sina mentions and Safi al-Din al-Urmawi, a 13th-century theorist, gives prominence to a fretting solution in which the third of Zalzal was placed exactly halfway between the standard Pythagorean frets at the 9/8 tone and 4/3 fourth. Here it's convenient to assume an open string length of 72: 72 64 59 54 |--------------|----------|----------| 1/1 9/8 72/59 4/3 0 204 345 498 9:8 64:59 59:54 204 141 153 After finding the familiar 9/8 and 4/3 frets, one would simply place the Zalzal fret so as to divide the distance between these frets (at lengths of 64 and 54) into two equal parts, arriving at a length of 59! Safi al-Din makes the resulting division of 72:64:59:54 one of his principal tetrachords, notable from a modern viewpoint for its use of the higher prime 59. It is also in musical terms a very attractive tuning if one prefers that the smaller Zalzalian step precede the larger (141-153 cents), and likes the rather subtle contrast between these steps at around 12 cents. As it happens, the resulting Zalzal third at 72/59 (344.7 cents) is quite close to the simple ratio of 11/9 (347.4 cents), with a difference of only 649:648 or 2.7 cents. More generally, as the range of 10th-15th century tunings illustrates, the region around 11/9 is simply one possible taste for Near Eastern Zalzalian or neutral thirds. -------------------------------------------------------------------- 2. Long chains of fifths and the 53-comma theory of Turkey and Syria -------------------------------------------------------------------- Near Eastern musicians are often, although not always, influenced by the Pythagorean technique of tuning stringed instruments, for example, in pure 3:2 fifths or 4:3 fourths. Scott Marcus has suggested that regular minor thirds on such instruments may often be around 32:27 (294 cents), the minor third produced by three pure fourths (e.g. D-F from D-G-C-F). This 32:27 minor third, and also the 9:8 tone, are generally assumed in 10th-15th century theory we have just surveyed. During those centuries, some musicians experimented with the idea of using longer chains of pure fifths or fourths to build a tuning system. As chronicled by Cris Forster in his book _Musical Mathematics: On the Art and Science of Acoustic Instruments_, al-Farabi described such a style of tuning on the Tunbur of Khorasan. In the 13th century, Safi al-Din described one approach to tuning the `oud as a system based on a chain of 17 notes in pure fifths or fourths. However, one complication of such an approach is that very large sets -- larger than the 17-note systems explored in both the Near Eastern and Western Europe around the 13th-15th centuries -- are needed to obtain the middle or Zalzalian steps and intervals so central to Near Eastern melody. Rather, middle second steps like 14:13, 13:12, 12:11, and 11:10 are treated as independent elements of music, not simply as derivatives from a chain of fifths. In modern times, however, Turkish and Syrian musicians have carried the approach of tuning in pure or virtually pure fifths further, at least in theory, to arrive at an elegant system for categorizing intervals and conceptualizing their approximate sizes. From a literal mathematical perspective, no number of pure 3/2 fifths can ever precisely equal any number of pure 2/1 octaves. However, in practice, 53 pure fifths will exceed 31 pure octave by a small factor called the comma of Mercator, about 3.615 cents. Thus a 53-note Pythagorean circle is possible, in which a chain of 52 pure fifths is tuned, followed by a pure 2/1 octave, leaving the "odd" 53rd fifth (which is not tuned directly) narrow by the small comma of 3.615 cents. Many historical European temperaments, as well as measured Near Eastern tunings on fixed-pitch instruments, have fifths at least this impure; and here the compromise applies to only one fifth out of a circle of 53. If this method were followed, then the octave would be divided into 53 steps of two slightly varying sizes. At 41 locations, we would have the traditional Pythagorean comma at 531441/524288 or 23.460 cents; this is the amount by which 12 pure fifths exceed 7 pure octaves. At the other 12 locations, we would have a small comma at 19.845 cents, smaller by the comma of Mercator. A different approach, often preferred for the sake of simplicity, is to regard each of the 53 fifths as tempered by a minute amount equal to 1/53 of the comma of Mercator, so that all 53 steps are at an identical 22.642 cents. More generally, in this simplified scheme, there will be 53 sizes of intervals as exact multiples of this standardized comma, named after the 17th-century English theorist John Holder as the "Holderian" (or sometimes "Holdrian") comma. From a Near Eastern perspective, the 53-comma system offers middle or Zalzalian second steps at 135.8 cents (close to 13/12, 138.6 cents); and 158.5 cents (a bit larger than Ibn Sina's 128/117 step at 155.6 cents). In some styles of Arab and Persian music, favoring steps at around 160 and 135 cents, the 53-comma system gives a rather close approximation to these sizes. Indeed for tunings like Ibn Sina's, or modern Persian variations such as those of Hormoz Farhat, a 53-comma system gives excellent approximations. However, many musicians understand the comma system not so much as a literal scheme of measuring interval sizes, as a useful way of categorizing intervals while leaving the exact intonation up to the performers. For example, an Egyptian, Lebanese, or Syrian Rast might be expressed as "9-7-6 commas," meaning that a usual whole-tone at around 9:8 (9 commas in this system) is followed by a larger and then a smaller Zalzalian step. If the steps are actually around 151-143 cents, as in the famous tuning of al-Farabi, then in fact we would have something more like 9-6.7-6.3 commas. However, 9-7-6 commas is understand more flexibly to say that the larger Zalzalian step precedes the smaller, whatever the exact tunings. In contrast, 9-6-7 commas might suggest Ibn Sina's Mustaqim, or a modern Persian counterpart where the smaller step precedes the larger. An interesting example where certain Turkish performers do seem very closely to approximate the precise values of the comma system is documented by Karl Signell. Here the interval in question is the large or "augmented" step often featured as the middle interval of Arab or Turkish Hijaz, a type of tetrachord named for a desert region on the Arabian Peninsula. Around 1300, Qutb al-Din al-Shirazi reported a tuning of 12:11-7:6-22:21 or 151-267-81 cents, a permutation or rearranged order for the step sizes of a tuning by Ptolemy, the Intense Chromatic at 22:21-12:11-7:6 or 81-151-267 cents. While there are many shadings of Hijaz (in a modern Turkish spelling, Hicaz) and related genera (e.g. Persian Chahargah) in use in the Near East, Turkish theory calls for a middle step of 12 commas, quite close to 7:6, but a bit larger (271.7 cents). In measuring the intervals used by the Turkish musician Necdet Ya[s-cedilla]ar, highly esteemed for his expert and precise intonation, Signell found that they were consistently in the range of 270-273 cents, with 272 cents as the average -- a veritable 12 commas, and a bit larger than 7:6 at 267 cents. This is not to say that the comma system is invariably a guide to exact Turkish intonation, which can be very flexible: Amine Beyhom, for example, measured a Turkish Hicaz with a middle step of around 265 cents, very close to 7:6 and in fact very slightly smaller. When used to describe middle, neutral, or Zalzalian thirds, the 53-comma system recognizes two general types. The smaller or 15-comma middle third, at a literal 339.6 cents, might more generally suggest the region of a third somewhere around Ibn Sina's 63/52 (332.2 cents) or 39/32 (342.5 cents), or possibly the slightly larger 11/9 (347.4 cents). In contrast, the 16-comma middle third, literally 362.3 cents, might suggest Ibn Sina's 16/13 (359.5 cents) or 26/21 (369.7 cents). In practice, there is an infinite variety of shadings, but the 15-comma and 16-comma categories can often indicate the general type of middle third expected. For example, a 16-comma third would be the norm in an Arab Rast (e.g. C-D-Ed-F); while a 15-comma third might be expected in Sika/Segah. which characteristically starts with the trichord Ed-F-G. Here C-G is presumably a pure or near-pure fifth at around 31 commas, with C-Ed at around 16 commas (the Rast third), and Ed-G a smaller third at around 15-commas (the Sika/Segah third). To sum up, while regular intervals such as major and minor thirds are often at or close to Pythagorean sizes determined by chains of pure fifths or fourths, the 53-comma system is more of a theoretical map to approximate interval sizes and categories than a method of tuning middle intervals. In that function, it does convey one very interesting side of the picture, but with personal and regional tastes taking priority over any precise mathematical scheme. ------------------------------------------------- 3. Musicianly taste, tarab, and "glissando zones" ------------------------------------------------- In addition to surveying the range of neutral or Zalzalian steps and third sizes in use around the early 11th century, Ibn Sina tells us that some people, for example, place the third of Zalzal higher or lower on the `oud than others, and that the ability of performers to distinguish different sizes of steps also varies greatly. He says that many musicians do not discriminate between steps of 14:13 (128 cents) and 13:12 (139 cents), but that the adept can and do. Can Akko[c-cedilla], a Turkish theorist, has measured flexible-pitch performances of Turkish music and demonstrated that the realization of a given step is in practice not fixed but variable by a comma or more, with a "cluster" of actual pitches representing such a modal step. Thus, as he commented at a conference summarized by Eric Ederer, much Western or Western-derived theory may be applying a "particle" model to what is a "wave" phenomenon. Indeed, one recent Turkish concept posits a "glissando zone" in which a middle step may be placed, with the interpretation highly variable, and the ability to make this variability as expressive and telling as possible one of the marks of a true artist. There is a concept that the middle or Zalzalian steps of a mode -- a middle third or seventh in Rast, for example, in contrast to the fourth or fifth -- tend to be more fluid, with apt pitches for these steps at different points in a piece resembling clouds or clusters. Between a regular minor third at around 32/27 or 13 commas, for example, and a regular major third at around 81/64 or 18 commas, there is a "three-comma glissando zone" which might range from around 14 commas or a large 6/5 minor third through a small middle third at 15 commas (e.g. 63/52, 17/14 at 336.1 cents, or 39/32, etc.) to a large middle third at 16 commas (e.g. 16/13, 21/17 at 365.8 cents, or 26/21) or a small major third at 5/4 or 17 commas. Where a note is placed at a given moment may depend on personal or regional tastes, musical context, and sheer inspiration. For example, the Turkish principle of cazibe or "attraction" suggests that a note will be slightly raised or lowered in approaching certain cadences. Thus a performer who places the Rast third at close to 5/4 (386 cents) or 17 commas (a placement favored in standard 20th-century Turkish theory and approximated in some modern practice) may lower it to around 16 commas (say 26/21, 370 cents) when descending to a final cadence. And a performer who favors a 16-comma Rast third (say 26/21, or perhaps 21/17 at 366 cents) may lower it to somewhere around 11/9 (347 cents). Such adjustments may be quite subtle, for example on the order of around half a comma, or 10-15 cents. As summarized by the Lebanese musician and teacher Ali Jihad Racy, the 20th-century Syrian theorist Tawfiq al-Sabbagh urges that to achieve the supreme quality in Arab Maqam music of tarab, ecstasy or enchantment, steps may often usefully be altered or inflected by around a comma. While al-Sabbagh, as a Syrian musician and an admirer of Turkish practice, uses the 53-comma system to indicate these approximate tunings and inflections, Scott Marcus has documented an often very flexible rather than mathematical understanding in the Arab world of a "comma": any small interval which makes a musical difference in a given context. For example, Racy showed Marcus how, in Maqam Bayyati, the third minor step is a bit lower than the Pythagorean minor third at 32/27 or 294 cents that would result from the pure fifths or fourths used in tuning the strings. While a reader oriented to small integer ratios like the 7/6 third, or to the 53-comma system, might take this to mean that a 7/6 third at 267 cents is intended, or perhaps the slightly larger third at a precise 12 commas or 272 cents, Marcus clarifies that the intent is a small and unspecified difference, left to the musician's judgment. Comparing some measured Persian tunings on tar or setar also leads to the idea of performer discretion within a certain understood range of variations, rather than any precise standard for each note, whether based on just ratios, the comma system, or some other criterion such as a given precisely equal division of the octave. Thus Nelly Caron and Dariuche Safvate in 1966 document a tuning of the Dastgah or modal form of Shur with steps of 136-140-224 cents, and a minor third at 276 cents. This is a bit higher than 7/6, just a bit wider than 12 commas (272 cents), and considerably narrower than the common 32/27 (294 cents). Another Persian musician, Hormoz Farhat, suggests a division for Shur of around 135-160-205 cents, with a third of 295 cents or a virtually just 32/27. A permutation of Ibn Sina's tetrachord included in his `oud tuning, using his steps of 9:8, 13:12, and 128:117 in ascending order of size, gives a very similar 13:12-128:117-9:8 at 139-156-204 cents. In short, the variability of tunings within a given tradition, and sometimes within a given piece, require that we regard fluidity, flexibility, and expressive variation not only as inevitable features of flexible-pitch performance, but as highly cultivated and desired refinements. The fact that while the `oud of 10th-15th century practice was generally fretted, modern `ouds are generally fretless, further accentuates this element of performerly discretion. Like performers of medieval European music such as Christopher Page who value a purely vocal performance with the liberty to tune a given interval aptly on each occasion, Near Eastern performers such as Amine Beyhom value instruments unconstrained by fixed tunings. Furthermore, the complexity and sophistication of fixed-pitch instruments such as the 79-tone qanun of Ozan Yarman built with a tuning system he designed intended for Turkish and other Near Eastern musics, or the just intonation qanun of Julien Jalal Ed-Dine Weiss as documented by the scholar Stefan Pohlit, also reflects the desire for many finely shaded nuances of expression. ------------- 4. Conclusion ------------- As a mainly melodic art, Near Eastern music features many finely graded sizes of neutral or Zalzalian steps and thirds. This was clearly true by the early 11th century, when Ibn Sina documented this variety, and has remained true in the dynamic setting of a constantly evolving set of related but sometimes divergent modal systems that have developed over the past millennium and more. Integer ratios, both simple and complex, play a central role in 10th-15th century theory. The middle third at 11/9 would sometimes occur, for example in al-Farabi's Mode of Zalzal between the third step at 27/22 and the 3/2 fifth (1/1-9/8-27/22-4/3-3/2 or 0-204-355-498-702 cents, with 347 cents or 11:9 as the difference of 3/2 and 27/22). However, this ratio does not seem especially privileged, but simply one possible point on a continuum. The 53-comma system, based on a 53-note musical circle with pure or virtually pure fifths, gives an elegant system for mapping regions or general categories of intervals, with some of the theoretical sizes (like a smaller or 6-comma middle second at 136 cents) in fact often quite close to other theoretical or practical values in use, such as Ibn Sina's 13:12 step at 139 cents, or Farhat's step at 135 cents, or the suggestion by Marcus that the lower step of an Egyptian Maqam Bayyati might be tuned at around 135-145 cents. However, variability is the rule, especially in flexible-pitch performance; and both personal and regional tastes may result in different tunings for fixed-pitch instruments as well. Another side of Near Eastern theory is the use, at times, of systems based on an equal division of the octave. The 53-comma theory, when premised on precisely equal comma steps at 1/53 octave, falls in this category, although it also very closely approximates the results of tuning in pure fifths, and could be applied (as in some Turkish interpretations) to a tuning in pure fifths, where slightly unequal commas at 23.460 and 19.845 cents would result if the tuning were carried to a full 53 notes. Other equal divisions proposed or used -- sometimes with the caution that in fact the steps should NOT be taken as equal in practice, but variable! -- include the 68-division of Byzantine theory (Chrysanthos of Madytos); divisions of 36, 72, or 144 based on the late Classic Greek theorist Cleonides as an interpreter of Aristoxenos (standard in Byzantine music theory starting in 1881); the 24-division (presented by the Syrian theorist Mikhail Mashaqa, although he found the 68-division more accurate); and the 17-division suggested as one option by Amine Beyhom. Sami Abu Shumays has suggested that recognizable local or regional styles of intonation, like speech communities with distinctive local dialects or "accents," may be part of the sophisticated musical cultures of the Arab world. A given tuning of the Rast third, for example, might mark a musician's style as typical of Egypt or Syria. The experience of Abu Shumays also fits an Arab musician's impression shared with Marcus that as you travel east from Egypt to Palestine, Syria, and finally Turkey, the note sika (the third of Rast) will tend gradually to get higher and higher. The experience of Julien Jalal Ed-Dine Weiss suggests that, within Syria, sika tends to be somewhat lower in Damascus (perhaps centering around 355-365 cents) than in Aleppo (perhaps around 365-375 cents). In sum, Near Eastern theorists from at least al-Farabi on have provided a range of sophisticated theoretical concepts, maps, and ratios which evocatively but imperfectly suggest the sovereign performer's discretion and taste which inform a yet more sophisticated practice. Margo Schulter 8 December 2014