------------------------------------------------------- The MET-24 temperament for Maqam music: Partitions or divisions of the apotome in context by Margo Schulter ------------------------------------------------------- The purpose of this paper is to document the MET-24 temperament which has grown out of a dialogue with Jacques Dudon (July 2011) in which he described some different shadings of the Tsaharuk system. Thus I was led to seek a milder temperament with extended or wide fifths (a "Milder Extended Temperament" or MET-24), by comparison to some previous temperaments I had used, approaching a bit more closely the ideal of pure fifths and fourths which is part of the tradition of Maqam. Just after designing the tuning, which I should emphasize is in many ways very close to the High Tolerance Temperament (HTT) family of George Secor (with regular fifths at 703.579 cents, and including, as one subset, two chains of these fifths at 58.090 cents apart), but differs subtly in certain objectives and compromises, I had an opportunity, thanks to Jacques, to see some partitions for qanun that Julien Jalal Ed-Dine Weiss and he had detailed, which gave me the idea of explaining the divisions of the tempered apotome (125.4 or 126.6 cents) in MET-24. This dialogue also led to my becoming aware of the French term _partage_ for a "partition" or "division" of an interval such as the fourth or octave, as would be found in a tuning of the qanun or some other musical instrument. Finding this a very attractive loan-word for the English language, also, I have accordingly borrowed it as the name of this computer file: met24-partage.txt. This borrowing is meant at once as a tribute to France for its efforts on behalf of world peace and humanity, and to Jacques Dudon and Julien Jalal Ed-Dine Weiss as ambassadors of musical creativity and goodwill. Additionally, I would like very warmly to thank Stefan Pohlit for his dissertation and articles very artfully chronicling and analyzing the qanun tunings and repertories of Julien, which seems a first name in this field, at least among musicians originally from regions other than that of the Near East, comparable to that of Josquin in Western European music, or Galileo in the history of physics or astronomy: a name so familiar that it serves in itself as a definitive identification. Having become aware of a range of quite ambitious fixed-pitch tuning systems developed for the qanun or other instruments, I would find apt for Julien a familiar adage of Martin Luther concerning Josquin: while other musicians do what the notes will permit, he has the notes do what he wills. Please let the reader be assured that nothing which follows will in any way alter this situation: MET-24 is neither a just system like Julien's, nor one with anything like its range and versatility, but merely an unequally tempered 24-note scheme which I hope is a bit better because of his influence upon me through Jacques Dudon. * * * Before presenting the diagrams for the division itself, followed by some possible gamuts for Arab, historical Ottoman, and Persian music, I wish to explain the layout of the keyboard, and also some important limitations as well as possibilities of a system seeking to provide a variety of steps and Maqam or Dastgah shadings in only 24 notes per octave. Most particularly, I am concerned that a reader might see that Zalzalian steps approximating 14/13, 13/12, 12/11, and 11/10 are all present in the system, and then assume that all four of these step sizes might be available from the same location, as it might happen on a qanun! Rather, as I illustrate below, at most two of these steps will be available from any single location, and at locations where they are available, such pairs will differ by a comma step at 23.4 or 24.6 cents: a tempered 14/13 and 12/11, or 13/12 and 11/10. One does not find, unfortunately, that happy combination for Arab music of a 12/11 above dugah for Rast or a 13/12 for Bayyati! In a small tuning set like MET-24, therefore, the principle of a variety of Zalzalian steps must combine with that of "modal color": choosing different locations or transpositions for the maqamat to arrive at different shadings and distinct intonational nuances for common or not so common modulations. These variations are a delight of unequal tuning, for example on a 17-note Persian tar or setar where Dariush Tala`i has spoken of the artful element of "temperament" or compromise in which the steps of a jins (which in Farsi he calls a _dang_) may somewhat diverge from their ideal forms as one moves about the instrument. An additional caution is that while MET-24 offers support for an "historical Ottoman" style where Maqam Rast has a high Zalzalian or neutral third step around 99/80 (368.9 cents) or 26/21 (369.7 cents), both being represented at a tempered 370.3 cents, it does not support a modern Turkish style calling for intervals at or near ratios of 5 (e.g. the 5/4 major third at 386.3 cents). However, a large Zalzalian or "submajor" third at 370 cents may fit not only some historical Ottoman styles, but also, for example, the modern practice of certain Eastern Arab localities such as Aleppo. It is from this view, with many thanks to Jacques Dudon, Julien Jalal Ed-Dine Weiss, and George Secor, that I quickly present first the MET-24 tuning and its possibilities and limitations; then the divisions of the tempered apotome; and finally a set of gamuts for Arab, historical Ottoman, and Persian music. I should add that I would like to know more about Kurdish maqamat and their intonation, having heard, for example, that a low Zalzalian step around 14/13 is often much prized among the Kurds as among the Iranians also. ------------------------------------------------------- 1. The MET-24 tuning: its possibilities and limitations ------------------------------------------------------- One approach for getting an overview of the tuning is to take the note C on the lower keyboard or chain of fifths as a point of reference: MET-24, "Milder Extended Temperament" for Maqam 183.984 346.875 679.688 887.109 1050.000 C#* Eb* F#* G#* Bb* C* D* E* F* G* A* B* C* 57.422 264.844 472.266 554.297 761.719 969.141 1176.562 1257.422 --------------------------------------------------------------------------- 126.563 289.453 622.266 829.688 992.578 C# Eb F# G# Bb C D E F G A B C 0 207.422 414.844 496.875 704.297 911.719 1119.141 1200 In 1024-EDO or 1024-ED2, the generators are fifths in alternating sizes of 703.125 cents (600 tuning units or TU) and 704.297 cents (601 TU). The two chains are tempered as follows, and placed at the distance of a small thirdtone of 57.422 cents (49 TU), so that the average size of the fifths is 703.711 cents, and all usual tones or major seconds are at 207.422 cents: ~13/11 16/9 4/3 1/1 3/2 9/8 22/13 14/11 21/11 63/44 14/13 21/13 289.5 992.6 496.9 0 704.3 207.4 911.7 414.8 1119.1 622.3 126.6 829.7 Eb Bb F C G D A E B F# C# G# 703.1 704.3 703.1 704.3 703.1 704.3 703.1 704.3 703.1 704.3 703.1 Placing the chains at 57.4 cents apart tempers 7/6 and 13/8 by almost exactly the same amount: 7/6 is approximated at 264.844 cents (2.027 cents narrow), and 13/8 at 842.578 cents (2.050 cents wide). This small thirdtone or large diesis step of 57.4 cents appears at 12 locations, and may represent 28/27, 121/117, 91/88, or 33/32. Two basic criteria for the tuning are to provide approximations of the four epimoric or superparticular Zalzalian steps of 14/13, 13/12, 12/11, and 11/10 all within 3 cents of just; while at the same time having a majority of the 11 fifths in each 12-note chain at the smaller size of 703.125 cents, identical to the fifth in a Tsaharuk shading with a generator of 15 steps in 128-ED2.[1] ---------------------------------------------------------- 1.1. Alternative perspectives: Intervals and comma choices ---------------------------------------------------------- While the above diagrams conventionally take C as the note of reference, or 1/1 step, three other perspectives may highlight some of the characteristic near-just intervals, especially the septimal ones. The first perspective is from C on the upper keyboard or chain of fifths, one likely placement for the step rast in a typical Arab style, with possible just interpretations suggested for the tempered interval sizes: 14/13 13/11 63/44 21/13 39/22 126.563 289.453 622.266 829.688 992.578 C#* Eb* F#* G#* Bb* C* D* E* F* G* A* B* C* 0 207.422 414.844 496.875 704.297 911.719 1119.141 1200 1/1 44/39 14/11 4/3 3/2 22/13 21/11 2/1 --------------------------------------------------------------------------- 176/169 8/7 18/13 264/169 12/7 69.141 232.031 564.844 772.266 935.156 C# Eb F# G# Bb C D E F G A B C 1142.578 150.000 357.422 439.453 646.875 854.297 1061.719 1142.578 176/91 12/11 16/13 9/7 16/11 18/11 24/13 176/91 In this perspective on the gamut with C* as rast, we often have choices between two steps a comma apart for a small or moderately large Zalzalian interval (127 or 150 cents, e.g. 14/13 or 12/11; 830 or 854 cents, e.g. 21/13 or 18/11); or between a regular or septimal major interval (207 or 232 cents, e.g. 9/8 or 8/7; 912 or 935 cents, e.g. 22/13 or 12/7). The tuning of Rast at 207-150-139 cents could represent al-Farabi's 9:8-12:11-88:81 (204-151-143 cents), or 9:8-128:117-13:12 (204-155-139 cents), with the actual sikah/segah at 357.4 cents about midway between al-Farabi's 27/22 (354.5 cents) and 16/13 (359.5 cents). Taking A on the lower keyboard as the point of reference might better fit a Persian style: 13/12 21/16 13/9 7/4 63/32 138.281 472.266 635.156 967.969 1175.391 Bb* C#* Eb* F#* G#* A* B* C* D* E* F* G* A* 57.422 264.844 345.703 553.125 760.547 842.578 1050.000 1257.422 91/88 7/6 11/9 11/8 14/9 13/8 11/6 91/44 --------------------------------------------------------------------------- 22/21 14/11 88/63 22/13 21/11 80.859 414.844 577.734 910.547 1117.969 Bb C# Eb F# G# A B C D E F G A 0 207.422 288.281 495.711 703.125 785.156 992.578 1200 1/1 44/39 13/11 4/3 3/2 11/7 39/22 2/1 Here the comma steps afford a choice between regular or septimal forms of minor intervals: thus 265 or 288 cents (7/6 or 13/11), 761 or 785 cents (14/9 or 11/7); and 968 or 993 cents (7/4 or 39/22). The modality corresponding with a moderate Arab Rast in the last example would be Ibn Sina's Mustaqim, as represented in the modern Persian Dastgah system, for example, by Gushe-ye Shekaste in Dastgah-e Mahur: 207.4-138-3-150.0 cents, with segah at 345.7 cents, which might represent either 72/59 (344.7 cents) or the simpler 11/9 (347.4 cents). Either ratio is slightly larger than Ibn Sina's 39/32 (342.5 cents), with 72/59 appearing in Safi al-Din al-Urmawi's division of 72:64:59:54 (204-141-153 cents). very close to Ibn Sina's Mustaqim. We can get still another perspective by setting an historical Ottoman rast at B on the lower keyboard, a good way to illustrate both this Ottoman or Aleppian shading of Rast, and the complications of having a system with diverse Zalzalian step sizes but only 12 notes in each chain of fifths. 13/12 169/132 13/9 13/8 169/88 139.453 427.734 635.156 842.578 1130.859 C* Eb* F* G Bb* B* C#* D* E* F#* G#* A* B* 57.422 264.844 345.703 553.125 760.547 967.969 1050.000 1257.422 91/88 7/6 11/9 11/8 14/9 7/4 11/6 91/44 --------------------------------------------------------------------------- 22/21 26/21 88/63 11/7 13/7 80.859 370.312 577.734 785.156 1074.6 C Eb F G Bb B C# D E F# G# A B 0 207.422 288.281 495.703 703.125 910.453 992.578 1200 1/1 44/39 13/11 4/3 3/2 22/13 39/22 2/1 At this location, the comma steps often afford a choice between a medium small and a large Zalzalian interval: thus 346 or 370 cents (11/9 or 26/21); 843 or 867 cents (13/8 or 104/63); 1050 or 1075 cents (11/6 or 13/7). In Rast, we can thus emulate a Turkish nuance by using 370 cents (Eb) as the usual segah, but 346 cents (D*) as a lowered version of this step in descending to a final cadence, as well as a fine step ushshaq for Maqam Ushshaq (or Arab Bayyati) on C#. The weakness of having only 12 notes in each chain of fifths, however, is shown by our lack of a high Zalzalian sixth at Ab (866.0 cents) for a conjunct Rast on B; a 17-note chain (Gb-A#) would make available this note. In an Arab style, the smaller Zalzalian sixth B-G* (842.6 cents) actually available is useful for a modulation from B Rast in an Aleppian shading to F# Bayyati (Bayyati on nawa/neva). In one of its developments, Bayyati may involve a configuration of ajnas (genera) calling for a rather low Zalzalian second but a high Zalzalian sixth above the final or _qarar_ F#, the "1/1" of the following diagram: 163 125 207 139 150 207 207 163 125 207 704 867 993 0 139 289 497 704 867 993 1200 C# Eb E F# G* A B C# Eb E F# |----------------|--------------|--------------|.....| Bayyati 1/1 Bayyati Rast tone This configuration has, above the final, a Bayyati tetrachord of 139-150-207 cents (close to 52:48:44:39 or 13:12-12:11-44:39 at 139-151-209 cents), followed by a conjunct Rast tetrachord at 207-163-125 cents, and then a tone completing the octave. The Bayyati tetrachord above the final is approximately 6-7-9 commas, with the smaller Zalzalian step preceding the larger; but the upper Rast tetrachord is approximately 9-7-6 commas, with the larger Zalzalian step below the smaller. Interestingly, the tetrachord _below_ the final, which could also be called Bayyati in Arab theory, is at 163-125-207 cents, or approximately 7-6-9 commas, using the same segah/sikah as in B Rast. Indeed, one possible interpretation would be to regard this "Bayyati below the final" as the upper tetrachord of a Rast pentachord on B, 9-7-6-9 commas.[2] These three gamuts found within MET-24, based on C* for a moderate Arab Rast, A for a Persian Mustaqim, or B for an historical Ottoman or Aleppian Rast, have their steps listed in Section 3. ----------------------------------------------- 1.2. A prelude to the partitions: some cautions ----------------------------------------------- The partitions or divisions in Section 2 show both Zalzalian and other steps actually found on the 24-note keyboard, and some intervals which measure the differences or spacings between these steps but do not themselves appear on the keyboard. The latter intervals are marked with brackets to show that they are not themselves available as steps or differences between two steps (e.g. higher and lower positions for segah) both present from the same location. This aspect of the system has some interesting musical implications. ----------------------------------------------- 1.2.1 The Rast/Ushshaq or Rast/Bayyati question ----------------------------------------------- For example, because a comma step of 23.4 or 24.6 cents is available at certain locations on the keyboard, it is possible from B on the lower keyboard to obtain a high segah at 370.3 cents (~26/21) for a bright Ottoman Rast; or a lower segah at 345.7 cents (~72/59) for a moderate Ottoman Ushshaq or Arab Bayyati. Ottoman Rast Ottoman Ushshaq 207.4 162.9 125.4 138.3 150.0 207.4 B C# Eb E C# D* E F# 0 207.4 370.3 495.7 0 138.3 288.3 495.7 Similarly, from G* on the upper keyboard, and taking this step as Dugah, it is possible to choose between a low segah for Persian Shur or Arab Bayyati with a step of 125.4 cents (~14/13), or a moderately high segah for an Arab Huseyni, for example, at 150.0 cents (~12/11). Low Persian Shur or Arab Bayyati Moderate Arab Huseyni 125.4 162.9 207.4 150.0 138.3 207.4 G* G#* Bb* C* G* A Bb* C* 0 125.4 288.3 495.7 0 150.0 288.3 495.7 In contrast, however, while the difference between the Zalzalian steps at 138.3 and 150.0 cents, or approximately 13/12 and 12/11, may be heard in either the Ottoman Ushshaq (~13:12:11) or Arab Huseyni (~11:12:13) ajnas above, there is not actually any step on the keyboard equal to the difference of these intervals, or 11.7 cents (almost exactly half a Pythagorean comma!). Thus one cannot, from any single location for Rast, obtain both a typical Arab Rast (Segah at 357.4 cents or ~59/48) and a moderate Arab Bayyati on step dugah (segah at 345.7 or 346.9 cents, near 72/59 or 11/9). However, this limitation can sometimes lead to adventurousness! For example, placing Rast on G*, one could play a moderate Arab Rast, a moderate Arab Huseyni (~11:12:13) on dugah at A*, or a very low "folk Bayyati" (~14:13:12) on a higher Dugah at Bb on the lower keyboard, 230.9 cents or a virtually just 8/7! Moderate Arab Rast (Disjunct or Conjunct) Rast ~9:8 Rast |------------------|.....|------------------| G* A* B C* D* E* F# G 0 207.4 357.4 495.7 703.1 910.5 1060.5 1200 207.4 150.0 138.3 207.4 207.4 150.0 139.5 Rast Rast ~9:8 |------------------|-----------------|......| G* A* B C* D* E F* G* 0 207.4 357.4 495.7 703.1 853.1 992.6 1200 207.4 150.0 138.3 207.4 150.0 139.5 207.4 Moderate Arab Huseyni on usual Dugah (~9/8) Huseyni Huseyni |------------------| |--------------------| A* B C* D* E* F# G* A* 0 150.0 288.3 495.7 703.1 853.1 992.6 1200 150.0 138.3 207.4 207.4 150.0 139.5 207.4 Low "Folk Bayyati" on high Dugah (~8/7) step [3] Bayyati Nahawand (~21:16) ~8:7 |------------------|-----------------|.........| Bb B C* Eb F F* G* Bb 0 126.6 264.8 496.9 704.3 761.7 969.1 1200 126.6 138.3 232.0 207.4 57.4 207.4 230.9 -------------------------------------------- 1.2.2. Half-comma strategies and refinements -------------------------------------------- Although two steps at an approximate Turkish/Syrian half-comma apart, for example the tempered 11/10 and 12/11, will not be available from the same step, at times a standard modulatory scheme may find these subtly different steps in exactly the desired places! Considerable ingenuity may be devoted both to identifying such situations, and devising musical schemes to make them fit into some more or less plausible historical scenario. For example, we might envision a subtle contrast between a dugah-segah step of 11/10 for Rast (as in a likely Aleppian or historical Ottoman setting), but 12/11 for Ushshaq or Bayyati. There is no one location from which we can obtain both positions of segah. However, in a typical Arab modulatory scheme, one modulates from Rast to Bayyati on nawa, or more generally the fifth step of the Rast transposition in use. This scheme, if we begin with Rast on B*, makes a half-comma contrast quite practical: Aleppian or Historical Ottoman Rast on B* Rast Rast |-------------------|......|-------------------| B* C#* Eb* E* F#* G#* Bb* B* 0 207.4 370.3 495.7 703.1 910.5 1073.4 1200 Bayyati/Ushshaq with 12/11 step on F#* 163 125 207 150 139 207 207 163 125 207 704 867 993 0 150 289 497 704 867 993 1200 C#* Eb* E* F#* G# A* B* C#* Eb* E* F#* |----------------|--------------|--------------|.....| Bayyati Bayyati Rast tone This Bayyati or Ushshaq on F#* is mostly identical to the version on F# presented near the end of Section 1.1 -- except that the final is located on the upper keyboard or chain of fifths, so that we have F#*-G# at 150 cents instead of F#-G* at 139 cents! The regular major and minor intervals of these two versions, and also the large and small Zalzalian seconds at 163 and 125 cents, are generated within a single chain of fifths, and so remain unaltered. However, while the earlier version has a contrast between a 163-125 division in B Rast and 139-150 in the root tetrachord above the final of F# Bayyati or Ushshaq, a difference of a full comma, here we have 163-125 and 150-139, a more nuanced half-comma distinction. ------------------------- 1.2.3. Varieties of Hijaz ------------------------- Sampling a few of the varieties of Hijaz and related tetrachords available in MET-24 may help to illustrate both the diversity of the system for Zalzalian and septimal or near-septimal intervals, and its utterly unsatisfactory nature from the viewpoint of a style like that of Julien Jalal Ed-Dine Weiss, favoring versions of Hijaz with the third step at or near 5/4. For example, for an Arab Hijazkar starting on C* as Rast, we have these alternatives: C* C# E F* C* C#* E* F* 0 69.1 357.4 496.7 0 126.6 414.8 496.9 69.1 288.4 139.5 126.6 288.4 82.0 C* D E* F* 0 150.0 414.8 1200 150.0 264.8 82.0 The first form, essentially a Rast tetrachord with a small semitone substituted for the usual tone of Rast, is like Amine Beyhom's Zirkula type, except that the initial semitone or thirdtone at 69 cents is smaller than a usual limma at 81 or 82 cents. While I find this tuning quite pleasing, it does have the arguably less than optimal trait of a regular minor third as the middle interval, i.e. in Turkish terms a 13-comma rather than 12-comma Hijaz. The second form at 127-288-82 cents approximates 28:26:22:21, and is not too far from the Karadeniz Hijaz at 5.5-13-3.5 commas (125-294-79 cents). The third form at 150-265-82 cents is very close to the classic Hijaz of Qutb al-Din al-Shirazi around 1300 at 84:77:66:63 or 12:11-7:6-22:21 (151-267-81 cents). This tuning might be considered more stylish than the others, because it does have for its middle interval a "plus-tone," as the Persian writer Hormoz Farhat terms it, notably smaller than a usual minor third (here around 13/11), and often close to 7/6. Another variant on Hijaz is also available from this location: C* C#* E F* 0 126.6 357.4 496.9 126.6 232.0 139.5 In classic Systematist terms around 1300, this is one common form of Buzurg, which could be defined as like Rast, but with a smallish Zalzalian second substituted for a tone, here more specifically 14:13-8:7-13:12 (128-231-139 cents). In modern terms, it might represent a shading of Hijaz Gharib, which the Syrian theorist Tawfiq al-Sabbagh, as summarized by Ali Jihad Racy, specifies as 6-10-6 commas. With C* as rast, there are two variations of Nikriz available. The first, described by Mikhail Mushaqa in terms of a 24-step system as 3-5-2 steps, has the same third step as in Rast: C* D* E F#* G* 0 207.4 357.4 622.3 704.3 207.4 150.0 264.8 82.0 Owen Wright shows Qutb al-Din's genus of Nirizi as likewise "204-150-267-81 cents," or 9:8-12:11-7:6-22:21, in other words a lower tone plus Qutb al-Din's Hijaz. Another form of Nikriz has a minor third step like that of Maqam Nahawand, with the "augmented fourth" step near 18/13, a small thirdtone lower than in the approximation of Qutb al-Din's Nirizi: C* D* Eb* F# G* 0 207.4 289.5 564.8 704.3 207.4 82.0 275.4 139.5 In this version, we have a large interval or "plus-tone" Eb*-F* at 275 cents, somewhat larger than the near-7:6 third of the previous example at 265 cents, but still notably smaller than a regular minor third at around 13/11. One JI interpretation of this tuning could be 1/1-44/39-13/11-18/13-3/2 (209-289-274-139 cents), with the large step representing 198/169 (274.173 cents, the difference between 13/11 and 18/13). It will be noted that none of these tunings of Hijaz and related ajnas have anything like a 5/4 major third! It is possible to find such a tuning, but mostly as the exception proving the rule: Bb B C#* Eb 0 126.6 391.4 496.9 126.6 264.8 105.5 This intonation has some resemblance to a Turkish Hijaz measured by Amine Beyhom at 130-265-90 cents -- but with the latter featuring a usual limma at around 256:243 or 4 commas, and thus a fourth at 485 cents, or about 1/2 comma narrow of 4/3. Emulating this nuance in MET-24 would produce a fourth at around 21/16, a full comma smaller than the tempered 4/3. Bb B C#* D* 0 126.6 391.4 472.3 126.6 264.8 80.9 For someone who desires more of these near-5 ratios and forms of Hijaz, expanding the MET-24 system to two 17-note or even 29-note chains would provide them in more abundance. However, their sparseness in the 24-note system seems, at least to this author, not a flaw but rather an artistic decision to focus mainly of approximate ratios of primes 2-3-7-11-13. A 24-note tuning such as that of the Syrian musician and theorist Tawfiq al-Sabah shows how it is quite possible to prioritize differently, obtaining a basic complement of Zalzalian intervals such as those which might fit the style of Damascus (e.g. rast-sikah at around 357 cents, expressed as an integer ratio of 102/83), plus a range of schismatic 5-limit approximations. Relatively modest tuning sizes such as 24 may have the virtue of parsimony, calling for decisions which may clarify a given musician's priorities, at least while designing a given system. MET-24 is offered in this spirit. ---------------------------------------------------------- 1.2.4. Half-comma dialectics of "Septimal Rast" and Buzurg ---------------------------------------------------------- Another kind of intonational dialectic or contrast involving a difference on the order of a half-comma may occur in modulating between two related modal forms, the first of which I rather freely term "Septimal Rast," and the second of which is Buzurg, the name of a pentachord and its related octave cycles or modes in the Systematist tradition of Safi al-Din al-Urmawi, Qutb al-Din al-Shirazi, and some of their successors and commentators. "Septimal Rast" is a possible modern description for what Ibn Sina calls a "most noble jins" with a lower step of 8:7, and upper Zalzalian steps arranged in either order of 13:12 and 14:13. There result the two permutations, which he considered musically equivalent, of either 104:91:88:84 (8:7-13:12-14:13 at 231-139-128 cents, i.e. 1/1-8/7-26/21-4/3 at 0-231-369-498 cents); or 16:14:13:12 (8:7-14:13-13:21 at 231-128-139 cents, i.e. 1/1-8/7-16/13-4/3 at 0-231-359-498 cents). Cris Forster, in his _Musical Mathematics_, emphasizes Ibn Sina's equal recognition of both permutations: 104 91 84 78 1/1 8/7 26/21 4/3 0 231.2 369.7 498.0 8:7 13:12 14:13 231.2 138.6 128.3 16 14 13 12 1/1 8/7 16/13 4/3 0 231.2 359.5 498.0 8:7 14:13 13:12 231.2 128.3 138.6 In MET-24, we have available modes using both conjunct and disjunct forms of these tetrachords. For example, here is a "Septimal Rast" using Ibn Sina's 16:14:13:12 division in a conjunct form of octave mode, with string lengths and cents shown for a JI version (above) as well as the tempered MET-24 values in cents (below): 64 56 52 48 42 39 36 32 1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1 0 231.2 359.5 498.0 729.2 857.5 996.1 1200 8:7 14:13 13:12 8:7 14:13 13:12 9:8 231.1 128.3 138.6 231.2 128.3 138.6 203.9 G* Bb B C* Eb E F* G* 0 230.9 357.4 495.7 727.4 853.1 992.6 1200 230.9 126.6 138.3 232.0 125.4 139.5 207.4 Ibn Sina's "most noble jins" became one of the principal ajnas also of Safi al-Din, as noted for example by Dr. Fazli Arslan, who gives a beautiful diagram of the conjunct 16:14:13:12 division that inspired the more humble text representation above. In the Systematist literature, the term Buzurg refers to a division of the 3/2 fifth into six steps or five intervals, and also to modal cycles adding an upper tetrachord to this genus, generally Rast or Hijaz. Two tunings for the basic Buzurg jins are the following: 336 312 273 252 234 224 0 128.3 359.5 498.0 626.3 702.0 1/1 14/13 16/13 4/3 56/39 3/2 14:13 8:7 13:12 14:13 117:112 128.3 231.2 138.6 128.3 75.6 156 144 126 117 108 104 0 138.6 369.7 498.0 636.6 702.0 1/1 13/12 26/21 4/3 13/9 3/2 13:12 8:7 14:13 13:12 27:26 138.6 231.2 128.3 128.3 65.3 In Buzurg, the step at 56/39 in the first tuning or 13/9 in the second tuning -- these two intonations are not exhaustive, but typical -- have evoked modern curiosity. However, these steps are easily explained if Buzurg as seen as a rotation of Ibn Sina's and Safi al-Din's jins I term "Septimal Rast," starting on the third degree of a conjunct Septimal Rast, as here with the 16:14:13:12 form: Septimal Rast 64 56 52 48 42 39 36 32 1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1 0 231.2 359.5 498.0 729.2 857.5 996.1 1200 8:7 14:13 13:12 8:7 14:13 13:12 9:8 231.1 128.3 138.6 231.2 128.3 138.6 203.9 Buzurg 52 48 42 39 36 1/1 13/12 26/21 4/3 13/9 (3/2) 0 138.6 369.7 498.0 636.6 (702.0) Here the entire Buzurg jins, except for its 3/2 step, can be derived from the rotation of Septimal Rast. An interesting aspect of this rotation is that our Septimal Rast has a Zalzalian third at 16/13 or 359 cents, while the resulting Buzurg rotation has a larger third at 26/21 cents or 370 cents -- a difference of 169:168 (10.274 cents), or close to a half-comma (e.g. the Pythagorean comma at 531441/524288 or 23.460 cents, with a half-comma at 11.730 cents). We may use the same procedure with Ibn Sina's other permutation of the "most noble" or Septimal Rast jins, again using a rotation on the third step of a conjunct cycle or octave mode, in order to arrive at the often-cited version of Buzurg beginning 14:13-8:7-16:13. Septimal Rast 416 364 336 312 273 252 234 208 1/1 8/7 26/21 4/3 32/21 104/63 16/9 2/1 0 231.2 369.7 498.0 729.2 867.8 996.1 1200 8:7 13:12 14:13 8:7 13:12 14:13 9:8 231.1 138.6 128.3 231.2 138.6 128.3 203.9 Buzurg 336 312 273 252 234 1/1 14/13 16/13 4/3 56/39 (3/2) 0 128.3 359.5 498.0 626.3 (702.0) Here the Septimal Rast has the larger Zalzalian third at 26/21, while the Buzurg has the smaller third at 16/13, exactly the converse of our first example. In MET-24, these same subtle half-comma nuances can be realized by modulating from either form of Septimal Rast to its Buzurg rotation on third step of a conjunct Septimal Rast, or also the seventh step of a disjunct form of Septimal Rast. The next two illustrations involve conjunct forms, with an upper Rast jins used to expand Buzurg from a pentachord to an octave cycle: Septimal Rast (conjunct ~16:14:13:12, ~16/13 third) 1/1 8/7 16/13 4/3 32/21 18/11 39/22 2/1 G* Bb B C* Eb E F* G* 0 230.9 357.4 495.7 727.7 853.1 992.6 1200 230.9 126.6 138.3 232.0 125.4 139.5 207.4 Buzurg (~26/21 third) 1/1 13/12 26/21 4/3 13/9 3/2 22/13 13/7 2/1 B C* Eb E F* F# G# Bb B 0 138.3 370.3 495.7 635.2 703.1 910.5 1073.4 1200 138.3 232.0 125.4 139.5 68.0 207.4 162.9 126.6 ------------- Septimal Rast (conjunct ~104:91:84:78, ~26/21 third) 1/1 8/7 26/21 4/3 32/21 104/63 39/22 2/1 F#* A Bb* B* D Eb* E* G* 0 232.0 370.3 496.9 727.7 867.2 992.6 1200 232.0 138.3 126.6 230.9 139.5 125.4 207.4 Buzurg (~16/13 third) 1/1 14/13 16/13 4/3 63/44 3/2 22/13 24/13 2/1 Bb* B* D Eb* E* F* G* A Bb* 0 126.6 357.4 496.9 622.3 704.3 911.7 1061.7 1200 126.6 232.0 139.5 125.4 82.0 207.4 150.0 138.3 Similar patterns occur in rotations from a disjunct Septimal Rast to Buzurg on the seventh step. To make the correspondences easier to follow, I have shown for Septimal Rast a Zalzalian second step below the final which serves also as the 1/1 of Buzurg: Buzurg (~26/21 third) 1/1 13/12 26/21 4/3 13/9 3/2 22/13 13/7 2/1 B C* Eb E F* F# G# Bb B 0 138.3 370.3 495.7 635.2 703.1 910.5 1073.4 1200 138.3 232.0 125.4 139.5 68.0 207.4 162.9 126.6 Septimal Rast (disjunct ~16:14:13:12, ~16/13 third) 24/13 1/1 8/7 16/13 4/3 3/2 12/7 24/13 2/1 B C* Eb E F* G* Bb B C* 1061.7 0 232.0 357.4 496.9 704.3 935.2 1061.7 1200 138.3 232.0 125.4 139.5 207.4 230.9 126.6 138.3 ------------- Buzurg (~16/13 third) 1/1 14/13 16/13 4/3 63/44 3/2 22/13 13/7 2/1 Bb* B* D Eb* E* F* G* A Bb* 0 126.6 357.4 496.9 622.3 704.3 911.7 1061.7 1200 126.6 232.0 139.5 125.4 82.0 207.4 150.0 138.3 Septimal Rast (disjunct ~104:91:84:78, ~26/21 third) 13/7 1/1 8/7 26/21 4/3 3/2 12/7 24/13 2/1 Bb* B* D Eb* E* F#* A Bb* B* 1073.4 0 232.0 370.3 495.7 704.3 935.2 1061.7 1200 126.6 232.0 139.5 125.4 207.4 230.9 126.6 138.3 These pairings illustrate both the kind of "half-comma" dialectic that occurs in Septimal Rast-Buzurg modulations, just or tempered, and some of the quirks of a temperament like MET-24. For example, a classic 16/9 step (996.1 cents) becomes more of a 39/22 (991.2 cents, with the tempered interval at 992.6 cents); and likewise a classic 56/39 (626.3 cents) may become more of a 63/44 (621.4 cents, or a tempered 622.3 cents). Also, while the idea of Septimal Rast-Buzurg rotations as a possible modulatory theme might fit in with the Systematist interest in modal cycles related by rotation as documented by such modern scholars as Anas Ghrab, it is a 21st-century approach which may or may not have precedent in the performance practices of the 13th-14th centuries. There is also the question of whether the semitone in Buzurg at 56/39-3/2 or 13/9-3/2 was often used as a direct melodic interval (at 117:112 or 27:26) in ascending to the 3/2 fifth; or whether the 56/39 or 13/9 step was used mainly as an "alternative" or lower position for the fifth, rather in the manner of an Iranian koron fifth, for example in Shur Dastgah. I tend to follow the latter usage, both as an expressive inflection in descending toward the final of Buzurg (as also is common in Shur), and as a pivotal step in modulating from Buzurg (56/39 or 13/9) to Septimal Rast (where the same step becomes the 16/9 of a conjunct form or 4/3 of a disjunct form). What these rotations, as a 21st-century device with or without precedent in the 13th-14th century treatment of Buzurg, illustrate is a way that one can explore half-comma nuances through modulation, and often in situations where two intervals such as 16/13 and 26/21 may not both be available from any single location in a just or tempered system. -------------------------------------------------- 2. Partitions or divisions of the tempered apotome -------------------------------------------------- In these divisions, as noted above, brackets are used to mark intervals which measure differences or spacings between step sizes but do not themselves occur on the keyboard, e.g. [12.9] tells us that the step sizes at 125.4 and 138.3 cents, representing 14/13 and 13/12, differ by 12.9 cents, but that there is no actual step of 12.9 cents to be found in the system. Thus one cannot, from a single location, choose between ~14/13 and ~13/12 steps, as might be very useful for different shadings of Persian Shur or Ottoman Ushshaq. Rather, one must seek these shadings at different locations.[4] The first division focuses on some basic steps and intervals found within a single 12-note chain of fifths, including the small Zalzalian step provided by the tempered apotome, e.g. C-C#, at 125.4 or 126.6 cents (near 14/13); and the large Zalzalian step provided by the double limma or diminished third (e.g. C#-Eb) at 162.9 cents (near 11/10). The enharmonic diesis of about 44.5 cents, not itself present as a step on the keyboard, measures the difference between limma and apotome, or double limma and usual tone; it is analogous to the Pythagorean comma in the 17-note system of Safi al-Din al-Urmawi based on a single chain of fifths. Division 1: Steps and differences in a single 12-note chain of fifths apotome ~14/13 126.6 _______________________________________________________________________ 82.0 126.6 80.9 125.4 162.9 207.4 ~22/21 ~14/13 ~11/10 ~9/8 limma apotome double limma tone | | | | 0 [44.5] 82.0 126.6 | [44.5] | [37.5] | [44.5] | enharmonic diesis enharmonic diesis The second division adds also the middle Zalzalian intervals approximating 13/12 and 12/11 which involve mixing notes from both chains of fifths. An approximate 13/12 at 138.3 or 139.5 cents is equal to a limma plus the small thirdtone at 57.4 cents between the two keyboards, e.g. E-F*; while an approximate 12/11 at 150.0 cents is equal to a usual tone less this small thirdtone (e.g. G*-A). Division 2: Includes all four Zalzalian steps apotome ~14/13 126.6 __________________________________________________________________________ 82.0 126.6 139.5 80.9 125.4 138.3 150.0 162.9 207.4 ~22/21 ~14/13 ~13/12 ~12/11 ~11/10 ~9/8 | [44.5] | [12.9] | [11.7] | [12.9] | [44.5] | limma apotome double limma tone | | | | 0 [44.5] 57.4 69.1 82.0 126.6 | [44.5] | [37.5] | [44.5] | enharmonic diesis enharmonic diesis Additionally, as a consequence rather than a primary feature of the tuning design, there are a few remote intervals at 105.5 cents (close to 17/16) and 182.8 or 184.0 cents (close to 10/9). The third division includes these intervals also for a complete survey of step sizes and differences within the apotome range between the limma and the usual ~9/8 tone. Division 3: Also includes remote steps near 17/16 and 10/9 apotome ~14/13 126.6 ___________________________________________________________________________ 82.0 126.6 139.5 80.9 105.5 125.4 138.4 150.0 162.9 182.8 207.4 ~22/21 ~17/16 ~14/13 ~13/12 ~12/11 ~11/10 ~10/9 ~9/8 limma apotome double limma tone | 24.6 | 19.9 | [12.9] | [11.7] | [12.9] | [19.9] | 24.5 | 0 24.6 [44.5] 57.4 69.1 82.0 [102.0] 126.6 | [44.5] | [37.5] | [44.5] | enharmonic diesis enharmonic diesis ------------------------------------------------------------ 3. Some gamuts for Arab, Ottoman/Aleppian, and Persian music ------------------------------------------------------------ The following tables show some possible 24-note gamuts in the MET-24 system for Arab, historical Ottoman, and Persian music. Intervals are shown in cents and Holderian commas (Hc) of 53-ED2, and as approximate ratios.[5] For keyboard diagrams of discussions of the three gamuts here presented, see Section 1.1 above. The main purpose of these gamuts is to illustrate some of the themes, possibilities, and limitations of the tuning system -- as compared to the illimitable diversity and subtlety of Maqam! ---------------------------------------------------------------------- MET-24 System: 24 notes for Arab style, C* = Perde Rast Intervals from Rast ---------------------------------------------------------------------- Bardah/step keyboard cents Hc approx. ratios ---------------------------------------------------------------------- RAST C* 0.0 0.00 1/1 ZIRKULA C# 69.1 3.05 27/26 126/121 176/169 Rahawi wadi C#* 126.6 5.59 14/13 RAHAWI D 150.0 6.62 12/11 DUKAH D* 207.4 9.16 9/8 273/242 44/39 Dukah `ali Eb 232.0 10.25 8/7 KURDI Eb* 289.5 12.78 13/11 SIKAH E 357.4 15.79 27/22 59/48 16/13 BUSELIK E* 414.8 18.32 33/26 80/63 14/11 Buselik `ali F 439.5 19.41 9/7 156/121 352/273 JAHARKAH F* 496.9 21.95 4/3 HIJAZ F# 564.8 24.95 18/13 SABA F#* 622.3 27.48 63/44 56/39 Saba `ali G 646.9 28.57 16/11 NAWA G* 704.3 31.11 3/2 SHURI G# 772.3 34.11 264/169 25/16 Hisar wadi G#* 829.7 36.64 21/13 HISAR A 854.3 37.73 18/11 64/39 HUSEYNI A* 911.7 40.27 22/13 Huseyni `ali Bb 935.2 41.30 12/7 `AJAM Bb* 992.6 43.84 39/22 484/273 16/9 AWJ B 1061.7 46.89 24/13 MAHUR B* 1119.1 49.43 21/11 Mahur `ali C 1142.6 50.46 27/14 176/91 64/33 KIRDAN C* 1200.0 53.00 2/1 ---------------------------------------------------------------------- To devise a note-naming system in the Arabic style with 17 basic degrees, the step or region a Zalzalian or middle second above Rast is here called Rahawi, a Persian name, although the usual Arabic name Tik Zirkula could also be used. Borrowing from the Syrian theorist Tawfiq al-Sabbagh, versions of a degree a comma lower or higher than the "usual" step are termed respectively wadi and `ali. There are thus 17 main steps plus 7 comma variants (wadi or `ali). ---------------------------------------------------------------------- ---------------------------------------------------------------------- ----------------------------------------------------------------------- MET-24 System: 24 notes for historical Ottoman style, B = Perde Rast This tuning may also fit certain Arab (e.g. Aleppian) tastes Intervals from Perde Rast ----------------------------------------------------------------------- Perde/step keyboard cents Hc approx. ratios ----------------------------------------------------------------------- RAST B 0.0 0.00 1/1 Nerm Shuri B* 57.4 2.54 33/32 91/88 28/27 SHURI C 80.9 3.57 22/21 Nerm Zengule C* 138.3 6.11 13/12 DUGAH C# 207.4 9.16 9/8 273/242 44/39 Nerm Kurdi C#* 264.8 11.70 7/6 KURDI D 288.3 12.73 13/11 Nerm Segah/Ushshaq D* 345.7 15.27 39/32 72/59 11/9 SEGAH Eb 370.3 16.36 99/80 26/21 Dik Buselik Eb* 427.7 18.89 32/25 169/132 242/189 CHARGAH E 495.7 21.89 4/3 Nerm Hicaz E* 553.1 24.43 11/8 HICAZ F 577.7 25.52 39/28 88/63 SABA F* 635.2 28.05 13/9 NEVA F# 703.1 31.05 3/2 Nerm Beyati F#* 760.5 33.59 273/176 14/9 BEYATI G 785.2 34.68 11/7 52/33 Nerm Hisar G* 842.6 37.21 13/8 96/59 44/27 HUSEYNI G# 910.5 40.21 22/13 Nerm ACEM G#* 968.0 42.75 7/4 ACEM A 992.6 43.84 39/22 484/273 16/9 EVDJ Bb 1073.4 47.41 13/7 Dik Mahur Bb* 1130.9 49.95 48/25 121/63 169/88 GERDANIYE B 1200.0 53.00 2/1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- A basic feature of this historical Ottoman gamut is its support for a bright Rast with Segah and Evdj around 26/21 and 13/7, along with many Zalzalian steps near 11/10 and 14/13 supporting various permutations of al-Farabi's or Safi al-Din al-Urmawi's "Medium Sundered (or Discontinuous)" with steps of 9:8-11:10-320:297 (a just 203.9-165.0-129.1 cents, and here 207.4-162.9-125.4 cents for the lower tetrachord of Rast at B-C#-Eb-E). When this gamut is used in an Arab (e.g. Aleppian) context, the low Zalzalian sixth step (Nerm Hisar) may expedite a modulation from Rast on B to Bayyati on F# with a Zalzalian second very close to 13/12. In Maqam Rast, one may emulate a modern Turkish nuance by placing the usual step sikah/segah at 370 cents (near 26/21) but lowering this step in a descent approaching a final cadence to 346 cents (near 11/9), a difference of a full comma. This may be rather less subtle than that typically practiced either by Turkish musicians or by Julien Jalal Ed-Dine Weiss, who often prefer a distinction closer to 1/2-comma. ---------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------------------------------- MET-24 System: 24 notes for Persian style, A = Dugah Intervals from Dugah ---------------------------------------------------------------------- keyboard cents Hc approx. ratios ---------------------------------------------------------------------- A 0.0 0.00 1/1 A* 57.4 2.54 33/32 91/88 28/27 Bb 80.9 3.57 22/21 Bb* 138.3 6.11 13/12 B 207.4 9.16 9/8 273/242 44/39 B* 264.8 11.70 7/6 C 288.3 12.73 13/11 C* 345.7 15.27 39/22 72/59 11/9 C# 414.8 18.32 33/26 14/11 C#* 472.3 20.86 21/16 D 495.7 21.89 4/3 D* 553.1 24.43 11/8 Eb 577.7 25.52 39/28 88/63 Eb* 635.2 28.05 13/9 E 703.1 31.05 3/2 E* 760.5 33.59 273/176 14/9 F 785.2 34.68 11/7 52/33 F* 842.6 37.21 13/8 96/59 44/27 F# 910.5 40.22 22/13 F#* 968.0 42.75 7/4 G 992.6 43.84 39/22 484/273 16/9 G* 1050.8 46.38 11/6 G# 1118.0 49.38 21/11 G#* 1175.4 51.91 63/32 77/39 A 1200.0 53.00 2/1 ---------------------------------------------------------------------- Since Persian names for the steps are not so often used, I have identified steps by keyboard locations, with A as the usual finalis in this arrangement for Shur Dastgah. An advantage of using A as a reference is that it shows the alternative regular and septimal minor steps and intervals often available from this location, e.g. B* (~7/6) and C (~13/11). ---------------------------------------------------------------------- ----------------------------------------------------------------------. ------- Notes ------- 1. This criterion of "a majority of smaller fifths" at 703.125 cents, the size closer to a just 3/2, can only apply to a system where each chain has an odd number of fifths, here 11 for a 24-note system. In fact, a system with two 17-note chains would be more versatile; each chain would have an equal number of smaller and larger fifths (8 of each). In my first draft of this partage, I mentioned that the "average" size of a fifth in a 12-note chain was 703.658 cents -- changing this figure here to 703.711 cents, the simple geometric mean of the two sizes (703.125 and 704.297 cents), and also the fifth for a regular temperament producing the system's regular tone at 207.422 cents. 2. Here I am much indebted to Scott Marcus for his case study of Maqam Bayyati in practice among Egyptian traditional musicians noting that while the second step of Bayyati might typically be placed at around 135-145 cents, the Zalzalian sixth step when it occurs is understood by "many" musicians to be a bit higher than its position of 850 cents in the theoretical model of 24 equal quartertones. See his "The Eastern Arab System of Melodic Modes In Theory and Practice: A Case Study of _Maqam Bayyati_" in _The Garland Encyclopedia of World Music, Volume 6: The Middle East_ (2002: 33-44 at 39). 3. Experience has taught me that in Bb Bayyati, the D* step at an approximate 21/16 above the finalis often comes into play, for example to obtain an upper jins with a 4/3 fourth such as a septimal color of Nahawand (D*-F-F*-G* or 232-57-207 cents, rather like the 8/7-28/27-9/8 of Archytas) or Rast (D*F-F#-G* or 232-125-139 cents, close to Ibn Sina's 16:14:13:12). 4. While using 12.9 cents for the difference between the two types of Zalzalian steps at 125.4/126.6 and 138.3/139.5 cents may be convenient, this precise degree of contrast does not occur in the most musically relevant situation: the division of a near-7/6 minor third at 264.844 cents into approximate steps of 14:13:12 or 12:13:14, as in Ibn Sina's 16:14: 13:12. Here we find the difference is either 125.4-139.5 (14.1 cents) or 126.6-138.3 (11.7 cents), so that 12.9 cents is more of a rough statistical average than a musical reality. 5. With the Arab gamut, the placement of the step rast at C* is in line with Arab (as opposed to Ottoman) notational conventions, and has the advantage of a convenient modulatory contrast between C Rast and Bayyati nawa on the step G* (nawa) with a "folk" color of G*-G#*-Bb*-C* or 125.4-162.9-207.4 cents. According to Amine Beyhom, such a folk Bayyati in Lebanon favors a step above the finalis of 130 cents or so. An alternative placement of rast at G* is ergonomic in a situation of playing on two standard 12-note MIDI keyboards and using a conventional keyboard shelf. Playing the conjunct form of G* Rast (G*-A*-B-C*-D*-E-F*-G*) is easier using a single hand since there is no need, as with Rast on F or C, to move from a natural on the lower keyboard to an accidental on the upper one (C*-D-Eb*-F* or G*-A-Bb*-C*). One partial solution, of course, is to reverse the MIDI channel mappings of the two keyboards, so that the one with the higher 12-note chain of fifths is physically in the lower position! This makes it easy to play a conjunct Rast on C* (with A-Bb*) or F* (with D-Eb*), but unfortunately harder to play, for example, Ibn Sina's beautiful 16:14:13:12 division on C (C*-Eb-E-F*), since the approximate 8:7 step C*-Eb will now move from a natural on the physically lower keyboard to an accidental on the physically upper one. With Rast on G*, and a usual MIDI channel mapping with the aurally higher chain of fifths on the physically higher manual, this problem does not arise for the conjunct and disjunct forms of Rast with a usual 207-cent step near 9/8 or 44/39, nor for the very attractive variant forms near 16:14:13:12. Margo Schulter Sacramento, California, U.S.A. 14 January 2013 Revised 26 December 2013