------------------------------------------------------- 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 in which he has described some different shadings of the Tsaharuk system. Thus I was led to seek a milder temperament with extended or wide fifths, 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, but differs subtly in certain objectives and compromises, I had an opportunity, thanks to Jacques, to see some partitions for qanun that Julien Jalaleddine Weiss and he had detailed. This gave me the idea of explaining the divisions of the tempered apotome (125.4 or 126.6 cents) in MET-24. Before presenting the diagrams for the division itself, followed by some possible gamuts for Arab, Turkish, 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 25.6 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. It is from this view, with many thanks to Jacques Dudon, Julien Jalaleddine 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, Turkish, 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 ------------------------------------------------------- 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] The partitions or divisions below 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. 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 Turkish Rast; or a lower Segah at 345.7 cents (~72/59) for a moderate Turkish Ushshaq or Arab Bayyati. Turkish Rast Turkish 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 Turkish 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 (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 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) Rast Rast |------------------| |--------------------| 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 [2] 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 -------------------------------------------------- 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 Turkish Ushshaq. Rather, one must seek these shadings at different locations.[3] 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 possible gamuts for Arab, Turkish, and Persian music ------------------------------------------------------------ The following tables show some possible 24-note gamuts in the MET-24 system for Arab, Turkish, and Persian music. Intervals are shown in cents and Holderian commas (Hc) of 53-ED2, and as approximate ratios.[4] 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, G* = Perde Rast Intervals from Rast ---------------------------------------------------------------------- Bardah/step keyboard cents Hc approx. ratios ---------------------------------------------------------------------- RAST G* 0.0 0.00 1/1 ZIRKULA G# 68.0 3.00 27/26 126/121 176/169 Rahawi wadi G#* 125.4 5.54 14/13 RAHAWI A 150.0 6.62 12/11 DUKAH A* 207.4 9.16 9/8 273/242 44/39 Dukah `ali Bb 230.9 10.20 8/7 KURDI Bb* 288.3 12.73 13/11 SIKAH B 357.4 15.79 27/22 59/48 16/13 BUSELIK B* 414.8 18.32 33/26 14/11 Buselik `ali C 438.3 19.36 9/7 156/121 352/273 JAHARKAH C* 495.7 21.89 4/3 HIJAZ C# 564.8 24.95 18/13 SABA C#* 622.3 27.48 63/44 56/39 Saba `ali D 645.7 28.52 16/11 NAWA D* 703.1 31.05 3/2 Nawa `ali Eb 727.7 32.14 32/21 SHURI Eb* 785.2 34.68 11/7 52/33 HISAR E 853.1 37.68 18/11 HUSEYNI E* 910.5 40.22 22/13 Huseyni `ali F 935.2 41.30 12/7 `AJAM F* 992.6 43.84 39/22 16/9 AWJ F# 1060.5 46.84 24/13 MAHUR F#* 1118.0 49.38 21/11 Mahur `ali G 1142.6 50.46 27/14 176/91 64/33 KIRDAN G* 1200.0 53.00 2/1 ---------------------------------------------------------------------- NOTE: 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" version 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 Ottoman style, B = Perde Rast 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 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 ----------------------------------------------------------------------- ----------------------------------------------------------------------- NOTE: A basic feature of this Turkish 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 Safi al-Din'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). The availability at this location of only a low Hisar or neutral sixth step at 842.6 cents (a near-just 96/59) is a quirk of the tuning system; but Baris Bozkurt, Ozan Yarman, M. Kemal Karaosmanoglu, and Can Akkoc have reported in a recent study that digital measurements of intervals in recorded performances of Maqam Rast actually show a peak at around 836 cents or 37 Holderian commas, so that this quirk may be a happy one. The terms nerm and dik, borrowed very freely from Ozan Yarman, here refer to a step about a comma lower or higher respectively than what one takes to be the "usual" position. ---------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------------------------------- 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 ---------------------------------------------------------------------- NOTE: 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. 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). 3. 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. 4. With the Arab gamut, the placement of the step rast at C* would be more in line with Arab (as opposed to Turkish) notational conventions, and would have 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. The advantage of placing rast at G* is in part 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 (upper jins 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*). A simple 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!