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Tuning systems for the qanun## Table of Contents

Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical University.

Online version of Stefan Pohlit's dissertation: see http://stefanpohlit.com/dissertation.engl..htm

The tuning tables on this page are specifically designed for the tuning system of the qanun (see the link for details on the system of tuning and playing a qanun with mandals/orabs). The logic behind the systems is as follows:

The empty strings of the qanun are tuned to a pythagorean diatonic scale, with a major third of 81/64, a major sixth of 27/16 and a major seventh of 243/128.

The possible pitches of a string obtained via raising/lowering the mandals lie within two apotomes (2187/2048, 113.7 cents). The base note is assumed in the middle. The mandals allow raising and lowering this note by maximally one apotome.

Each apotome is divided into 7 unequal parts, which requires 14 mandals per string. The first rough subdivision of the apotome is always into one syntonic comma (81/80, 21.5 cents), one Zarlinian semitone (25/24, 70.7 cents) and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.

The tuning systems are all described by a series of cent values, which describe the subdivision of one apotome. According to the system sketched above, the first and the last value are always 22 cents (or 21.5 cents). This subdivision pattern occurs twice on each string, altogether 14 times per octave. This is followed by listings of some important rational intervals that are possible in this tuning, mainly in the range of a fourth (the range where the ajnas - maqam tetrachords - reside),

An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the major limma), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the pythagorean limma, the same interval as between E and F) - we have two notes differing by one schisma (2 cents). So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.

## Notation

The notes without accidentals stand for the pythagorean intervals of the base tuning of the qanun. Raising a pitch by an apotome is notated with "#", lowering a pitch by the same amount is notated with "b". Sharps are higher than flats (unlike in meantone systems): C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Both properties indicate that the framework is essentially pythagorean. The tuning system as a whole, however, is not.For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.

This gives 15 potential different pitches per base note, corresponding to the mandals. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si), corresponding to the strings, lead to a notation system of 7*15=105 pitches, in accordance with the real playing capabilities of the qanun. See the following document, which also gives all the pitches in one octave (in ratios and cents) that can be played by system 1 and 2.

(used with permission J. J. Weiss/S. Pohlit)

## System 1

© J.J.Weiss. Luthier: Ejder Gulec.Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).

65/64 and 55/54 are each split into two roughly equal parts.

This gives the following rational intervals between the mandals:

81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80

In cents (approximations):

22, 13, 13, 12, 16, 16, 22

Rational intervals each string can be detuned (approximations in cents in parentheses):

81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114)

Intervals ratios, ascending from C:

256/243 (90), 16/15 (112),

784/729 (126),13/12 (138),12/11 (150),11/10 (166), 10/9 (182), 9/8 (204)32/27 (294), 6/5 (316),

98/81 (330),39/32 (342),27/22 (354),99/80 (370), 5/4 (386), 81/64 (408)Interval ratios, descending from F:

9/8 (204), 10/9 (182),

54/49 (169),128/117 (156),88/81 (144),320/297 (129), 16/15 (112), 256/243 (90)81/64 (408), 5/4 (386),

243/196 (372),16/13 (360),11/9 (348),40/33 (333), 6/5 (316), 32/27 (294)A complete list of all intervals available within one octave can be found in the above-mentioned document (on the first page).

## System 2, better suited for ottoman maqams

© J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten.Mandal positions in ratios:

81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80

In cents (approximations):

22|16|15|12|13|14|22

Rational intervals each string can be detuned (approximations in cents in parentheses):

81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114)

Intervals ratios, ascending from C:

256/243 (90), 16/15 (112),

14/13 (128), 88/81 (144), 128/117 or 35/32 (156), 119/108 (168), 10/9 (182), 9/8 (204)32/27 (294), 6/5 (316),

63/52 (332), 11/9 (348), 16/13 or 315/256 (360), 119/96 (372), 5/4 (386), 81/64 (408)Interval ratios descending from F:

9/8 (204), 10/9 (182),

208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126), 16/15 (112), 256/243 (90)81/64 (408), 5/4 (386),

26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330), 6/5 (316), 32/27 (294)A complete list of all intervals available within one octave can be found in the above-mentioned document (on the second page).

## Other models

Julien Weiss has developed a number of other systems besides the two described above. A notable class of these are so-called super-symmetrical systems, which have the property that the intervals ascending from C and the intervals descending from F (which show slight differences in the previous two systems, marked inboldabove) are the same.3 examples are described below. For more and detailed descriptions see chapter 3.4 and appendix I in Stefan Pohlit's dissertation .

## Super-symmetric model with non-aliquot division of 65/64

© J.J. WeissSimilar to system 1, but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13).

Mandal positions in ratios:

81/80 (22), 120/119 (14), 1547/1536 (12), 512/507 (17), 1547/1536 (12), 120/119 (14), 81/80 (22)

Table of pitches from C to F (approximations in cents):

Interval ratios, ascending from C:

245/243 (90), 16/15 (112),

128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),10/9 (182), 9/8 (204)On the E string (from Eb to E):

32/27 (294), 6/5 (316),

144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372),5/4 (386), 81/64 (408)Interval ratios, descending from F:

9(8 /204), 10/9 (182),

119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126), 16/15 (112), 256/243 (90)X81/64 (408), 5/4 (386),

119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330), 6/5 (316), 32/27 (294)Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context.

## Equal division of the Zarlinian semitone

© J.J. WeissThis is the simplest variant for luthiers...

Mandal positions (cents): 22|14|14|14|14|14|22

Mandal positions in ratios:

81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80

Table of pitches from C to F (approximations in cents):

Interval ratios, ascending from C:

256/243 (90), 16/15 (112),

100/93 (126), 400/369 (140), 200/183 (153.78), 400/363 (168).10/9 (182), 9/8 (204)32/27 (294), 6/5 (316),

75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94),5/4 (386), 81/64 (408)Interval ratios descending from F:

9/8, 10/9,

248/225 (168.49), 82/75 (154.47), 244/225 (140.34), 242/225 (126.09), 16/15, 256/243or approximating ratios: XXX81/64, 5/4,

31/25 (372.40), 49/40 (351.33), 61/50 (344.25), 121/100 (330), 6/5, 32/27Or approximatiing ratios: XXX## Super-symmetrical model with 14/13

© J.J. WeissThe idea behind this system is as follows:

Dividing the apotome (114 cents) into 3 equal parts gives 38 cents, and adding this to the pythagorean limma (90 cents) gives 128 cents, which is an approximation for 14/13 (two-third tone, a favorite interval of Avicenna/Ibn Sina).

The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotome with an equal division into 3 parts, which yields the following mandal positions (cents):

22, 16, 13, 12, 13, 16, 22

(Observe that 22+16 = 38, as well as 13+12+13.)

Mandal positions in ratios:

81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80

Since the pythagorean limma appears prominently in the basic framework anyway (as semitone from E to F and from B to C as well as one apotome minus a syntonic comma several times on each string), 14/13 also appears at various positions.

Table of pitches from C to F (approximations in cents):

Interval ratios, ascending from C:

256/243 (90), 16/15/112), 14/13 (128), 64/59 (141), 59/54 /153), 209/189 (166), 10(9 (182), 9/8 (204)

32/27 (294), 6/5 (316), 63/52 (332), 72/59 (345), 59/48 (357), 26/21 (370), 5/4 (386), 81/64 (408)