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The fretting of the Arab lute

[Joseph Monzo]

© 1998, 2004 by Joseph L. Monzo


The lute became the most popular instrument in Europe during the 1500s, similar to the guitar's popularity today. The European instrument was derived from the Arab 'ud ("al-'ud" --> "lute"), whose first flowering occurred during the 600s and 700s in Syria and Iraq.

It is worth remembering thru-out this discussion that lute frets were always made of catgut and tied around the neck of the instrument, and could thus be moved at will, unlike the embedded metal frets on today's fretted string instruments.


The Arab lute's open strings and ancient fretting were entirely Pythagorean (that is, 3-limit), derived most likely from Sumerian/Babylonian, Indian, and Greek precedents.

The ancient tuning formed a 13-tone Pythagorean chain 3(-6 ... +6). This gave a scale which is audibly nearly the same as our modern standard 12edo tuning, but with two varieties of tritone.

The ancient Arab lute fretting:


 2,3,5,7,11-monzo       ---------- ratio ---------   string & finger       string-length Semitones Ellis's cents

[  5  -2,  0  0  0 >      32 / 9        3.555555556  4 little                   3 / 4         9.96   792
[ -3   3,  0  0  0 >      27 / 8        3.375        4 ring                    64 / 81        9.06   702
[  8  -4,  0  0  0 >     256 / 81       3.160493827  4 middle (ancient)        27 / 32        7.92   588
[  0   1,  0  0  0 >       3 / 1        3.0          4 index                    8 / 9         7.02   498
[ 11  -6,  0  0  0 >    2048 / 729      2.809327846  4 near-index (ancient)   243 / 256       5.88   384
[  3  -1,  0  0  0 >       8 / 3        2.666666667  4 open = 3 little          1 / 1         4.98   294
[ -5   4,  0  0  0 >      81 / 32       2.53125      3 ring                    64 / 81        4.08   204
[  6  -3,  0  0  0 >      64 / 27       2.37037037   3 middle (ancient)        27 / 32        2.94    90
[ -2   2,  0  0  0 >       9 / 4        2.25         3 index                    8 / 9         2.04     0
[  9  -5,  0  0  0 >     512 / 243      2.106995885  3 near-index (ancient)   243 / 256       0.90  1086
[  1   0,  0  0  0 >       2 / 1        2.0          3 open = 2 little          1 / 1         0.00   996
[ -7   5,  0  0  0 >     243 / 128      1.8984375    2 ring                    64 / 81       11.10   906
[  4  -2,  0  0  0 >      16 / 9        1.777777778  2 middle (ancient)        27 / 32        9.96   792
[ -4   3,  0  0  0 >      27 / 16       1.6875       2 index                    8 / 9         9.06   702
[  7  -4,  0  0  0 >     128 / 81       1.580246914  2 near-index (ancient)   243 / 256       7.92   588
[ -1   1,  0  0  0 >       3 / 2        1.5          2 open = 1 little          1 / 1         7.02   498
[ -9   6,  0  0  0 >     729 / 512      1.423828125  1 ring                    64 / 81        6.12   408
[  2  -1,  0  0  0 >       4 / 3        1.333333333  1 middle (ancient)        27 / 32        4.98   294
[ -6   4,  0  0  0 >      81 / 64       1.265625     1 index                    8 / 9         4.08   204
[  5  -3,  0  0  0 >      32 / 27       1.185185185  1 near-index (ancient)   243 / 256       2.94    90
[ -3   2,  0  0  0 >       9 / 8        1.125        1 open                     1 / 1         2.04     0


ChartObject Arab Lute Ancient Fretting


(Note that regular Pythagorean notation is sufficient for the ancient fretting.)


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During the 8th century AD, Zalzal (d. 791) specified new higher fret-placements for the "near-index" and "middle-finger" frets, using respectively ratios of 7 and of 11. His refrettings resulted from equidistant string-length divisions.

Zalzal's 22/27 "middle-finger" fret results from a division of the string-length between 8/9 (the whole-tone) and 2/3 (the perfect-5th) into 3 equal parts: 8/9 = 24/27, 1/3 of the string distance above that = 22/27, and 2/3 of the string distance above that = 18/27 = 2/3. As the lute string-lengths cover a distance of only 3/4, this obviously would have been calculated on a monochord.

Zalzal's 49/54 fret results from a division of the string-length between the open string and his 22/27 "middle-finger" fret exactly in half: 1/1 = 54/54, and 22/27 = 44/54, so 1/2 of the string distance between them = 49/54. This fret also has the result of dividing the string-length between the open string and 8/9 (the whole-tone or major-2nd) into 6 equal parts: 1/1 = 54/54, 5/6 of the string distance above that = 49/54, and 1/6 of the string distance above that = 48/54 = 8/9.

Despite the fact that later theorists revised the placement of these frets to render them within a Pythagorean framework, their acoustic gestalts have persisted in music of the "middle-east". This resulted eventually in the controversy over the official adoption by the countries of the Arab world of 24edo ("quarter-tones") in 1932, since 24edo gives good approximations to all of the 11-limit ratios posited by Zalzal.

Zalzal's descending tetrachord-division has the proportion 243 : 256 : 264 : 288 : 294 : 324. Some EDOs which approximate it well are:

Zalzal's fret measurements:


 2,3,5,7,11-monzo      ---------- ratio ---------   string & finger         string-length Semitones Ellis's cents

[  5  -2,  0  0  0 >      32 / 9        3.555555556  4 little                   3 / 4         9.96   792
[ -3   3,  0  0  0 >      27 / 8        3.375        4 ring                    64 / 81        9.06   702
[  2   2,  0  0 -1 >      36 / 11       3.272727273  4 middle (Zalzal)         22 / 27        8.53   649
[  0   1,  0  0  0 >       3 / 1        3.0          4 index                    8 / 9         7.02   498
[  4   2,  0 -2  0 >     144 / 49       2.93877551   4 near-index (Zalzal)     49 / 54        6.66   462
[  3  -1,  0  0  0 >       8 / 3        2.666666667  4 open = 3 little          1 / 1         4.98   294
[ -5   4,  0  0  0 >      81 / 32       2.53125      3 ring                    64 / 81        4.08   204
[  0   3,  0  0 -1 >      27 / 11       2.454545455  3 middle (Zalzal)         22 / 27        3.55   151
[ -2   2,  0  0  0 >       9 / 4        2.25         3 index                    8 / 9         2.04     0
[  2   3,  0 -2  0 >     108 / 49       2.204081633  3 near-index (Zalzal)     49 / 54        1.68  1164
[  1   0,  0  0  0 >       2 / 1        2.0          3 open = 2 little          1 / 1         0.00   996
[ -7   5,  0  0  0 >     243 / 128      1.8984375    2 ring                    64 / 81       11.10   906
[ -2   4,  0  0 -1 >      81 / 44       1.840909091  2 middle (Zalzal)         22 / 27       10.57   853
[ -4   3,  0  0  0 >      27 / 16       1.6875       2 index                    8 / 9         9.06   702
[  0   4,  0 -2  0 >      81 / 49       1.653061224  2 near-index (Zalzal)     49 / 54        8.70   666
[ -1   1,  0  0  0 >       3 / 2        1.5          2 open = 1 little          1 / 1         7.02   498
[ -9   6,  0  0  0 >     729 / 512      1.423828125  1 ring                    64 / 81        6.12   408
[ -4   5,  0  0 -1 >     243 / 176      1.380681818  1 middle (Zalzal)         22 / 27        5.58   355
[ -6   4,  0  0  0 >      81 / 64       1.265625     1 index                    8 / 9         4.08   204
[ -2   5,  0 -2  0 >     243 / 196      1.239795918  1 near-index (Zalzal)     49 / 54        3.72   168
[ -3   2,  0  0  0 >       9 / 8        1.125        1 open                     1 / 1         2.04     0


ChartObject Arab Lute with Zalzal's Replacements


(I use HEWM notation here, in both the JI and 72edo forms, for the ratios of 7 and 11.)


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Abu Nasr al-Farabi (870 - 950 AD) described a lute fretting which encompassed both the old Pythagorean fretting and approximations to the two new intervals of Zalzal.

During the 1300s, Mahmoud and Abdulqadir both (independently) reinstated some of the ancient Pythagorean notes, and also moved the placement of Zalzal's frets to nearby notes from an extended Pythagorean chain. The entire system is an 18-tone Pythagorean chain 3(-11 ... +6).

(The "Arabic scale" is always described as a 17-tone scale, as in both Helmholtz and Partch -- see my tabulation of Partch's description. It is not clear to me how one note was lost.)

The adjusted fret placements specified by Mahmoud and Abdulqadir:


 2,3,5,7,11-monzo     ---------- ratio ----------  string & finger         string-length Semitones Ellis's cents

[  5  -2,  0  0  0 >      32 / 9        3.555555556  4 little                   3 / 4         9.96   792
[ -3   3,  0  0  0 >      27 / 8        3.375        4 ring                    64 / 81        9.06   702
[ 16  -9,  0  0  0 >   65536 / 19683    3.329573744  4 middle (M&A)          6561 / 8192      8.82   678
[  8  -4,  0  0  0 >     256 / 81       3.160493827  4 middle (ancient)        27 / 32        7.92   588
[  0   1,  0  0  0 >       3 / 1        3.0          4 index                    8 / 9         7.02   498
[ 19 -11,  0  0  0 >  524288 / 177147   2.959621106  4 near-index (M&A)     59049 / 65536     6.78   475
[ 11  -6,  0  0  0 >    2048 / 729      2.809327846  4 near-index (ancient)   243 / 256       5.88   384
[  3  -1,  0  0  0 >       8 / 3        2.666666667  4 open = 3 little          1 / 1         4.98   294
[ -5   4,  0  0  0 >      81 / 32       2.53125      3 ring                    64 / 81        4.08   204
[ 14  -8,  0  0  0 >   16384 / 6561     2.497180308  3 middle (M&A)          6561 / 8192      3.84   180
[  6  -3,  0  0  0 >      64 / 27       2.37037037   3 middle (ancient)        27 / 32        2.94    90
[ -2   2,  0  0  0 >       9 / 4        2.25         3 index                    8 / 9         2.04     0
[ 17 -10,  0  0  0 >  131072 / 59049    2.219715829  3 near-index (M&A)     59049 / 65536     1.80  1177
[  9  -5,  0  0  0 >     512 / 243      2.106995885  3 near-index (ancient)   243 / 256       0.90  1086
[  1   0,  0  0  0 >       2 / 1        2.0          3 open = 2 little          1 / 1         0.00   996
[ -7   5,  0  0  0 >     243 / 128      1.8984375    2 ring                    64 / 81       11.10   906
[ 12  -7,  0  0  0 >    4096 / 2187     1.872885231  2 middle (M&A)          6561 / 8192     10.86   882
[  4  -2,  0  0  0 >      16 / 9        1.777777778  2 middle (ancient)        27 / 32        9.96   792
[ -4   3,  0  0  0 >      27 / 16       1.6875       2 index                    8 / 9         9.06   702
[ 15  -9,  0  0  0 >   32768 / 19683    1.664786872  2 near-index (M&A)     59049 / 65536     8.82   678
[  7  -4,  0  0  0 >     128 / 81       1.580246914  2 near-index (ancient)   243 / 256       7.92   588
[ -1   1,  0  0  0 >       3 / 2        1.5          2 open = 1 little          1 / 1         7.02   498
[ -9   6,  0  0  0 >     729 / 512      1.423828125  1 ring                    64 / 81        6.12   408
[ 10  -6,  0  0  0 >    1024 / 729      1.404663923  1 middle (M&A)          6561 / 8192      5.88   384
[  2  -1,  0  0  0 >       4 / 3        1.333333333  1 middle (ancient)        27 / 32        4.98   294
[ -6   4,  0  0  0 >      81 / 64       1.265625     1 index                    8 / 9         4.08   204
[ 13  -8,  0  0  0 >    8192 / 6561     1.248590154  1 near-index (M&A)     59049 / 65536     3.84   180
[  5  -3,  0  0  0 >      32 / 27       1.185185185  1 near-index (ancient)   243 / 256       2.94    90
[ -3   2,  0  0  0 >       9 / 8        1.125        1 open                     1 / 1         2.04     0


ChartObject Arab Lute Mahmoud & Abdulqadir


ChartObject Arab Lute Mahmoud & Abdulqadir

(Note that, again, regular Pythagorean notation is sufficient for the frettings of Mahmoud and Abdulqadir.)


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Lastly, graphs showing all frettings together, for comparison. (I have also included some Persian frets which i derived from Ellis's cents measurements in Helmholtz 1954, but an explanation of that must wait for a future webpage.)


 2,3,5,7,11-monzo     ---------- ratio ----------   string & finger        string-length Semitones Ellis's cents

[  5  -2,  0  0  0 >      32 / 9        3.555555556  4 little                    3 / 4         9.96   792
[ -3   3,  0  0  0 >      27 / 8        3.375        4 ring                     64 / 81        9.06   702
[ 16  -9,  0  0  0 >   65536 / 19683    3.329573744  4 middle (M&A)           6561 / 8192      8.82   678
[  2   2,  0  0 -1 >      36 / 11       3.272727273  4 middle (Zalzal)          22 / 27        8.53   649
[ 16  -1, -4  0 -1 >   65536 / 20625    3.17750303   4 middle (Persian)       6875 / 8192      8.01   598
[  8  -4,  0  0  0 >     256 / 81       3.160493827  4 middle (ancient)         27 / 32        7.92   588
[  0   1,  0  0  0 >       3 / 1        3.0          4 index                     8 / 9         7.02   498
[ 19 -11,  0  0  0 >  524288 / 177147   2.959621106  4 near-index (M&A)      59049 / 65536     6.78   475
[  4   2,  0 -2  0 >     144 / 49       2.93877551   4 near-index (Zalzal)      49 / 54        6.66   462
[ 23 -10,  0 -2  0 > 8388608 / 2893401  2.899220675  4 near-index (Persian) 964467 / 1048576   6.43   439
[ 11  -6,  0  0  0 >    2048 / 729      2.809327846  4 near-index (ancient)    243 / 256       5.88   384
[  3  -1,  0  0  0 >       8 / 3        2.666666667  4 open = 3 little           1 / 1         4.98   294
[ -5   4,  0  0  0 >      81 / 32       2.53125      3 ring                     64 / 81        4.08   204
[ 14  -8,  0  0  0 >   16384 / 6561     2.497180308  3 middle (M&A)           6561 / 8192      3.84   180
[  0   3,  0  0 -1 >      27 / 11       2.454545455  3 middle (Zalzal)          22 / 27        3.55   151
[ 14   0, -4  0 -1 >   16384 / 6875     2.383127273  3 middle (Persian)       6875 / 8192      3.03   100
[  6  -3,  0  0  0 >      64 / 27       2.37037037   3 middle (ancient)         27 / 32        2.94    90
[ -2   2,  0  0  0 >       9 / 4        2.25         3 index                     8 / 9         2.04     0
[ 17 -10,  0  0  0 >  131072 / 59049    2.219715829  3 near-index (M&A)      59049 / 65536     1.80  1177
[  2   3,  0 -2  0 >     108 / 49       2.204081633  3 near-index (Zalzal)      49 / 54        1.68  1164
[ 21  -9,  0 -2  0 > 2097152 / 964467   2.174415506  3 near-index (Persian) 964467 / 1048576   1.45  1141
[  9  -5,  0  0  0 >     512 / 243      2.106995885  3 near-index (ancient)    243 / 256       0.90  1086
[  1   0,  0  0  0 >       2 / 1        2.0          3 open = 2 little           1 / 1         0.00   996
[ -7   5,  0  0  0 >     243 / 128      1.8984375    2 ring                     64 / 81       11.10   906
[ 12  -7,  0  0  0 >    4096 / 2187     1.872885231  2 middle (M&A)           6561 / 8192     10.86   882
[ -2   4,  0  0 -1 >      81 / 44       1.840909091  2 middle (Zalzal)          22 / 27       10.57   853
[ 12   1, -4  0 -1 >   12288 / 6875     1.787345455  2 middle (Persian)       6875 / 8192     10.05   801
[  4  -2,  0  0  0 >      16 / 9        1.777777778  2 middle (ancient)         27 / 32        9.96   792
[ -4   3,  0  0  0 >      27 / 16       1.6875       2 index                     8 / 9         9.06   702
[ 15  -9,  0  0  0 >   32768 / 19683    1.664786872  2 near-index (M&A)      59049 / 65536     8.82   678
[  0   4,  0 -2  0 >      81 / 49       1.653061224  2 near-index (Zalzal)      49 / 54        8.70   666
[ 19  -8,  0 -2  0 >  524288 / 321489   1.63081163   2 near-index (Persian) 964467 / 1048576   8.47   643
[  7  -4,  0  0  0 >     128 / 81       1.580246914  2 near-index (ancient)    243 / 256       7.92   588
[ -1   1,  0  0  0 >       3 / 2        1.5          2 open = 1 little           1 / 1         7.02   498
[ -9   6,  0  0  0 >     729 / 512      1.423828125  1 ring                     64 / 81        6.12   408
[ 10  -6,  0  0  0 >    1024 / 729      1.404663923  1 middle (M&A)           6561 / 8192      5.88   384
[ -4   5,  0  0 -1 >     243 / 176      1.380681818  1 middle (Zalzal)          22 / 27        5.58   355
[ 10   2, -4  0 -1 >    9216 / 6875     1.340509091  1 middle (Persian)       6875 / 8192      5.07   303
[  2  -1,  0  0  0 >       4 / 3        1.333333333  1 middle (ancient)         27 / 32        4.98   294
[ -6   4,  0  0  0 >      81 / 64       1.265625     1 index                     8 / 9         4.08   204
[ 13  -8,  0  0  0 >    8192 / 6561     1.248590154  1 near-index (M&A)      59049 / 65536     3.84   180
[ -2   5,  0 -2  0 >     243 / 196      1.239795918  1 near-index (Zalzal)      49 / 54        3.72   168
[ 17  -7,  0 -2  0 >  131072 / 107163   1.223108722  1 near-index (Persian) 964467 / 1048576   3.49   145
[  5  -3,  0  0  0 >      32 / 27       1.185185185  1 near-index (ancient)    243 / 256       2.94    90
[ -3   2,  0  0  0 >       9 / 8        1.125        1 open                      1 / 1         2.04     0


ChartObject Arab Lute
Ancient Fretting,
Persian & Zalzal's Replacements
Mahmoud &Abdulquidir's Additions


ChartObject Arab Lute - Semitones


Relevant analyses of Arabic lute-fretting appear on p 280-284 and 515-517 of Helmholtz 1954; and descriptions appear on p 370-371, 373, 404, 425, and 430 of Partch 1974.


REFERENCES

Helmholtz, Hermann L.F. von. 1877 (1954).


The 1911 Edition Encyclopaedia, "lute"


Hefny, Mahmoud el. 1931.


Partch, Harry. 1974.


Racy, A. J. Arab Music.


Touma, Habib Hassan. 1995.


Yahoo tuning list posts, 2002.


Forster, Cris. 2004. essay on al-Farabi's lute fretting.


updated:

2000.01.24 -- thanks to Paul Erlich for pointing out errors
2004.07.10 -- entire page completely redone

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