“It is striking that ar-râst does not have a just fifth!”, I pondered almost 21 years ago. That was about an Arabic music scale, for which I had reconstructed integer ratios for the note frequencies. At the fifth, one would expect nearly exactly 700 cents, which is seven equidistant semitone steps, but it turned out to be only 680.45. However could that happen?
Just yesterday I suddenly saw the solution: by changing 80 to 81, or 13⅓ to 13½, it does work out! How could I not have seen that before? Because I focused too much on the similarity with other scales, al-bayâtî en al-sîkâh, which can be interpreted as consisting of the same intervals, only shifted.
The change of 80 to 81 causes the intervals 10:9 and 9:8, between fourth and fifth, fifth and sixth, to change places. Below I repeat the table I had then, followed by one with the change incorporated.
Sullam maqâmât ar-râst (سلّم مقامات الراست)
| 1 | 9 | 54 | 1:1 | - | 0.00 |
| 2 | 10 | 60 | 10:9 | 10:9 | 182.40 |
| 3 | 11 | 66 | 11:9 | 11:10 | 347.41 |
| 4 | 12 | 72 | 4:3 | 12:11 | 498.04 |
| 5 | 13⅓ | 80 | 40:27 | 10:9 | 680.45 |
| 6 | 15 | 90 | 5:3 | 9:8 | 884.36 |
| 7 | 16½ | 99 | 11:6 | 11:10 | 1049.36 |
| 8 | 18 | 108 | 2:1 | 12:11 | 1200.00 |
Now instead it becomes:
| 1 | 9 | 54 | 1:1 | - | 0.00 |
| 2 | 10 | 60 | 10:9 | 10:9 | 182.40 |
| 3 | 11 | 66 | 11:9 | 11:10 | 347.41 |
| 4 | 12 | 72 | 4:3 | 12:11 | 498.04 |
| 5 | 13½ | 81 | 3:2 | 9:8 | 701.96 |
| 6 | 15 | 90 | 5:3 | 10:9 | 884.36 |
| 7 | 16½ | 99 | 11:6 | 11:10 | 1049.36 |
| 8 | 18 | 108 | 2:1 | 12:11 | 1200.00 |
In my then article I only analysed the ascending scale. The source I used specifies the descending scale as slightly different, the penultimate, seventh step then having a full flat, not a half flat. In Dutch note notation, the note is beh, not bes. That too, in hindsight, is easy to fit in into my scheme: just change 99 to 96. The 12:10 or 6:5 minor third, which in ascending mode consists of 11:10 and 12:11, in the descending scale is split into 9:8 and 16:15.
| 1 | 9 | 54 | 1:1 | - | 0.00 |
| 2 | 10 | 60 | 10:9 | 10:9 | 182.40 |
| 3 | 11 | 66 | 11:9 | 11:10 | 347.41 |
| 4 | 12 | 72 | 4:3 | 12:11 | 498.04 |
| 5 | 13½ | 81 | 3:2 | 9:8 | 701.96 |
| 6 | 15 | 90 | 5:3 | 10:9 | 884.36 |
| 7 | 16 | 96 | 16:9 | 16:15 | 996.09 |
| 8 | 18 | 108 | 2:1 | 9:8 | 1200.00 |
The as-sîkâh scale too, in which I missed a nearly 500 cents just fourth, can be easily fixed: 80 become 81, and 90 will be 88. This is the old situation:
Sullam maqâmât as-sîkâh (سلّم مقامات السيكاه)
| 1 | 11 | 66 | 1:1 | - | 0.00 |
| 2 | 12 | 72 | 12:11 | 12:11 | 150.64 |
| 3 | 13⅓ | 80 | 40:33 | 10:9 | 333.04 |
| 4 | 15 | 90 | 15:11 | 9:8 | 536.95 |
| 5 | 16½ | 99 | 3:2 | 11:10 | 701.96 |
| 6 | 18 | 108 | 18:11 | 12:11 | 852.59 |
| 7 | 20 | 120 | 20:11 | 10:9 | 1035.00 |
| 8 | 22 | 132 | 2:1 | 11:10 | 1200.00 |
And this is the new:
| 1 | 11 | 66 | 1:1 | - | 0.00 |
| 2 | 12 | 72 | 12:11 | 12:11 | 150.64 |
| 3 | 13½ | 81 | 27:22 | 9:8 | 354.55 |
| 4 | 14⅔ | 88 | 4:3 | 88:81 | 498.05 |
| 5 | 16½ | 99 | 3:2 | 9:8 | 701.96 |
| 6 | 18 | 108 | 18:11 | 12:11 | 852.59 |
| 7 | 20 | 120 | 20:11 | 10:9 | 1035.00 |
| 8 | 22 | 132 | 2:1 | 11:10 | 1200.00 |
These solutions occurred to me while for a
new articles series,
I researched how Lilypond (nice name, by the way) supports Turkish music.
The include file called “turkish-makam.ly” also sports a “Rast”
scale, which however is very similar to the ordinary Western major scale,
of course with a fifth close to 700 cents.
Wikipedia too ventures into equalling Rast to the Western major scale, but comments: “In common Arabic and Turkish practice, the third note e𝄳 and the seventh note b𝄳 in Rast are even lower than in this theory, [...]”. I think I know why that is: it follows from the sequence of relatively small integers to define the notes’ frequency ratios, as specified in my tables. Nine, ten, eleven. That makes the Rast third 347.408 cents (ratio 11:9), not 386.314 cents (ratio 5:4), and not 384.906 (17 commas of 1/53 of an octave each, where 17 is 8 plus 9).
The name and scale Rast, by the way, doesn’t have an Arabic origin, nor a Turkish one, but rather is Persian. That language, other than the other two, is related to English, and the cognate of ‘rast’ is ‘right’. The right, simple, pure, just scale. Perhaps that is the meaning of the word in this context.