To appreciate the dimensions of the problem, a closer look at the tuning system would be needed. Let us start with a string that is tuned to, say, middle C when it is free. Stopped at its exact middle, it would produce C', one octave above its fundamental. Stopped at its third, the remaining two-thirds would give G, its perfect fifth (dominant), and at its exact quarter, we would get its perfect fourth (subdominant), namely F. At its fifth, we would get the natural third, E. But -here just comes the juicy part- neither the sixth, the seventh nor the eighth would get us anywhere near the second of the fundamental the way it used to sound from our already established experiences with the woodwind; only the ninth starts to give the right acoustic character of our long-sought D. Now, where should we place A and B? Symmetry being the order of the day, A was automatically located at the same distance from G as D from C, a ninth away. The beauty of the symmetry was revealed on closer inspection; the distance from F to G is exactly a ninth, too! Then the same distance from D to E was applied from A to B. This latter distance is rather interesting; it was a tenth, and the difference between a ninth and a tenth is 1/81 (=1/9 × 1/9), i.e. a "comma". The distance thus remaining from B to C' is indeed the same as from E to F.
That magic symmetry was thus complete: both the C-D-E-F and the G-A-B-C' groups had the same set of distances, namely 1/9 - 1/10 - 1/15. The 1/9 between F and G came to be called "interval of resonance" since the second (upper) half (=tetrachord) echoed the first (lower) one. [In Arabic, a tetrachord (=four strings, in Greek) was sometimes called "wing"; an octave is thus much like a bird.] Now, if we were a 1/9 away from D, and not a 1/10, i.e. a comma away from natural E, F would still be a 1/21 away; another magic distance, later to be called "limma". Further calculations revealed that a 1/10 is evenly divisible into two limmas, meaning that a 1/9 (a whole tone in our modern parlance) is made of two limmas and one comma. This is precisely why F# and Gb were not considered to be enharmonic: F# is one limma above F, Gb is one limma below G, and there exists a comma between F# and Gb.
The next step in perfecting the system was to enable a lutist to determine where to stop a given string to get a given note as easily, and accurately, as possible. Enter "drawn frets". The system I am going to cite here was mentioned in old, authentic manuscripts, but my direct source was a book written decades ago by a Syrian researcher with the name of Mikhaïl Khalilallah Werdi.
To finish off, let me explain how one can convert all this commas/limmas soup into modern-age cents. As you can see from the above illustration, an octave, say from C to C', contains twelve limmas and five commas. Now, as mentioned before, a limma equals three commas and three-quarters of a comma; thus, twelve limmas equal forty-five commas. So, an octave is really made up of fifty commas in all. Now, bearing in mind that the Equally Tempered Scale stipulates that all twelve semitones of an octave are created equal, each having one hundred cents, it follows that an octave must be made of twelve hundred cents. It's simple, then:
1 octave = 50 commas = 1200 cents, thusThat is all there is to it, really. Have I ever said it was difficult?!
1 comma = 24 cents
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