to Commas and Limmas

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,That is all there is to it, really. Have I ever said it was difficult?!thus

1 comma = 24 cents

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