| Safi
al-Dîn starts the third discourse with the processes of the addition,
division and subtraction of the intervals. |
| |
| If
two intervals are equal to each other the numerator and the denominator
are multiplied by each other while being added. The biggest (uzma) and the
smallest (suðra) number of the interval are obtained. Then in order
to find the medium (vasat) number, the numerator of the one is multiplied
by the denominator of the other as these two are of equal ratios. If we
add the two tetrachord, it is 4x4=16 which is "uzma" side. 3x3=9
number is "sugra". 4x3=12 is medium number. So, three numbers
are formed, which are 16, 12, and 9. As it is seen, 16/12=4/3. |
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| If
we want to add a third tetrachord to these we should do the process of 16/9x4/3.
Depending on this we get 64, 48, 36, 27 numbers whose ratios to each other
subsequently are 4/3. |
| |
| If
one of the two intervals is more and the other is small, we multiply the
numerator and the denominator by each other. Let's add 4/3 and 9/8. The
result is 4/3 x9/8=36/24 and these are the big and the small sides. Later
on, if we want to leave the tetrachord at the beginning side or in Safi
al-Dîn's words "if we want to do the addition in the high in
pitch (tîz) side", we multiply the denominator of the tetrachord
by the numerator of the other ratio. That is to say, it is 3x9=27 which
is one of the vasat numbers. In other words, in the ratios between the three
numbers 36, 27, 24, pentachord comes at the beginning side and tanînî
comes after that. 36/27=4/3, 27/24=4/3. |
| |
| If
we want to do the addition at the low in pitch (pest) side, that is, if
we want the tanînî we want to add to leave at the beginning,
then we multiply the denominator of the tanînî by the numerator
of the tetrachord. In this case medium number is 32. In other words, as
it is seen between the numbers of 36, 32, 24, 36/32=9/8 is at the beginning
side, 32/24=4/3 is at the end. |