IV. 3) The Arrangement of

Tetrachords in

two Octaves with the

Intervals at Kinds

Safi al-Dîn arranged "munfasýl al-ahad" at the "lower octave (zu'l-kull al-athkal)" with the intervals of "kawî kinds" up to now. But now, he will perform the same process in two octaves with again the arrangement of munfasýl al-ahad and with the types of kawî kinds mentioned above. These are 12. Let's take the first table. The others will go on at the same order. He does the same arrangement of munfasýl al-ahad in the lower octave in two octaves.

The ratios are:

I. With the intervals of "first non-conjunct kind/gayr-i muttasil ".

double octave

img0000000629

II. With the intervals of "second non-conjunct";

9/8x64/59x59/54x9/8x64/59x59/54x9/8x9/8x64/59x59/54x9/8x64/59x59/54x9/8=4/1=4.

III. With the intervals of "third non-conjunct ";

10/9x12/11x 11/10x10/9x12/11x11/10x9/8x10/9x12/11x11/10x10/9x12/11x11/10x9/8=4.

IV. With the intervals of "first conjunct/muttasil";

8/7x9/8x28/27x8/7x 9/8x28/27x9/8x8/7x9/8x28/27x8/7x9/8x28/27x9/8=4.

V. With the intervals of "second conjunct";

9/8x10/9x16/15x9/8x10/9x16/15x9/8x9/8x10/9x16/15x9/8x10/9x16/15x9/8=4.

VI. With the intervals of "third conjunct";

10/9x11/10x 12/11x10/9x11/10x12/11x9/8x10/9x11/10x12/11x10/9x11/10x12/11x9/8=4.

VII. With the intervals of "first doubling/zü't-tadî´f ";

VIII. With the intervals of "second doubling";

9/8x9/8x256/243x9/8x9/8x256/243x9/8x9/8x9/8x256/243x9/8x9/8x256/243x9/8=4.

IX. With the intervals of "third doubling ";

10/9x 10/9x27/25x10/9x10/9x27/25x9/8x10/9x10/9x27/25x10/9x10/9x27/25x9/8=4.

X. With the intervals of "first disjunct /munfasil";

8/7x10/9x21/20x8/7x 10/9x21/20x9/8x8/7x10/9x21/20x8/7x10/9x21/20x9/8=4.

XI. With the intervals of "second disjunct";

9/8x11/10x320/297x9/8x11/10x320/297x9/8x9/8x11/10x320/297x9/8x11/10x320/297x 9/8=4.

XII. With the intervals of "third disjunct";

10/9x12/11x11/10x10/9x12/11x11/10x9/8x10/9x12/11x11/10x10/9x12/11x11/10x9/8=4.

Safi al-Dîn indicates that when examined the most consonant one and the best one to the human nature of these scales are two scales of "second doubling kind/zü't-tad'îf al-thanî" (9/8x9/8x256/243=4/3 and 10/9x10/9x27/25=4/3) "second conjunct kind/muttasýl; (9/8x10/9x16/15=4/3).