CLAUDIUS PTOLEMY:

TETRABIBLOS

OR THE QUADRIPARTITE MATHEMATICAL TREATISE

13. Of the Aspects of the Signs.

Of the parts of the zodiac those first are familiar one to another which are in aspect. These are the ones which are in opposition, enclosing two right angles, six signs, and 180 degrees; those which are in trine, enclosing one and one-third right angles, four signs, and 120 degrees; those which are said to be in quartile, enclosing one right angle, three signs, and 90 degrees, and finally those that occupy the sextile position, enclosing two-thirds of a right angle, two signs, and 60 degrees.
We may learn from the following why only these intervals have been taken into consideration. The explanation of opposition is immediately obvious, because it causes the signs to meet on one straight line. But if we take the two fractions and the two superparticulars most important in music, and if the fractions one-half and one-third be applied to opposition, composed of two right angles, the half makes the quartile and the third the sextile and trine. Of the superparticulars, if the sesquialter and sesquitertian be applied to the quartile interval of one right angle, which lies between them, the sesquialter makes the ratio of the quartile to the sextile and the sesquitertian that of trine to quartile. Of these aspects trine and sextile are called harmonious because they ara composed of signs of the same kind, either entirely of feminine or entirely of masculine signs; while quartile and opposition are disharmonious because they are composed of signs of opposite kinds.



Ptolemy's Tetrabiblos

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