13.21 - Principle of Proportion - increases and decreases are in inverse or direct proportion to rate of change (delta). The ratio of two numbers to each other. Proportion is in three kinds: (1) multiplex. (2) Superparticularis. (3) Superpartiens. Proportio multiplex is when the larger number contains the smaller so many times without a remainder, as 2:1 (dupla), 3:1 (tripla), 4:1 (quadrupla). Proportio superparticularis is when the larger number exceeds the smaller by one only as 3:2 (sesquialtera), 4:3 (sesquitertia), 5:4 (sesquiquarta). Proportio superpartiens is when the larger number exceeds the smaller by more than one, as 5:3 (superbipartienstertias), 7:4 (supertripartiensquartas), 9:5 (superquadripartiensquintas).
Thus, it will be understood, that instead of giving simply the ratio between two numbers, early writers on arithmetic and geometry, as well as music, coined a single word to express that ratio; for example, 17:5 was said to be Triplasuperbipartiensquintas, i.e., that the larger number contained the smaller number three times (tripla) with two remainder (bipariens). Again, Triplasupertripartiensquartas proportio, signified that the larger contained the smaller three times and three over, as 15:4, 27:8, etc., the last part of the compound word always pointing out the smaller of the numbers compared, or an exact multiple of it. Lastly, the addition of sub showed that the smaller number was compared to the larger, e.g., 4:15 would be called Subtriplasupertripartiensquartas proportio. This system of proportion was used not only with reference to intervals but also to the comparative length of notes (time). [A Dictionary of Musical Terms]