Fundamentals of Arithmetic; Fundamentals of Music

But if you remove the square and the triangle, and all geometry were consumed, the names of three, four, and other numbers will not perish. Conversely, when I name any geometric form, the name of numbers is simultaneously implied; when I name numbers, I have not yet named any geometric form. That the power of numbers is even more prior to music can be proven most strongly from this: not only are those things prior by nature which exist of themselves compared to those which are referred to something else, but even that very musical modulation is annotated by the names of numbers, and the same thing can happen in this that was predicted regarding geometry. For the diatessaron fourth, diapente fifth, and diapason octave are named from the numbers that precede them. The proportion of the sounds themselves against one another is found in these numbers alone and no others. For the sound that is a diapason octave consonance is collected by the proportion of a double number; the modulation that is a diatessaron fourth is composed by an epitrita ratio of 4:3 comparison; the consonance they call diapente fifth is joined by a hemiolia ratio of 3:2 mean; what is an epogdous ratio of 9:8 in numbers is the same as a tone in music. And, not to labor in pursuing each detail, the subsequent parts of this work will demonstrate without any doubt how much more prior arithmetic is. Indeed, it precedes spherical science and astronomy as much as the two remaining disciplines arithmetic and music precede this third one geometry by nature. For in astronomy there are circles, a sphere, a center, parallel circles, and a central axis, all of which are the concerns of the geometric discipline. Therefore, it is also possible to show from this that the power of geometry is older, because all motion is after rest, and a standing position is always prior by nature, while astronomy is the doctrine of mobile things, and geometry is the doctrine of immobile things; or because the very motion of the stars is celebrated by harmonic modulations.