. . . when he sees that, to simplify procedure and analysis, sounds are designated by figures, and intervals by numbers, he will boast of having tracked down our mathematical vices.

What would the average person say if he were familiar with the formulae of our good old thorough bass, studied by all the recognized masters of imagination and emotional expression—Bach, Mozart, Schubert, et al.? . . .

The mere fact that something can be expressed in figures does not prove that it is “mathematical” by nature. Let us suppose that we have set up a tabulation in which we express, in measurements of twelve inches to the foot, the heights of various mountains in a range. Now, the discovery or establishment of certain relationships between the figures will not explain why the peaks are famous for their beauty. But it would be foolish to contend that measurement of the Alps robs them of their splendor.

Average persons will readily agree that music has, indeed, some connection with mathematics through physics. Probably we all have heard, at one time or another, about Pythagoras’s “music of the spheres” and about vibration ratios and overtones. It is easy to see that the length of strings or pipes used in the production of tones is related to the pitch of the tones thus produced. Certainly this is very interesting, and something which instrument makers need to know; but one can hardly call the knowledge a decisive factor in the art of composition.

When we learn that the special nature of the octave interval is the result of the vibration ratio 1:2 of the two tones which constitute the octave—a very simple relationship—we are quite impressed; but the ear needs no such scientific instruction to find out that the octave is the purest form of consonance. When I am composing, it is enough for me to bear in mind what my ear tells me. Yet, how I shall use the octave, whether I prefer it or even avoid it, must depend on my artistic intention and not on the interval’s physico-mathematical nature.

Ernst Krenek