When the figures deduced from time are mixed in a calculation with those deduced from space to form part of an unforeseen mathematical species, we are left without the remotest possibility of imagining what really happens. Yet one would expect to imagine this, since we have to deal, not with abstract calculations, but with calculations which are supposed to tell us something about the universe. Still, the result of abstractly constructed calculations can be retransformed into images of the actual relationships of natural phenomena, and later tested in experiments. Can it be that, viewed objectively, the universe has “been designed by a pure mathematician” (as Sir James Jeans indicates in The Mysterious Universe), so that a mathematical interpretation would, in a certain measure, correspond to the world of facts?

However that may be, the twelve-tone technique, too, seems to be based on a quite diffuse and abstract method of thinking, miles away from the nature of music. When, for instance, I decide to build a movement in a string quartet, using the inversions of all the series found in the preceding movement, I may appear to have embarked on an absurd mechanistic project, to be guilty of pretentious caprice. But suppose my mind is “structured twelve-tonelike,” analogous to Sir James’s “mathematically designed” universe, or suppose that thinking in series relations is a natural habit of mine; one could then well imagine that this “abstract” method would aid me in getting results susceptible of being smoothly retransformed into the “nature of music.”

Ernst Krenek