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Physicists and mathematicians are far in advance of musicians in realizing that their respective sciences do not serve to establish a concept of the universe conforming to an objectively existent nature. They are fully cognizant of the fact that, conversely, their task is to make an existing conception of the universe conform to the largest possible number of observations demonstrable by scientific experiments. . . .

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Physical and, above all, mathematical theories owe their independence from so-called reality to their logical coherence. . . . All the facts and operations of ordinary or Euclidean geometry can be reduced to a series of first principles called axioms; as, for instance, a straight line is the shortest distance between two points, or, only one parallel can be drawn to a straight line through any given point outside it, and the like. . . .

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. . . Musical systems, languages of sound, or sound languages as we call them in our terminology, have not been created by nature or by some mystical Supreme Being, but have been produced by man to render music possible within a certain sphere. The fundamental facts of a sound language may have many formal points of similarity with the axiomatic system. In his Foundations of Geometry, David Hilbert introduced a discussion of Euclidean axioms with the following monumental propositions:

“We imagine three distinct categories of things. The things in the first category we shall call points and designate by A, B, C . . . ; the things in the second category we shall call straight lines and designate by a, b, c . . . ; the things in the third category we shall call planes and designate by α, β, γ. . . . We conceive of these points, straight lines, and planes as having certain mutual relations which we indicate by means of such words as ‘are situated,’ ‘between,’ ‘parallel,’ ‘congruent,’ ‘continuous.’ The exact, and for mathematical purposes complete, description of these relationships is achieved through the axioms of geometry.”

The importance of these propositions lies in the fact that Hilbert did not write, “There are points, straight lines, and planes which one can handle thus and so if one has explored their natural qualities, which proceedings can be called geometry.” Instead, he characterizes the points, straight lines, and planes as thought-concepts, as products of the mind; and whatever we can do with these thought-concepts depends on the definitions which we impose on them.

One can very well imagine a general theoretical interpretation of music which would start with the following proposition:

“We imagine three different categories of things. The things in the first category we shall call tones, the things in the second category chords, and the things in the third category melodies. We conceive of the tones, chords, and melodies as having certain mutual relationships which we indicate by means of such words as ‘high,’ ‘low,’ ‘interval,’ ‘consonance,’ ‘dissonance,’ ‘contrary motion,’ ‘inversion,’ and so forth.”

These propositions would form the basis of a musical theory including all sound languages of whatever nature. As the study of axioms eliminates the idea that axioms are something absolute, conceiving them instead as free propositions of the human mind, just so would this musical theory free us from the concept of major-minor tonality (or any other systematized form of musical material) as an irrevocable law of nature. The theory would make way for the concept that the facts of a sound language, too, are products of the human mind, created freely by musical thought.

Ernst Krenek