The only segment of our conventional body of theoretical musical knowledge which the dodecaphonists have deigned to admit and which, in fact, alone makes their speculations possible, is the twelve-tone tempered scale. We have already been told of this scale’s weakness: because of its basic impurity it can be used only as a supplementary regulative to a tone system containing natural intervals—at least, so long as we want to save our music from total instrumental mechanization and have human voices participate in its execution. True, some kind of a restricted technique of composition can be developed on a foundation of compromise scales and arbitrary working rules, but doubtless the general result will always be one similar to the kind of poetry that is created by pouring written words out of a tumbler without calling in grammar and syntax. A higher tonal organization is not attempted and cannot be achieved, especially if one permits the technical working rules to slip off into the aforementioned set of supplementary statutes which are nothing but stylistic whims and, as such, not subject to any controlling power of general validity. Of course, there are those superrefined prophets who proudly claim that they can, by the rules of this stylistic method, write pieces in C major, which seems to be a procedure as direct as leaving one’s house in New England through the front door and entering the back door by a little detour via Chicago.

Twelve-tone operations are not the only nightmares that haunt the composing zealot who wants to be up to date. Are there not city sky lines whose ragged contours demand to be reproduced in melodic lines? Some other composers invent, with the aid of addition, subtraction, and other numerical operations, ways of combining tones mechanically; and finally, there are always colors as organizing agents. It is easy to recognize the underlying principle in all these and similar methods: it is a simple equation between a given number of tones and anything else that consists of an equal number of constituent parts. We could go on counting such methods of tonal equations, but only to enter a sphere in which there is almost nothing that could not be brought into direct equational relationship with harmony and melody: fever curves, cooking recipes, railroad timetables (the music resulting from them may be rather monotonous, though), catalogues of country fairs, the depth of the ocean between Halifax and Ireland, and so on.

If the inventors of such systems had looked into music history, they would have found that their methods are by no means as modern as they think. Moreover, their predecessors’ lack of lasting success should have made them suspicious. The earliest attempts at composing by a method of this kind can be found in several treatises of the eleventh and twelfth centuries, in which an equation of the five vowels of the Latin language and five successive tones of a church mode is used. The melodies thus constructed must, even to the inventor of this system, have sounded trivial enough, because we see an additional, transposed equation recommended to heighten the poor melodic effect somewhat. (I was never quite sure that this invention was not contrived with tongue in cheek.) Obviously this method did not appeal to contemporary composers, since in spite of the medieval theorists’ fondness for plagiarizing each other, it did not reappear in later treatises.

Other equations were devised with the spots of dice, a pastime very popular in Mozart’s time—in fact, some of the methods of composing with this recipe are published under Mozart’s or Haydn’s name, one even in Boston, Massachusetts.

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. . . with this [twelve-tone] method no pieces can be produced which could fill big spaces with broad symphonic colors, or which could satisfy many people’s demands for simplicity, directness, and personal sympathy.

Paul Hindemith