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The reflection that intervals represented in
the series will have primary importance in the harmonic atmosphere of the
opus, because they are repeated again and again, is one of the agents leading
to an attempt to find a series containing all the eleven possible intervals
between the twelve different tones. It is clear that such a series, under
favorable circumstances, could least affect the composition’s harmonic
color, as no one interval would outweigh
another. . . . an examination
of the all-interval-series (of which there are many more than one might
imagine) opens up a large number of fascinating problems.
The puzzling question of how many series of this kind can be formed is, of course, negligible for musical purposes, and interesting only from a mathematical point of view. I have presented the problem to numerous mathematicians; so far, none of them have found a general and satisfactory solution. Nicolas Slonimsky, who has gone into the matter a little more deeply from a musical viewpoint, partly at my suggestion, has made interesting discoveries along this line.
Ernst Krenek
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