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By Ernst Levy, Siegmund Levarie

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Extra resources for A theory of harmony

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We are now ready for a series of experiments whose purpose is to show what happens to a tone when the length of the string is varied. Page 5 It is at once apparent that we need a guiding principle, for the "glissando" effect produced by a continuous shortening of the string does not lead us beyond the not very revealing discovery that pitch rises with the shortening and falls with the lengthening of the string. A glissando is a pitch motion, a continual becoming. A tone, however, is a being, an individual.

The result is the following series: We write this series on graph paper, starting from 1/1 c and proceeding downward. It will be noticed that the term overtones has not been mentioned. Our first, ascending series is, of course, identical with that produced by the natural phenomenon of the overtones, but we disregard this fact to which we assign the meaning of a coincidence. The results of number operations (division and multiplication) applied to the string are independent of the existence or nonexistence of parallel natural phenomena.

The unfolding of a tone in the two chords forms a stable whole: A unilateral realization will disrupt the balance and make the chord tend toward its complement. For instance, C major will tend toward F minor, and vice versa. Hence we may say: a major triad tends to become dominant; a minor triad, subdominant. The oscillation thus produced is, theo- Page 24 retically, perpetual. The original trends can never be completely eliminated; no cadence is ever absolutely closed: A stoppage of tile "perpetual motion" may be effected by changing the mode of the intended last chord, thus introducing a sudden countercurrent: A still more definite cadence is obtained by a combination of such a "counterblow" and an oscillation to the opposite side of the tonic: Passing to the opposite side by a skip, the middle term being omitted, results in the classic cadence: Probing a little further into tonality (tonality: the relationship of tones and chords to a central tone or chord called tonic) we add the dominants of Page 25 the dominants.

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