By Ernst Levy, Siegmund Levarie
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Extra info for A theory of harmony
The howling of a siren, the glissando on a string, are examples embodying that concept. * The development of mathematics offers a case in point. We see it starting from units (integers) and slowly making its way towards the continuum (calculus). If we state, "There exists an infinity of tones," we have already separated the continuum into discrete quantities. More so: we have in fact taken one tone as a starting point, and we imagine now that tone lying on an infinity of different pitches. The tone with which we started corresponds to the monas, the One, and the infinity of tones is then represented by the infinite series of integers.
We now turn to the study of the primary cadences. Two triads spring from a generator: a major triad and a minor triad. 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.
Above all, the very concept of a "contracted third'' is difficult to accept. Goethe has lucidly formulated the objections. He says: "If the third is an interval provided by nature, how can it be flatted without being destroyed? How much or how little may one flat or sharp it in order that it may no more be a major third, and yet still be a third? " Now as to the polarity theory, it must be admitted at once that while its principle is by far more satisfactory than that of the turbidity theory, yet Page 15 the difficulties are no less formidable.