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By Ernst Levy, Siegmund Levarie
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Extra resources for A theory of harmony
For the sake of convenience, we shall in the following always refer to Å, except where is expressly indicated. * We obtained the primary cadences by observing the latent dynamism of the triad. Dynamically considered, no chord, no cadence is every completely final. Now relations between tones or chords are to be viewed not only dynamically but also hierarchically. " Instead of dynamic trends, we shall consider fixed relations. The two concepts are complementary: dynamism without hierarchy is boundless, hierarchy without dynamism is empty.
The results of number operations (division and multiplication) applied to the string are independent of the existence or nonexistence of parallel natural phenomena. * The two series are reciprocal. Musically, reciprocation means reproducing an interval in the opposite directionan operation clearly distinguished from inversion, which is the reproduction of a tone in the opposite direction. In inversion the interval changes, but the tones remain. In reciprocation, the interval remains, but one of the tones changes.
The system turns to a vertical position, and the tones at once assume the quality of pitch. At the same time, the two triads assume the qualities of major and minor as we know them. Looking at the major triad, we find that nothing has been changed through the influence of gravity. The generator is also the fundamental of the chord. Turning to the minor triad, we notice that generator and fundamental are divorced. Absolute conception and telluric adaptation (symbol +) are in contradiction to each other.