By Ernst Levy, Siegmund Levarie
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Nor were the senaric values all admitted at once. First to gain recognition was the fifth; much later only came the third. Perhaps the first scale, or at least the first fixed tones of a scale, resulted from flanking a generator by its two reciprocal fifths: Next came the tetratonic scale. The fifth being considered a unit, adding another fifth above resulted in this: Later, a symmetrical arrangement of four fifths around the generator gave the pentatonic scale: Finally, by adding two more reciprocal fifths, the Pythagorean heptatonic scale was obtained: Such may have been, roughly, the genesis of the diatonic scale.
A theory based on the natural phenomenon of the overtones cannot explain the break between the consonant senarius and the following, dissonant ratios. Second difficulty: the minor triad above the fundamental does not exist in the overtones. Moreover, if the viewpoint of the consonance of major as a "con-sonance" with the overtones be maintained, it can be shown that through clashes between the minor third and certain overtones we should expect the minor chord to be an outspoken dissonance rather that a disturbed consonance.
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. A subdominant is the reciprocation of a dominant, and vice versa; but the inversion of a dominant is still a dominant, though in position of a fourth instead of a fifth. The so-called senarius, comprising the first six ratios, forms two mutually reciprocal triads, one major, the other minor, about which more will be said in the next chapter.
A theory of harmony by Ernst Levy, Siegmund Levarie