By Ernst Levy, Siegmund Levarie
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Extra info for A theory of harmony
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. ). The corresponding numbers are 2, 3, 5. ). Numbers multiplied or divided by 2 or its powers represent octave-identical tones.
Hence he regards the minor triad as a modified, "turbid" form of the major triad; and the minor third, characteristic of the minor triad, as a contracted major third. Before proceeding to give a critical account of both theories, we have to elucidate a point not mentioned in the foregoing statements. If one admits that the triad is a consonant unit, then the question arises as to why the senarius forms such a unit, and why a break occurs after the sixth ratio (or, more correctly as we shall see later, after the eighth ratio).
The following diagram will illustrate the relationship between the two concept pairs. Let me offer the following explanation of the diagram. The realm of dissonance is illimited. The radius of the outer circle should be imagined as infinite, r= ¥. In he midst of that immense mass of dissonance lies the one consonant chord, the triad, not unlike a crystal within a mountain. Being the immediate product of the senarius, it occupies a central position. The two circles, then, symbolize the natural consonances and disonances.
A theory of harmony by Ernst Levy, Siegmund Levarie