Proto-tonality

A “tertial element” is a primary interval (y, z) such that z is a generic “klang generator” e of a non-empty harmonic tone system (Sect. 8.1). A “proto-tonal system” is a harmonic tone system such that the quintic element (c, d) satisfies c is as close as possible to alog2(3/2); moreover, assuming a non-empty system, there exists a tertial element (y, z) such that y is as close as possible to alog2(5/4). It is proven that a proto-tonal system satisfies a = 12, b = 7 = c, d = 4, and (if the system is not empty) e = 2. Section 8.2 revisits the theory of categorical equal temperament from a historical and historical-theoretical perspective, now that a = 12 has been established. Finally, Sect. 8.3 considers an alternative to the theory of proto-tonality, based on Clough and Douthett’s (1991) theory of “maximally even sets.” It is shown that the alternative theory is weaker than the proposed theory in terms of explanatory power.