Communicating the Primary Intervals

A “semi-efficient” tone system consists of a composite tone system and a set of “primary” intervals generated by a privileged note interval (c, d), the “quintic element” (Sect. 5.1). In such a system, the note-interval transmission function, restricted to the primary intervals, is assumed to be one-to-one. A semi-efficient system is efficient if, in addition, the difference between the pitch images of two primary intervals (under the transmission function) is not arbitrarily small. It is proven that efficient tone systems are not type-3 systems (of which the complete “Pythagorean” system is a familiar example); on the other hand, equal-tempered systems, which are special types of naturally oriented type-1 or 2 tone systems, are efficient. An efficient tone system is “coherent” if a certain algorithm, by which a primary interval may easily be computed from its transmitted image, exists (Sect. 5.2). It is proven that coherent tone systems satisfy cb − ad = ±1, where (a, b) is the cognitive octave and (c, d) is the quintic element. Finally (Sect. 5.3), since efficient tone systems are equal tempered and not dense, one may relax the unrealistic assumption of absolutely accurate note transmission. By the theory of categorical equal temperament deviations from strict ET of up to one-half of one equal-tempered increment are allowed, in either direction.