Communicating Pitches and Transmitting Notes

Section 3.1 focuses on notes and note intervals, defined as integer pairs, and on the octave relationship, defined in terms of a privileged note interval (a, b), the “cognitive octave.” Section 3.2 focuses on pairs of note intervals and a sense in which they may be said to “generate” all note intervals. In Sect. 3.3 “pitch” and “pitch interval” are defined as real numbers. The “signal” by which a pitch is communicated is a periodic wave relative to a frequency f that relates exponentially to the pitch. A theorem is proven, by which a “pitch-communication system” is equivalent to a logarithmic transformation of f. Section 3.4 establishes a hierarchy of privileged pitch intervals, first of which is the psychoacoustical octave log2. A “Phi-centered” pitch-communication system is defined, where a transmitter-privileged pitch φ is assumed privileged for the receiver as well. The system is either “absolute,” “relative,” or “reflexive,” the latter modeling a person engaged in self-communication. Finally, Sect. 3.5 combines the note system of 3.1 with the pitch-communication system of 3.4. “Transmission functions” from notes to pitches and from note intervals to pitch intervals, are posited. It is proven that the transmission function for intervals is a homomorphism from “note-interval space” into “pitch-interval space.” It is proven further that a “composite tone system,” where the cognitive octave (a, b) is transmitted as the psychoacoustical octave, falls into one of three mutually exclusive types. An example of a “type-1” system is the usual tone system, (a, b) = (12, 7), under 12-tone equal temperament; the usual tone system under Pythagorean intonation is a “type-3” system.