Receiving Notes

Following a lead found in Bartók’s Music for String Instruments, Percussion, and Celesta (Sect. 6.1), a “note-reception system” is constructed in Sect. 6.2, as follows. Relative to a referential note selected from a set of notes termed “core,” a received pitch is paired with the note a primary interval away. Section 6.3 then explores the idea of self-communicating the core in a potentially endless feedback cycle, such that the “received message” of one cycle becomes the “transmitted message” of the next. “Stability” is reached if at some iteration and onwards the received message always equals the original transmitted message, namely the core. It is proven that if stability is reached at all, it is reached at the very first iteration. In Sect. 6.4 “diatonic note” is defined as a transmitted note the reflexive image of which (at some iteration) is constant relative to every core element. It is shown that every note (at some iteration) is diatonic if and only if the system is stable, and thus the core itself is a set of diatonic notes. It is shown further that a diatonic core consists of exactly ⌊a/2⌋ + 1 elements that may be ordered quintically. Finally, Sect. 6.5 revisits interesting properties of diatonic systems as previously studied by Balzano, Agmon, Clough and Douthett, and Carey and Clampitt.