Rothenberg Propriety - Definition of Propriety

Definition of Propriety

Rothenberg defined propriety in a very general context; however for nearly all purposes it suffices to consider what in musical contexts is often called a periodic scale, though in fact these correspond to what mathematicians call a quasiperiodic function. These are scales which repeat at a certain fixed interval higher each note in a certain finite set of notes. The fixed interval is typically an octave, and so the scale consists of all notes belonging to a finite number of pitch classes. If β_i denotes a scale element for each integer i, then β_i+℘ = β_i + Ω, where Ω is typically an octave of 1200 cents, though it could be any fixed amount of cents; and ℘ is the number of scale elements in the Ω period, which is sometimes termed the size of the scale.

For any i one can consider the set of all differences by i steps between scale elements class(i) = {β_n+i − β_n}. We may in the usual way extend the ordering on the elements of a set to the sets themselves, saying A < B if and only if for every a ∈ A and b ∈ B we have a < b. Then a scale is strictly proper if i < j implies class(i) < class(j). It is proper if i ≤ j implies class(i) ≤ class(j). Strict propriety implies propriety but a proper scale need not be strictly proper; an example is the diatonic scale in equal temperament, where the tritone interval belongs both to the class of the fourth (as an augmented fourth) and to the class of the fifth (as a diminished fifth). Strict propriety is the same as coherence in the sense of Balzano.