from to , three step intervals from to . Rothenberg hypothesizes that proper scales provide a point or frame of reference which aids perception ("stable gestalt") and that improper scales contradictions require a drone or ostinato to provide a point of reference. An example of an improper scale is the Japanese Hirajōshi scale. Its steps in semitones are 2, 1, 4, 1, 4. The single step intervals vary from the semitone from G to A to the major third from A to C. Two step intervals vary from the minor third from C to E and the tritone, from A to D. There the minor third as a two step interval is smaller than the major third which occurs as a one step interval, creating contradiction ("a contradiction occurs...when the ordering of two specific intervals is the opposite of the ordering of their corresponding generic intervals."). Mathematical 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 βi denotes a scale element for each integer i, then βi+℘ = βi + 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) = {βn+i − βn}. 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