Rothenberg propriety

Concept in diatonic set theory

In diatonic set theory, Rothenberg propriety is an important concept, lack of contradiction and ambiguity, in the general theory of musical scales which was introduced by David Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context by Gerald Balzano, who termed it coherence. "Rothenberg calls a scale 'strictly proper' if it possesses a generic ordering, 'proper' if it admits ambiguities but no contradictions, and 'improper' if it admits contradictions." A scale is strictly proper if all two step intervals are larger than any one step interval, all three step intervals are larger than any two step interval and so on. For instance with the diatonic scale, the one step intervals are the semitone (1) and tone (2), the two step intervals are the minor (3) and major (4) third, the three step intervals are the fourth (5) and tritone (6), the four step intervals are the fifth (7) and tritone (6), the five step intervals are the minor (8) and major (9) sixth, and the six step intervals are the minor (t) and major (e) seventh. So it's not strictly proper because the three step intervals and the four step intervals share an interval size (the tritone), causing ambiguity ("two [specific] intervals, that sound the same, map onto different codes [general intervals]"). Such a scale is just called "proper". For example, the major pentatonic scale is strictly proper:
1C2D2E3G2A3C2C4E5A5D5G5C3C7G7D7A7E8C4C9AtG9EtDtC

The pentatonic scales which are proper, but not strictly, are:

An example of an improper scale is the Japanese Hirajōshi scale. 1C2D1E4G1A4C2C3E5A6D5G5C3C7G7D6A7E9C4C8AeG8EeDtC

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. ").

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 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.

The interval class class(i) modulo Ω depends only on i modulo ℘, hence we may also define a version of class, Class(i), for pitch classes modulo Ω, which are called generic intervals. The specific pitch classes belonging to Class(i) are then called specific intervals. The class of the unison, Class(0), consists solely of multiples of Ω and is typically excluded from consideration, so that the number of generic intervals is ℘ − 1. Hence the generic intervals are numbered from 1 to ℘ − 1, and a scale is proper if for any two generic intervals i < j implies class(i) < class(j). If we represent the elements of Class(i) by intervals reduced to those between the unison and Ω, we may order them as usual, and so define propriety by stating that i < j for generic classes entails Class(i) < Class(j). This procedure, while a good deal more convoluted than the definition as originally stated, is how the matter is normally approached in diatonic set theory. Consider the diatonic (major) scale in the common 12 tone equal temperament, which follows the pattern (in semitones) 2 2 1 2 2 2 1. No interval in this scale, spanning any given number of scale steps, is narrower (consisting of fewer semitones) than an interval spanning fewer scale steps. For example, one cannot find a fourth in this scale that is smaller than a third: the smallest fourths are five semitones wide, and the largest thirds are four semitones. Therefore, the diatonic scale is proper. However, there is an interval that contains the same number of semitones as an interval spanning fewer scale degrees: the augmented fourth (F G A B) and the diminished fifth (B C D E F) are both six semitones wide. Therefore, the diatonic scale is proper but not strictly proper. On the other hand, consider the enigmatic scale, which follows the pattern 1 3 2 2 2 1 1. It is possible to find intervals in this scale that are narrower than other intervals in the scale spanning fewer scale steps: for example, the fourth built on the 6th scale step is three semitones wide, while the third built on the 2nd scale step is five semitones wide. Therefore, the enigmatic scale is not proper.