Rothenberg propriety is a concept in the theory of musical scales developed by David Rothenberg. It classifies scales as proper, strictly proper, and improper.
A scale is "strictly proper" if every "second" is smaller than every "third," every "third" smaller than every "fourth," etc. The terms "third" and "fourth," in Rothenberg's paper, refer to "generic interval classes" within the scale rather than the familiar diatonic interval categories. The diatonic scale in 31-EDO is strictly proper; the double harmonic scale (C Db E F G Ab B C) in 26-EDO is strictly proper (and is a very interesting listen!) as the B-Db "third" is now smaller than the Db-E "second" (unlike in 12, 31, etc).
A scale is "proper" if there is some interval class (e.g. "fourth") which is the *same* size as the next-larger one (e.g. a "fifth"), but nothing which is strictly larger. The diatonic scale in 12-EDO is proper, since the augmented fourth is the same size as the diminished fifth; the double harmonic scale in 19-EDO is proper.
A scale is "improper" if it isn't strictly proper or proper; e.g. there is some interval class (e.g. 'fourth') that is *larger* than the next-larger class (e.g. 'fifth'). The diatonic scale in 17-EDO is improper as the augmented fourth is now larger than the diminished fifth. The double harmonic scale is improper in 12-EDO and 31-EDO.
If there are two generic interval classes which share some specific interval (such as the 12-EDO diatonic scale with the 600 cent interval), e.g. the scale is "proper," the resulting intervals are called ambiguous. In an improper scale, the interval classes that are "mis-ordered" relative to one another are called contradictions.
This metric has been extended several ways - see also:
Carey 1998 writes, “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.”
It's easy to see the concept in action at the 7-step diatonic scale (
L-L-s-L-L-L-s) as rendered in three different EDOs:
- 12EDO (
2-2-1-2-2-2-1) is proper but not strictly proper because of the ambiguities of d5 (1+2+2+1=6) and A4 (2+2+2=6) in three-step and four-step intervals.
- 17EDO (
3-3-1-3-3-3-1) is improper because of the contradiction in d5 (1+3+3+1=8) being smaller than A4 (3+3+3=9).
- 19EDO (
3-3-2-3-3-3-2) is strictly proper.
- Carey, Norman (1998). Distribution Modulo One and Musical Scales, p.103, n.19. University of Rochester. Ph.D. dissertation.