# Rothenberg propriety

**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 larger 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), the resulting intervals are called *ambiguous*. In an improper scale, the interval classes that are misordered relative to one another are called *contradictions*.

This metric has been extended several ways - see also:

- Lumma stability and impropriety factor
- Rothenberg efficiency on Tonalsoft encyclopedia
- Rothenberg redundancy on Tonalsoft encyclopedia
- Rothenberg stability on Tonalsoft encyclopedia

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.”*^{[1]}

Strictly proper scales are constant structures, and proper but not strictly proper scales are not. Improper scales usually do, but see the 22edo example below for a counter-example.

## Examples

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*.

The 22EDO scale C D E vF# G ^Ab B C (`4-4-3-2-2-6-1`

) has both ambiguity (C-vF# 4th equals vF#-C 5th) and contradiction (^Ab-B 2nd exceeds E-G 3rd). Such a scale is classified as improper.

## See also

## References

- ↑ Carey, Norman (1998).
*Distribution Modulo One and Musical Scales*, p.103, n.19. University of Rochester. Ph.D. dissertation.

## External links

- Rothenberg, David. (1978). "A Model for Pattern Perception with Musical Applications. Part I: Pitch structures as order-preserving maps; Part II: The Information Content of Pitch Structures; Part III: The Graph Embedding of Pitch Structures".
*Mathematical Systems Theory*11, pp. 199-234, 353-372; 12, pp. 73-101. - Lumma, Carl. (2011?).
*A Quick Tour of Rothenberg's Musical Pattern Recognition Model*.