Rothenberg propriety information

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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."^[1] 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]"^[2]). Such a scale is just called "proper".

For example, the major pentatonic scale is strictly proper:

The pentatonic scales which are proper, but not strictly, are:^[2]

• {0,1,4,6,8} (Lydian chord)
• {0,2,4,6,8} (whole tone scale)
• {0,1,4,6,9} (gamma chord)
• {0,2,4,6,9} (dominant ninth chord)
• {0,1,3,6,9} (dominant minor ninth chord)

The one strictly proper pentatonic scale:

• {0,2,4,7,9} (major pentatonic)

The heptatonic scales which are proper, but not strictly, are:^[2]

• {0,1,3,4,6,8,9} (harmonic minor scale)
• {0,1,3,5,6,8,t} (diatonic scale)
• {0,1,3,4,6,8,t} (Altered scale)
• {0,1,2,4,6,8,t} (Major Neapolitan scale)

Propriety may also be considered as scales whose stability = 1, with stability defined as, "the ratio of the number of non-ambiguous undirected intervals...to the total number of undirected intervals," in which case the diatonic scale has a stability of 20⁄21.^[2]

The twelve equal scale is strictly proper as is any equal tempered scale because it has only one interval size for each number of steps Most tempered scales are proper too. As another example, the otonal harmonic fragment 5⁄4, 6⁄4, 7⁄4, 8⁄4 is strictly proper, with the one step intervals varying in size from 8⁄7 to 5⁄4, two step intervals vary from 4⁄3 to 3⁄2, three step intervals from 8⁄5 to 7⁄4.

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.^[3]

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."^[2]).