Rothenberg propriety

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 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.

The fixed interval is typically an octave, and so the scale consists of all notes belonging to a finite number of pitch classes.

For any i one can consider the set of all differences by i steps between scale elements class(i) = {βn+i − βn}.

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).

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.

Balzano introduced the idea of attempting to characterize the diatonic scale in terms of propriety.

Each of these scales, if spelled correctly, has a version in any meantone tuning, and when the fifth is flatter than 700 cents, they all become strictly proper.