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Indiatonic set theory,Rothenberg propriety is an important concept, lack of contradiction and ambiguity, in the general theory ofmusical scales which was introduced byDavid Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context byGerald Balzano, who termed itcoherence.
"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] Ascale is strictly proper if all two stepintervals are larger than any one step interval, all three step intervals are larger than any two step interval and so on. For instance with thediatonic 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 majorpentatonic scale is strictly proper:
| 1 | C | 2 | D | 2 | E | 3 | G | 2 | A | 3 | C | |
| 2 | C | 4 | E | 5 | A | 5 | D | 5 | G | 5 | C | |
| 3 | C | 7 | G | 7 | D | 7 | A | 7 | E | 8 | C | |
| 4 | C | 9 | A | t | G | 9 | E | t | D | t | C |
The pentatonic scales which are proper, but not strictly, are:[2]
The one strictly proper pentatonic scale:
The heptatonic scales which are proper, but not strictly, are:[2]
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 of20⁄21.[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, theotonal harmonic fragment5⁄4,6⁄4,7⁄4,8⁄4 is strictly proper, with the one step intervals varying in size from8⁄7 to5⁄4, two step intervals vary from4⁄3 to3⁄2, three step intervals from8⁄5 to7⁄4.
Rothenberg hypothesizes that proper scales provide a point or frame of reference which aids perception ("stablegestalt") and that improper scales contradictions require adrone orostinato to provide a point of reference.[3]
An example of an improper scale is the JapaneseHirajōshi scale.
| 1 | C | 2 | D | 1 | E♭ | 4 | G | 1 | A♭ | 4 | C | ||
| 2 | C | 3 | E♭ | 5 | A♭ | 6 | D | 5 | G | 5 | C | ||
| 3 | C | 7 | G | 7 | D | 6 | A♭ | 7 | E♭ | 9 | C | ||
| 4 | C | 8 | A♭ | e | G | 8 | E♭ | e | D | t | C |
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]).
Rothenberg defined propriety in a very general context; however for nearly all purposes it suffices to consider what in musical contexts is often called aperiodic scale, though in fact these correspond to what mathematicians call aquasiperiodic 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 anoctave, and so the scale consists of all notes belonging to a finite number ofpitch classes. Ifβi denotes a scale element for each integer i, thenβi+℘ =βi + Ω, whereΩ 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 anyi one can consider the set of all differences byi 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, sayingA <B if and only if for everya ∈A andb ∈B we havea <b. Then a scale isstrictly proper ifi <j implies class(i) < class(j). It isproper ifi ≤j implies class(i) ≤ class(j). Strict propriety implies propriety but a proper scale need not be strictly proper; an example is thediatonic scale inequal temperament, where thetritone interval belongs both to the class of the fourth (as anaugmented fourth) and to the class of the fifth (as adiminished fifth). Strict propriety is the same ascoherence in the sense of Balzano.
Theinterval class class(i) modulo Ω depends only onimodulo ℘, hence we may also define a version of class, Class(i), forpitch classes moduloΩ, which are calledgeneric intervals. The specific pitch classes belonging to Class(i) are then calledspecific intervals. The class of theunison, 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 intervalsi <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 thati <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 indiatonic 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 theenigmatic 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.
Balzano introduced the idea of attempting to characterize thediatonic scale in terms of propriety. There are no strictly proper seven-note scales in12 equal temperament; however, thereare five proper scales, one of which is the diatonic scale. Here transposition and modes are not counted separately, so thatdiatonic scale encompasses both themajor diatonic scale and thenatural minor scale beginning with any pitch. Each of these scales, if spelled correctly, has a version in anymeantone tuning, and when the fifth is flatter than 700cents, they all become strictly proper. In particular, five of the seven strictly proper seven-note scales in19 equal temperament are one of these scales. The five scales are:
In any meantone system with fifths flatter than 700 cents, one also has the following strictly proper scale: C D♭ E F♭ G A♭ B♭ (which isPhrygian Dominant♭4 scale).
The diatonic, ascending minor, harmonic minor, harmonic major and this last unnamed scale all contain complete circles of three major and four minor thirds, variously arranged. The Locrian major scale has a circle of four major and two minor thirds, along with adiminished third, which inseptimal meantone temperament approximates aseptimal major second of ratio8⁄7. The other scales are all of the scales with a complete circle of three major and four minor thirds, which since (5⁄4)3 (6⁄5)4 =81⁄20, tempered to two octaves in meantone, is indicative of meantone.
The first three scales are of basic importance tocommon practice music, and the harmonic major scale often used, and that the diatonic scale is not singled out by propriety is perhaps less interesting[according to whom?] than that the backbone scales of diatonic practice all are.
