Rothenberg propriety
Encyclopedia
In music, Rothenberg propriety denotes an important concept in the general theory of 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. The notion belongs to the class of concepts often, but misleadingly, termed diatonic set theory
; in fact, as with most concepts of diatonic set theory, it applies far more widely than simply to the diatonic scale
.
. 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 an octave
, and so the scale consists of all notes belonging to a finite number of pitch class
es. If βi denotes a scale element for each integer i, then βi+℘ = βi + Ω, where Ω 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 any i one can consider the set of all differences by i 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, saying A < B if and only if for every a ∈ A and b ∈ B we have a < b. Then a scale is strictly proper if i < j implies class(i) < class(j). It is proper if i ≤ j implies class(i) ≤ class(j). 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). Strict propriety is the same as coherence in the sense of Balzano.
class(i) modulo Ω depends only on i modulo
℘, hence we may also define a version of class, Class(i), for pitch class
es modulo Ω, which are called generic interval
s. The specific pitch classes belonging to Class(i) are then called specific interval
s. The class of the unison
, 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 intervals i < 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 that i < 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 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. 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 the enigmatic 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.
in terms of propriety. There are no strictly proper seven-note scales in 12 equal temperament
; however, there are five proper scales, one of which is the diatonic scale. Here we are not counting transposition separately, so that diatonic scale encompasses both the major diatonic scale and the natural minor scale
. Each of these scales, if spelled correctly, has a version in any meantone
tuning, and when the fifth is flatter than 700 cent
s, they all become strictly proper. In particular, five of the seven strictly proper seven-note scales in 19 equal temperament
are one of these scales. The five scales are:
In any meantone system with fifths flatter than 700 cents, we also have the following strictly proper scale: C D♭ E F♭ G A♭ B♭.
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 a diminished third
, which in septimal meantone temperament
approximates a septimal major second of ratio 8/7. 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 to common practice music, and the harmonic major scale often used, and the fact that the diatonic scale is not singled out by propriety is perhaps less interesting than the fact that the backbone scales of diatonic practice all are.
David Rothenberg
David Rothenberg is a professor of philosophy and music at the New Jersey Institute of Technology, with a special interest in animal sounds as music...
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. The notion belongs to the class of concepts often, but misleadingly, termed diatonic set theory
Diatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and...
; in fact, as with most concepts of diatonic set theory, it applies far more widely than simply to the diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
.
Definition of propriety
Rothenberg defined propriety in a very general context; however for nearly all purposes it suffices to consider what in musical contexts is often called a periodic scale, though in fact these correspond to what mathematicians call a quasiperiodic functionQuasiperiodic function
In mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition.A simple case is if the function obeys the equation:...
. 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 an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, and so the scale consists of all notes belonging to a finite number of pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
es. If βi denotes a scale element for each integer i, then βi+℘ = βi + Ω, where Ω 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 any i one can consider the set of all differences by i 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, saying A < B if and only if for every a ∈ A and b ∈ B we have a < b. Then a scale is strictly proper if i < j implies class(i) < class(j). It is proper if i ≤ j implies class(i) ≤ class(j). Strict propriety implies propriety but a proper scale need not be strictly proper; an example is the diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
in equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, where the tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
interval belongs both to the class of the fourth (as an augmented fourth) and to the class of the fifth (as a diminished fifth). Strict propriety is the same as coherence in the sense of Balzano.
Generic and specific intervals
The interval classInterval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
class(i) modulo Ω depends only on i modulo
Modulo operation
In computing, the modulo operation finds the remainder of division of one number by another.Given two positive numbers, and , a modulo n can be thought of as the remainder, on division of a by n...
℘, hence we may also define a version of class, Class(i), for pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
es modulo Ω, which are called generic interval
Generic interval
In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members...
s. The specific pitch classes belonging to Class(i) are then called specific interval
Specific interval
In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal...
s. The class of the unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
, 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 intervals i < 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 that i < 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 in diatonic set theory
Diatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and...
.
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 the enigmatic scale
Enigmatic scale
The enigmatic scale is an unusual musical scale, with elements of both major and minor scales, as well as the whole-tone 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.
Diatonic scale theory
Balzano introduced the idea of attempting to characterize the diatonic scaleDiatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
in terms of propriety. There are no strictly proper seven-note scales in 12 equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
; however, there are five proper scales, one of which is the diatonic scale. Here we are not counting transposition separately, so that diatonic scale encompasses both the major diatonic scale and the natural minor scale
Minor scale
A minor scale in Western music theory includes any scale that contains, in its tonic triad, at least three essential scale degrees: 1) the tonic , 2) a minor-third, or an interval of a minor third above the tonic, and 3) a perfect-fifth, or an interval of a perfect fifth above the tonic, altogether...
. Each of these scales, if spelled correctly, has a version in any meantone
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
tuning, and when the fifth is flatter than 700 cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
s, they all become strictly proper. In particular, five of the seven strictly proper seven-note scales in 19 equal temperament
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
are one of these scales. The five scales are:
- DiatonicDiatonic scaleIn music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
/majorMajor scaleIn music theory, the major scale or Ionian scale is one of the diatonic scales. It is made up of seven distinct notes, plus an eighth which duplicates the first an octave higher. In solfege these notes correspond to the syllables "Do, Re, Mi, Fa, Sol, La, Ti/Si, ", the "Do" in the parenthesis at...
/IonianIonian modeIonian mode is the name assigned by Heinrich Glarean in 1547 to his new authentic mode on C , which uses the diatonic octave species from C to the C an octave higher, divided at G into a fourth species of perfect fifth plus a third species of perfect fourth : C D...
: C D E F G A B
- Melodic/Ascending minorMinor scaleA minor scale in Western music theory includes any scale that contains, in its tonic triad, at least three essential scale degrees: 1) the tonic , 2) a minor-third, or an interval of a minor third above the tonic, and 3) a perfect-fifth, or an interval of a perfect fifth above the tonic, altogether...
: C D E♭ F G A B
- Harmonic minorMinor scaleA minor scale in Western music theory includes any scale that contains, in its tonic triad, at least three essential scale degrees: 1) the tonic , 2) a minor-third, or an interval of a minor third above the tonic, and 3) a perfect-fifth, or an interval of a perfect fifth above the tonic, altogether...
: C D E♭ F G A♭ B
- Harmonic majorHarmonic major scaleIn music theory, the harmonic major scale is a musical scale which found occasional use during the common practice era and is now occasionally employed, most often in jazz. It was named by Rimsky-Korsakov...
: C D E F G A♭ B
- Major locrianMajor locrian scaleIn music, the major Locrian scale, also called the Locrian major scale, is the scale obtained by sharpening the second and third degrees of the Locrian mode. With a tonic of C, it consists of the notes C D E F G♭ A♭ B♭. It can be described as a whole tone scale extending from G♭ to E, with F...
: C D E F G♭ A♭ B♭
In any meantone system with fifths flatter than 700 cents, we also have the following strictly proper scale: C D♭ E F♭ G A♭ B♭.
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 a diminished third
Diminished third
In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,...
, which in septimal meantone temperament
Septimal meantone temperament
In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as...
approximates a septimal major second of ratio 8/7. 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 to common practice music, and the harmonic major scale often used, and the fact that the diatonic scale is not singled out by propriety is perhaps less interesting than the fact that the backbone scales of diatonic practice all are.