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Scale 531: "Degian"

Scale 531: Degian, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names




Cardinality is the count of how many pitches are in the scale.

4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11


Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.



A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

1 (unhemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)


An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.



Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.


Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

prime: 153


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

A deep scale is one where the interval vector has 6 different digits.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 3, 5, 3]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<1, 0, 2, 2, 1, 0>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.


Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,3,5}
<2> = {4,8}
<3> = {7,9,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.


Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.


Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.


Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.



Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".


Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(4, 0, 14)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsA{9,1,4}110.5
Minor Triadsam{9,0,4}110.5
Parsimonious Voice Leading Between Common Triads of Scale 531. Created by Ian Ring ©2019 am am A A am->A

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.



Modes are the rotational transformation of this scale. Scale 531 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 2313
Scale 2313: Osrian, Ian Ring Music TheoryOsrian
3rd mode:
Scale 801
Scale 801: Fahian, Ian Ring Music TheoryFahian
4th mode:
Scale 153
Scale 153: Bajian, Ian Ring Music TheoryBajianThis is the prime mode


The prime form of this scale is Scale 153

Scale 153Scale 153: Bajian, Ian Ring Music TheoryBajian


The tetratonic modal family [531, 2313, 801, 153] (Forte: 4-17) is the complement of the octatonic modal family [891, 1647, 1971, 2493, 2871, 3033, 3483, 3789] (Forte: 8-17)


The inverse of a scale is a reflection using the root as its axis. The inverse of 531 is 2313

Scale 2313Scale 2313: Osrian, Ian Ring Music TheoryOsrian


In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 531       T0I <11,0> 2313
T1 <1,1> 1062      T1I <11,1> 531
T2 <1,2> 2124      T2I <11,2> 1062
T3 <1,3> 153      T3I <11,3> 2124
T4 <1,4> 306      T4I <11,4> 153
T5 <1,5> 612      T5I <11,5> 306
T6 <1,6> 1224      T6I <11,6> 612
T7 <1,7> 2448      T7I <11,7> 1224
T8 <1,8> 801      T8I <11,8> 2448
T9 <1,9> 1602      T9I <11,9> 801
T10 <1,10> 3204      T10I <11,10> 1602
T11 <1,11> 2313      T11I <11,11> 3204
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 801      T0MI <7,0> 153
T1M <5,1> 1602      T1MI <7,1> 306
T2M <5,2> 3204      T2MI <7,2> 612
T3M <5,3> 2313      T3MI <7,3> 1224
T4M <5,4> 531       T4MI <7,4> 2448
T5M <5,5> 1062      T5MI <7,5> 801
T6M <5,6> 2124      T6MI <7,6> 1602
T7M <5,7> 153      T7MI <7,7> 3204
T8M <5,8> 306      T8MI <7,8> 2313
T9M <5,9> 612      T9MI <7,9> 531
T10M <5,10> 1224      T10MI <7,10> 1062
T11M <5,11> 2448      T11MI <7,11> 2124

The transformations that map this set to itself are: T0, T1I, T4M, T9MI

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 529Scale 529: Raga Bilwadala, Ian Ring Music TheoryRaga Bilwadala
Scale 533Scale 533: Dehian, Ian Ring Music TheoryDehian
Scale 535Scale 535: Dejian, Ian Ring Music TheoryDejian
Scale 539Scale 539: Delian, Ian Ring Music TheoryDelian
Scale 515Scale 515: Depian, Ian Ring Music TheoryDepian
Scale 523Scale 523: Debian, Ian Ring Music TheoryDebian
Scale 547Scale 547: Pyrric, Ian Ring Music TheoryPyrric
Scale 563Scale 563: Thacritonic, Ian Ring Music TheoryThacritonic
Scale 595Scale 595: Sogitonic, Ian Ring Music TheorySogitonic
Scale 659Scale 659: Raga Rasika Ranjani, Ian Ring Music TheoryRaga Rasika Ranjani
Scale 787Scale 787: Aeolapritonic, Ian Ring Music TheoryAeolapritonic
Scale 19Scale 19: Acuian, Ian Ring Music TheoryAcuian
Scale 275Scale 275: Dalic, Ian Ring Music TheoryDalic
Scale 1043Scale 1043: Gizian, Ian Ring Music TheoryGizian
Scale 1555Scale 1555: Jotian, Ian Ring Music TheoryJotian
Scale 2579Scale 2579: Pupian, Ian Ring Music TheoryPupian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler ( used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.