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Scale 3597: "Wijian"

Scale 3597: Wijian, 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.

6 (hexatonic)

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.

enantiomorph: 1551


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

4 (multihemitonic)


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

2 (dicohemitonic)


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: 111


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.

[2, 1, 6, 1, 1, 1]

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.

<4, 3, 3, 2, 2, 1>

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> = {1,2,6}
<2> = {2,3,7}
<3> = {3,4,8,9}
<4> = {5,9,10}
<5> = {6,10,11}

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.

(34, 9, 55)

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
Diminished Triads{9,0,3}000

The following pitch classes are not present in any of the common triads: {2,10,11}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 3597 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1923
Scale 1923: Lulian, Ian Ring Music TheoryLulian
3rd mode:
Scale 3009
Scale 3009: Suvian, Ian Ring Music TheorySuvian
4th mode:
Scale 111
Scale 111: Aroian, Ian Ring Music TheoryAroianThis is the prime mode
5th mode:
Scale 2103
Scale 2103: Murian, Ian Ring Music TheoryMurian
6th mode:
Scale 3099
Scale 3099: Tixian, Ian Ring Music TheoryTixian


The prime form of this scale is Scale 111

Scale 111Scale 111: Aroian, Ian Ring Music TheoryAroian


The hexatonic modal family [3597, 1923, 3009, 111, 2103, 3099] (Forte: 6-Z3) is the complement of the hexatonic modal family [159, 993, 2127, 3111, 3603, 3849] (Forte: 6-Z36)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3597 is 1551

Scale 1551Scale 1551: Jorian, Ian Ring Music TheoryJorian


Only scales that are chiral will have an enantiomorph. Scale 3597 is chiral, and its enantiomorph is scale 1551

Scale 1551Scale 1551: Jorian, Ian Ring Music TheoryJorian


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> 3597       T0I <11,0> 1551
T1 <1,1> 3099      T1I <11,1> 3102
T2 <1,2> 2103      T2I <11,2> 2109
T3 <1,3> 111      T3I <11,3> 123
T4 <1,4> 222      T4I <11,4> 246
T5 <1,5> 444      T5I <11,5> 492
T6 <1,6> 888      T6I <11,6> 984
T7 <1,7> 1776      T7I <11,7> 1968
T8 <1,8> 3552      T8I <11,8> 3936
T9 <1,9> 3009      T9I <11,9> 3777
T10 <1,10> 1923      T10I <11,10> 3459
T11 <1,11> 3846      T11I <11,11> 2823
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1677      T0MI <7,0> 1581
T1M <5,1> 3354      T1MI <7,1> 3162
T2M <5,2> 2613      T2MI <7,2> 2229
T3M <5,3> 1131      T3MI <7,3> 363
T4M <5,4> 2262      T4MI <7,4> 726
T5M <5,5> 429      T5MI <7,5> 1452
T6M <5,6> 858      T6MI <7,6> 2904
T7M <5,7> 1716      T7MI <7,7> 1713
T8M <5,8> 3432      T8MI <7,8> 3426
T9M <5,9> 2769      T9MI <7,9> 2757
T10M <5,10> 1443      T10MI <7,10> 1419
T11M <5,11> 2886      T11MI <7,11> 2838

The transformations that map this set to itself are: T0

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 3599Scale 3599: Heptatonic Chromatic 4, Ian Ring Music TheoryHeptatonic Chromatic 4
Scale 3593Scale 3593: Wigian, Ian Ring Music TheoryWigian
Scale 3595Scale 3595: Wihian, Ian Ring Music TheoryWihian
Scale 3589Scale 3589: Widian, Ian Ring Music TheoryWidian
Scale 3605Scale 3605: Olkian, Ian Ring Music TheoryOlkian
Scale 3613Scale 3613: Wosian, Ian Ring Music TheoryWosian
Scale 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian
Scale 3725Scale 3725: Kyrian, Ian Ring Music TheoryKyrian
Scale 3853Scale 3853: Yomian, Ian Ring Music TheoryYomian
Scale 3085Scale 3085: Tepian, Ian Ring Music TheoryTepian
Scale 3341Scale 3341: Vahian, Ian Ring Music TheoryVahian
Scale 2573Scale 2573: Pulian, Ian Ring Music TheoryPulian
Scale 1549Scale 1549: Joqian, Ian Ring Music TheoryJoqian

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.