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Scale 93: "Anuian"

Scale 93: Anuian, 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.

5 (pentatonic)

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.

2 (dihemitonic)


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

1 (uncohemitonic)


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.



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, 1, 2, 6]

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.

<2, 3, 2, 2, 0, 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> = {2,3,8}
<3> = {4,9,10}
<4> = {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.

(14, 5, 32)

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{0,3,6}000

The following pitch classes are not present in any of the common triads: {2,4}

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 93 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1047
Scale 1047: Gician, Ian Ring Music TheoryGician
3rd mode:
Scale 2571
Scale 2571: Pukian, Ian Ring Music TheoryPukian
4th mode:
Scale 3333
Scale 3333: Vacian, Ian Ring Music TheoryVacian
5th mode:
Scale 1857
Scale 1857: Liwian, Ian Ring Music TheoryLiwian


This is the prime form of this scale.


The pentatonic modal family [93, 1047, 2571, 3333, 1857] (Forte: 5-8) is the complement of the heptatonic modal family [381, 1119, 2001, 2607, 3351, 3723, 3909] (Forte: 7-8)


The inverse of a scale is a reflection using the root as its axis. The inverse of 93 is 1857

Scale 1857Scale 1857: Liwian, Ian Ring Music TheoryLiwian


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> 93       T0I <11,0> 1857
T1 <1,1> 186      T1I <11,1> 3714
T2 <1,2> 372      T2I <11,2> 3333
T3 <1,3> 744      T3I <11,3> 2571
T4 <1,4> 1488      T4I <11,4> 1047
T5 <1,5> 2976      T5I <11,5> 2094
T6 <1,6> 1857      T6I <11,6> 93
T7 <1,7> 3714      T7I <11,7> 186
T8 <1,8> 3333      T8I <11,8> 372
T9 <1,9> 2571      T9I <11,9> 744
T10 <1,10> 1047      T10I <11,10> 1488
T11 <1,11> 2094      T11I <11,11> 2976
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1353      T0MI <7,0> 597
T1M <5,1> 2706      T1MI <7,1> 1194
T2M <5,2> 1317      T2MI <7,2> 2388
T3M <5,3> 2634      T3MI <7,3> 681
T4M <5,4> 1173      T4MI <7,4> 1362
T5M <5,5> 2346      T5MI <7,5> 2724
T6M <5,6> 597      T6MI <7,6> 1353
T7M <5,7> 1194      T7MI <7,7> 2706
T8M <5,8> 2388      T8MI <7,8> 1317
T9M <5,9> 681      T9MI <7,9> 2634
T10M <5,10> 1362      T10MI <7,10> 1173
T11M <5,11> 2724      T11MI <7,11> 2346

The transformations that map this set to itself are: T0, T6I

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 95Scale 95: Arkian, Ian Ring Music TheoryArkian
Scale 89Scale 89: Aggian, Ian Ring Music TheoryAggian
Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian
Scale 85Scale 85: Segian, Ian Ring Music TheorySegian
Scale 77Scale 77: Alvian, Ian Ring Music TheoryAlvian
Scale 109Scale 109: Amsian, Ian Ring Music TheoryAmsian
Scale 125Scale 125: Atwian, Ian Ring Music TheoryAtwian
Scale 29Scale 29: Aduian, Ian Ring Music TheoryAduian
Scale 61Scale 61: Ajuian, Ian Ring Music TheoryAjuian
Scale 157Scale 157: Balian, Ian Ring Music TheoryBalian
Scale 221Scale 221: Biyian, Ian Ring Music TheoryBiyian
Scale 349Scale 349: Borimic, Ian Ring Music TheoryBorimic
Scale 605Scale 605: Dycrimic, Ian Ring Music TheoryDycrimic
Scale 1117Scale 1117: Raptimic, Ian Ring Music TheoryRaptimic
Scale 2141Scale 2141: Nanian, Ian Ring Music TheoryNanian

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.