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Scale 91: "Anoian"

Scale 91: Anoian, 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.

enantiomorph: 2881


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.

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.



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, 2, 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, 2, 3, 1, 1, 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> = {3,7,8}
<3> = {4,5,9}
<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, 1, 30)

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: {1,4}

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

2nd mode:
Scale 2093
Scale 2093: Mulian, Ian Ring Music TheoryMulian
3rd mode:
Scale 1547
Scale 1547: Jopian, Ian Ring Music TheoryJopian
4th mode:
Scale 2821
Scale 2821: Rukian, Ian Ring Music TheoryRukian
5th mode:
Scale 1729
Scale 1729: Kowian, Ian Ring Music TheoryKowian


This is the prime form of this scale.


The pentatonic modal family [91, 2093, 1547, 2821, 1729] (Forte: 5-10) is the complement of the heptatonic modal family [607, 761, 1993, 2351, 3223, 3659, 3877] (Forte: 7-10)


The inverse of a scale is a reflection using the root as its axis. The inverse of 91 is 2881

Scale 2881Scale 2881: Satian, Ian Ring Music TheorySatian


Only scales that are chiral will have an enantiomorph. Scale 91 is chiral, and its enantiomorph is scale 2881

Scale 2881Scale 2881: Satian, Ian Ring Music TheorySatian


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> 91       T0I <11,0> 2881
T1 <1,1> 182      T1I <11,1> 1667
T2 <1,2> 364      T2I <11,2> 3334
T3 <1,3> 728      T3I <11,3> 2573
T4 <1,4> 1456      T4I <11,4> 1051
T5 <1,5> 2912      T5I <11,5> 2102
T6 <1,6> 1729      T6I <11,6> 109
T7 <1,7> 3458      T7I <11,7> 218
T8 <1,8> 2821      T8I <11,8> 436
T9 <1,9> 1547      T9I <11,9> 872
T10 <1,10> 3094      T10I <11,10> 1744
T11 <1,11> 2093      T11I <11,11> 3488
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 361      T0MI <7,0> 721
T1M <5,1> 722      T1MI <7,1> 1442
T2M <5,2> 1444      T2MI <7,2> 2884
T3M <5,3> 2888      T3MI <7,3> 1673
T4M <5,4> 1681      T4MI <7,4> 3346
T5M <5,5> 3362      T5MI <7,5> 2597
T6M <5,6> 2629      T6MI <7,6> 1099
T7M <5,7> 1163      T7MI <7,7> 2198
T8M <5,8> 2326      T8MI <7,8> 301
T9M <5,9> 557      T9MI <7,9> 602
T10M <5,10> 1114      T10MI <7,10> 1204
T11M <5,11> 2228      T11MI <7,11> 2408

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 89Scale 89: Aggian, Ian Ring Music TheoryAggian
Scale 93Scale 93: Anuian, Ian Ring Music TheoryAnuian
Scale 95Scale 95: Arkian, Ian Ring Music TheoryArkian
Scale 83Scale 83: Amuian, Ian Ring Music TheoryAmuian
Scale 87Scale 87: Asrian, Ian Ring Music TheoryAsrian
Scale 75Scale 75: Iloian, Ian Ring Music TheoryIloian
Scale 107Scale 107: Ansian, Ian Ring Music TheoryAnsian
Scale 123Scale 123: Asuian, Ian Ring Music TheoryAsuian
Scale 27Scale 27: Adoian, Ian Ring Music TheoryAdoian
Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian
Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 347Scale 347: Barimic, Ian Ring Music TheoryBarimic
Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic
Scale 1115Scale 1115: Superlocrian Hexamirror, Ian Ring Music TheorySuperlocrian Hexamirror
Scale 2139Scale 2139: Namian, Ian Ring Music TheoryNamian

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.