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Scale 61: "Ajuian"

Scale 61: Ajuian, 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: 1921


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

3 (trihemitonic)


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: 47


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, 1, 7]

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.

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

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.

(15, 6, 32)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 61 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1039
Scale 1039: Gixian, Ian Ring Music TheoryGixian
3rd mode:
Scale 2567
Scale 2567: Puhian, Ian Ring Music TheoryPuhian
4th mode:
Scale 3331
Scale 3331: Vabian, Ian Ring Music TheoryVabian
5th mode:
Scale 3713
Scale 3713: Xibian, Ian Ring Music TheoryXibian


The prime form of this scale is Scale 47

Scale 47Scale 47: Agoian, Ian Ring Music TheoryAgoian


The pentatonic modal family [61, 1039, 2567, 3331, 3713] (Forte: 5-2) is the complement of the heptatonic modal family [191, 2017, 2143, 3119, 3607, 3851, 3973] (Forte: 7-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 61 is 1921

Scale 1921Scale 1921: Lukian, Ian Ring Music TheoryLukian


Only scales that are chiral will have an enantiomorph. Scale 61 is chiral, and its enantiomorph is scale 1921

Scale 1921Scale 1921: Lukian, Ian Ring Music TheoryLukian


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> 61       T0I <11,0> 1921
T1 <1,1> 122      T1I <11,1> 3842
T2 <1,2> 244      T2I <11,2> 3589
T3 <1,3> 488      T3I <11,3> 3083
T4 <1,4> 976      T4I <11,4> 2071
T5 <1,5> 1952      T5I <11,5> 47
T6 <1,6> 3904      T6I <11,6> 94
T7 <1,7> 3713      T7I <11,7> 188
T8 <1,8> 3331      T8I <11,8> 376
T9 <1,9> 2567      T9I <11,9> 752
T10 <1,10> 1039      T10I <11,10> 1504
T11 <1,11> 2078      T11I <11,11> 3008
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1291      T0MI <7,0> 2581
T1M <5,1> 2582      T1MI <7,1> 1067
T2M <5,2> 1069      T2MI <7,2> 2134
T3M <5,3> 2138      T3MI <7,3> 173
T4M <5,4> 181      T4MI <7,4> 346
T5M <5,5> 362      T5MI <7,5> 692
T6M <5,6> 724      T6MI <7,6> 1384
T7M <5,7> 1448      T7MI <7,7> 2768
T8M <5,8> 2896      T8MI <7,8> 1441
T9M <5,9> 1697      T9MI <7,9> 2882
T10M <5,10> 3394      T10MI <7,10> 1669
T11M <5,11> 2693      T11MI <7,11> 3338

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 63Scale 63: Hexatonic Chromatic, Ian Ring Music TheoryHexatonic Chromatic
Scale 57Scale 57: Ahoian, Ian Ring Music TheoryAhoian
Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian
Scale 53Scale 53: Absian, Ian Ring Music TheoryAbsian
Scale 45Scale 45: Aprian, Ian Ring Music TheoryAprian
Scale 29Scale 29: Aduian, Ian Ring Music TheoryAduian
Scale 93Scale 93: Anuian, Ian Ring Music TheoryAnuian
Scale 125Scale 125: Atwian, Ian Ring Music TheoryAtwian
Scale 189Scale 189: Befian, Ian Ring Music TheoryBefian
Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic
Scale 573Scale 573: Saptimic, Ian Ring Music TheorySaptimic
Scale 1085Scale 1085: Gozian, Ian Ring Music TheoryGozian
Scale 2109Scale 2109: Muvian, Ian Ring Music TheoryMuvian

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.