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Scale 2091: "Mukian"

Scale 2091: Mukian, 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: 2691


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.

prime: 87


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, 2, 6, 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.

<2, 3, 1, 2, 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> = {2,3,4,7,8}
<3> = {4,5,8,9,10}
<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, 7, 36)

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 2091 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3093
Scale 3093: Buqian, Ian Ring Music TheoryBuqian
3rd mode:
Scale 1797
Scale 1797: Lalian, Ian Ring Music TheoryLalian
4th mode:
Scale 1473
Scale 1473: Javian, Ian Ring Music TheoryJavian
5th mode:
Scale 87
Scale 87: Asrian, Ian Ring Music TheoryAsrianThis is the prime mode


The prime form of this scale is Scale 87

Scale 87Scale 87: Asrian, Ian Ring Music TheoryAsrian


The pentatonic modal family [2091, 3093, 1797, 1473, 87] (Forte: 5-9) is the complement of the heptatonic modal family [351, 1521, 1989, 2223, 3159, 3627, 3861] (Forte: 7-9)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2091 is 2691

Scale 2691Scale 2691: Rahian, Ian Ring Music TheoryRahian


Only scales that are chiral will have an enantiomorph. Scale 2091 is chiral, and its enantiomorph is scale 2691

Scale 2691Scale 2691: Rahian, Ian Ring Music TheoryRahian


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> 2091       T0I <11,0> 2691
T1 <1,1> 87      T1I <11,1> 1287
T2 <1,2> 174      T2I <11,2> 2574
T3 <1,3> 348      T3I <11,3> 1053
T4 <1,4> 696      T4I <11,4> 2106
T5 <1,5> 1392      T5I <11,5> 117
T6 <1,6> 2784      T6I <11,6> 234
T7 <1,7> 1473      T7I <11,7> 468
T8 <1,8> 2946      T8I <11,8> 936
T9 <1,9> 1797      T9I <11,9> 1872
T10 <1,10> 3594      T10I <11,10> 3744
T11 <1,11> 3093      T11I <11,11> 3393
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 171      T0MI <7,0> 2721
T1M <5,1> 342      T1MI <7,1> 1347
T2M <5,2> 684      T2MI <7,2> 2694
T3M <5,3> 1368      T3MI <7,3> 1293
T4M <5,4> 2736      T4MI <7,4> 2586
T5M <5,5> 1377      T5MI <7,5> 1077
T6M <5,6> 2754      T6MI <7,6> 2154
T7M <5,7> 1413      T7MI <7,7> 213
T8M <5,8> 2826      T8MI <7,8> 426
T9M <5,9> 1557      T9MI <7,9> 852
T10M <5,10> 3114      T10MI <7,10> 1704
T11M <5,11> 2133      T11MI <7,11> 3408

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 2089Scale 2089: Mujian, Ian Ring Music TheoryMujian
Scale 2093Scale 2093: Mulian, Ian Ring Music TheoryMulian
Scale 2095Scale 2095: Mumian, Ian Ring Music TheoryMumian
Scale 2083Scale 2083: Mofian, Ian Ring Music TheoryMofian
Scale 2087Scale 2087: Muhian, Ian Ring Music TheoryMuhian
Scale 2099Scale 2099: Raga Megharanji, Ian Ring Music TheoryRaga Megharanji
Scale 2107Scale 2107: Mutian, Ian Ring Music TheoryMutian
Scale 2059Scale 2059: Moqian, Ian Ring Music TheoryMoqian
Scale 2075Scale 2075: Mozian, Ian Ring Music TheoryMozian
Scale 2123Scale 2123: Nacian, Ian Ring Music TheoryNacian
Scale 2155Scale 2155: Newian, Ian Ring Music TheoryNewian
Scale 2219Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic
Scale 2347Scale 2347: Raga Viyogavarali, Ian Ring Music TheoryRaga Viyogavarali
Scale 2603Scale 2603: Gadimic, Ian Ring Music TheoryGadimic
Scale 3115Scale 3115: Tihian, Ian Ring Music TheoryTihian
Scale 43Scale 43: Alfian, Ian Ring Music TheoryAlfian
Scale 1067Scale 1067: Gopian, Ian Ring Music TheoryGopian

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.