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Scale 2115: "Muyian"

Scale 2115: Muyian, 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.

4 (tetratonic)

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 Starr/Rahn algorithm.

prime: 135


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, an indicator of maximum hierarchization.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 5, 5, 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, 1, 0, 0, 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> = {1,5}
<2> = {2,6,10}
<3> = {7,11}

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 Distribution Spectra have 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.

(4, 0, 13)

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 2115 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 3105
Scale 3105: Tibian, Ian Ring Music TheoryTibian
3rd mode:
Scale 225
Scale 225: Bibian, Ian Ring Music TheoryBibian
4th mode:
Scale 135
Scale 135: Attian, Ian Ring Music TheoryAttianThis is the prime mode


The prime form of this scale is Scale 135

Scale 135Scale 135: Attian, Ian Ring Music TheoryAttian


The tetratonic modal family [2115, 3105, 225, 135] (Forte: 4-6) is the complement of the octatonic modal family [495, 1935, 2295, 3015, 3195, 3555, 3645, 3825] (Forte: 8-6)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2115 is itself, because it is a palindromic scale!

Scale 2115Scale 2115: Muyian, Ian Ring Music TheoryMuyian


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> 2115       T0I <11,0> 2115
T1 <1,1> 135      T1I <11,1> 135
T2 <1,2> 270      T2I <11,2> 270
T3 <1,3> 540      T3I <11,3> 540
T4 <1,4> 1080      T4I <11,4> 1080
T5 <1,5> 2160      T5I <11,5> 2160
T6 <1,6> 225      T6I <11,6> 225
T7 <1,7> 450      T7I <11,7> 450
T8 <1,8> 900      T8I <11,8> 900
T9 <1,9> 1800      T9I <11,9> 1800
T10 <1,10> 3600      T10I <11,10> 3600
T11 <1,11> 3105      T11I <11,11> 3105
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 225      T0MI <7,0> 225
T1M <5,1> 450      T1MI <7,1> 450
T2M <5,2> 900      T2MI <7,2> 900
T3M <5,3> 1800      T3MI <7,3> 1800
T4M <5,4> 3600      T4MI <7,4> 3600
T5M <5,5> 3105      T5MI <7,5> 3105
T6M <5,6> 2115       T6MI <7,6> 2115
T7M <5,7> 135      T7MI <7,7> 135
T8M <5,8> 270      T8MI <7,8> 270
T9M <5,9> 540      T9MI <7,9> 540
T10M <5,10> 1080      T10MI <7,10> 1080
T11M <5,11> 2160      T11MI <7,11> 2160

The transformations that map this set to itself are: T0, T0I, T6M, T6MI

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 2113Scale 2113: Muxian, Ian Ring Music TheoryMuxian
Scale 2117Scale 2117: Raga Sumukam, Ian Ring Music TheoryRaga Sumukam
Scale 2119Scale 2119: Mubian, Ian Ring Music TheoryMubian
Scale 2123Scale 2123: Nacian, Ian Ring Music TheoryNacian
Scale 2131Scale 2131: Nahian, Ian Ring Music TheoryNahian
Scale 2147Scale 2147: Narian, Ian Ring Music TheoryNarian
Scale 2051Scale 2051: Tritonic Chromatic 2, Ian Ring Music TheoryTritonic Chromatic 2
Scale 2083Scale 2083: Mofian, Ian Ring Music TheoryMofian
Scale 2179Scale 2179, Ian Ring Music Theory
Scale 2243Scale 2243: Noyian, Ian Ring Music TheoryNoyian
Scale 2371Scale 2371: Omoian, Ian Ring Music TheoryOmoian
Scale 2627Scale 2627: Qerian, Ian Ring Music TheoryQerian
Scale 3139Scale 3139: Towian, Ian Ring Music TheoryTowian
Scale 67Scale 67: Abrian, Ian Ring Music TheoryAbrian
Scale 1091Scale 1091: Pedian, Ian Ring Music TheoryPedian

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.