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Scale 2095: "MUMian"

Scale 2095: MUMian, 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).



Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

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: 3715


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

4 (multihemitonic)


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

3 (tricohemitonic)


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: 95


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, 1, 1, 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.

<4, 4, 3, 2, 1, 1>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.8, 0.667, 0.6, 0, 0.2, 0.333>

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

(34, 13, 55)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.


Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.



This scale has no generator.

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{11,2,5}000

The following pitch classes are not present in any of the common triads: {0,1,3}

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 2095 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3095
Scale 3095: TIVian, Ian Ring Music TheoryTIVian
3rd mode:
Scale 3595
Scale 3595: WIHian, Ian Ring Music TheoryWIHian
4th mode:
Scale 3845
Scale 3845: YIHian, Ian Ring Music TheoryYIHian
5th mode:
Scale 1985
Scale 1985: MEWian, Ian Ring Music TheoryMEWian
6th mode:
Scale 95
Scale 95: ARKian, Ian Ring Music TheoryARKianThis is the prime mode


The prime form of this scale is Scale 95

Scale 95Scale 95: ARKian, Ian Ring Music TheoryARKian


The hexatonic modal family [2095, 3095, 3595, 3845, 1985, 95] (Forte: 6-2) is the complement of the hexatonic modal family [95, 1985, 2095, 3095, 3595, 3845] (Forte: 6-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2095 is 3715

Scale 3715Scale 3715: XICian, Ian Ring Music TheoryXICian


Only scales that are chiral will have an enantiomorph. Scale 2095 is chiral, and its enantiomorph is scale 3715

Scale 3715Scale 3715: XICian, Ian Ring Music TheoryXICian


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> 2095       T0I <11,0> 3715
T1 <1,1> 95      T1I <11,1> 3335
T2 <1,2> 190      T2I <11,2> 2575
T3 <1,3> 380      T3I <11,3> 1055
T4 <1,4> 760      T4I <11,4> 2110
T5 <1,5> 1520      T5I <11,5> 125
T6 <1,6> 3040      T6I <11,6> 250
T7 <1,7> 1985      T7I <11,7> 500
T8 <1,8> 3970      T8I <11,8> 1000
T9 <1,9> 3845      T9I <11,9> 2000
T10 <1,10> 3595      T10I <11,10> 4000
T11 <1,11> 3095      T11I <11,11> 3905
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1195      T0MI <7,0> 2725
T1M <5,1> 2390      T1MI <7,1> 1355
T2M <5,2> 685      T2MI <7,2> 2710
T3M <5,3> 1370      T3MI <7,3> 1325
T4M <5,4> 2740      T4MI <7,4> 2650
T5M <5,5> 1385      T5MI <7,5> 1205
T6M <5,6> 2770      T6MI <7,6> 2410
T7M <5,7> 1445      T7MI <7,7> 725
T8M <5,8> 2890      T8MI <7,8> 1450
T9M <5,9> 1685      T9MI <7,9> 2900
T10M <5,10> 3370      T10MI <7,10> 1705
T11M <5,11> 2645      T11MI <7,11> 3410

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 2093Scale 2093: MULian, Ian Ring Music TheoryMULian
Scale 2091Scale 2091: MUKian, Ian Ring Music TheoryMUKian
Scale 2087Scale 2087: MUHian, Ian Ring Music TheoryMUHian
Scale 2103Scale 2103: MURian, Ian Ring Music TheoryMURian
Scale 2111Scale 2111: Heptatonic Chromatic 2, Ian Ring Music TheoryHeptatonic Chromatic 2
Scale 2063Scale 2063: Pentatonic Chromatic 2, Ian Ring Music TheoryPentatonic Chromatic 2
Scale 2079Scale 2079: Hexatonic Chromatic 4, Ian Ring Music TheoryHexatonic Chromatic 4
Scale 2127Scale 2127: NAFian, Ian Ring Music TheoryNAFian
Scale 2159Scale 2159: NEYian, Ian Ring Music TheoryNEYian
Scale 2223Scale 2223: Konian, Ian Ring Music TheoryKonian
Scale 2351Scale 2351: Gynian, Ian Ring Music TheoryGynian
Scale 2607Scale 2607: Aerolian, Ian Ring Music TheoryAerolian
Scale 3119Scale 3119: TIKian, Ian Ring Music TheoryTIKian
Scale 47Scale 47: AGOian, Ian Ring Music TheoryAGOian
Scale 1071Scale 1071: GORian, Ian Ring Music TheoryGORian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.