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Scale 55: "Aspian"

Scale 55: Aspian, 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: 3457


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.

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.



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, 1, 2, 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, 2, 2, 2, 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}
<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, 2, 30)

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

2nd mode:
Scale 2075
Scale 2075: Mozian, Ian Ring Music TheoryMozian
3rd mode:
Scale 3085
Scale 3085: Tepian, Ian Ring Music TheoryTepian
4th mode:
Scale 1795
Scale 1795: Lakian, Ian Ring Music TheoryLakian
5th mode:
Scale 2945
Scale 2945: Sihian, Ian Ring Music TheorySihian


This is the prime form of this scale.


The pentatonic modal family [55, 2075, 3085, 1795, 2945] (Forte: 5-3) is the complement of the heptatonic modal family [319, 1009, 2207, 3151, 3623, 3859, 3977] (Forte: 7-3)


The inverse of a scale is a reflection using the root as its axis. The inverse of 55 is 3457

Scale 3457Scale 3457: Vobian, Ian Ring Music TheoryVobian


Only scales that are chiral will have an enantiomorph. Scale 55 is chiral, and its enantiomorph is scale 3457

Scale 3457Scale 3457: Vobian, Ian Ring Music TheoryVobian


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> 55       T0I <11,0> 3457
T1 <1,1> 110      T1I <11,1> 2819
T2 <1,2> 220      T2I <11,2> 1543
T3 <1,3> 440      T3I <11,3> 3086
T4 <1,4> 880      T4I <11,4> 2077
T5 <1,5> 1760      T5I <11,5> 59
T6 <1,6> 3520      T6I <11,6> 118
T7 <1,7> 2945      T7I <11,7> 236
T8 <1,8> 1795      T8I <11,8> 472
T9 <1,9> 3590      T9I <11,9> 944
T10 <1,10> 3085      T10I <11,10> 1888
T11 <1,11> 2075      T11I <11,11> 3776
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1315      T0MI <7,0> 2197
T1M <5,1> 2630      T1MI <7,1> 299
T2M <5,2> 1165      T2MI <7,2> 598
T3M <5,3> 2330      T3MI <7,3> 1196
T4M <5,4> 565      T4MI <7,4> 2392
T5M <5,5> 1130      T5MI <7,5> 689
T6M <5,6> 2260      T6MI <7,6> 1378
T7M <5,7> 425      T7MI <7,7> 2756
T8M <5,8> 850      T8MI <7,8> 1417
T9M <5,9> 1700      T9MI <7,9> 2834
T10M <5,10> 3400      T10MI <7,10> 1573
T11M <5,11> 2705      T11MI <7,11> 3146

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 53Scale 53: Absian, Ian Ring Music TheoryAbsian
Scale 51Scale 51: Arfian, Ian Ring Music TheoryArfian
Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian
Scale 63Scale 63: Hexatonic Chromatic, Ian Ring Music TheoryHexatonic Chromatic
Scale 39Scale 39: Afuian, Ian Ring Music TheoryAfuian
Scale 47Scale 47: Agoian, Ian Ring Music TheoryAgoian
Scale 23Scale 23: Aphian, Ian Ring Music TheoryAphian
Scale 87Scale 87: Asrian, Ian Ring Music TheoryAsrian
Scale 119Scale 119: Smoian, Ian Ring Music TheorySmoian
Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian
Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic
Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
Scale 1079Scale 1079: Gowian, Ian Ring Music TheoryGowian
Scale 2103Scale 2103: Murian, Ian Ring Music TheoryMurian

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.