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Scale 21: "Wholetone Trichord"

Scale 21: Wholetone Trichord, 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

Western Modern
Wholetone Trichord



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

3 (tritonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11


Forte Number

A code assigned by theorist Alan 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.

0 (anhemitonic)


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

0 (ancohemitonic)


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.


Deep Scale

A deep scale is one where the interval vector has 6 different digits.


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.

[0, 2, 0, 1, 0, 0]

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hansen.


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> = {2,8}
<2> = {4,10}

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".


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 21 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 1029
Scale 1029: AMPIAN, Ian Ring Music TheoryAMPIAN
3rd mode:
Scale 1281
Scale 1281: HUQIAN, Ian Ring Music TheoryHUQIAN


This is the prime form of this scale.


The tritonic modal family [21, 1029, 1281] (Forte: 3-6) is the complement of the nonatonic modal family [1407, 1533, 2037, 2751, 3423, 3759, 3927, 4011, 4053] (Forte: 9-6)


The inverse of a scale is a reflection using the root as its axis. The inverse of 21 is 1281

Scale 1281Scale 1281: HUQIAN, Ian Ring Music TheoryHUQIAN


T0 21  T0I 1281
T1 42  T1I 2562
T2 84  T2I 1029
T3 168  T3I 2058
T4 336  T4I 21
T5 672  T5I 42
T6 1344  T6I 84
T7 2688  T7I 168
T8 1281  T8I 336
T9 2562  T9I 672
T10 1029  T10I 1344
T11 2058  T11I 2688

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 23Scale 23: APHIAN, Ian Ring Music TheoryAPHIAN
Scale 17Scale 17: Major Third Ditone, Ian Ring Music TheoryMajor Third Ditone
Scale 19Scale 19: Hexatonic Trichord, Ian Ring Music TheoryHexatonic Trichord
Scale 25Scale 25: ACKIAN, Ian Ring Music TheoryACKIAN
Scale 29Scale 29: ADUIAN, Ian Ring Music TheoryADUIAN
Scale 5Scale 5: Vietnamese Ditonic, Ian Ring Music TheoryVietnamese Ditonic
Scale 13Scale 13: DIJIAN, Ian Ring Music TheoryDIJIAN
Scale 37Scale 37: Minor Seventh Trichord, Ian Ring Music TheoryMinor Seventh Trichord
Scale 53Scale 53: ABSIAN, Ian Ring Music TheoryABSIAN
Scale 85Scale 85: SEGIAN, Ian Ring Music TheorySEGIAN
Scale 149Scale 149: Eskimo Tetratonic, Ian Ring Music TheoryEskimo Tetratonic
Scale 277Scale 277: Mixolyric, Ian Ring Music TheoryMixolyric
Scale 533Scale 533: DEHIAN, Ian Ring Music TheoryDEHIAN
Scale 1045Scale 1045: GIBIAN, Ian Ring Music TheoryGIBIAN
Scale 2069Scale 2069: MOWIAN, Ian Ring Music TheoryMOWIAN

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.