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Scale 1539: "JIKIAN"

Scale 1539: JIKIAN, 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.

4 (tetratonic)

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.

2 (dihemitonic)


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.

prime: 27

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.

[2, 1, 2, 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> = {1,2,8}
<2> = {3,9}
<3> = {4,10,11}

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".


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 1539 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 2817
Scale 2817, Ian Ring Music Theory
3rd mode:
Scale 27
Scale 27: ADOIAN, Ian Ring Music TheoryADOIANThis is the prime mode
4th mode:
Scale 2061
Scale 2061: MORIAN, Ian Ring Music TheoryMORIAN


The prime form of this scale is Scale 27

Scale 27Scale 27: ADOIAN, Ian Ring Music TheoryADOIAN


The tetratonic modal family [1539, 2817, 27, 2061] (Forte: 4-3) is the complement of the octatonic modal family [639, 1017, 2367, 3231, 3663, 3879, 3987, 4041] (Forte: 8-3)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1539 is 2061

Scale 2061Scale 2061: MORIAN, Ian Ring Music TheoryMORIAN


T0 1539  T0I 2061
T1 3078  T1I 27
T2 2061  T2I 54
T3 27  T3I 108
T4 54  T4I 216
T5 108  T5I 432
T6 216  T6I 864
T7 432  T7I 1728
T8 864  T8I 3456
T9 1728  T9I 2817
T10 3456  T10I 1539
T11 2817  T11I 3078

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 1537Scale 1537: JIJIAN, Ian Ring Music TheoryJIJIAN
Scale 1541Scale 1541: JILIAN, Ian Ring Music TheoryJILIAN
Scale 1543Scale 1543: JOMIAN, Ian Ring Music TheoryJOMIAN
Scale 1547Scale 1547: JOPIAN, Ian Ring Music TheoryJOPIAN
Scale 1555Scale 1555: JOTIAN, Ian Ring Music TheoryJOTIAN
Scale 1571Scale 1571: Lagitonic, Ian Ring Music TheoryLagitonic
Scale 1603Scale 1603: JUXIAN, Ian Ring Music TheoryJUXIAN
Scale 1667Scale 1667: KEKIAN, Ian Ring Music TheoryKEKIAN
Scale 1795Scale 1795: LAKIAN, Ian Ring Music TheoryLAKIAN
Scale 1027Scale 1027: GEQIAN, Ian Ring Music TheoryGEQIAN
Scale 1283Scale 1283: HURIAN, Ian Ring Music TheoryHURIAN
Scale 515Scale 515: DEPIAN, Ian Ring Music TheoryDEPIAN
Scale 2563Scale 2563: POFIAN, Ian Ring Music TheoryPOFIAN
Scale 3587Scale 3587: Pentatonic Chromatic 4, Ian Ring Music TheoryPentatonic Chromatic 4

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.