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Scale 3587: "Pentatonic Chromatic 4"

Scale 3587: Pentatonic Chromatic 4, 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
Pentatonic Chromatic 4



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.



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 Rahn/Ring formula.

prime: 31


Indicates if the scale can be constructed using a generator, and an origin.

generator: 1
origin: 9

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, 8, 1, 1, 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, 3, 2, 1, 0, 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> = {1,8}
<2> = {2,9}
<3> = {3,10}
<4> = {4,11}

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, 0, 20)


This scale has a generator of 1, originating on 9.

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 3587 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3841
Scale 3841: Pentatonic Chromatic Descending, Ian Ring Music TheoryPentatonic Chromatic Descending
3rd mode:
Scale 31
Scale 31: Pentatonic Chromatic, Ian Ring Music TheoryPentatonic ChromaticThis is the prime mode
4th mode:
Scale 2063
Scale 2063: Pentatonic Chromatic 2, Ian Ring Music TheoryPentatonic Chromatic 2
5th mode:
Scale 3079
Scale 3079: Pentatonic Chromatic 3, Ian Ring Music TheoryPentatonic Chromatic 3


The prime form of this scale is Scale 31

Scale 31Scale 31: Pentatonic Chromatic, Ian Ring Music TheoryPentatonic Chromatic


The pentatonic modal family [3587, 3841, 31, 2063, 3079] (Forte: 5-1) is the complement of the heptatonic modal family [127, 2111, 3103, 3599, 3847, 3971, 4033] (Forte: 7-1)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3587 is 2063

Scale 2063Scale 2063: Pentatonic Chromatic 2, Ian Ring Music TheoryPentatonic Chromatic 2


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> 3587       T0I <11,0> 2063
T1 <1,1> 3079      T1I <11,1> 31
T2 <1,2> 2063      T2I <11,2> 62
T3 <1,3> 31      T3I <11,3> 124
T4 <1,4> 62      T4I <11,4> 248
T5 <1,5> 124      T5I <11,5> 496
T6 <1,6> 248      T6I <11,6> 992
T7 <1,7> 496      T7I <11,7> 1984
T8 <1,8> 992      T8I <11,8> 3968
T9 <1,9> 1984      T9I <11,9> 3841
T10 <1,10> 3968      T10I <11,10> 3587
T11 <1,11> 3841      T11I <11,11> 3079
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 677      T0MI <7,0> 1193
T1M <5,1> 1354      T1MI <7,1> 2386
T2M <5,2> 2708      T2MI <7,2> 677
T3M <5,3> 1321      T3MI <7,3> 1354
T4M <5,4> 2642      T4MI <7,4> 2708
T5M <5,5> 1189      T5MI <7,5> 1321
T6M <5,6> 2378      T6MI <7,6> 2642
T7M <5,7> 661      T7MI <7,7> 1189
T8M <5,8> 1322      T8MI <7,8> 2378
T9M <5,9> 2644      T9MI <7,9> 661
T10M <5,10> 1193      T10MI <7,10> 1322
T11M <5,11> 2386      T11MI <7,11> 2644

The transformations that map this set to itself are: T0, T10I

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 3585Scale 3585: Tetratonic Chromatic Descending, Ian Ring Music TheoryTetratonic Chromatic Descending
Scale 3589Scale 3589: Widian, Ian Ring Music TheoryWidian
Scale 3591Scale 3591: Wifian, Ian Ring Music TheoryWifian
Scale 3595Scale 3595: Wihian, Ian Ring Music TheoryWihian
Scale 3603Scale 3603: Womian, Ian Ring Music TheoryWomian
Scale 3619Scale 3619: Thanimic, Ian Ring Music TheoryThanimic
Scale 3651Scale 3651: Wuqian, Ian Ring Music TheoryWuqian
Scale 3715Scale 3715: Xician, Ian Ring Music TheoryXician
Scale 3843Scale 3843: Hexatonic Chromatic 5, Ian Ring Music TheoryHexatonic Chromatic 5
Scale 3075Scale 3075: Tetratonic Chromatic 3, Ian Ring Music TheoryTetratonic Chromatic 3
Scale 3331Scale 3331: Vabian, Ian Ring Music TheoryVabian
Scale 2563Scale 2563: Pofian, Ian Ring Music TheoryPofian
Scale 1539Scale 1539: Jikian, Ian Ring Music TheoryJikian

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.