The Exciting Universe Of Music Theory

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Scale 3523

Scale 3523, 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.

7 (heptatonic)

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: 2167


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

5 (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: 239


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, 5, 1, 1, 2, 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.

<5, 4, 3, 3, 4, 2>

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> = {1,2,5}
<2> = {2,3,6}
<3> = {3,4,7}
<4> = {5,8,9}
<5> = {6,9,10}
<6> = {7,10,11}

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.

(55, 20, 84)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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
Major TriadsF♯{6,10,1}110.5
Diminished Triads{7,10,1}110.5

The following pitch classes are not present in any of the common triads: {0,8,11}

Parsimonious Voice Leading Between Common Triads of Scale 3523. Created by Ian Ring ©2019 F# F# F#->g°

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.



Modes are the rotational transformation of this scale. Scale 3523 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3809
Scale 3809: Yelian, Ian Ring Music TheoryYelian
3rd mode:
Scale 247
Scale 247: Bopian, Ian Ring Music TheoryBopian
4th mode:
Scale 2171
Scale 2171: Negian, Ian Ring Music TheoryNegian
5th mode:
Scale 3133
Scale 3133: Tosian, Ian Ring Music TheoryTosian
6th mode:
Scale 1807
Scale 1807: Larian, Ian Ring Music TheoryLarian
7th mode:
Scale 2951
Scale 2951: Silian, Ian Ring Music TheorySilian


The prime form of this scale is Scale 239

Scale 239Scale 239: Bikian, Ian Ring Music TheoryBikian


The heptatonic modal family [3523, 3809, 247, 2171, 3133, 1807, 2951] (Forte: 7-5) is the complement of the pentatonic modal family [143, 481, 2119, 3107, 3601] (Forte: 5-5)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3523 is 2167

Scale 2167Scale 2167: Nedian, Ian Ring Music TheoryNedian


Only scales that are chiral will have an enantiomorph. Scale 3523 is chiral, and its enantiomorph is scale 2167

Scale 2167Scale 2167: Nedian, Ian Ring Music TheoryNedian


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> 3523       T0I <11,0> 2167
T1 <1,1> 2951      T1I <11,1> 239
T2 <1,2> 1807      T2I <11,2> 478
T3 <1,3> 3614      T3I <11,3> 956
T4 <1,4> 3133      T4I <11,4> 1912
T5 <1,5> 2171      T5I <11,5> 3824
T6 <1,6> 247      T6I <11,6> 3553
T7 <1,7> 494      T7I <11,7> 3011
T8 <1,8> 988      T8I <11,8> 1927
T9 <1,9> 1976      T9I <11,9> 3854
T10 <1,10> 3952      T10I <11,10> 3613
T11 <1,11> 3809      T11I <11,11> 3131
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2293      T0MI <7,0> 1507
T1M <5,1> 491      T1MI <7,1> 3014
T2M <5,2> 982      T2MI <7,2> 1933
T3M <5,3> 1964      T3MI <7,3> 3866
T4M <5,4> 3928      T4MI <7,4> 3637
T5M <5,5> 3761      T5MI <7,5> 3179
T6M <5,6> 3427      T6MI <7,6> 2263
T7M <5,7> 2759      T7MI <7,7> 431
T8M <5,8> 1423      T8MI <7,8> 862
T9M <5,9> 2846      T9MI <7,9> 1724
T10M <5,10> 1597      T10MI <7,10> 3448
T11M <5,11> 3194      T11MI <7,11> 2801

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 3521Scale 3521: Wanian, Ian Ring Music TheoryWanian
Scale 3525Scale 3525: Zocrian, Ian Ring Music TheoryZocrian
Scale 3527Scale 3527: Ronyllic, Ian Ring Music TheoryRonyllic
Scale 3531Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
Scale 3539Scale 3539: Aeoryllic, Ian Ring Music TheoryAeoryllic
Scale 3555Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
Scale 3459Scale 3459: Vocian, Ian Ring Music TheoryVocian
Scale 3491Scale 3491: Tharian, Ian Ring Music TheoryTharian
Scale 3395Scale 3395: Vepian, Ian Ring Music TheoryVepian
Scale 3267Scale 3267: Urfian, Ian Ring Music TheoryUrfian
Scale 3779Scale 3779, Ian Ring Music Theory
Scale 4035Scale 4035: Octatonic Chromatic 7, Ian Ring Music TheoryOctatonic Chromatic 7
Scale 2499Scale 2499: Pirian, Ian Ring Music TheoryPirian
Scale 3011Scale 3011, Ian Ring Music Theory
Scale 1475Scale 1475: Uffian, Ian Ring Music TheoryUffian

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.