The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 539: "Delian"

Scale 539: Delian, 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: 2825


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: 155


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, 2, 1, 5, 3]

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, 3, 2, 1, 1>

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

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.

(11, 7, 36)

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 TriadsA{9,1,4}121
Minor Triadsam{9,0,4}210.67
Diminished Triads{9,0,3}121
Parsimonious Voice Leading Between Common Triads of Scale 539. Created by Ian Ring ©2019 am am a°->am A A am->A

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.

Central Verticesam
Peripheral Verticesa°, A


Modes are the rotational transformation of this scale. Scale 539 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2317
Scale 2317: Odoian, Ian Ring Music TheoryOdoian
3rd mode:
Scale 1603
Scale 1603: Juxian, Ian Ring Music TheoryJuxian
4th mode:
Scale 2849
Scale 2849: Rubian, Ian Ring Music TheoryRubian
5th mode:
Scale 217
Scale 217: Biwian, Ian Ring Music TheoryBiwian


The prime form of this scale is Scale 155

Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian


The pentatonic modal family [539, 2317, 1603, 2849, 217] (Forte: 5-16) is the complement of the heptatonic modal family [623, 889, 1939, 2359, 3017, 3227, 3661] (Forte: 7-16)


The inverse of a scale is a reflection using the root as its axis. The inverse of 539 is 2825

Scale 2825Scale 2825: Rumian, Ian Ring Music TheoryRumian


Only scales that are chiral will have an enantiomorph. Scale 539 is chiral, and its enantiomorph is scale 2825

Scale 2825Scale 2825: Rumian, Ian Ring Music TheoryRumian


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> 539       T0I <11,0> 2825
T1 <1,1> 1078      T1I <11,1> 1555
T2 <1,2> 2156      T2I <11,2> 3110
T3 <1,3> 217      T3I <11,3> 2125
T4 <1,4> 434      T4I <11,4> 155
T5 <1,5> 868      T5I <11,5> 310
T6 <1,6> 1736      T6I <11,6> 620
T7 <1,7> 3472      T7I <11,7> 1240
T8 <1,8> 2849      T8I <11,8> 2480
T9 <1,9> 1603      T9I <11,9> 865
T10 <1,10> 3206      T10I <11,10> 1730
T11 <1,11> 2317      T11I <11,11> 3460
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 809      T0MI <7,0> 665
T1M <5,1> 1618      T1MI <7,1> 1330
T2M <5,2> 3236      T2MI <7,2> 2660
T3M <5,3> 2377      T3MI <7,3> 1225
T4M <5,4> 659      T4MI <7,4> 2450
T5M <5,5> 1318      T5MI <7,5> 805
T6M <5,6> 2636      T6MI <7,6> 1610
T7M <5,7> 1177      T7MI <7,7> 3220
T8M <5,8> 2354      T8MI <7,8> 2345
T9M <5,9> 613      T9MI <7,9> 595
T10M <5,10> 1226      T10MI <7,10> 1190
T11M <5,11> 2452      T11MI <7,11> 2380

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 537Scale 537: Atuian, Ian Ring Music TheoryAtuian
Scale 541Scale 541: Demian, Ian Ring Music TheoryDemian
Scale 543Scale 543: Denian, Ian Ring Music TheoryDenian
Scale 531Scale 531: Degian, Ian Ring Music TheoryDegian
Scale 535Scale 535: Dejian, Ian Ring Music TheoryDejian
Scale 523Scale 523: Debian, Ian Ring Music TheoryDebian
Scale 555Scale 555: Aeolycritonic, Ian Ring Music TheoryAeolycritonic
Scale 571Scale 571: Kynimic, Ian Ring Music TheoryKynimic
Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic
Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic
Scale 795Scale 795: Aeologimic, Ian Ring Music TheoryAeologimic
Scale 27Scale 27: Adoian, Ian Ring Music TheoryAdoian
Scale 283Scale 283: Aerylitonic, Ian Ring Music TheoryAerylitonic
Scale 1051Scale 1051: Gifian, Ian Ring Music TheoryGifian
Scale 1563Scale 1563: Joyian, Ian Ring Music TheoryJoyian
Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian

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.