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Scale 529: "Raga Bilwadala"

Scale 529: Raga Bilwadala, 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

Raga Bilwadala



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 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: 265


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.

prime: 137


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.

[4, 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.

<0, 0, 1, 1, 1, 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> = {3,4,5}
<2> = {7,8,9}

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".

Strictly Proper

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.

(0, 0, 6)

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
Minor Triadsam{9,0,4}000

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 529 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 289
Scale 289: Valian, Ian Ring Music TheoryValian
3rd mode:
Scale 137
Scale 137: Ute Tritonic, Ian Ring Music TheoryUte TritonicThis is the prime mode


The prime form of this scale is Scale 137

Scale 137Scale 137: Ute Tritonic, Ian Ring Music TheoryUte Tritonic


The tritonic modal family [529, 289, 137] (Forte: 3-11) is the complement of the enneatonic modal family [1775, 1915, 1975, 2935, 3005, 3035, 3515, 3565, 3805] (Forte: 9-11)


The inverse of a scale is a reflection using the root as its axis. The inverse of 529 is 265

Scale 265Scale 265: Boxian, Ian Ring Music TheoryBoxian


Only scales that are chiral will have an enantiomorph. Scale 529 is chiral, and its enantiomorph is scale 265

Scale 265Scale 265: Boxian, Ian Ring Music TheoryBoxian


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> 529       T0I <11,0> 265
T1 <1,1> 1058      T1I <11,1> 530
T2 <1,2> 2116      T2I <11,2> 1060
T3 <1,3> 137      T3I <11,3> 2120
T4 <1,4> 274      T4I <11,4> 145
T5 <1,5> 548      T5I <11,5> 290
T6 <1,6> 1096      T6I <11,6> 580
T7 <1,7> 2192      T7I <11,7> 1160
T8 <1,8> 289      T8I <11,8> 2320
T9 <1,9> 578      T9I <11,9> 545
T10 <1,10> 1156      T10I <11,10> 1090
T11 <1,11> 2312      T11I <11,11> 2180
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 769      T0MI <7,0> 25
T1M <5,1> 1538      T1MI <7,1> 50
T2M <5,2> 3076      T2MI <7,2> 100
T3M <5,3> 2057      T3MI <7,3> 200
T4M <5,4> 19      T4MI <7,4> 400
T5M <5,5> 38      T5MI <7,5> 800
T6M <5,6> 76      T6MI <7,6> 1600
T7M <5,7> 152      T7MI <7,7> 3200
T8M <5,8> 304      T8MI <7,8> 2305
T9M <5,9> 608      T9MI <7,9> 515
T10M <5,10> 1216      T10MI <7,10> 1030
T11M <5,11> 2432      T11MI <7,11> 2060

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 531Scale 531: Degian, Ian Ring Music TheoryDegian
Scale 533Scale 533: Dehian, Ian Ring Music TheoryDehian
Scale 537Scale 537: Atuian, Ian Ring Music TheoryAtuian
Scale 513Scale 513: Major Sixth Ditone, Ian Ring Music TheoryMajor Sixth Ditone
Scale 521Scale 521: Astian, Ian Ring Music TheoryAstian
Scale 545Scale 545: Dewian, Ian Ring Music TheoryDewian
Scale 561Scale 561: Phratic, Ian Ring Music TheoryPhratic
Scale 593Scale 593: Saric, Ian Ring Music TheorySaric
Scale 657Scale 657: Epathic, Ian Ring Music TheoryEpathic
Scale 785Scale 785: Aeoloric, Ian Ring Music TheoryAeoloric
Scale 17Scale 17: Major Third Ditone, Ian Ring Music TheoryMajor Third Ditone
Scale 273Scale 273: Augmented Triad, Ian Ring Music TheoryAugmented Triad
Scale 1041Scale 1041: Hitian, Ian Ring Music TheoryHitian
Scale 1553Scale 1553: Josian, Ian Ring Music TheoryJosian
Scale 2577Scale 2577: Punian, Ian Ring Music TheoryPunian

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.