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Scale 2721: "Raga Puruhutika"

Scale 2721: Raga Puruhutika, 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 Puruhutika
Unknown / Unsorted



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: 171


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

1 (unhemitonic)


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: 171


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.

[5, 2, 2, 2, 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.

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

(9, 3, 32)

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{5,9,0}000

The following pitch classes are not present in any of the common triads: {7,11}

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 2721 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 213
Scale 213: Bitian, Ian Ring Music TheoryBitian
3rd mode:
Scale 1077
Scale 1077: Govian, Ian Ring Music TheoryGovian
4th mode:
Scale 1293
Scale 1293: Huxian, Ian Ring Music TheoryHuxian
5th mode:
Scale 1347
Scale 1347: Igoian, Ian Ring Music TheoryIgoian


The prime form of this scale is Scale 171

Scale 171Scale 171: Pruian, Ian Ring Music TheoryPruian


The pentatonic modal family [2721, 213, 1077, 1293, 1347] (Forte: 5-24) is the complement of the heptatonic modal family [687, 1401, 1509, 1941, 2391, 3243, 3669] (Forte: 7-24)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2721 is 171

Scale 171Scale 171: Pruian, Ian Ring Music TheoryPruian


Only scales that are chiral will have an enantiomorph. Scale 2721 is chiral, and its enantiomorph is scale 171

Scale 171Scale 171: Pruian, Ian Ring Music TheoryPruian


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> 2721       T0I <11,0> 171
T1 <1,1> 1347      T1I <11,1> 342
T2 <1,2> 2694      T2I <11,2> 684
T3 <1,3> 1293      T3I <11,3> 1368
T4 <1,4> 2586      T4I <11,4> 2736
T5 <1,5> 1077      T5I <11,5> 1377
T6 <1,6> 2154      T6I <11,6> 2754
T7 <1,7> 213      T7I <11,7> 1413
T8 <1,8> 426      T8I <11,8> 2826
T9 <1,9> 852      T9I <11,9> 1557
T10 <1,10> 1704      T10I <11,10> 3114
T11 <1,11> 3408      T11I <11,11> 2133
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2691      T0MI <7,0> 2091
T1M <5,1> 1287      T1MI <7,1> 87
T2M <5,2> 2574      T2MI <7,2> 174
T3M <5,3> 1053      T3MI <7,3> 348
T4M <5,4> 2106      T4MI <7,4> 696
T5M <5,5> 117      T5MI <7,5> 1392
T6M <5,6> 234      T6MI <7,6> 2784
T7M <5,7> 468      T7MI <7,7> 1473
T8M <5,8> 936      T8MI <7,8> 2946
T9M <5,9> 1872      T9MI <7,9> 1797
T10M <5,10> 3744      T10MI <7,10> 3594
T11M <5,11> 3393      T11MI <7,11> 3093

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 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 2725Scale 2725: Raga Nagagandhari, Ian Ring Music TheoryRaga Nagagandhari
Scale 2729Scale 2729: Aeragimic, Ian Ring Music TheoryAeragimic
Scale 2737Scale 2737: Raga Hari Nata, Ian Ring Music TheoryRaga Hari Nata
Scale 2689Scale 2689: Ragian, Ian Ring Music TheoryRagian
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 2753Scale 2753: Ritian, Ian Ring Music TheoryRitian
Scale 2785Scale 2785: Ronian, Ian Ring Music TheoryRonian
Scale 2593Scale 2593: Puxian, Ian Ring Music TheoryPuxian
Scale 2657Scale 2657: Qokian, Ian Ring Music TheoryQokian
Scale 2849Scale 2849: Rubian, Ian Ring Music TheoryRubian
Scale 2977Scale 2977: Sobian, Ian Ring Music TheorySobian
Scale 2209Scale 2209: Nidian, Ian Ring Music TheoryNidian
Scale 2465Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani
Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian
Scale 3745Scale 3745: Xuvian, Ian Ring Music TheoryXuvian
Scale 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 1697Scale 1697: Raga Kuntvarali, Ian Ring Music TheoryRaga Kuntvarali

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.