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Scale 2261: "Raga Caturangini"

Scale 2261: Raga Caturangini, 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

Carnatic
Raga Caturangini
Unknown / Unsorted
Ratnakanthi
Zeitler
Phrolimic
Dozenal
Nukian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,2,4,6,7,11}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z26

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

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.

[3]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

no

Hemitonia

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

2 (dihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

Imperfections

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.

2

Modes

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.

5

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 427

Generator

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

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[2, 2, 2, 1, 4, 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.

<2, 3, 2, 3, 4, 1>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p4m3n2s3d2t

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

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

2

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

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.

no

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.

2.232

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.767

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

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.

no

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.

[6]

Propriety

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".

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.

(4, 12, 58)

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 TriadsC{0,4,7}131.5
G{7,11,2}221
Minor Triadsem{4,7,11}221
bm{11,2,6}131.5
Parsimonious Voice Leading Between Common Triads of Scale 2261. Created by Ian Ring ©2019 C C em em C->em Parsimonious Voice Leading Between Common Triads of Scale 2261. Created by Ian Ring ©2019 G em->G bm bm G->bm

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesem, G
Peripheral VerticesC, bm

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Major: {0, 4, 7}
Minor: {11, 2, 6}

Modes

Modes are the rotational transformation of this scale. Scale 2261 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1589
Scale 1589: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
3rd mode:
Scale 1421
Scale 1421: Raga Trimurti, Ian Ring Music TheoryRaga Trimurti
4th mode:
Scale 1379
Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic
5th mode:
Scale 2737
Scale 2737: Raga Hari Nata, Ian Ring Music TheoryRaga Hari Nata
6th mode:
Scale 427
Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha SimantiniThis is the prime mode

Prime

The prime form of this scale is Scale 427

Scale 427Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha Simantini

Complement

The hexatonic modal family [2261, 1589, 1421, 1379, 2737, 427] (Forte: 6-Z26) is the complement of the hexatonic modal family [679, 917, 1253, 1337, 2387, 3241] (Forte: 6-Z48)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2261 is 1379

Scale 1379Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic

Transformations:

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> 2261       T0I <11,0> 1379
T1 <1,1> 427      T1I <11,1> 2758
T2 <1,2> 854      T2I <11,2> 1421
T3 <1,3> 1708      T3I <11,3> 2842
T4 <1,4> 3416      T4I <11,4> 1589
T5 <1,5> 2737      T5I <11,5> 3178
T6 <1,6> 1379      T6I <11,6> 2261
T7 <1,7> 2758      T7I <11,7> 427
T8 <1,8> 1421      T8I <11,8> 854
T9 <1,9> 2842      T9I <11,9> 1708
T10 <1,10> 1589      T10I <11,10> 3416
T11 <1,11> 3178      T11I <11,11> 2737
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3521      T0MI <7,0> 119
T1M <5,1> 2947      T1MI <7,1> 238
T2M <5,2> 1799      T2MI <7,2> 476
T3M <5,3> 3598      T3MI <7,3> 952
T4M <5,4> 3101      T4MI <7,4> 1904
T5M <5,5> 2107      T5MI <7,5> 3808
T6M <5,6> 119      T6MI <7,6> 3521
T7M <5,7> 238      T7MI <7,7> 2947
T8M <5,8> 476      T8MI <7,8> 1799
T9M <5,9> 952      T9MI <7,9> 3598
T10M <5,10> 1904      T10MI <7,10> 3101
T11M <5,11> 3808      T11MI <7,11> 2107

The transformations that map this set to itself are: T0, T6I

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 2263Scale 2263: Lycrian, Ian Ring Music TheoryLycrian
Scale 2257Scale 2257: Lydian Pentatonic, Ian Ring Music TheoryLydian Pentatonic
Scale 2259Scale 2259: Raga Mandari, Ian Ring Music TheoryRaga Mandari
Scale 2265Scale 2265: Raga Rasamanjari, Ian Ring Music TheoryRaga Rasamanjari
Scale 2269Scale 2269: Pygian, Ian Ring Music TheoryPygian
Scale 2245Scale 2245: Raga Vaijayanti, Ian Ring Music TheoryRaga Vaijayanti
Scale 2253Scale 2253: Raga Amarasenapriya, Ian Ring Music TheoryRaga Amarasenapriya
Scale 2277Scale 2277: Kagimic, Ian Ring Music TheoryKagimic
Scale 2293Scale 2293: Gorian, Ian Ring Music TheoryGorian
Scale 2197Scale 2197: Raga Hamsadhvani, Ian Ring Music TheoryRaga Hamsadhvani
Scale 2229Scale 2229: Raga Nalinakanti, Ian Ring Music TheoryRaga Nalinakanti
Scale 2133Scale 2133: Raga Kumurdaki, Ian Ring Music TheoryRaga Kumurdaki
Scale 2389Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
Scale 2517Scale 2517: Harmonic Lydian, Ian Ring Music TheoryHarmonic Lydian
Scale 2773Scale 2773: Lydian, Ian Ring Music TheoryLydian
Scale 3285Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
Scale 213Scale 213: Bitian, Ian Ring Music TheoryBitian
Scale 1237Scale 1237: Salimic, Ian Ring Music TheorySalimic

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 (http://allthescales.org) 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.