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Scale 931: "Raga Kalakanthi"

Scale 931: Raga Kalakanthi, 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 Kalakanthi
Dozenal
Fukian
Zeitler
Bacrimic

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

{0,1,5,7,8,9}

Forte Number

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

6-16

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.

none

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.

yes
enantiomorph: 2233

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

3

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: 371

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.

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

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

Interval Spectrum

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

p3m4n2s2d3t

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

Spectra Variation

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

3

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.

2.116

Polygon Perimeter

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

5.699

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.

none

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.

(14, 19, 65)

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♯{1,5,8}221
F{5,9,0}221
Minor Triadsfm{5,8,0}221
Augmented TriadsC♯+{1,5,9}221

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

Parsimonious Voice Leading Between Common Triads of Scale 931. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F fm->F

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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 931 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2513
Scale 2513: Aerycrimic, Ian Ring Music TheoryAerycrimic
3rd mode:
Scale 413
Scale 413: Ganimic, Ian Ring Music TheoryGanimic
4th mode:
Scale 1127
Scale 1127: Eparimic, Ian Ring Music TheoryEparimic
5th mode:
Scale 2611
Scale 2611: Raga Vasanta, Ian Ring Music TheoryRaga Vasanta
6th mode:
Scale 3353
Scale 3353: Phraptimic, Ian Ring Music TheoryPhraptimic

Prime

The prime form of this scale is Scale 371

Scale 371Scale 371: Rythimic, Ian Ring Music TheoryRythimic

Complement

The hexatonic modal family [931, 2513, 413, 1127, 2611, 3353] (Forte: 6-16) is the complement of the hexatonic modal family [371, 791, 1841, 2233, 2443, 3269] (Forte: 6-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 931 is 2233

Scale 2233Scale 2233: Donimic, Ian Ring Music TheoryDonimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 931 is chiral, and its enantiomorph is scale 2233

Scale 2233Scale 2233: Donimic, Ian Ring Music TheoryDonimic

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> 931       T0I <11,0> 2233
T1 <1,1> 1862      T1I <11,1> 371
T2 <1,2> 3724      T2I <11,2> 742
T3 <1,3> 3353      T3I <11,3> 1484
T4 <1,4> 2611      T4I <11,4> 2968
T5 <1,5> 1127      T5I <11,5> 1841
T6 <1,6> 2254      T6I <11,6> 3682
T7 <1,7> 413      T7I <11,7> 3269
T8 <1,8> 826      T8I <11,8> 2443
T9 <1,9> 1652      T9I <11,9> 791
T10 <1,10> 3304      T10I <11,10> 1582
T11 <1,11> 2513      T11I <11,11> 3164
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2611      T0MI <7,0> 2443
T1M <5,1> 1127      T1MI <7,1> 791
T2M <5,2> 2254      T2MI <7,2> 1582
T3M <5,3> 413      T3MI <7,3> 3164
T4M <5,4> 826      T4MI <7,4> 2233
T5M <5,5> 1652      T5MI <7,5> 371
T6M <5,6> 3304      T6MI <7,6> 742
T7M <5,7> 2513      T7MI <7,7> 1484
T8M <5,8> 931       T8MI <7,8> 2968
T9M <5,9> 1862      T9MI <7,9> 1841
T10M <5,10> 3724      T10MI <7,10> 3682
T11M <5,11> 3353      T11MI <7,11> 3269

The transformations that map this set to itself are: T0, T8M

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 929Scale 929: Fujian, Ian Ring Music TheoryFujian
Scale 933Scale 933: Dadimic, Ian Ring Music TheoryDadimic
Scale 935Scale 935: Chromatic Dorian, Ian Ring Music TheoryChromatic Dorian
Scale 939Scale 939: Mela Senavati, Ian Ring Music TheoryMela Senavati
Scale 947Scale 947: Mela Gayakapriya, Ian Ring Music TheoryMela Gayakapriya
Scale 899Scale 899: Foqian, Ian Ring Music TheoryFoqian
Scale 915Scale 915: Raga Kalagada, Ian Ring Music TheoryRaga Kalagada
Scale 963Scale 963: Gacian, Ian Ring Music TheoryGacian
Scale 995Scale 995: Phrathian, Ian Ring Music TheoryPhrathian
Scale 803Scale 803: Loritonic, Ian Ring Music TheoryLoritonic
Scale 867Scale 867: Phrocrimic, Ian Ring Music TheoryPhrocrimic
Scale 675Scale 675: Altered Pentatonic, Ian Ring Music TheoryAltered Pentatonic
Scale 419Scale 419: Hon-kumoi-joshi, Ian Ring Music TheoryHon-kumoi-joshi
Scale 1443Scale 1443: Raga Phenadyuti, Ian Ring Music TheoryRaga Phenadyuti
Scale 1955Scale 1955: Sonian, Ian Ring Music TheorySonian
Scale 2979Scale 2979: Gyptian, Ian Ring Music TheoryGyptian

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.