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Scale 331: "Raga Chhaya Todi"

Scale 331: Raga Chhaya Todi, 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 Chhaya Todi
Western Modern
Locrian Pentatonic 1
Myanmar
Hsikci
Zeitler
Byptitonic
Dozenal
CANian

Analysis

Cardinality

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

{0,1,3,6,8}

Forte Number

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

5-29

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: 2641

Hemitonia

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

1 (unhemitonic)

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.

4

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

yes

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, an indicator of maximum hierarchization.

no

Interval Structure

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

[1, 2, 3, 2, 4]

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, 2, 2, 1, 3, 1>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.25, 0.5, 0.5, 0, 0.75, 0.5>

Interval Spectrum

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

p3mn2s2dt

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

Spectra Variation

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

2.4

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.

2.049

Polygon Perimeter

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

5.664

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(2, 3, 32)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.8

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.2

Generator

This scale has no generator.

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 TriadsG♯{8,0,3}110.5
Diminished Triads{0,3,6}110.5

The following pitch classes are not present in any of the common triads: {1}

Parsimonious Voice Leading Between Common Triads of Scale 331. Created by Ian Ring ©2019 G# G# c°->G#

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 331 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2213
Scale 2213: Raga Desh, Ian Ring Music TheoryRaga Desh
3rd mode:
Scale 1577
Scale 1577: Raga Chandrakauns, Ian Ring Music TheoryRaga Chandrakauns
4th mode:
Scale 709
Scale 709: Raga Shri Kalyan, Ian Ring Music TheoryRaga Shri Kalyan
5th mode:
Scale 1201
Scale 1201: Mixolydian Pentatonic, Ian Ring Music TheoryMixolydian Pentatonic

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [331, 2213, 1577, 709, 1201] (Forte: 5-29) is the complement of the heptatonic modal family [727, 1483, 1721, 1837, 2411, 2789, 3253] (Forte: 7-29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 331 is 2641

Scale 2641Scale 2641: Raga Hindol, Ian Ring Music TheoryRaga Hindol

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 331 is chiral, and its enantiomorph is scale 2641

Scale 2641Scale 2641: Raga Hindol, Ian Ring Music TheoryRaga Hindol

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> 331       T0I <11,0> 2641
T1 <1,1> 662      T1I <11,1> 1187
T2 <1,2> 1324      T2I <11,2> 2374
T3 <1,3> 2648      T3I <11,3> 653
T4 <1,4> 1201      T4I <11,4> 1306
T5 <1,5> 2402      T5I <11,5> 2612
T6 <1,6> 709      T6I <11,6> 1129
T7 <1,7> 1418      T7I <11,7> 2258
T8 <1,8> 2836      T8I <11,8> 421
T9 <1,9> 1577      T9I <11,9> 842
T10 <1,10> 3154      T10I <11,10> 1684
T11 <1,11> 2213      T11I <11,11> 3368
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 121      T0MI <7,0> 961
T1M <5,1> 242      T1MI <7,1> 1922
T2M <5,2> 484      T2MI <7,2> 3844
T3M <5,3> 968      T3MI <7,3> 3593
T4M <5,4> 1936      T4MI <7,4> 3091
T5M <5,5> 3872      T5MI <7,5> 2087
T6M <5,6> 3649      T6MI <7,6> 79
T7M <5,7> 3203      T7MI <7,7> 158
T8M <5,8> 2311      T8MI <7,8> 316
T9M <5,9> 527      T9MI <7,9> 632
T10M <5,10> 1054      T10MI <7,10> 1264
T11M <5,11> 2108      T11MI <7,11> 2528

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 329Scale 329: Lonic, Ian Ring Music TheoryLonic
Scale 333Scale 333: Bogitonic, Ian Ring Music TheoryBogitonic
Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic
Scale 323Scale 323: CAJian, Ian Ring Music TheoryCAJian
Scale 327Scale 327: Syptitonic, Ian Ring Music TheorySyptitonic
Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic
Scale 347Scale 347: Barimic, Ian Ring Music TheoryBarimic
Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 267Scale 267: BOBian, Ian Ring Music TheoryBOBian
Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini
Scale 395Scale 395: Phrygian Pentatonic, Ian Ring Music TheoryPhrygian Pentatonic
Scale 459Scale 459: Zaptimic, Ian Ring Music TheoryZaptimic
Scale 75Scale 75: ILOian, Ian Ring Music TheoryILOian
Scale 203Scale 203: MiaoYao 5 Tone Type 3, Ian Ring Music TheoryMiaoYao 5 Tone Type 3
Scale 587Scale 587: Pathitonic, Ian Ring Music TheoryPathitonic
Scale 843Scale 843: Molimic, Ian Ring Music TheoryMolimic
Scale 1355Scale 1355: Aeolorimic, Ian Ring Music TheoryAeolorimic
Scale 2379Scale 2379: Raga Gurjari Todi, Ian Ring Music TheoryRaga Gurjari Todi

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.