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Scale 1585: "Raga Khamaji Durga"

Scale 1585: Raga Khamaji Durga, 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 Khamaji Durga
Zeitler
Phraditonic
Dozenal
Jumian

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

{0,4,5,9,10}

Forte Number

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

5-20

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: 397

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.

4

Prime Form

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

no
prime: 355

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.

[4, 1, 4, 1, 2]

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, 1, 1, 2, 3, 1>

Interval Spectrum

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

p3m2nsd2t

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

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.

1.799

Polygon Perimeter

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

5.499

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.

(4, 1, 30)

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}110.5
Minor Triadsam{9,0,4}110.5

The following pitch classes are not present in any of the common triads: {10}

Parsimonious Voice Leading Between Common Triads of Scale 1585. Created by Ian Ring ©2019 F F am am F->am

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 1585 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 355
Scale 355: Aeoloritonic, Ian Ring Music TheoryAeoloritonicThis is the prime mode
3rd mode:
Scale 2225
Scale 2225: Ionian Pentatonic, Ian Ring Music TheoryIonian Pentatonic
4th mode:
Scale 395
Scale 395: Phrygian Pentatonic, Ian Ring Music TheoryPhrygian Pentatonic
5th mode:
Scale 2245
Scale 2245: Raga Vaijayanti, Ian Ring Music TheoryRaga Vaijayanti

Prime

The prime form of this scale is Scale 355

Scale 355Scale 355: Aeoloritonic, Ian Ring Music TheoryAeoloritonic

Complement

The pentatonic modal family [1585, 355, 2225, 395, 2245] (Forte: 5-20) is the complement of the heptatonic modal family [743, 919, 1849, 2419, 2507, 3257, 3301] (Forte: 7-20)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1585 is 397

Scale 397Scale 397: Aeolian Pentatonic, Ian Ring Music TheoryAeolian Pentatonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1585 is chiral, and its enantiomorph is scale 397

Scale 397Scale 397: Aeolian Pentatonic, Ian Ring Music TheoryAeolian Pentatonic

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> 1585       T0I <11,0> 397
T1 <1,1> 3170      T1I <11,1> 794
T2 <1,2> 2245      T2I <11,2> 1588
T3 <1,3> 395      T3I <11,3> 3176
T4 <1,4> 790      T4I <11,4> 2257
T5 <1,5> 1580      T5I <11,5> 419
T6 <1,6> 3160      T6I <11,6> 838
T7 <1,7> 2225      T7I <11,7> 1676
T8 <1,8> 355      T8I <11,8> 3352
T9 <1,9> 710      T9I <11,9> 2609
T10 <1,10> 1420      T10I <11,10> 1123
T11 <1,11> 2840      T11I <11,11> 2246
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 775      T0MI <7,0> 3097
T1M <5,1> 1550      T1MI <7,1> 2099
T2M <5,2> 3100      T2MI <7,2> 103
T3M <5,3> 2105      T3MI <7,3> 206
T4M <5,4> 115      T4MI <7,4> 412
T5M <5,5> 230      T5MI <7,5> 824
T6M <5,6> 460      T6MI <7,6> 1648
T7M <5,7> 920      T7MI <7,7> 3296
T8M <5,8> 1840      T8MI <7,8> 2497
T9M <5,9> 3680      T9MI <7,9> 899
T10M <5,10> 3265      T10MI <7,10> 1798
T11M <5,11> 2435      T11MI <7,11> 3596

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 1587Scale 1587: Raga Rudra Pancama, Ian Ring Music TheoryRaga Rudra Pancama
Scale 1589Scale 1589: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
Scale 1593Scale 1593: Zogimic, Ian Ring Music TheoryZogimic
Scale 1569Scale 1569: Jocian, Ian Ring Music TheoryJocian
Scale 1577Scale 1577: Raga Chandrakauns (Kafi), Ian Ring Music TheoryRaga Chandrakauns (Kafi)
Scale 1553Scale 1553: Josian, Ian Ring Music TheoryJosian
Scale 1617Scale 1617: Phronitonic, Ian Ring Music TheoryPhronitonic
Scale 1649Scale 1649: Bolimic, Ian Ring Music TheoryBolimic
Scale 1713Scale 1713: Raga Khamas, Ian Ring Music TheoryRaga Khamas
Scale 1841Scale 1841: Thogimic, Ian Ring Music TheoryThogimic
Scale 1073Scale 1073: Gosian, Ian Ring Music TheoryGosian
Scale 1329Scale 1329: Epygitonic, Ian Ring Music TheoryEpygitonic
Scale 561Scale 561: Phratic, Ian Ring Music TheoryPhratic
Scale 2609Scale 2609: Raga Bhinna Shadja, Ian Ring Music TheoryRaga Bhinna Shadja
Scale 3633Scale 3633: Daptimic, Ian Ring Music TheoryDaptimic

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.