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Scale 2509: "Double Harmonic Minor"

Scale 2509: Double Harmonic Minor, 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

Western Modern
Double Harmonic Minor
Modern Greek
Niavent
Zeitler
Bogian
Dozenal
Pixian
Exoticisms
Hungarian Minor
Egyptian Heptatonic
Flamenco Mode
Carnatic
Mela Simhendramadhyama
Raga Madhava Manohari
Arabic
Maqam Nawa Athar
Turkish
Hisar
Carnatic Melakarta
Simhendramadhyamam
Carnatic Numbered Melakarta
57th Melakarta raga

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,2,3,6,7,8,11}

Forte Number

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

7-22

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.

[1]

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.

4 (multihemitonic)

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.

6

Prime Form

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

no
prime: 871

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, 1, 3, 1, 1, 3, 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.

<4, 2, 4, 5, 4, 2>

Interval Spectrum

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

p4m5n4s2d4t2

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

Spectra Variation

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

1.714

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.

2.433

Polygon Perimeter

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

5.899

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.

yes

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.

[2]

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, 22, 84)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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{7,11,2}331.67
G♯{8,0,3}242
B{11,3,6}331.67
Minor Triadscm{0,3,7}331.67
g♯m{8,11,3}331.67
bm{11,2,6}242
Augmented TriadsD♯+{3,7,11}421.33
Diminished Triads{0,3,6}242
g♯°{8,11,2}242
Parsimonious Voice Leading Between Common Triads of Scale 2509. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ G# G# cm->G# Parsimonious Voice Leading Between Common Triads of Scale 2509. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m D#+->B g#° g#° G->g#° bm bm G->bm g#°->g#m g#m->G# bm->B

view full size

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesD♯+
Peripheral Verticesc°, g♯°, G♯, bm

Modes

Modes are the rotational transformation of this scale. Scale 2509 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1651
Scale 1651: Asian, Ian Ring Music TheoryAsian
3rd mode:
Scale 2873
Scale 2873: Ionian Augmented Sharp 2, Ian Ring Music TheoryIonian Augmented Sharp 2
4th mode:
Scale 871
Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7This is the prime mode
5th mode:
Scale 2483
Scale 2483: Double Harmonic, Ian Ring Music TheoryDouble Harmonic
6th mode:
Scale 3289
Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6
7th mode:
Scale 923
Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian

Prime

The prime form of this scale is Scale 871

Scale 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7

Complement

The heptatonic modal family [2509, 1651, 2873, 871, 2483, 3289, 923] (Forte: 7-22) is the complement of the pentatonic modal family [403, 611, 793, 2249, 2353] (Forte: 5-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2509 is 1651

Scale 1651Scale 1651: Asian, Ian Ring Music TheoryAsian

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> 2509       T0I <11,0> 1651
T1 <1,1> 923      T1I <11,1> 3302
T2 <1,2> 1846      T2I <11,2> 2509
T3 <1,3> 3692      T3I <11,3> 923
T4 <1,4> 3289      T4I <11,4> 1846
T5 <1,5> 2483      T5I <11,5> 3692
T6 <1,6> 871      T6I <11,6> 3289
T7 <1,7> 1742      T7I <11,7> 2483
T8 <1,8> 3484      T8I <11,8> 871
T9 <1,9> 2873      T9I <11,9> 1742
T10 <1,10> 1651      T10I <11,10> 3484
T11 <1,11> 3302      T11I <11,11> 2873
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3289      T0MI <7,0> 871
T1M <5,1> 2483      T1MI <7,1> 1742
T2M <5,2> 871      T2MI <7,2> 3484
T3M <5,3> 1742      T3MI <7,3> 2873
T4M <5,4> 3484      T4MI <7,4> 1651
T5M <5,5> 2873      T5MI <7,5> 3302
T6M <5,6> 1651      T6MI <7,6> 2509
T7M <5,7> 3302      T7MI <7,7> 923
T8M <5,8> 2509       T8MI <7,8> 1846
T9M <5,9> 923      T9MI <7,9> 3692
T10M <5,10> 1846      T10MI <7,10> 3289
T11M <5,11> 3692      T11MI <7,11> 2483

The transformations that map this set to itself are: T0, T2I, T8M, T6MI

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 2511Scale 2511: Aeroptyllic, Ian Ring Music TheoryAeroptyllic
Scale 2505Scale 2505: Mydimic, Ian Ring Music TheoryMydimic
Scale 2507Scale 2507: Todi That, Ian Ring Music TheoryTodi That
Scale 2501Scale 2501: Ralimic, Ian Ring Music TheoryRalimic
Scale 2517Scale 2517: Harmonic Lydian, Ian Ring Music TheoryHarmonic Lydian
Scale 2525Scale 2525: Aeolaryllic, Ian Ring Music TheoryAeolaryllic
Scale 2541Scale 2541: Algerian, Ian Ring Music TheoryAlgerian
Scale 2445Scale 2445: Zadimic, Ian Ring Music TheoryZadimic
Scale 2477Scale 2477: Harmonic Minor, Ian Ring Music TheoryHarmonic Minor
Scale 2381Scale 2381: Takemitsu Linea Mode 1, Ian Ring Music TheoryTakemitsu Linea Mode 1
Scale 2253Scale 2253: Raga Amarasenapriya, Ian Ring Music TheoryRaga Amarasenapriya
Scale 2765Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
Scale 3021Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
Scale 3533Scale 3533: Thadyllic, Ian Ring Music TheoryThadyllic
Scale 461Scale 461: Raga Syamalam, Ian Ring Music TheoryRaga Syamalam
Scale 1485Scale 1485: Minor Romani, Ian Ring Music TheoryMinor Romani

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.