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Scale 2477: "Harmonic Minor"

Scale 2477: 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
Harmonic Minor
Schenkerian
Mischung 4
Hindustani
Pilu That
Carnatic
Mela Kiravani
Raga Kiranavali
Unknown / Unsorted
Kirvani
Kalyana Vasantha
Deshi(3)
Sultani Yakah
Zhalibny Minor
Armoniko minore
Mohammedan
Arabic
Maqam Bayat-e-Esfahan
Zeitler
Mydian
Dozenal
Harian
Carnatic Melakarta
Keeravani
Carnatic Numbered Melakarta
21st 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

{0,2,3,5,7,8,11}

Forte Number

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

7-32

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: 1715

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.

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.

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: 859

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, 2, 2, 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.

<3, 3, 5, 4, 4, 2>

Interval Spectrum

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

p4m4n5s3d3t2

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

Spectra Variation

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

1.429

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.

2.549

Polygon Perimeter

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

5.967

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".

Proper

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.

(0, 18, 82)

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}331.8
G♯{8,0,3}331.7
Minor Triadscm{0,3,7}232
fm{5,8,0}331.8
g♯m{8,11,3}431.6
Augmented TriadsD♯+{3,7,11}331.7
Diminished Triads{2,5,8}232
{5,8,11}231.9
g♯°{8,11,2}231.9
{11,2,5}232
Parsimonious Voice Leading Between Common Triads of Scale 2477. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# fm fm d°->fm d°->b° Parsimonious Voice Leading Between Common Triads of Scale 2477. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m f°->fm f°->g#m fm->G# g#° g#° G->g#° G->b° g#°->g#m g#m->G#

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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2477 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1643
Scale 1643: Locrian Natural 6, Ian Ring Music TheoryLocrian Natural 6
3rd mode:
Scale 2869
Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
4th mode:
Scale 1741
Scale 1741: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
5th mode:
Scale 1459
Scale 1459: Phrygian Dominant, Ian Ring Music TheoryPhrygian Dominant
6th mode:
Scale 2777
Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic
7th mode:
Scale 859
Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrianThis is the prime mode

Prime

The prime form of this scale is Scale 859

Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian

Complement

The heptatonic modal family [2477, 1643, 2869, 1741, 1459, 2777, 859] (Forte: 7-32) is the complement of the pentatonic modal family [595, 665, 805, 1225, 2345] (Forte: 5-32)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2477 is 1715

Scale 1715Scale 1715: Harmonic Minor Inverse, Ian Ring Music TheoryHarmonic Minor Inverse

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2477 is chiral, and its enantiomorph is scale 1715

Scale 1715Scale 1715: Harmonic Minor Inverse, Ian Ring Music TheoryHarmonic Minor Inverse

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> 2477       T0I <11,0> 1715
T1 <1,1> 859      T1I <11,1> 3430
T2 <1,2> 1718      T2I <11,2> 2765
T3 <1,3> 3436      T3I <11,3> 1435
T4 <1,4> 2777      T4I <11,4> 2870
T5 <1,5> 1459      T5I <11,5> 1645
T6 <1,6> 2918      T6I <11,6> 3290
T7 <1,7> 1741      T7I <11,7> 2485
T8 <1,8> 3482      T8I <11,8> 875
T9 <1,9> 2869      T9I <11,9> 1750
T10 <1,10> 1643      T10I <11,10> 3500
T11 <1,11> 3286      T11I <11,11> 2905
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3227      T0MI <7,0> 2855
T1M <5,1> 2359      T1MI <7,1> 1615
T2M <5,2> 623      T2MI <7,2> 3230
T3M <5,3> 1246      T3MI <7,3> 2365
T4M <5,4> 2492      T4MI <7,4> 635
T5M <5,5> 889      T5MI <7,5> 1270
T6M <5,6> 1778      T6MI <7,6> 2540
T7M <5,7> 3556      T7MI <7,7> 985
T8M <5,8> 3017      T8MI <7,8> 1970
T9M <5,9> 1939      T9MI <7,9> 3940
T10M <5,10> 3878      T10MI <7,10> 3785
T11M <5,11> 3661      T11MI <7,11> 3475

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 2479Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed
Scale 2473Scale 2473: Raga Takka, Ian Ring Music TheoryRaga Takka
Scale 2475Scale 2475: Neapolitan Minor, Ian Ring Music TheoryNeapolitan Minor
Scale 2469Scale 2469: Raga Bhinna Pancama, Ian Ring Music TheoryRaga Bhinna Pancama
Scale 2485Scale 2485: Harmonic Major, Ian Ring Music TheoryHarmonic Major
Scale 2493Scale 2493: Manyllic, Ian Ring Music TheoryManyllic
Scale 2445Scale 2445: Zadimic, Ian Ring Music TheoryZadimic
Scale 2461Scale 2461: Sagian, Ian Ring Music TheorySagian
Scale 2509Scale 2509: Double Harmonic Minor, Ian Ring Music TheoryDouble Harmonic Minor
Scale 2541Scale 2541: Algerian, Ian Ring Music TheoryAlgerian
Scale 2349Scale 2349: Raga Ghantana, Ian Ring Music TheoryRaga Ghantana
Scale 2413Scale 2413: Locrian Natural 2, Ian Ring Music TheoryLocrian Natural 2
Scale 2221Scale 2221: Raga Sindhura Kafi, Ian Ring Music TheoryRaga Sindhura Kafi
Scale 2733Scale 2733: Melodic Minor Ascending, Ian Ring Music TheoryMelodic Minor Ascending
Scale 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 3501Scale 3501: Maqam Nahawand, Ian Ring Music TheoryMaqam Nahawand
Scale 429Scale 429: Koptimic, Ian Ring Music TheoryKoptimic
Scale 1453Scale 1453: Aeolian, Ian Ring Music TheoryAeolian

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.