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Scale 2327: "Epalimic"

Scale 2327: Epalimic, 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

Zeitler
Epalimic
Dozenal
Ofoian

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

{0,1,2,4,8,11}

Forte Number

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

6-Z39

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: 3347

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.

2 (dicohemitonic)

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.

4

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: 317

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

Interval Spectrum

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

p2m3n3s3d3t

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

3.667

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.

(29, 19, 67)

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 TriadsE{4,8,11}221
Minor Triadsc♯m{1,4,8}131.5
Augmented TriadsC+{0,4,8}221
Diminished Triadsg♯°{8,11,2}131.5
Parsimonious Voice Leading Between Common Triads of Scale 2327. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E g#° g#° E->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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC+, E
Peripheral Verticesc♯m, g♯°

Modes

Modes are the rotational transformation of this scale. Scale 2327 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3211
Scale 3211: Epacrimic, Ian Ring Music TheoryEpacrimic
3rd mode:
Scale 3653
Scale 3653: Sathimic, Ian Ring Music TheorySathimic
4th mode:
Scale 1937
Scale 1937: Galimic, Ian Ring Music TheoryGalimic
5th mode:
Scale 377
Scale 377: Kathimic, Ian Ring Music TheoryKathimic
6th mode:
Scale 559
Scale 559: Lylimic, Ian Ring Music TheoryLylimic

Prime

The prime form of this scale is Scale 317

Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic

Complement

The hexatonic modal family [2327, 3211, 3653, 1937, 377, 559] (Forte: 6-Z39) is the complement of the hexatonic modal family [187, 1559, 1889, 2141, 2827, 3461] (Forte: 6-Z10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2327 is 3347

Scale 3347Scale 3347: Synimic, Ian Ring Music TheorySynimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2327 is chiral, and its enantiomorph is scale 3347

Scale 3347Scale 3347: Synimic, Ian Ring Music TheorySynimic

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> 2327       T0I <11,0> 3347
T1 <1,1> 559      T1I <11,1> 2599
T2 <1,2> 1118      T2I <11,2> 1103
T3 <1,3> 2236      T3I <11,3> 2206
T4 <1,4> 377      T4I <11,4> 317
T5 <1,5> 754      T5I <11,5> 634
T6 <1,6> 1508      T6I <11,6> 1268
T7 <1,7> 3016      T7I <11,7> 2536
T8 <1,8> 1937      T8I <11,8> 977
T9 <1,9> 3874      T9I <11,9> 1954
T10 <1,10> 3653      T10I <11,10> 3908
T11 <1,11> 3211      T11I <11,11> 3721
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1457      T0MI <7,0> 437
T1M <5,1> 2914      T1MI <7,1> 874
T2M <5,2> 1733      T2MI <7,2> 1748
T3M <5,3> 3466      T3MI <7,3> 3496
T4M <5,4> 2837      T4MI <7,4> 2897
T5M <5,5> 1579      T5MI <7,5> 1699
T6M <5,6> 3158      T6MI <7,6> 3398
T7M <5,7> 2221      T7MI <7,7> 2701
T8M <5,8> 347      T8MI <7,8> 1307
T9M <5,9> 694      T9MI <7,9> 2614
T10M <5,10> 1388      T10MI <7,10> 1133
T11M <5,11> 2776      T11MI <7,11> 2266

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 2325Scale 2325: Pynitonic, Ian Ring Music TheoryPynitonic
Scale 2323Scale 2323: Doptitonic, Ian Ring Music TheoryDoptitonic
Scale 2331Scale 2331: Dylimic, Ian Ring Music TheoryDylimic
Scale 2335Scale 2335: Epydian, Ian Ring Music TheoryEpydian
Scale 2311Scale 2311: Raga Kumarapriya, Ian Ring Music TheoryRaga Kumarapriya
Scale 2319Scale 2319: Oduian, Ian Ring Music TheoryOduian
Scale 2343Scale 2343: Tharimic, Ian Ring Music TheoryTharimic
Scale 2359Scale 2359: Gadian, Ian Ring Music TheoryGadian
Scale 2391Scale 2391: Molian, Ian Ring Music TheoryMolian
Scale 2455Scale 2455: Bothian, Ian Ring Music TheoryBothian
Scale 2071Scale 2071: Moxian, Ian Ring Music TheoryMoxian
Scale 2199Scale 2199: Dyptimic, Ian Ring Music TheoryDyptimic
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 2839Scale 2839: Lyptian, Ian Ring Music TheoryLyptian
Scale 3351Scale 3351: Crater Scale, Ian Ring Music TheoryCrater Scale
Scale 279Scale 279: Poditonic, Ian Ring Music TheoryPoditonic
Scale 1303Scale 1303: Epolimic, Ian Ring Music TheoryEpolimic

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.