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Scale 3211: "Epacrimic"

Scale 3211: Epacrimic, 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

41161837294116105072918310504116183
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
Epacrimic

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

{0,1,3,7,10,11}

Forte Number

A code assigned by theorist Alan 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: 2599

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

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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 Hansen.

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

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 TriadsD♯{3,7,10}221
Minor Triadscm{0,3,7}131.5
Augmented TriadsD♯+{3,7,11}221
Diminished Triads{7,10,1}131.5
Parsimonious Voice Leading Between Common Triads of Scale 3211. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ D# D# D#->D#+ D#->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 VerticesD♯, D♯+
Peripheral Verticescm, g°

Modes

Modes are the rotational transformation of this scale. Scale 3211 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3653
Scale 3653: Sathimic, Ian Ring Music TheorySathimic
3rd mode:
Scale 1937
Scale 1937: Galimic, Ian Ring Music TheoryGalimic
4th mode:
Scale 377
Scale 377: Kathimic, Ian Ring Music TheoryKathimic
5th mode:
Scale 559
Scale 559: Lylimic, Ian Ring Music TheoryLylimic
6th mode:
Scale 2327
Scale 2327: Epalimic, Ian Ring Music TheoryEpalimic

Prime

The prime form of this scale is Scale 317

Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic

Complement

The hexatonic modal family [3211, 3653, 1937, 377, 559, 2327] (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 3211 is 2599

Scale 2599Scale 2599: Malimic, Ian Ring Music TheoryMalimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3211 is chiral, and its enantiomorph is scale 2599

Scale 2599Scale 2599: Malimic, Ian Ring Music TheoryMalimic

Transformations:

T0 3211  T0I 2599
T1 2327  T1I 1103
T2 559  T2I 2206
T3 1118  T3I 317
T4 2236  T4I 634
T5 377  T5I 1268
T6 754  T6I 2536
T7 1508  T7I 977
T8 3016  T8I 1954
T9 1937  T9I 3908
T10 3874  T10I 3721
T11 3653  T11I 3347

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 3209Scale 3209: Aeraphitonic, Ian Ring Music TheoryAeraphitonic
Scale 3213Scale 3213: Eponimic, Ian Ring Music TheoryEponimic
Scale 3215Scale 3215: Katydian, Ian Ring Music TheoryKatydian
Scale 3203Scale 3203, Ian Ring Music Theory
Scale 3207Scale 3207, Ian Ring Music Theory
Scale 3219Scale 3219: Ionaphimic, Ian Ring Music TheoryIonaphimic
Scale 3227Scale 3227: Aeolocrian, Ian Ring Music TheoryAeolocrian
Scale 3243Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
Scale 3275Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani
Scale 3083Scale 3083, Ian Ring Music Theory
Scale 3147Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic
Scale 3339Scale 3339, Ian Ring Music Theory
Scale 3467Scale 3467: Katonian, Ian Ring Music TheoryKatonian
Scale 3723Scale 3723: Myptian, Ian Ring Music TheoryMyptian
Scale 2187Scale 2187: Ionothitonic, Ian Ring Music TheoryIonothitonic
Scale 2699Scale 2699: Sythimic, Ian Ring Music TheorySythimic
Scale 1163Scale 1163: Raga Rukmangi, Ian Ring Music TheoryRaga Rukmangi

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.