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Scale 3213: "Eponimic"

Scale 3213: Eponimic, 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

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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
Eponimic

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

{0,2,3,7,10,11}

Forte Number

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

6-14

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: 1575

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.

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.

5

Prime Form

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

no
prime: 315

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, 2, 3, 4, 3, 0]

Interval Spectrum

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

p3m4n3s2d3

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

3.333

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}221.2
G{7,11,2}221.2
Minor Triadscm{0,3,7}131.6
gm{7,10,2}231.4
Augmented TriadsD♯+{3,7,11}321
Parsimonious Voice Leading Between Common Triads of Scale 3213. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ D# D# D#->D#+ gm gm D#->gm Parsimonious Voice Leading Between Common Triads of Scale 3213. Created by Ian Ring ©2019 G D#+->G gm->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
Radius2
Self-Centeredno
Central VerticesD♯, D♯+, G
Peripheral Verticescm, gm

Modes

Modes are the rotational transformation of this scale. Scale 3213 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1827
Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic
3rd mode:
Scale 2961
Scale 2961: Bygimic, Ian Ring Music TheoryBygimic
4th mode:
Scale 441
Scale 441: Thycrimic, Ian Ring Music TheoryThycrimic
5th mode:
Scale 567
Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
6th mode:
Scale 2331
Scale 2331: Dylimic, Ian Ring Music TheoryDylimic

Prime

The prime form of this scale is Scale 315

Scale 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic

Complement

The hexatonic modal family [3213, 1827, 2961, 441, 567, 2331] (Forte: 6-14) is the complement of the hexatonic modal family [315, 945, 1575, 2205, 2835, 3465] (Forte: 6-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3213 is 1575

Scale 1575Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3213 is chiral, and its enantiomorph is scale 1575

Scale 1575Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic

Transformations:

T0 3213  T0I 1575
T1 2331  T1I 3150
T2 567  T2I 2205
T3 1134  T3I 315
T4 2268  T4I 630
T5 441  T5I 1260
T6 882  T6I 2520
T7 1764  T7I 945
T8 3528  T8I 1890
T9 2961  T9I 3780
T10 1827  T10I 3465
T11 3654  T11I 2835

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 3215Scale 3215: Katydian, Ian Ring Music TheoryKatydian
Scale 3209Scale 3209: Aeraphitonic, Ian Ring Music TheoryAeraphitonic
Scale 3211Scale 3211: Epacrimic, Ian Ring Music TheoryEpacrimic
Scale 3205Scale 3205, Ian Ring Music Theory
Scale 3221Scale 3221: Bycrimic, Ian Ring Music TheoryBycrimic
Scale 3229Scale 3229: Aeolaptian, Ian Ring Music TheoryAeolaptian
Scale 3245Scale 3245: Mela Varunapriya, Ian Ring Music TheoryMela Varunapriya
Scale 3277Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati
Scale 3085Scale 3085, Ian Ring Music Theory
Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
Scale 3341Scale 3341, Ian Ring Music Theory
Scale 3469Scale 3469: Monian, Ian Ring Music TheoryMonian
Scale 3725Scale 3725: Kyrian, Ian Ring Music TheoryKyrian
Scale 2189Scale 2189: Zagitonic, Ian Ring Music TheoryZagitonic
Scale 2701Scale 2701: Hawaiian, Ian Ring Music TheoryHawaiian
Scale 1165Scale 1165: Gycritonic, Ian Ring Music TheoryGycritonic

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.