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Scale 3321: "Epagyllic"

Scale 3321: Epagyllic, 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
Epagyllic
Dozenal
Utoian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,3,4,5,6,7,10,11}

Forte Number

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

8-8

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.

[5]

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.

6 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

4 (multicohemitonic)

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.

2

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.

7

Prime Form

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

no
prime: 927

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.

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

<6, 4, 4, 5, 6, 3>

Interval Spectrum

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

p6m5n4s4d6t3

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> = {1,3}
<2> = {2,4}
<3> = {3,5}
<4> = {4,6,8}
<5> = {7,9}
<6> = {8,10}
<7> = {9,11}

Spectra Variation

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

2

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.

2.5

Polygon Perimeter

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

5.934

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.

[10]

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.

(44, 16, 93)

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 TriadsC{0,4,7}242
D♯{3,7,10}331.67
B{11,3,6}331.67
Minor Triadscm{0,3,7}331.67
d♯m{3,6,10}242
em{4,7,11}331.67
Augmented TriadsD♯+{3,7,11}421.33
Diminished Triads{0,3,6}242
{4,7,10}242

The following pitch classes are not present in any of the common triads: {5}

Parsimonious Voice Leading Between Common Triads of Scale 3321. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ em em C->em d#m d#m D# D# d#m->D# d#m->B D#->D#+ D#->e° D#+->em D#+->B e°->em

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°, C, d♯m, e°

Modes

Modes are the rotational transformation of this scale. Scale 3321 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 927
Scale 927: Gaptyllic, Ian Ring Music TheoryGaptyllicThis is the prime mode
3rd mode:
Scale 2511
Scale 2511: Aeroptyllic, Ian Ring Music TheoryAeroptyllic
4th mode:
Scale 3303
Scale 3303: Mylyllic, Ian Ring Music TheoryMylyllic
5th mode:
Scale 3699
Scale 3699: Galyllic, Ian Ring Music TheoryGalyllic
6th mode:
Scale 3897
Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic
7th mode:
Scale 999
Scale 999: Ionodyllic, Ian Ring Music TheoryIonodyllic
8th mode:
Scale 2547
Scale 2547: Raga Ramkali, Ian Ring Music TheoryRaga Ramkali

Prime

The prime form of this scale is Scale 927

Scale 927Scale 927: Gaptyllic, Ian Ring Music TheoryGaptyllic

Complement

The octatonic modal family [3321, 927, 2511, 3303, 3699, 3897, 999, 2547] (Forte: 8-8) is the complement of the tetratonic modal family [99, 387, 2097, 2241] (Forte: 4-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3321 is 999

Scale 999Scale 999: Ionodyllic, Ian Ring Music TheoryIonodyllic

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> 3321       T0I <11,0> 999
T1 <1,1> 2547      T1I <11,1> 1998
T2 <1,2> 999      T2I <11,2> 3996
T3 <1,3> 1998      T3I <11,3> 3897
T4 <1,4> 3996      T4I <11,4> 3699
T5 <1,5> 3897      T5I <11,5> 3303
T6 <1,6> 3699      T6I <11,6> 2511
T7 <1,7> 3303      T7I <11,7> 927
T8 <1,8> 2511      T8I <11,8> 1854
T9 <1,9> 927      T9I <11,9> 3708
T10 <1,10> 1854      T10I <11,10> 3321
T11 <1,11> 3708      T11I <11,11> 2547
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2511      T0MI <7,0> 3699
T1M <5,1> 927      T1MI <7,1> 3303
T2M <5,2> 1854      T2MI <7,2> 2511
T3M <5,3> 3708      T3MI <7,3> 927
T4M <5,4> 3321       T4MI <7,4> 1854
T5M <5,5> 2547      T5MI <7,5> 3708
T6M <5,6> 999      T6MI <7,6> 3321
T7M <5,7> 1998      T7MI <7,7> 2547
T8M <5,8> 3996      T8MI <7,8> 999
T9M <5,9> 3897      T9MI <7,9> 1998
T10M <5,10> 3699      T10MI <7,10> 3996
T11M <5,11> 3303      T11MI <7,11> 3897

The transformations that map this set to itself are: T0, T10I, T4M, 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 3323Scale 3323: Lacrygic, Ian Ring Music TheoryLacrygic
Scale 3325Scale 3325: Mixolygic, Ian Ring Music TheoryMixolygic
Scale 3313Scale 3313: Aeolacrian, Ian Ring Music TheoryAeolacrian
Scale 3317Scale 3317: Katynyllic, Ian Ring Music TheoryKatynyllic
Scale 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian
Scale 3289Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6
Scale 3257Scale 3257: Mela Calanata, Ian Ring Music TheoryMela Calanata
Scale 3193Scale 3193: Zathian, Ian Ring Music TheoryZathian
Scale 3449Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
Scale 3577Scale 3577: Loptygic, Ian Ring Music TheoryLoptygic
Scale 3833Scale 3833: Dycrygic, Ian Ring Music TheoryDycrygic
Scale 2297Scale 2297: Thylian, Ian Ring Music TheoryThylian
Scale 2809Scale 2809: Gythyllic, Ian Ring Music TheoryGythyllic
Scale 1273Scale 1273: Ronian, Ian Ring Music TheoryRonian

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.