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Scale 3789: "Eporyllic"

Scale 3789: Eporyllic, 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
Eporyllic
Dozenal
Yeyian

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

{0,2,3,6,7,9,10,11}

Forte Number

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

8-17

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.

[4.5]

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.

5 (multihemitonic)

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.

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.

7

Prime Form

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

no
prime: 891

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, 3, 1, 2, 1, 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.

<5, 4, 6, 6, 5, 2>

Interval Spectrum

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

p5m6n6s4d5t2

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

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.

2.616

Polygon Perimeter

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

6.002

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.

[9]

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.

(24, 54, 132)

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{2,6,9}342.21
D♯{3,7,10}342.07
G{7,11,2}342.21
B{11,3,6}441.93
Minor Triadscm{0,3,7}342.21
d♯m{3,6,10}441.93
gm{7,10,2}342.21
bm{11,2,6}342.07
Augmented TriadsD+{2,6,10}441.93
D♯+{3,7,11}441.93
Diminished Triads{0,3,6}242.36
d♯°{3,6,9}242.36
f♯°{6,9,0}242.5
{9,0,3}242.5
Parsimonious Voice Leading Between Common Triads of Scale 3789. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ cm->a° D D D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° d#m d#m D+->d#m gm gm D+->gm bm bm D+->bm d#°->d#m D# D# d#m->D# d#m->B D#->D#+ D#->gm Parsimonious Voice Leading Between Common Triads of Scale 3789. Created by Ian Ring ©2019 G D#+->G D#+->B f#°->a° gm->G G->bm bm->B

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3789 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1971
Scale 1971: Aerynyllic, Ian Ring Music TheoryAerynyllic
3rd mode:
Scale 3033
Scale 3033: Doptyllic, Ian Ring Music TheoryDoptyllic
4th mode:
Scale 891
Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllicThis is the prime mode
5th mode:
Scale 2493
Scale 2493: Manyllic, Ian Ring Music TheoryManyllic
6th mode:
Scale 1647
Scale 1647: Polyllic, Ian Ring Music TheoryPolyllic
7th mode:
Scale 2871
Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic
8th mode:
Scale 3483
Scale 3483: Mixotharyllic, Ian Ring Music TheoryMixotharyllic

Prime

The prime form of this scale is Scale 891

Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic

Complement

The octatonic modal family [3789, 1971, 3033, 891, 2493, 1647, 2871, 3483] (Forte: 8-17) is the complement of the tetratonic modal family [153, 531, 801, 2313] (Forte: 4-17)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3789 is 1647

Scale 1647Scale 1647: Polyllic, Ian Ring Music TheoryPolyllic

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> 3789       T0I <11,0> 1647
T1 <1,1> 3483      T1I <11,1> 3294
T2 <1,2> 2871      T2I <11,2> 2493
T3 <1,3> 1647      T3I <11,3> 891
T4 <1,4> 3294      T4I <11,4> 1782
T5 <1,5> 2493      T5I <11,5> 3564
T6 <1,6> 891      T6I <11,6> 3033
T7 <1,7> 1782      T7I <11,7> 1971
T8 <1,8> 3564      T8I <11,8> 3942
T9 <1,9> 3033      T9I <11,9> 3789
T10 <1,10> 1971      T10I <11,10> 3483
T11 <1,11> 3942      T11I <11,11> 2871
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3789       T0MI <7,0> 1647
T1M <5,1> 3483      T1MI <7,1> 3294
T2M <5,2> 2871      T2MI <7,2> 2493
T3M <5,3> 1647      T3MI <7,3> 891
T4M <5,4> 3294      T4MI <7,4> 1782
T5M <5,5> 2493      T5MI <7,5> 3564
T6M <5,6> 891      T6MI <7,6> 3033
T7M <5,7> 1782      T7MI <7,7> 1971
T8M <5,8> 3564      T8MI <7,8> 3942
T9M <5,9> 3033      T9MI <7,9> 3789
T10M <5,10> 1971      T10MI <7,10> 3483
T11M <5,11> 3942      T11MI <7,11> 2871

The transformations that map this set to itself are: T0, T9I, T0M, T9MI

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 3791Scale 3791: Stodygic, Ian Ring Music TheoryStodygic
Scale 3785Scale 3785: Epagian, Ian Ring Music TheoryEpagian
Scale 3787Scale 3787: Kagyllic, Ian Ring Music TheoryKagyllic
Scale 3781Scale 3781: Gyphian, Ian Ring Music TheoryGyphian
Scale 3797Scale 3797: Rocryllic, Ian Ring Music TheoryRocryllic
Scale 3805Scale 3805: Moptygic, Ian Ring Music TheoryMoptygic
Scale 3821Scale 3821: Epyrygic, Ian Ring Music TheoryEpyrygic
Scale 3725Scale 3725: Kyrian, Ian Ring Music TheoryKyrian
Scale 3757Scale 3757: Raga Mian Ki Malhar, Ian Ring Music TheoryRaga Mian Ki Malhar
Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian
Scale 3917Scale 3917: Katoptyllic, Ian Ring Music TheoryKatoptyllic
Scale 4045Scale 4045: Gyptygic, Ian Ring Music TheoryGyptygic
Scale 3277Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati
Scale 3533Scale 3533: Thadyllic, Ian Ring Music TheoryThadyllic
Scale 2765Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
Scale 1741Scale 1741: Lydian Diminished, Ian Ring Music TheoryLydian Diminished

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.