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Scale 687: "Aeolythian"

Scale 687: Aeolythian, 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
Aeolythian
Dozenal
Efoian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,3,5,7,9}

Forte Number

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

7-24

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: 3753

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.

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.

6

Prime Form

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

yes

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, 1, 2, 2, 2, 3]

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, 5, 3, 4, 4, 2>

Interval Spectrum

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

p4m4n3s5d3t2

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

Spectra Variation

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

2.571

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.

2.549

Polygon Perimeter

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

5.967

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.

(19, 38, 102)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsF{5,9,0}221.2
Minor Triadscm{0,3,7}142
dm{2,5,9}142
Augmented TriadsC♯+{1,5,9}231.4
Diminished Triads{9,0,3}231.4
Parsimonious Voice Leading Between Common Triads of Scale 687. Created by Ian Ring ©2019 cm cm cm->a° C#+ C#+ dm dm C#+->dm F F C#+->F F->a°

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
Radius2
Self-Centeredno
Central VerticesF
Peripheral Verticescm, dm

Modes

Modes are the rotational transformation of this scale. Scale 687 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2391
Scale 2391: Molian, Ian Ring Music TheoryMolian
3rd mode:
Scale 3243
Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
4th mode:
Scale 3669
Scale 3669: Mothian, Ian Ring Music TheoryMothian
5th mode:
Scale 1941
Scale 1941: Aeranian, Ian Ring Music TheoryAeranian
6th mode:
Scale 1509
Scale 1509: Ragian, Ian Ring Music TheoryRagian
7th mode:
Scale 1401
Scale 1401: Pagian, Ian Ring Music TheoryPagian

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [687, 2391, 3243, 3669, 1941, 1509, 1401] (Forte: 7-24) is the complement of the pentatonic modal family [171, 1377, 1413, 1557, 2133] (Forte: 5-24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 687 is 3753

Scale 3753Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 687 is chiral, and its enantiomorph is scale 3753

Scale 3753Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian

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> 687       T0I <11,0> 3753
T1 <1,1> 1374      T1I <11,1> 3411
T2 <1,2> 2748      T2I <11,2> 2727
T3 <1,3> 1401      T3I <11,3> 1359
T4 <1,4> 2802      T4I <11,4> 2718
T5 <1,5> 1509      T5I <11,5> 1341
T6 <1,6> 3018      T6I <11,6> 2682
T7 <1,7> 1941      T7I <11,7> 1269
T8 <1,8> 3882      T8I <11,8> 2538
T9 <1,9> 3669      T9I <11,9> 981
T10 <1,10> 3243      T10I <11,10> 1962
T11 <1,11> 2391      T11I <11,11> 3924
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3627      T0MI <7,0> 2703
T1M <5,1> 3159      T1MI <7,1> 1311
T2M <5,2> 2223      T2MI <7,2> 2622
T3M <5,3> 351      T3MI <7,3> 1149
T4M <5,4> 702      T4MI <7,4> 2298
T5M <5,5> 1404      T5MI <7,5> 501
T6M <5,6> 2808      T6MI <7,6> 1002
T7M <5,7> 1521      T7MI <7,7> 2004
T8M <5,8> 3042      T8MI <7,8> 4008
T9M <5,9> 1989      T9MI <7,9> 3921
T10M <5,10> 3978      T10MI <7,10> 3747
T11M <5,11> 3861      T11MI <7,11> 3399

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 685Scale 685: Raga Suddha Bangala, Ian Ring Music TheoryRaga Suddha Bangala
Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 695Scale 695: Sarian, Ian Ring Music TheorySarian
Scale 703Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllic
Scale 655Scale 655: Kataptimic, Ian Ring Music TheoryKataptimic
Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian
Scale 719Scale 719: Kanian, Ian Ring Music TheoryKanian
Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian
Scale 559Scale 559: Lylimic, Ian Ring Music TheoryLylimic
Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian
Scale 815Scale 815: Bolian, Ian Ring Music TheoryBolian
Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic
Scale 175Scale 175: Bewian, Ian Ring Music TheoryBewian
Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian
Scale 1199Scale 1199: Magian, Ian Ring Music TheoryMagian
Scale 1711Scale 1711: Adonai Malakh, Ian Ring Music TheoryAdonai Malakh
Scale 2735Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic

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.