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Scale 1639: "Aeolothian"

Scale 1639: Aeolothian, 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
Aeolothian
Dozenal
Kasian

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

{0,1,2,5,6,9,10}

Forte Number

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

7-21

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: 3277

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

4 (multihemitonic)

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.

6

Prime Form

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

no
prime: 823

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, 3, 1, 3, 1, 2]

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.

<4, 2, 4, 6, 4, 1>

Interval Spectrum

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

p4m6n4s2d4t

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> = {4,5,6,7}
<4> = {5,6,7,8}
<5> = {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

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.

2.433

Polygon Perimeter

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

5.899

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.

(7, 33, 90)

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 TriadsD{2,6,9}331.7
F{5,9,0}242.1
F♯{6,10,1}331.7
A♯{10,2,5}341.9
Minor Triadsdm{2,5,9}331.7
f♯m{6,9,1}431.5
a♯m{10,1,5}331.7
Augmented TriadsC♯+{1,5,9}431.5
D+{2,6,10}341.9
Diminished Triadsf♯°{6,9,0}242.1
Parsimonious Voice Leading Between Common Triads of Scale 1639. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m a#m a#m C#+->a#m D D dm->D A# A# dm->A# D+ D+ D->D+ D->f#m F# F# D+->F# D+->A# f#° f#° F->f#° f#°->f#m f#m->F# F#->a#m a#m->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
Radius3
Self-Centeredno
Central VerticesC♯+, dm, D, f♯m, F♯, a♯m
Peripheral VerticesD+, F, f♯°, A♯

Modes

Modes are the rotational transformation of this scale. Scale 1639 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2867
Scale 2867: Socrian, Ian Ring Music TheorySocrian
3rd mode:
Scale 3481
Scale 3481: Katathian, Ian Ring Music TheoryKatathian
4th mode:
Scale 947
Scale 947: Mela Gayakapriya, Ian Ring Music TheoryMela Gayakapriya
5th mode:
Scale 2521
Scale 2521: Mela Dhatuvardhani, Ian Ring Music TheoryMela Dhatuvardhani
6th mode:
Scale 827
Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian
7th mode:
Scale 2461
Scale 2461: Sagian, Ian Ring Music TheorySagian

Prime

The prime form of this scale is Scale 823

Scale 823Scale 823: Stodian, Ian Ring Music TheoryStodian

Complement

The heptatonic modal family [1639, 2867, 3481, 947, 2521, 827, 2461] (Forte: 7-21) is the complement of the pentatonic modal family [307, 787, 817, 2201, 2441] (Forte: 5-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1639 is 3277

Scale 3277Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1639 is chiral, and its enantiomorph is scale 3277

Scale 3277Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati

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> 1639       T0I <11,0> 3277
T1 <1,1> 3278      T1I <11,1> 2459
T2 <1,2> 2461      T2I <11,2> 823
T3 <1,3> 827      T3I <11,3> 1646
T4 <1,4> 1654      T4I <11,4> 3292
T5 <1,5> 3308      T5I <11,5> 2489
T6 <1,6> 2521      T6I <11,6> 883
T7 <1,7> 947      T7I <11,7> 1766
T8 <1,8> 1894      T8I <11,8> 3532
T9 <1,9> 3788      T9I <11,9> 2969
T10 <1,10> 3481      T10I <11,10> 1843
T11 <1,11> 2867      T11I <11,11> 3686
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1639       T0MI <7,0> 3277
T1M <5,1> 3278      T1MI <7,1> 2459
T2M <5,2> 2461      T2MI <7,2> 823
T3M <5,3> 827      T3MI <7,3> 1646
T4M <5,4> 1654      T4MI <7,4> 3292
T5M <5,5> 3308      T5MI <7,5> 2489
T6M <5,6> 2521      T6MI <7,6> 883
T7M <5,7> 947      T7MI <7,7> 1766
T8M <5,8> 1894      T8MI <7,8> 3532
T9M <5,9> 3788      T9MI <7,9> 2969
T10M <5,10> 3481      T10MI <7,10> 1843
T11M <5,11> 2867      T11MI <7,11> 3686

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

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 1637Scale 1637: Syptimic, Ian Ring Music TheorySyptimic
Scale 1635Scale 1635: Sygimic, Ian Ring Music TheorySygimic
Scale 1643Scale 1643: Locrian Natural 6, Ian Ring Music TheoryLocrian Natural 6
Scale 1647Scale 1647: Polyllic, Ian Ring Music TheoryPolyllic
Scale 1655Scale 1655: Katygyllic, Ian Ring Music TheoryKatygyllic
Scale 1607Scale 1607: Epytimic, Ian Ring Music TheoryEpytimic
Scale 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian
Scale 1575Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic
Scale 1703Scale 1703: Mela Vanaspati, Ian Ring Music TheoryMela Vanaspati
Scale 1767Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic
Scale 1895Scale 1895: Salyllic, Ian Ring Music TheorySalyllic
Scale 1127Scale 1127: Eparimic, Ian Ring Music TheoryEparimic
Scale 1383Scale 1383: Pynian, Ian Ring Music TheoryPynian
Scale 615Scale 615: Schoenberg Hexachord, Ian Ring Music TheorySchoenberg Hexachord
Scale 2663Scale 2663: Lalian, Ian Ring Music TheoryLalian
Scale 3687Scale 3687: Zonyllic, Ian Ring Music TheoryZonyllic

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.