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Scale 3371: "Aeolylian"

Scale 3371: Aeolylian, 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
Aeolylian
Dozenal
Vezian

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

{0,1,3,5,8,10,11}

Forte Number

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

7-23

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: 2711

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.

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.

6

Prime Form

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

no
prime: 701

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, 2, 2, 3, 2, 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.

<3, 5, 4, 3, 5, 1>

Interval Spectrum

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

p5m3n4s5d3t

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.

(26, 43, 104)

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 TriadsC♯{1,5,8}231.5
G♯{8,0,3}231.5
Minor Triadsfm{5,8,0}321.17
g♯m{8,11,3}241.83
a♯m{10,1,5}142.17
Diminished Triads{5,8,11}231.5
Parsimonious Voice Leading Between Common Triads of Scale 3371. Created by Ian Ring ©2019 C# C# fm fm C#->fm a#m a#m C#->a#m f°->fm g#m g#m f°->g#m G# G# fm->G# g#m->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.

Diameter4
Radius2
Self-Centeredno
Central Verticesfm
Peripheral Verticesg♯m, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 3371 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3733
Scale 3733: Gycrian, Ian Ring Music TheoryGycrian
3rd mode:
Scale 1957
Scale 1957: Pyrian, Ian Ring Music TheoryPyrian
4th mode:
Scale 1513
Scale 1513: Stathian, Ian Ring Music TheoryStathian
5th mode:
Scale 701
Scale 701: Mixonyphian, Ian Ring Music TheoryMixonyphianThis is the prime mode
6th mode:
Scale 1199
Scale 1199: Magian, Ian Ring Music TheoryMagian
7th mode:
Scale 2647
Scale 2647: Dadian, Ian Ring Music TheoryDadian

Prime

The prime form of this scale is Scale 701

Scale 701Scale 701: Mixonyphian, Ian Ring Music TheoryMixonyphian

Complement

The heptatonic modal family [3371, 3733, 1957, 1513, 701, 1199, 2647] (Forte: 7-23) is the complement of the pentatonic modal family [173, 1067, 1441, 1669, 2581] (Forte: 5-23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3371 is 2711

Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3371 is chiral, and its enantiomorph is scale 2711

Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian

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> 3371       T0I <11,0> 2711
T1 <1,1> 2647      T1I <11,1> 1327
T2 <1,2> 1199      T2I <11,2> 2654
T3 <1,3> 2398      T3I <11,3> 1213
T4 <1,4> 701      T4I <11,4> 2426
T5 <1,5> 1402      T5I <11,5> 757
T6 <1,6> 2804      T6I <11,6> 1514
T7 <1,7> 1513      T7I <11,7> 3028
T8 <1,8> 3026      T8I <11,8> 1961
T9 <1,9> 1957      T9I <11,9> 3922
T10 <1,10> 3914      T10I <11,10> 3749
T11 <1,11> 3733      T11I <11,11> 3403
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 191      T0MI <7,0> 4001
T1M <5,1> 382      T1MI <7,1> 3907
T2M <5,2> 764      T2MI <7,2> 3719
T3M <5,3> 1528      T3MI <7,3> 3343
T4M <5,4> 3056      T4MI <7,4> 2591
T5M <5,5> 2017      T5MI <7,5> 1087
T6M <5,6> 4034      T6MI <7,6> 2174
T7M <5,7> 3973      T7MI <7,7> 253
T8M <5,8> 3851      T8MI <7,8> 506
T9M <5,9> 3607      T9MI <7,9> 1012
T10M <5,10> 3119      T10MI <7,10> 2024
T11M <5,11> 2143      T11MI <7,11> 4048

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 3369Scale 3369: Mixolimic, Ian Ring Music TheoryMixolimic
Scale 3373Scale 3373: Lodian, Ian Ring Music TheoryLodian
Scale 3375Scale 3375: Vecian, Ian Ring Music TheoryVecian
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic
Scale 3367Scale 3367: Moptian, Ian Ring Music TheoryMoptian
Scale 3379Scale 3379: Verdi's Scala Enigmatica Descending, Ian Ring Music TheoryVerdi's Scala Enigmatica Descending
Scale 3387Scale 3387: Aeryptyllic, Ian Ring Music TheoryAeryptyllic
Scale 3339Scale 3339: Smuian, Ian Ring Music TheorySmuian
Scale 3355Scale 3355: Bagian, Ian Ring Music TheoryBagian
Scale 3403Scale 3403: Bylian, Ian Ring Music TheoryBylian
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3499Scale 3499: Hamel, Ian Ring Music TheoryHamel
Scale 3115Scale 3115: Tihian, Ian Ring Music TheoryTihian
Scale 3243Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
Scale 3627Scale 3627: Kalian, Ian Ring Music TheoryKalian
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 2347Scale 2347: Raga Viyogavarali, Ian Ring Music TheoryRaga Viyogavarali
Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
Scale 1323Scale 1323: Ritsu, Ian Ring Music TheoryRitsu

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.