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Scale 367: "Aerodian"

Scale 367: Aerodian, 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
Aerodian
Dozenal
Cekian

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

{0,1,2,3,5,6,8}

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-Z36

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: 3793

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.

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, 1, 2, 4]

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, 4, 4, 3, 4, 2>

Interval Spectrum

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

p4m3n4s4d4t2

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

Spectra Variation

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

3.143

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.

2.299

Polygon Perimeter

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

5.803

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.

(38, 34, 100)

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}231.4
G♯{8,0,3}231.4
Minor Triadsfm{5,8,0}221.2
Diminished Triads{0,3,6}142
{2,5,8}142
Parsimonious Voice Leading Between Common Triads of Scale 367. Created by Ian Ring ©2019 G# G# c°->G# C# C# C#->d° fm fm C#->fm fm->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 Verticesc°, d°

Modes

Modes are the rotational transformation of this scale. Scale 367 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2231
Scale 2231: Macrian, Ian Ring Music TheoryMacrian
3rd mode:
Scale 3163
Scale 3163: Rogian, Ian Ring Music TheoryRogian
4th mode:
Scale 3629
Scale 3629: Boptian, Ian Ring Music TheoryBoptian
5th mode:
Scale 1931
Scale 1931: Stogian, Ian Ring Music TheoryStogian
6th mode:
Scale 3013
Scale 3013: Thynian, Ian Ring Music TheoryThynian
7th mode:
Scale 1777
Scale 1777: Saptian, Ian Ring Music TheorySaptian

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [367, 2231, 3163, 3629, 1931, 3013, 1777] (Forte: 7-Z36) is the complement of the pentatonic modal family [151, 737, 1801, 2123, 3109] (Forte: 5-Z36)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 367 is 3793

Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 367 is chiral, and its enantiomorph is scale 3793

Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian

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> 367       T0I <11,0> 3793
T1 <1,1> 734      T1I <11,1> 3491
T2 <1,2> 1468      T2I <11,2> 2887
T3 <1,3> 2936      T3I <11,3> 1679
T4 <1,4> 1777      T4I <11,4> 3358
T5 <1,5> 3554      T5I <11,5> 2621
T6 <1,6> 3013      T6I <11,6> 1147
T7 <1,7> 1931      T7I <11,7> 2294
T8 <1,8> 3862      T8I <11,8> 493
T9 <1,9> 3629      T9I <11,9> 986
T10 <1,10> 3163      T10I <11,10> 1972
T11 <1,11> 2231      T11I <11,11> 3944
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1147      T0MI <7,0> 3013
T1M <5,1> 2294      T1MI <7,1> 1931
T2M <5,2> 493      T2MI <7,2> 3862
T3M <5,3> 986      T3MI <7,3> 3629
T4M <5,4> 1972      T4MI <7,4> 3163
T5M <5,5> 3944      T5MI <7,5> 2231
T6M <5,6> 3793      T6MI <7,6> 367
T7M <5,7> 3491      T7MI <7,7> 734
T8M <5,8> 2887      T8MI <7,8> 1468
T9M <5,9> 1679      T9MI <7,9> 2936
T10M <5,10> 3358      T10MI <7,10> 1777
T11M <5,11> 2621      T11MI <7,11> 3554

The transformations that map this set to itself are: T0, 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 365Scale 365: Marimic, Ian Ring Music TheoryMarimic
Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic
Scale 375Scale 375: Sodian, Ian Ring Music TheorySodian
Scale 383Scale 383: Logyllic, Ian Ring Music TheoryLogyllic
Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic
Scale 351Scale 351: Epanian, Ian Ring Music TheoryEpanian
Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic
Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian
Scale 495Scale 495: Bocryllic, Ian Ring Music TheoryBocryllic
Scale 111Scale 111: Aroian, Ian Ring Music TheoryAroian
Scale 239Scale 239: Bikian, Ian Ring Music TheoryBikian
Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian
Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic
Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic
Scale 2415Scale 2415: Lothyllic, Ian Ring Music TheoryLothyllic

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.