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Scale 1139: "Aerygimic"

Scale 1139: Aerygimic, 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
Aerygimic
Dozenal
Hagian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,1,4,5,6,10}

Forte Number

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

6-Z43

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: 2501

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.

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.

5

Prime Form

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

no
prime: 359

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, 3, 1, 1, 4, 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.

<3, 2, 2, 3, 3, 2>

Interval Spectrum

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

p3m3n2s2d3t2

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

Spectra Variation

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

2.667

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.

2.116

Polygon Perimeter

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

5.699

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.

(10, 17, 64)

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♯{6,10,1}121
Minor Triadsa♯m{10,1,5}210.67
Diminished Triadsa♯°{10,1,4}121

The following pitch classes are not present in any of the common triads: {0}

Parsimonious Voice Leading Between Common Triads of Scale 1139. Created by Ian Ring ©2019 F# F# a#m a#m F#->a#m a#° a#° a#°->a#m

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.

Diameter2
Radius1
Self-Centeredno
Central Verticesa♯m
Peripheral VerticesF♯, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 1139 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2617
Scale 2617: Pylimic, Ian Ring Music TheoryPylimic
3rd mode:
Scale 839
Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
4th mode:
Scale 2467
Scale 2467: Raga Padi, Ian Ring Music TheoryRaga Padi
5th mode:
Scale 3281
Scale 3281: Raga Vijayavasanta, Ian Ring Music TheoryRaga Vijayavasanta
6th mode:
Scale 461
Scale 461: Raga Syamalam, Ian Ring Music TheoryRaga Syamalam

Prime

The prime form of this scale is Scale 359

Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic

Complement

The hexatonic modal family [1139, 2617, 839, 2467, 3281, 461] (Forte: 6-Z43) is the complement of the hexatonic modal family [407, 739, 1817, 2251, 2417, 3173] (Forte: 6-Z17)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1139 is 2501

Scale 2501Scale 2501: Ralimic, Ian Ring Music TheoryRalimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1139 is chiral, and its enantiomorph is scale 2501

Scale 2501Scale 2501: Ralimic, Ian Ring Music TheoryRalimic

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> 1139       T0I <11,0> 2501
T1 <1,1> 2278      T1I <11,1> 907
T2 <1,2> 461      T2I <11,2> 1814
T3 <1,3> 922      T3I <11,3> 3628
T4 <1,4> 1844      T4I <11,4> 3161
T5 <1,5> 3688      T5I <11,5> 2227
T6 <1,6> 3281      T6I <11,6> 359
T7 <1,7> 2467      T7I <11,7> 718
T8 <1,8> 839      T8I <11,8> 1436
T9 <1,9> 1678      T9I <11,9> 2872
T10 <1,10> 3356      T10I <11,10> 1649
T11 <1,11> 2617      T11I <11,11> 3298
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 359      T0MI <7,0> 3281
T1M <5,1> 718      T1MI <7,1> 2467
T2M <5,2> 1436      T2MI <7,2> 839
T3M <5,3> 2872      T3MI <7,3> 1678
T4M <5,4> 1649      T4MI <7,4> 3356
T5M <5,5> 3298      T5MI <7,5> 2617
T6M <5,6> 2501      T6MI <7,6> 1139
T7M <5,7> 907      T7MI <7,7> 2278
T8M <5,8> 1814      T8MI <7,8> 461
T9M <5,9> 3628      T9MI <7,9> 922
T10M <5,10> 3161      T10MI <7,10> 1844
T11M <5,11> 2227      T11MI <7,11> 3688

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 1137Scale 1137: Stonitonic, Ian Ring Music TheoryStonitonic
Scale 1141Scale 1141: Rynimic, Ian Ring Music TheoryRynimic
Scale 1143Scale 1143: Styrian, Ian Ring Music TheoryStyrian
Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian
Scale 1123Scale 1123: Iwato, Ian Ring Music TheoryIwato
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
Scale 1075Scale 1075: Gotian, Ian Ring Music TheoryGotian
Scale 1203Scale 1203: Pagimic, Ian Ring Music TheoryPagimic
Scale 1267Scale 1267: Katynian, Ian Ring Music TheoryKatynian
Scale 1395Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
Scale 1651Scale 1651: Asian, Ian Ring Music TheoryAsian
Scale 115Scale 115: Ashian, Ian Ring Music TheoryAshian
Scale 627Scale 627: Mogimic, Ian Ring Music TheoryMogimic
Scale 2163Scale 2163: Nebian, Ian Ring Music TheoryNebian
Scale 3187Scale 3187: Koptian, Ian Ring Music TheoryKoptian

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.