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Scale 821: "Aeranimic"

Scale 821: Aeranimic, 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
Aeranimic
Dozenal
Fatian

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

{0,2,4,5,8,9}

Forte Number

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

6-31

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: 1433

Hemitonia

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

2 (dihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

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: 691

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.

[2, 2, 1, 3, 1, 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.

<2, 2, 3, 4, 3, 1>

Interval Spectrum

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

p3m4n3s2d2t

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

Spectra Variation

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

1.667

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.

2.366

Polygon Perimeter

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

5.864

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".

Proper

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.

(0, 8, 54)

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}321.17
Minor Triadsdm{2,5,9}231.5
fm{5,8,0}321.17
am{9,0,4}231.5
Augmented TriadsC+{0,4,8}231.5
Diminished Triads{2,5,8}231.5
Parsimonious Voice Leading Between Common Triads of Scale 821. Created by Ian Ring ©2019 C+ C+ fm fm C+->fm am am C+->am dm dm d°->dm d°->fm F F dm->F fm->F F->am

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesfm, F
Peripheral VerticesC+, d°, dm, am

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There are 2 ways that this hexatonic scale can be split into two common triads.


Augmented: {0, 4, 8}
Minor: {2, 5, 9}

Diminished: {2, 5, 8}
Minor: {9, 0, 4}

Modes

Modes are the rotational transformation of this scale. Scale 821 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1229
Scale 1229: Raga Simharava, Ian Ring Music TheoryRaga Simharava
3rd mode:
Scale 1331
Scale 1331: Raga Vasantabhairavi, Ian Ring Music TheoryRaga Vasantabhairavi
4th mode:
Scale 2713
Scale 2713: Porimic, Ian Ring Music TheoryPorimic
5th mode:
Scale 851
Scale 851: Raga Hejjajji, Ian Ring Music TheoryRaga Hejjajji
6th mode:
Scale 2473
Scale 2473: Raga Takka, Ian Ring Music TheoryRaga Takka

Prime

The prime form of this scale is Scale 691

Scale 691Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga Kalavati

Complement

The hexatonic modal family [821, 1229, 1331, 2713, 851, 2473] (Forte: 6-31) is the complement of the hexatonic modal family [691, 811, 1433, 1637, 2393, 2453] (Forte: 6-31)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 821 is 1433

Scale 1433Scale 1433: Dynimic, Ian Ring Music TheoryDynimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 821 is chiral, and its enantiomorph is scale 1433

Scale 1433Scale 1433: Dynimic, Ian Ring Music TheoryDynimic

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> 821       T0I <11,0> 1433
T1 <1,1> 1642      T1I <11,1> 2866
T2 <1,2> 3284      T2I <11,2> 1637
T3 <1,3> 2473      T3I <11,3> 3274
T4 <1,4> 851      T4I <11,4> 2453
T5 <1,5> 1702      T5I <11,5> 811
T6 <1,6> 3404      T6I <11,6> 1622
T7 <1,7> 2713      T7I <11,7> 3244
T8 <1,8> 1331      T8I <11,8> 2393
T9 <1,9> 2662      T9I <11,9> 691
T10 <1,10> 1229      T10I <11,10> 1382
T11 <1,11> 2458      T11I <11,11> 2764
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1811      T0MI <7,0> 2333
T1M <5,1> 3622      T1MI <7,1> 571
T2M <5,2> 3149      T2MI <7,2> 1142
T3M <5,3> 2203      T3MI <7,3> 2284
T4M <5,4> 311      T4MI <7,4> 473
T5M <5,5> 622      T5MI <7,5> 946
T6M <5,6> 1244      T6MI <7,6> 1892
T7M <5,7> 2488      T7MI <7,7> 3784
T8M <5,8> 881      T8MI <7,8> 3473
T9M <5,9> 1762      T9MI <7,9> 2851
T10M <5,10> 3524      T10MI <7,10> 1607
T11M <5,11> 2953      T11MI <7,11> 3214

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 823Scale 823: Stodian, Ian Ring Music TheoryStodian
Scale 817Scale 817: Zothitonic, Ian Ring Music TheoryZothitonic
Scale 819Scale 819: Augmented Inverse, Ian Ring Music TheoryAugmented Inverse
Scale 825Scale 825: Thyptimic, Ian Ring Music TheoryThyptimic
Scale 829Scale 829: Lygian, Ian Ring Music TheoryLygian
Scale 805Scale 805: Rothitonic, Ian Ring Music TheoryRothitonic
Scale 813Scale 813: Larimic, Ian Ring Music TheoryLarimic
Scale 789Scale 789: Zogitonic, Ian Ring Music TheoryZogitonic
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 885Scale 885: Sathian, Ian Ring Music TheorySathian
Scale 949Scale 949: Mela Mararanjani, Ian Ring Music TheoryMela Mararanjani
Scale 565Scale 565: Aeolyphritonic, Ian Ring Music TheoryAeolyphritonic
Scale 693Scale 693: Arezzo Major Diatonic Hexachord, Ian Ring Music TheoryArezzo Major Diatonic Hexachord
Scale 309Scale 309: Palitonic, Ian Ring Music TheoryPalitonic
Scale 1333Scale 1333: Lyptimic, Ian Ring Music TheoryLyptimic
Scale 1845Scale 1845: Lagian, Ian Ring Music TheoryLagian
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented

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.