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Scale 1359: "Aerygian"

Scale 1359: Aerygian, 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
Aerygian

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

{0,1,2,3,6,8,10}

Forte Number

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

7-24

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: 3669

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.

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: 687

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 1, 3, 2, 2, 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, 5, 3, 4, 4, 2>

Interval Spectrum

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

p4m4n3s5d3t2

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.

(19, 38, 102)

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 TriadsF♯{6,10,1}142
G♯{8,0,3}142
Minor Triadsd♯m{3,6,10}221.2
Augmented TriadsD+{2,6,10}231.4
Diminished Triads{0,3,6}231.4
Parsimonious Voice Leading Between Common Triads of Scale 1359. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# D+ D+ D+->d#m F# F# D+->F#

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 Verticesd♯m
Peripheral VerticesF♯, G♯

Modes

Modes are the rotational transformation of this scale. Scale 1359 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2727
Scale 2727: Mela Manavati, Ian Ring Music TheoryMela Manavati
3rd mode:
Scale 3411
Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
4th mode:
Scale 3753
Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian
5th mode:
Scale 981
Scale 981: Mela Kantamani, Ian Ring Music TheoryMela Kantamani
6th mode:
Scale 1269
Scale 1269: Katythian, Ian Ring Music TheoryKatythian
7th mode:
Scale 1341
Scale 1341: Madian, Ian Ring Music TheoryMadian

Prime

The prime form of this scale is Scale 687

Scale 687Scale 687: Aeolythian, Ian Ring Music TheoryAeolythian

Complement

The heptatonic modal family [1359, 2727, 3411, 3753, 981, 1269, 1341] (Forte: 7-24) is the complement of the pentatonic modal family [171, 1377, 1413, 1557, 2133] (Forte: 5-24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1359 is 3669

Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1359 is chiral, and its enantiomorph is scale 3669

Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian

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> 1359       T0I <11,0> 3669
T1 <1,1> 2718      T1I <11,1> 3243
T2 <1,2> 1341      T2I <11,2> 2391
T3 <1,3> 2682      T3I <11,3> 687
T4 <1,4> 1269      T4I <11,4> 1374
T5 <1,5> 2538      T5I <11,5> 2748
T6 <1,6> 981      T6I <11,6> 1401
T7 <1,7> 1962      T7I <11,7> 2802
T8 <1,8> 3924      T8I <11,8> 1509
T9 <1,9> 3753      T9I <11,9> 3018
T10 <1,10> 3411      T10I <11,10> 1941
T11 <1,11> 2727      T11I <11,11> 3882
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1149      T0MI <7,0> 1989
T1M <5,1> 2298      T1MI <7,1> 3978
T2M <5,2> 501      T2MI <7,2> 3861
T3M <5,3> 1002      T3MI <7,3> 3627
T4M <5,4> 2004      T4MI <7,4> 3159
T5M <5,5> 4008      T5MI <7,5> 2223
T6M <5,6> 3921      T6MI <7,6> 351
T7M <5,7> 3747      T7MI <7,7> 702
T8M <5,8> 3399      T8MI <7,8> 1404
T9M <5,9> 2703      T9MI <7,9> 2808
T10M <5,10> 1311      T10MI <7,10> 1521
T11M <5,11> 2622      T11MI <7,11> 3042

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 1357Scale 1357: Takemitsu Linea Mode 2, Ian Ring Music TheoryTakemitsu Linea Mode 2
Scale 1355Scale 1355: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 1351Scale 1351: Aeraptimic, Ian Ring Music TheoryAeraptimic
Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse
Scale 1375Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllic
Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic
Scale 1295Scale 1295, Ian Ring Music Theory
Scale 1327Scale 1327: Zalian, Ian Ring Music TheoryZalian
Scale 1423Scale 1423: Doptian, Ian Ring Music TheoryDoptian
Scale 1487Scale 1487: Mothyllic, Ian Ring Music TheoryMothyllic
Scale 1103Scale 1103: Lynimic, Ian Ring Music TheoryLynimic
Scale 1231Scale 1231: Logian, Ian Ring Music TheoryLogian
Scale 1615Scale 1615: Sydian, Ian Ring Music TheorySydian
Scale 1871Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic
Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic
Scale 847Scale 847: Ganian, Ian Ring Music TheoryGanian
Scale 2383Scale 2383: Katorian, Ian Ring Music TheoryKatorian
Scale 3407Scale 3407: Katocryllic, Ian Ring Music TheoryKatocryllic

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.