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Scale 1019: "Aeranygic"

Scale 1019: Aeranygic, 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
Aeranygic
Dozenal
GELian

Analysis

Cardinality

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

 9 (enneatonic)

Pitch Class Set

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

 {0,1,3,4,5,6,7,8,9}

Forte Number

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

 9-3

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: 3065

Hemitonia

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

 7 (multihemitonic)

Cohemitonia

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

 5 (multicohemitonic)

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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 9

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 895

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, an indicator of maximum hierarchization.

 no

Interval Structure

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

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

 <7, 6, 7, 7, 6, 3>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.5, 0, 0.5, 0.333, 0, 0>

Interval Spectrum

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

 p6m7n7s6d7t3

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

Spectra Variation

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

 2.222

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.

2.683

Polygon Perimeter

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

6.038

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(64, 107, 194)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.63

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.326

Generator

This scale has no generator.

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{0,4,7}342.35
C♯{1,5,8}342.35
F{5,9,0}442.24
G♯{8,0,3}342.35
A{9,1,4}342.24
Minor Triadscm{0,3,7}342.53
c♯m{1,4,8}442.24
fm{5,8,0}342.24
f♯m{6,9,1}342.53
am{9,0,4}442.12
Augmented TriadsC+{0,4,8}542
C♯+{1,5,9}442.24
Diminished Triadsc°{0,3,6}242.76
c♯°{1,4,7}252.71
d♯°{3,6,9}242.76
f♯°{6,9,0}252.71
a°{9,0,3}242.59
Parsimonious Voice Leading Between Common Triads of Scale 1019. Created by Ian Ring ©2019 c° c° cm cm c°->cm d#° d#° c°->d#° C C cm->C G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° c#m c#m C+->c#m fm fm C+->fm C+->G# am am C+->am c#°->c#m C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F f#m f#m C#+->f#m C#+->A d#°->f#m fm->F f#° f#° F->f#° F->am f#°->f#m a° a° G#->a° a°->am am->A

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.

Diameter5
Radius4
Self-Centeredno
Central Verticesc°, cm, C, C+, c♯m, C♯, C♯+, d♯°, fm, F, f♯m, G♯, a°, am, A
Peripheral Verticesc♯°, f♯°

Modes

Modes are the rotational transformation of this scale.Scale 1019 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 2557		Scale 2557: Dothygic, Ian Ring Music TheoryDothygic
3rd mode:
Scale 1663		Scale 1663: Lydygic, Ian Ring Music TheoryLydygic
4th mode:
Scale 2879		Scale 2879: Stadygic, Ian Ring Music TheoryStadygic
5th mode:
Scale 3487		Scale 3487: Byptygic, Ian Ring Music TheoryByptygic
6th mode:
Scale 3791		Scale 3791: Stodygic, Ian Ring Music TheoryStodygic
7th mode:
Scale 3943		Scale 3943: Zynygic, Ian Ring Music TheoryZynygic
8th mode:
Scale 4019		Scale 4019: Lonygic, Ian Ring Music TheoryLonygic
9th mode:
Scale 4057		Scale 4057: Phrygic, Ian Ring Music TheoryPhrygic

Prime

The prime form of this scale is Scale 895

Scale 895Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygic

Complement

The enneatonic modal family [1019, 2557, 1663, 2879, 3487, 3791, 3943, 4019, 4057] (Forte: 9-3) is the complement of the tritonic modal family [19, 769, 2057] (Forte: 3-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1019 is 3065

Scale 3065Scale 3065: Zothygic, Ian Ring Music TheoryZothygic

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111101111111001019k = 1h = 1
211101111111001019k = 2h = 1
32(1)(1)0(1)(1)(1)(1)(1)(1)(1)001019k = 3h = 2
42(11)0(11)(11)(11)1001019k = 4h = 2
52(11)0(11)(11)(11)1001019k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1019 is chiral, and its enantiomorph is scale 3065

Scale 3065Scale 3065: Zothygic, Ian Ring Music TheoryZothygic

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 1019       T0I <11,0> 3065
T1 <1,1> 2038      T1I <11,1> 2035
T2 <1,2> 4076      T2I <11,2> 4070
T3 <1,3> 4057      T3I <11,3> 4045
T4 <1,4> 4019      T4I <11,4> 3995
T5 <1,5> 3943      T5I <11,5> 3895
T6 <1,6> 3791      T6I <11,6> 3695
T7 <1,7> 3487      T7I <11,7> 3295
T8 <1,8> 2879      T8I <11,8> 2495
T9 <1,9> 1663      T9I <11,9> 895
T10 <1,10> 3326      T10I <11,10> 1790
T11 <1,11> 2557      T11I <11,11> 3580
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2939      T0MI <7,0> 3035
T1M <5,1> 1783      T1MI <7,1> 1975
T2M <5,2> 3566      T2MI <7,2> 3950
T3M <5,3> 3037      T3MI <7,3> 3805
T4M <5,4> 1979      T4MI <7,4> 3515
T5M <5,5> 3958      T5MI <7,5> 2935
T6M <5,6> 3821      T6MI <7,6> 1775
T7M <5,7> 3547      T7MI <7,7> 3550
T8M <5,8> 2999      T8MI <7,8> 3005
T9M <5,9> 1903      T9MI <7,9> 1915
T10M <5,10> 3806      T10MI <7,10> 3830
T11M <5,11> 3517      T11MI <7,11> 3565

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 1017Scale 1017: Dythyllic, Ian Ring Music TheoryDythyllic
Scale 1021Scale 1021: Ladygic, Ian Ring Music TheoryLadygic
Scale 1023Scale 1023: Chromatic Decamode, Ian Ring Music TheoryChromatic Decamode
Scale 1011Scale 1011: Kycryllic, Ian Ring Music TheoryKycryllic
Scale 1015Scale 1015: Ionodygic, Ian Ring Music TheoryIonodygic
Scale 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 987Scale 987: Aeraptyllic, Ian Ring Music TheoryAeraptyllic
Scale 955Scale 955: Ionogyllic, Ian Ring Music TheoryIonogyllic
Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic
Scale 763Scale 763: Doryllic, Ian Ring Music TheoryDoryllic
Scale 507Scale 507: Moryllic, Ian Ring Music TheoryMoryllic
Scale 1531Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic
Scale 2043Scale 2043: Lythyllian, Ian Ring Music TheoryLythyllian
Scale 3067Scale 3067: Goptyllian, Ian Ring Music TheoryGoptyllian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.

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