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Scale 991: "Aeolygic"

Scale 991: Aeolygic, 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
Aeolygic
Dozenal
Gatian

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

{0,1,2,3,4,6,7,8,9}

Forte Number

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

9-5

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: 3961

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.

2

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.

8

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, 1, 2, 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, 6, 6, 7, 4>

Interval Spectrum

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

p7m6n6s6d7t4

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}
<4> = {4,5,6}
<5> = {6,7,8}
<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.

1.778

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 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.

(28, 94, 180)

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}342.21
D{2,6,9}242.64
G♯{8,0,3}342.21
A{9,1,4}342.14
Minor Triadscm{0,3,7}342.29
c♯m{1,4,8}342.21
f♯m{6,9,1}342.36
am{9,0,4}442.07
Augmented TriadsC+{0,4,8}442
Diminished Triads{0,3,6}242.57
c♯°{1,4,7}242.57
d♯°{3,6,9}242.71
f♯°{6,9,0}242.43
{9,0,3}242.43
Parsimonious Voice Leading Between Common Triads of Scale 991. Created by Ian Ring ©2019 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 C+->G# am am C+->am c#°->c#m A A c#m->A D D D->d#° f#m f#m D->f#m f#° f#° f#°->f#m f#°->am f#m->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 991 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 2543
Scale 2543: Dydygic, Ian Ring Music TheoryDydygic
3rd mode:
Scale 3319
Scale 3319: Tholygic, Ian Ring Music TheoryTholygic
4th mode:
Scale 3707
Scale 3707: Rynygic, Ian Ring Music TheoryRynygic
5th mode:
Scale 3901
Scale 3901: Bycrygic, Ian Ring Music TheoryBycrygic
6th mode:
Scale 1999
Scale 1999: Zacrygic, Ian Ring Music TheoryZacrygic
7th mode:
Scale 3047
Scale 3047: Panygic, Ian Ring Music TheoryPanygic
8th mode:
Scale 3571
Scale 3571: Dyrygic, Ian Ring Music TheoryDyrygic
9th mode:
Scale 3833
Scale 3833: Dycrygic, Ian Ring Music TheoryDycrygic

Prime

This is the prime form of this scale.

Complement

The enneatonic modal family [991, 2543, 3319, 3707, 3901, 1999, 3047, 3571, 3833] (Forte: 9-5) is the complement of the tritonic modal family [67, 193, 2081] (Forte: 3-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 991 is 3961

Scale 3961Scale 3961: Zathygic, Ian Ring Music TheoryZathygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 991 is chiral, and its enantiomorph is scale 3961

Scale 3961Scale 3961: Zathygic, Ian Ring Music TheoryZathygic

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> 991       T0I <11,0> 3961
T1 <1,1> 1982      T1I <11,1> 3827
T2 <1,2> 3964      T2I <11,2> 3559
T3 <1,3> 3833      T3I <11,3> 3023
T4 <1,4> 3571      T4I <11,4> 1951
T5 <1,5> 3047      T5I <11,5> 3902
T6 <1,6> 1999      T6I <11,6> 3709
T7 <1,7> 3998      T7I <11,7> 3323
T8 <1,8> 3901      T8I <11,8> 2551
T9 <1,9> 3707      T9I <11,9> 1007
T10 <1,10> 3319      T10I <11,10> 2014
T11 <1,11> 2543      T11I <11,11> 4028
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3961      T0MI <7,0> 991
T1M <5,1> 3827      T1MI <7,1> 1982
T2M <5,2> 3559      T2MI <7,2> 3964
T3M <5,3> 3023      T3MI <7,3> 3833
T4M <5,4> 1951      T4MI <7,4> 3571
T5M <5,5> 3902      T5MI <7,5> 3047
T6M <5,6> 3709      T6MI <7,6> 1999
T7M <5,7> 3323      T7MI <7,7> 3998
T8M <5,8> 2551      T8MI <7,8> 3901
T9M <5,9> 1007      T9MI <7,9> 3707
T10M <5,10> 2014      T10MI <7,10> 3319
T11M <5,11> 4028      T11MI <7,11> 2543

The transformations that map this set to itself are: T0, T0MI

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 989Scale 989: Phrolyllic, Ian Ring Music TheoryPhrolyllic
Scale 987Scale 987: Aeraptyllic, Ian Ring Music TheoryAeraptyllic
Scale 983Scale 983: Thocryllic, Ian Ring Music TheoryThocryllic
Scale 975Scale 975: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4
Scale 1007Scale 1007: Epitygic, Ian Ring Music TheoryEpitygic
Scale 1023Scale 1023: Chromatic Decamode, Ian Ring Music TheoryChromatic Decamode
Scale 927Scale 927: Gaptyllic, Ian Ring Music TheoryGaptyllic
Scale 959Scale 959: Katylygic, Ian Ring Music TheoryKatylygic
Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic
Scale 735Scale 735: Sylyllic, Ian Ring Music TheorySylyllic
Scale 479Scale 479: Kocryllic, Ian Ring Music TheoryKocryllic
Scale 1503Scale 1503: Padygic, Ian Ring Music TheoryPadygic
Scale 2015Scale 2015: Messiaen Mode 7, Ian Ring Music TheoryMessiaen Mode 7
Scale 3039Scale 3039: Godyllian, Ian Ring Music TheoryGodyllian

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.