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Scale 987: "Aeraptyllic"

Scale 987: Aeraptyllic, 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
Aeraptyllic
Dozenal
Garian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

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

Forte Number

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

8-18

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: 2937

Hemitonia

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

5 (multihemitonic)

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.

7

Prime Form

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

no
prime: 879

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, 2, 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.

<5, 4, 6, 5, 5, 3>

Interval Spectrum

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

p5m5n6s4d5t3

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

Spectra Variation

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

2

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.

2.616

Polygon Perimeter

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

6.002

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.

(12, 59, 138)

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}342.08
G♯{8,0,3}342.08
A{9,1,4}342
Minor Triadscm{0,3,7}342.15
c♯m{1,4,8}342.08
f♯m{6,9,1}342.23
am{9,0,4}441.92
Augmented TriadsC+{0,4,8}441.85
Diminished Triads{0,3,6}242.38
c♯°{1,4,7}242.46
d♯°{3,6,9}242.46
f♯°{6,9,0}242.31
{9,0,3}242.31
Parsimonious Voice Leading Between Common Triads of Scale 987. 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 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 987 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2541
Scale 2541: Algerian, Ian Ring Music TheoryAlgerian
3rd mode:
Scale 1659
Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban
4th mode:
Scale 2877
Scale 2877: Phrylyllic, Ian Ring Music TheoryPhrylyllic
5th mode:
Scale 1743
Scale 1743: Epigyllic, Ian Ring Music TheoryEpigyllic
6th mode:
Scale 2919
Scale 2919: Molyllic, Ian Ring Music TheoryMolyllic
7th mode:
Scale 3507
Scale 3507: Maqam Hijaz, Ian Ring Music TheoryMaqam Hijaz
8th mode:
Scale 3801
Scale 3801: Maptyllic, Ian Ring Music TheoryMaptyllic

Prime

The prime form of this scale is Scale 879

Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic

Complement

The octatonic modal family [987, 2541, 1659, 2877, 1743, 2919, 3507, 3801] (Forte: 8-18) is the complement of the tetratonic modal family [147, 609, 777, 2121] (Forte: 4-18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 987 is 2937

Scale 2937Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 987 is chiral, and its enantiomorph is scale 2937

Scale 2937Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic

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> 987       T0I <11,0> 2937
T1 <1,1> 1974      T1I <11,1> 1779
T2 <1,2> 3948      T2I <11,2> 3558
T3 <1,3> 3801      T3I <11,3> 3021
T4 <1,4> 3507      T4I <11,4> 1947
T5 <1,5> 2919      T5I <11,5> 3894
T6 <1,6> 1743      T6I <11,6> 3693
T7 <1,7> 3486      T7I <11,7> 3291
T8 <1,8> 2877      T8I <11,8> 2487
T9 <1,9> 1659      T9I <11,9> 879
T10 <1,10> 3318      T10I <11,10> 1758
T11 <1,11> 2541      T11I <11,11> 3516
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2937      T0MI <7,0> 987
T1M <5,1> 1779      T1MI <7,1> 1974
T2M <5,2> 3558      T2MI <7,2> 3948
T3M <5,3> 3021      T3MI <7,3> 3801
T4M <5,4> 1947      T4MI <7,4> 3507
T5M <5,5> 3894      T5MI <7,5> 2919
T6M <5,6> 3693      T6MI <7,6> 1743
T7M <5,7> 3291      T7MI <7,7> 3486
T8M <5,8> 2487      T8MI <7,8> 2877
T9M <5,9> 879      T9MI <7,9> 1659
T10M <5,10> 1758      T10MI <7,10> 3318
T11M <5,11> 3516      T11MI <7,11> 2541

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 985Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
Scale 989Scale 989: Phrolyllic, Ian Ring Music TheoryPhrolyllic
Scale 991Scale 991: Aeolygic, Ian Ring Music TheoryAeolygic
Scale 979Scale 979: Mela Dhavalambari, Ian Ring Music TheoryMela Dhavalambari
Scale 983Scale 983: Thocryllic, Ian Ring Music TheoryThocryllic
Scale 971Scale 971: Mela Gavambodhi, Ian Ring Music TheoryMela Gavambodhi
Scale 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 1019Scale 1019: Aeranygic, Ian Ring Music TheoryAeranygic
Scale 923Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian
Scale 955Scale 955: Ionogyllic, Ian Ring Music TheoryIonogyllic
Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode
Scale 475Scale 475: Aeolygian, Ian Ring Music TheoryAeolygian
Scale 1499Scale 1499: Bebop Locrian, Ian Ring Music TheoryBebop Locrian
Scale 2011Scale 2011: Raphygic, Ian Ring Music TheoryRaphygic
Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic

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.