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Scale 1751: "Aeolyryllic"

Scale 1751: Aeolyryllic, 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
Aeolyryllic
Dozenal
Kukian

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

{0,1,2,4,6,7,9,10}

Forte Number

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

8-27

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: 3437

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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: 1463

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

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

Interval Spectrum

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

p5m5n6s5d4t3

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}
<2> = {2,3,4}
<3> = {4,5}
<4> = {5,6,7}
<5> = {7,8}
<6> = {8,9,10}
<7> = {10,11}

Spectra Variation

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

1.25

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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, 24, 103)

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}342.23
D{2,6,9}242.23
F♯{6,10,1}441.92
A{9,1,4}441.92
Minor Triadsf♯m{6,9,1}441.85
gm{7,10,2}342.23
am{9,0,4}342.15
Augmented TriadsD+{2,6,10}342.15
Diminished Triadsc♯°{1,4,7}242.31
{4,7,10}242.31
f♯°{6,9,0}242.23
{7,10,1}242.31
a♯°{10,1,4}242.15
Parsimonious Voice Leading Between Common Triads of Scale 1751. Created by Ian Ring ©2019 C C c#° c#° C->c#° C->e° am am C->am A A c#°->A D D D+ D+ D->D+ f#m f#m D->f#m F# F# D+->F# gm gm D+->gm e°->gm f#° f#° f#°->f#m f#°->am f#m->F# f#m->A F#->g° a#° a#° F#->a#° g°->gm am->A A->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 1751 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2923
Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
3rd mode:
Scale 3509
Scale 3509: Stogyllic, Ian Ring Music TheoryStogyllic
4th mode:
Scale 1901
Scale 1901: Ionidyllic, Ian Ring Music TheoryIonidyllic
5th mode:
Scale 1499
Scale 1499: Bebop Locrian, Ian Ring Music TheoryBebop Locrian
6th mode:
Scale 2797
Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
7th mode:
Scale 1723
Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic
8th mode:
Scale 2909
Scale 2909: Mocryllic, Ian Ring Music TheoryMocryllic

Prime

The prime form of this scale is Scale 1463

Scale 1463Scale 1463: Ugrian, Ian Ring Music TheoryUgrian

Complement

The octatonic modal family [1751, 2923, 3509, 1901, 1499, 2797, 1723, 2909] (Forte: 8-27) is the complement of the tetratonic modal family [293, 593, 649, 1097] (Forte: 4-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1751 is 3437

Scale 3437Scale 3437: Vopian, Ian Ring Music TheoryVopian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1751 is chiral, and its enantiomorph is scale 3437

Scale 3437Scale 3437: Vopian, Ian Ring Music TheoryVopian

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> 1751       T0I <11,0> 3437
T1 <1,1> 3502      T1I <11,1> 2779
T2 <1,2> 2909      T2I <11,2> 1463
T3 <1,3> 1723      T3I <11,3> 2926
T4 <1,4> 3446      T4I <11,4> 1757
T5 <1,5> 2797      T5I <11,5> 3514
T6 <1,6> 1499      T6I <11,6> 2933
T7 <1,7> 2998      T7I <11,7> 1771
T8 <1,8> 1901      T8I <11,8> 3542
T9 <1,9> 3802      T9I <11,9> 2989
T10 <1,10> 3509      T10I <11,10> 1883
T11 <1,11> 2923      T11I <11,11> 3766
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3941      T0MI <7,0> 1247
T1M <5,1> 3787      T1MI <7,1> 2494
T2M <5,2> 3479      T2MI <7,2> 893
T3M <5,3> 2863      T3MI <7,3> 1786
T4M <5,4> 1631      T4MI <7,4> 3572
T5M <5,5> 3262      T5MI <7,5> 3049
T6M <5,6> 2429      T6MI <7,6> 2003
T7M <5,7> 763      T7MI <7,7> 4006
T8M <5,8> 1526      T8MI <7,8> 3917
T9M <5,9> 3052      T9MI <7,9> 3739
T10M <5,10> 2009      T10MI <7,10> 3383
T11M <5,11> 4018      T11MI <7,11> 2671

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 1749Scale 1749: Acoustic, Ian Ring Music TheoryAcoustic
Scale 1747Scale 1747: Mela Ramapriya, Ian Ring Music TheoryMela Ramapriya
Scale 1755Scale 1755: Octatonic, Ian Ring Music TheoryOctatonic
Scale 1759Scale 1759: Pylygic, Ian Ring Music TheoryPylygic
Scale 1735Scale 1735: Mela Navanitam, Ian Ring Music TheoryMela Navanitam
Scale 1743Scale 1743: Epigyllic, Ian Ring Music TheoryEpigyllic
Scale 1767Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic
Scale 1783Scale 1783: Youlan Scale, Ian Ring Music TheoryYoulan Scale
Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian
Scale 1719Scale 1719: Lyryllic, Ian Ring Music TheoryLyryllic
Scale 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian
Scale 1879Scale 1879: Mixoryllic, Ian Ring Music TheoryMixoryllic
Scale 2007Scale 2007: Stonygic, Ian Ring Music TheoryStonygic
Scale 1239Scale 1239: Epaptian, Ian Ring Music TheoryEpaptian
Scale 1495Scale 1495: Messiaen Mode 6, Ian Ring Music TheoryMessiaen Mode 6
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 2775Scale 2775: Godyllic, Ian Ring Music TheoryGodyllic
Scale 3799Scale 3799: Aeralygic, Ian Ring Music TheoryAeralygic

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.