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Scale 4027: "Ragyllian"

Scale 4027: Ragyllian, 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
Ragyllian
Dozenal
Zopian

Analysis

Cardinality

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

10 (decatonic)

Pitch Class Set

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

{0,1,3,4,5,7,8,9,10,11}

Forte Number

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

10-4

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.

[4]

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.

no

Hemitonia

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

8 (multihemitonic)

Cohemitonia

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

6 (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.

9

Prime Form

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

no
prime: 1919

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, 1, 2, 1, 1, 1, 1, 1]

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.

<8, 8, 8, 9, 8, 4>

Interval Spectrum

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

p8m9n8s8d8t4

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

Spectra Variation

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

1.2

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.

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.

2.866

Polygon Perimeter

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

6.141

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.

[8]

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.

(4, 129, 210)

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}452.43
C♯{1,5,8}352.74
D♯{3,7,10}353
E{4,8,11}452.43
F{5,9,0}352.74
G♯{8,0,3}452.52
A{9,1,4}452.65
Minor Triadscm{0,3,7}352.74
c♯m{1,4,8}452.43
em{4,7,11}452.65
fm{5,8,0}452.52
g♯m{8,11,3}352.74
am{9,0,4}452.43
a♯m{10,1,5}353
Augmented TriadsC+{0,4,8}652.13
C♯+{1,5,9}452.78
D♯+{3,7,11}452.78
Diminished Triadsc♯°{1,4,7}252.83
{4,7,10}253.13
{5,8,11}252.91
{7,10,1}253.13
{9,0,3}252.91
a♯°{10,1,4}253.13
Parsimonious Voice Leading Between Common Triads of Scale 4027. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° em em C->em c#m c#m C+->c#m E E C+->E 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 C#+->A a#m a#m C#+->a#m D# D# D#->D#+ D#->e° D#->g° D#+->em g#m g#m D#+->g#m e°->em em->E E->f° E->g#m f°->fm fm->F F->am g°->a#m g#m->G# G#->a° a°->am am->A a#° a#° A->a#° a#°->a#m

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
Radius5
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 4027 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 4061
Scale 4061: Staptyllian, Ian Ring Music TheoryStaptyllian
3rd mode:
Scale 2039
Scale 2039: Danyllian, Ian Ring Music TheoryDanyllian
4th mode:
Scale 3067
Scale 3067: Goptyllian, Ian Ring Music TheoryGoptyllian
5th mode:
Scale 3581
Scale 3581: Epocryllian, Ian Ring Music TheoryEpocryllian
6th mode:
Scale 1919
Scale 1919: Rocryllian, Ian Ring Music TheoryRocryllianThis is the prime mode
7th mode:
Scale 3007
Scale 3007: Zyryllian, Ian Ring Music TheoryZyryllian
8th mode:
Scale 3551
Scale 3551: Sagyllian, Ian Ring Music TheorySagyllian
9th mode:
Scale 3823
Scale 3823: Epinyllian, Ian Ring Music TheoryEpinyllian
10th mode:
Scale 3959
Scale 3959: Katagyllian, Ian Ring Music TheoryKatagyllian

Prime

The prime form of this scale is Scale 1919

Scale 1919Scale 1919: Rocryllian, Ian Ring Music TheoryRocryllian

Complement

The decatonic modal family [4027, 4061, 2039, 3067, 3581, 1919, 3007, 3551, 3823, 3959] (Forte: 10-4) is the complement of the ditonic modal family [17, 257] (Forte: 2-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4027 is 3007

Scale 3007Scale 3007: Zyryllian, Ian Ring Music TheoryZyryllian

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> 4027       T0I <11,0> 3007
T1 <1,1> 3959      T1I <11,1> 1919
T2 <1,2> 3823      T2I <11,2> 3838
T3 <1,3> 3551      T3I <11,3> 3581
T4 <1,4> 3007      T4I <11,4> 3067
T5 <1,5> 1919      T5I <11,5> 2039
T6 <1,6> 3838      T6I <11,6> 4078
T7 <1,7> 3581      T7I <11,7> 4061
T8 <1,8> 3067      T8I <11,8> 4027
T9 <1,9> 2039      T9I <11,9> 3959
T10 <1,10> 4078      T10I <11,10> 3823
T11 <1,11> 4061      T11I <11,11> 3551
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3007      T0MI <7,0> 4027
T1M <5,1> 1919      T1MI <7,1> 3959
T2M <5,2> 3838      T2MI <7,2> 3823
T3M <5,3> 3581      T3MI <7,3> 3551
T4M <5,4> 3067      T4MI <7,4> 3007
T5M <5,5> 2039      T5MI <7,5> 1919
T6M <5,6> 4078      T6MI <7,6> 3838
T7M <5,7> 4061      T7MI <7,7> 3581
T8M <5,8> 4027       T8MI <7,8> 3067
T9M <5,9> 3959      T9MI <7,9> 2039
T10M <5,10> 3823      T10MI <7,10> 4078
T11M <5,11> 3551      T11MI <7,11> 4061

The transformations that map this set to itself are: T0, T8I, T8M, 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 4025Scale 4025: Kalygic, Ian Ring Music TheoryKalygic
Scale 4029Scale 4029: Major/Minor Mixed, Ian Ring Music TheoryMajor/Minor Mixed
Scale 4031Scale 4031: Chromatic Undecamode 6, Ian Ring Music TheoryChromatic Undecamode 6
Scale 4019Scale 4019: Lonygic, Ian Ring Music TheoryLonygic
Scale 4023Scale 4023: Styptyllian, Ian Ring Music TheoryStyptyllian
Scale 4011Scale 4011: Styrygic, Ian Ring Music TheoryStyrygic
Scale 3995Scale 3995: Ionygic, Ian Ring Music TheoryIonygic
Scale 4059Scale 4059: Zolyllian, Ian Ring Music TheoryZolyllian
Scale 4091Scale 4091: Chromatic Undecamode 10, Ian Ring Music TheoryChromatic Undecamode 10
Scale 3899Scale 3899: Katorygic, Ian Ring Music TheoryKatorygic
Scale 3963Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian
Scale 3771Scale 3771: Stophygic, Ian Ring Music TheoryStophygic
Scale 3515Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian
Scale 3003Scale 3003: Genus Chromaticum, Ian Ring Music TheoryGenus Chromaticum
Scale 1979Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic

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.