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Scale 3039: "Godyllian"

Scale 3039: Godyllian, 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
Godyllian
Dozenal
Tamian

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

{0,1,2,3,4,6,7,8,9,11}

Forte Number

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

10-5

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.

[1.5]

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.

1

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: 1983

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 5
origin: 3

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, 2, 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, 8, 9, 4>

Interval Spectrum

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

p9m8n8s8d8t4

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}
<5> = {5,6,7}
<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

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.

[3]

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, 96, 177)

Generator

This scale has a generator of 5, originating on 3.

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}452.5
D{2,6,9}352.95
E{4,8,11}352.59
G{7,11,2}352.77
G♯{8,0,3}452.41
A{9,1,4}352.86
B{11,3,6}452.68
Minor Triadscm{0,3,7}452.41
c♯m{1,4,8}352.77
em{4,7,11}352.59
f♯m{6,9,1}352.95
g♯m{8,11,3}452.5
am{9,0,4}452.68
bm{11,2,6}352.86
Augmented TriadsC+{0,4,8}552.41
D♯+{3,7,11}552.41
Diminished Triads{0,3,6}252.86
c♯°{1,4,7}253.05
d♯°{3,6,9}253.05
f♯°{6,9,0}253.05
g♯°{8,11,2}253.05
{9,0,3}252.86
Parsimonious Voice Leading Between Common Triads of Scale 3039. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B 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 C+->G# am am C+->am c#°->c#m A A c#m->A D D d#° d#° D->d#° f#m f#m D->f#m bm bm D->bm d#°->B D#+->em Parsimonious Voice Leading Between Common Triads of Scale 3039. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m D#+->B em->E E->g#m f#° f#° f#°->f#m f#°->am f#m->A g#° g#° G->g#° G->bm g#°->g#m g#m->G# G#->a° a°->am am->A bm->B

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 3039 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 3567
Scale 3567: Epityllian, Ian Ring Music TheoryEpityllian
3rd mode:
Scale 3831
Scale 3831: Ionyllian, Ian Ring Music TheoryIonyllian
4th mode:
Scale 3963
Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian
5th mode:
Scale 4029
Scale 4029: Major/Minor Mixed, Ian Ring Music TheoryMajor/Minor Mixed
6th mode:
Scale 2031
Scale 2031: Gadyllian, Ian Ring Music TheoryGadyllian
7th mode:
Scale 3063
Scale 3063: Solyllian, Ian Ring Music TheorySolyllian
8th mode:
Scale 3579
Scale 3579: Zyphyllian, Ian Ring Music TheoryZyphyllian
9th mode:
Scale 3837
Scale 3837: Minor Pentatonic With Leading Tones, Ian Ring Music TheoryMinor Pentatonic With Leading Tones
10th mode:
Scale 1983
Scale 1983: Soryllian, Ian Ring Music TheorySoryllianThis is the prime mode

Prime

The prime form of this scale is Scale 1983

Scale 1983Scale 1983: Soryllian, Ian Ring Music TheorySoryllian

Complement

The decatonic modal family [3039, 3567, 3831, 3963, 4029, 2031, 3063, 3579, 3837, 1983] (Forte: 10-5) is the complement of the ditonic modal family [33, 129] (Forte: 2-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3039 is 3963

Scale 3963Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian

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> 3039       T0I <11,0> 3963
T1 <1,1> 1983      T1I <11,1> 3831
T2 <1,2> 3966      T2I <11,2> 3567
T3 <1,3> 3837      T3I <11,3> 3039
T4 <1,4> 3579      T4I <11,4> 1983
T5 <1,5> 3063      T5I <11,5> 3966
T6 <1,6> 2031      T6I <11,6> 3837
T7 <1,7> 4062      T7I <11,7> 3579
T8 <1,8> 4029      T8I <11,8> 3063
T9 <1,9> 3963      T9I <11,9> 2031
T10 <1,10> 3831      T10I <11,10> 4062
T11 <1,11> 3567      T11I <11,11> 4029
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4089      T0MI <7,0> 1023
T1M <5,1> 4083      T1MI <7,1> 2046
T2M <5,2> 4071      T2MI <7,2> 4092
T3M <5,3> 4047      T3MI <7,3> 4089
T4M <5,4> 3999      T4MI <7,4> 4083
T5M <5,5> 3903      T5MI <7,5> 4071
T6M <5,6> 3711      T6MI <7,6> 4047
T7M <5,7> 3327      T7MI <7,7> 3999
T8M <5,8> 2559      T8MI <7,8> 3903
T9M <5,9> 1023      T9MI <7,9> 3711
T10M <5,10> 2046      T10MI <7,10> 3327
T11M <5,11> 4092      T11MI <7,11> 2559

The transformations that map this set to itself are: T0, T3I

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 3037Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale
Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic
Scale 3031Scale 3031: Epithygic, Ian Ring Music TheoryEpithygic
Scale 3023Scale 3023: Mothygic, Ian Ring Music TheoryMothygic
Scale 3055Scale 3055: Messiaen Mode 7, Ian Ring Music TheoryMessiaen Mode 7
Scale 3071Scale 3071: Chromatic Undecamode 2, Ian Ring Music TheoryChromatic Undecamode 2
Scale 2975Scale 2975: Aeroptygic, Ian Ring Music TheoryAeroptygic
Scale 3007Scale 3007: Zyryllian, Ian Ring Music TheoryZyryllian
Scale 2911Scale 2911: Katygic, Ian Ring Music TheoryKatygic
Scale 2783Scale 2783: Gothygic, Ian Ring Music TheoryGothygic
Scale 2527Scale 2527: Phradygic, Ian Ring Music TheoryPhradygic
Scale 3551Scale 3551: Sagyllian, Ian Ring Music TheorySagyllian
Scale 4063Scale 4063: Chromatic Undecamode 7, Ian Ring Music TheoryChromatic Undecamode 7
Scale 991Scale 991: Aeolygic, Ian Ring Music TheoryAeolygic
Scale 2015Scale 2015: Messiaen Mode 7, Ian Ring Music TheoryMessiaen Mode 7

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.