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Scale 3021: "Stodyllic"

Scale 3021: Stodyllic, 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
Stodyllic
Dozenal
Tabian

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

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

Forte Number

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

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: 1659

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.

[2, 1, 3, 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.

<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 TriadsD{2,6,9}342.23
G{7,11,2}342.08
G♯{8,0,3}342.15
B{11,3,6}441.92
Minor Triadscm{0,3,7}342.08
g♯m{8,11,3}342.08
bm{11,2,6}342
Augmented TriadsD♯+{3,7,11}441.85
Diminished Triads{0,3,6}242.31
d♯°{3,6,9}242.31
f♯°{6,9,0}242.46
g♯°{8,11,2}242.46
{9,0,3}242.38
Parsimonious Voice Leading Between Common Triads of Scale 3021. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ G# G# cm->G# D D d#° d#° D->d#° f#° f#° D->f#° bm bm D->bm d#°->B Parsimonious Voice Leading Between Common Triads of Scale 3021. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m D#+->B f#°->a° g#° g#° G->g#° G->bm g#°->g#m g#m->G# G#->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3021 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1779
Scale 1779: Zynyllic, Ian Ring Music TheoryZynyllic
3rd mode:
Scale 2937
Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic
4th mode:
Scale 879
Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllicThis is the prime mode
5th mode:
Scale 2487
Scale 2487: Dothyllic, Ian Ring Music TheoryDothyllic
6th mode:
Scale 3291
Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic
7th mode:
Scale 3693
Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic
8th mode:
Scale 1947
Scale 1947: Byptyllic, Ian Ring Music TheoryByptyllic

Prime

The prime form of this scale is Scale 879

Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic

Complement

The octatonic modal family [3021, 1779, 2937, 879, 2487, 3291, 3693, 1947] (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 3021 is 1659

Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3021 is chiral, and its enantiomorph is scale 1659

Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban

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

The transformations that map this set to itself are: T0, T6MI

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 3023Scale 3023: Mothygic, Ian Ring Music TheoryMothygic
Scale 3017Scale 3017: Gacrian, Ian Ring Music TheoryGacrian
Scale 3019Scale 3019: Subian, Ian Ring Music TheorySubian
Scale 3013Scale 3013: Thynian, Ian Ring Music TheoryThynian
Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
Scale 3037Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale
Scale 3053Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic
Scale 2957Scale 2957: Thygian, Ian Ring Music TheoryThygian
Scale 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 2893Scale 2893: Lylian, Ian Ring Music TheoryLylian
Scale 2765Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
Scale 2509Scale 2509: Double Harmonic Minor, Ian Ring Music TheoryDouble Harmonic Minor
Scale 3533Scale 3533: Thadyllic, Ian Ring Music TheoryThadyllic
Scale 4045Scale 4045: Gyptygic, Ian Ring Music TheoryGyptygic
Scale 973Scale 973: Mela Syamalangi, Ian Ring Music TheoryMela Syamalangi
Scale 1997Scale 1997: Raga Cintamani, Ian Ring Music TheoryRaga Cintamani

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.