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Scale 4039: "Nonatonic Chromatic 7"

Scale 4039: Nonatonic Chromatic 7, 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

Western Modern
Nonatonic Chromatic 7
Dozenal
Zowian
Zeitler
Ionogygic

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,1,2,6,7,8,9,10,11}

Forte Number

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

9-1

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.

7 (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.

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.

8

Prime Form

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

no
prime: 511

Generator

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

generator: 1
origin: 6

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, 4, 1, 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, 7, 6, 6, 6, 3>

Interval Spectrum

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

p6m6n6s7d8t3

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> = {1,4}
<2> = {2,5}
<3> = {3,6}
<4> = {4,7}
<5> = {5,8}
<6> = {6,9}
<7> = {7,10}
<8> = {8,11}

Spectra Variation

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

2.667

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.

2.433

Polygon Perimeter

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

5.873

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

yes

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.

(140, 35, 120)

Generator

This scale has a generator of 1, originating on 6.

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}242
F♯{6,10,1}341.8
G{7,11,2}352.2
Minor Triadsf♯m{6,9,1}352.2
gm{7,10,2}341.8
bm{11,2,6}242
Augmented TriadsD+{2,6,10}431.6
Diminished Triadsf♯°{6,9,0}163
{7,10,1}232
g♯°{8,11,2}163
Parsimonious Voice Leading Between Common Triads of Scale 4039. Created by Ian Ring ©2019 D D D+ D+ D->D+ f#m f#m D->f#m F# F# D+->F# gm gm D+->gm bm bm D+->bm f#° f#° f#°->f#m f#m->F# F#->g° g°->gm Parsimonious Voice Leading Between Common Triads of Scale 4039. Created by Ian Ring ©2019 G gm->G g#° g#° G->g#° G->bm

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.

Diameter6
Radius3
Self-Centeredno
Central VerticesD+, g°
Peripheral Verticesf♯°, g♯°

Modes

Modes are the rotational transformation of this scale. Scale 4039 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 4067
Scale 4067: Nonatonic Chromatic 8, Ian Ring Music TheoryNonatonic Chromatic 8
3rd mode:
Scale 4081
Scale 4081: Nonatonic Chromatic Descending, Ian Ring Music TheoryNonatonic Chromatic Descending
4th mode:
Scale 511
Scale 511: Chromatic Nonamode, Ian Ring Music TheoryChromatic NonamodeThis is the prime mode
5th mode:
Scale 2303
Scale 2303: Nonatonic Chromatic 2, Ian Ring Music TheoryNonatonic Chromatic 2
6th mode:
Scale 3199
Scale 3199: Nonatonic Chromatic 3, Ian Ring Music TheoryNonatonic Chromatic 3
7th mode:
Scale 3647
Scale 3647: Nonatonic Chromatic 4, Ian Ring Music TheoryNonatonic Chromatic 4
8th mode:
Scale 3871
Scale 3871: Nonatonic Chromatic 5, Ian Ring Music TheoryNonatonic Chromatic 5
9th mode:
Scale 3983
Scale 3983: Nonatonic Chromatic 6, Ian Ring Music TheoryNonatonic Chromatic 6

Prime

The prime form of this scale is Scale 511

Scale 511Scale 511: Chromatic Nonamode, Ian Ring Music TheoryChromatic Nonamode

Complement

The enneatonic modal family [4039, 4067, 4081, 511, 2303, 3199, 3647, 3871, 3983] (Forte: 9-1) is the complement of the tritonic modal family [7, 2051, 3073] (Forte: 3-1)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4039 is 3199

Scale 3199Scale 3199: Nonatonic Chromatic 3, Ian Ring Music TheoryNonatonic Chromatic 3

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> 4039       T0I <11,0> 3199
T1 <1,1> 3983      T1I <11,1> 2303
T2 <1,2> 3871      T2I <11,2> 511
T3 <1,3> 3647      T3I <11,3> 1022
T4 <1,4> 3199      T4I <11,4> 2044
T5 <1,5> 2303      T5I <11,5> 4088
T6 <1,6> 511      T6I <11,6> 4081
T7 <1,7> 1022      T7I <11,7> 4067
T8 <1,8> 2044      T8I <11,8> 4039
T9 <1,9> 4088      T9I <11,9> 3983
T10 <1,10> 4081      T10I <11,10> 3871
T11 <1,11> 4067      T11I <11,11> 3647
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3829      T0MI <7,0> 1519
T1M <5,1> 3563      T1MI <7,1> 3038
T2M <5,2> 3031      T2MI <7,2> 1981
T3M <5,3> 1967      T3MI <7,3> 3962
T4M <5,4> 3934      T4MI <7,4> 3829
T5M <5,5> 3773      T5MI <7,5> 3563
T6M <5,6> 3451      T6MI <7,6> 3031
T7M <5,7> 2807      T7MI <7,7> 1967
T8M <5,8> 1519      T8MI <7,8> 3934
T9M <5,9> 3038      T9MI <7,9> 3773
T10M <5,10> 1981      T10MI <7,10> 3451
T11M <5,11> 3962      T11MI <7,11> 2807

The transformations that map this set to itself are: T0, T8I

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 4037Scale 4037: Ionyllic, Ian Ring Music TheoryIonyllic
Scale 4035Scale 4035: Octatonic Chromatic 7, Ian Ring Music TheoryOctatonic Chromatic 7
Scale 4043Scale 4043: Phrocrygic, Ian Ring Music TheoryPhrocrygic
Scale 4047Scale 4047: Decatonic Chromatic 7, Ian Ring Music TheoryDecatonic Chromatic 7
Scale 4055Scale 4055: Dagyllian, Ian Ring Music TheoryDagyllian
Scale 4071Scale 4071: Decatonic Chromatic 8, Ian Ring Music TheoryDecatonic Chromatic 8
Scale 3975Scale 3975: Octatonic Chromatic 6, Ian Ring Music TheoryOctatonic Chromatic 6
Scale 4007Scale 4007: Doptygic, Ian Ring Music TheoryDoptygic
Scale 3911Scale 3911: Katyryllic, Ian Ring Music TheoryKatyryllic
Scale 3783Scale 3783: Phrygyllic, Ian Ring Music TheoryPhrygyllic
Scale 3527Scale 3527: Ronyllic, Ian Ring Music TheoryRonyllic
Scale 3015Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic
Scale 1991Scale 1991: Phryptyllic, Ian Ring Music TheoryPhryptyllic

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.