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Scale 2559: "Decatonic Chromatic 2"

Scale 2559: Decatonic Chromatic 2, 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
Decatonic Chromatic 2
Zeitler
Zogyllian
Dozenal
Pocian

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

{0,1,2,3,4,5,6,7,8,11}

Forte Number

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

10-1

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.

[3.5]

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.

9 (multihemitonic)

Cohemitonia

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

8 (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: 1023

Generator

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

generator: 1
origin: 11

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, 1, 1, 1, 1, 3, 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.

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

Interval Spectrum

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

p8m8n8s8d9t4

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

Spectra Variation

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

1.8

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.

2.75

Polygon Perimeter

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

6.073

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.

[7]

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.

(120, 84, 165)

Generator

This scale has a generator of 1, originating on 11.

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}452.4
C♯{1,5,8}352.9
E{4,8,11}452.4
G{7,11,2}352.7
G♯{8,0,3}352.4
B{11,3,6}352.7
Minor Triadscm{0,3,7}452.4
c♯m{1,4,8}352.7
em{4,7,11}352.4
fm{5,8,0}352.7
g♯m{8,11,3}452.4
bm{11,2,6}352.9
Augmented TriadsC+{0,4,8}552.3
D♯+{3,7,11}552.3
Diminished Triads{0,3,6}252.9
c♯°{1,4,7}252.9
{2,5,8}253.1
{5,8,11}252.9
g♯°{8,11,2}252.9
{11,2,5}253.1
Parsimonious Voice Leading Between Common Triads of Scale 2559. 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 fm fm C+->fm C+->G# c#°->c#m C# C# c#m->C# C#->d° C#->fm d°->b° D#+->em Parsimonious Voice Leading Between Common Triads of Scale 2559. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m D#+->B em->E E->f° E->g#m f°->fm g#° g#° G->g#° bm bm G->bm g#°->g#m g#m->G# b°->bm 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 2559 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 3327
Scale 3327: Decatonic Chromatic 3, Ian Ring Music TheoryDecatonic Chromatic 3
3rd mode:
Scale 3711
Scale 3711: Decatonic Chromatic 4, Ian Ring Music TheoryDecatonic Chromatic 4
4th mode:
Scale 3903
Scale 3903: Decatonic Chromatic 5, Ian Ring Music TheoryDecatonic Chromatic 5
5th mode:
Scale 3999
Scale 3999: Decatonic Chromatic 6, Ian Ring Music TheoryDecatonic Chromatic 6
6th mode:
Scale 4047
Scale 4047: Decatonic Chromatic 7, Ian Ring Music TheoryDecatonic Chromatic 7
7th mode:
Scale 4071
Scale 4071: Decatonic Chromatic 8, Ian Ring Music TheoryDecatonic Chromatic 8
8th mode:
Scale 4083
Scale 4083: Decatonic Chromatic 9, Ian Ring Music TheoryDecatonic Chromatic 9
9th mode:
Scale 4089
Scale 4089: Decatonic Chromatic Descending, Ian Ring Music TheoryDecatonic Chromatic Descending
10th mode:
Scale 1023
Scale 1023: Chromatic Decamode, Ian Ring Music TheoryChromatic DecamodeThis is the prime mode

Prime

The prime form of this scale is Scale 1023

Scale 1023Scale 1023: Chromatic Decamode, Ian Ring Music TheoryChromatic Decamode

Complement

The decatonic modal family [2559, 3327, 3711, 3903, 3999, 4047, 4071, 4083, 4089, 1023] (Forte: 10-1) is the complement of the ditonic modal family [3, 2049] (Forte: 2-1)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2559 is 4083

Scale 4083Scale 4083: Decatonic Chromatic 9, Ian Ring Music TheoryDecatonic Chromatic 9

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

The transformations that map this set to itself are: T0, T7I

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 2557Scale 2557: Dothygic, Ian Ring Music TheoryDothygic
Scale 2555Scale 2555: Bythygic, Ian Ring Music TheoryBythygic
Scale 2551Scale 2551: Thocrygic, Ian Ring Music TheoryThocrygic
Scale 2543Scale 2543: Dydygic, Ian Ring Music TheoryDydygic
Scale 2527Scale 2527: Phradygic, Ian Ring Music TheoryPhradygic
Scale 2495Scale 2495: Aeolocrygic, Ian Ring Music TheoryAeolocrygic
Scale 2431Scale 2431: Gythygic, Ian Ring Music TheoryGythygic
Scale 2303Scale 2303: Nonatonic Chromatic 2, Ian Ring Music TheoryNonatonic Chromatic 2
Scale 2815Scale 2815: Aeradyllian, Ian Ring Music TheoryAeradyllian
Scale 3071Scale 3071: Chromatic Undecamode 2, Ian Ring Music TheoryChromatic Undecamode 2
Scale 3583Scale 3583: Chromatic Undecamode 3, Ian Ring Music TheoryChromatic Undecamode 3
Scale 511Scale 511: Chromatic Nonamode, Ian Ring Music TheoryChromatic Nonamode
Scale 1535Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllian

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.