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# Scale 2047: "Chromatic Undecamode"

### 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
Chromatic Undecamode
Undecatonic Chromatic
Zeitler
Monatic
Dozenal
ANTian

## Analysis

#### Cardinality

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

11 (hendecatonic)

#### Pitch Class Set

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

{0,1,2,3,4,5,6,7,8,9,10}

#### Forte Number

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

11-1

#### 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.

[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.

10 (multihemitonic)

#### Cohemitonia

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

9 (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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

11

#### Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

yes

#### Generator

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

generator: 1
origin: 0

#### Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.

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, 1, 1, 2]

#### 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.

<10, 10, 10, 10, 10, 5>

#### Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<1, 1, 1, 1, 1, 1>

#### Interval Spectrum

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

p10m10n10s10d10t5

#### 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}
<6> = {6,7}
<7> = {7,8}
<8> = {8,9}
<9> = {9,10}
<10> = {10,11}

#### Spectra Variation

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

0.909

#### Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

yes

#### 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.

2.933

#### Polygon Perimeter

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

6.176

#### Myhill Property

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

yes

#### Centre of Gravity Distance

When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.

0.09091

#### 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.

[10]

#### 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, 165, 220)

#### Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.857

#### Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.6

## Generator

This scale has a generator of 1, originating on 0.

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

C♯{1,5,8}452.87
D{2,6,9}452.93
D♯{3,7,10}453.1
F{5,9,0}452.87
F♯{6,10,1}452.93
G♯{8,0,3}353.13
A{9,1,4}452.8
A♯{10,2,5}353
c♯m{1,4,8}452.93
dm{2,5,9}452.87
d♯m{3,6,10}453.07
fm{5,8,0}353
f♯m{6,9,1}452.8
gm{7,10,2}353.13
am{9,0,4}452.93
a♯m{10,1,5}452.87
C♯+{1,5,9}652.6
D+{2,6,10}552.9
c♯°{1,4,7}253.3
{2,5,8}253.27
d♯°{3,6,9}253.3
{4,7,10}253.3
f♯°{6,9,0}253.2
{7,10,1}253.37
{9,0,3}253.37
a♯°{10,1,4}253.2

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.

Diameter 5 5 yes

## Modes

Modes are the rotational transformation of this scale. Scale 2047 can be rotated to make 10 other scales. The 1st mode is itself.

 2nd mode:Scale 3071 Chromatic Undecamode 2 3rd mode:Scale 3583 Chromatic Undecamode 3 4th mode:Scale 3839 Chromatic Undecamode 4 5th mode:Scale 3967 Chromatic Undecamode 5 6th mode:Scale 4031 Chromatic Undecamode 6 7th mode:Scale 4063 Chromatic Undecamode 7 8th mode:Scale 4079 Chromatic Undecamode 8 9th mode:Scale 4087 Chromatic Undecamode 9 10th mode:Scale 4091 Chromatic Undecamode 10 11th mode:Scale 4093 Chromatic Undecamode 11

## Prime

This is the prime form of this scale.

## Complement

The hendecatonic modal family [2047, 3071, 3583, 3839, 3967, 4031, 4063, 4079, 4087, 4091, 4093] (Forte: 11-1) is the complement of the monotonic modal family [1] (Forte: 1-1)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2047 is 4093

 Scale 4093 Chromatic Undecamode 11

## Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

## Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
12(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)0
22(11111)(11111)10
32(11111)(11111)10
42(11111)(11111)10
52(11111)(11111)10

## Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center (0.045455, 0.07873) 0.09091 150 500

## 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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2047       T0I <11,0> 4093
T1 <1,1> 4094      T1I <11,1> 4091
T2 <1,2> 4093      T2I <11,2> 4087
T3 <1,3> 4091      T3I <11,3> 4079
T4 <1,4> 4087      T4I <11,4> 4063
T5 <1,5> 4079      T5I <11,5> 4031
T6 <1,6> 4063      T6I <11,6> 3967
T7 <1,7> 4031      T7I <11,7> 3839
T8 <1,8> 3967      T8I <11,8> 3583
T9 <1,9> 3839      T9I <11,9> 3071
T10 <1,10> 3583      T10I <11,10> 2047
T11 <1,11> 3071      T11I <11,11> 4094
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3967      T0MI <7,0> 4063
T1M <5,1> 3839      T1MI <7,1> 4031
T2M <5,2> 3583      T2MI <7,2> 3967
T3M <5,3> 3071      T3MI <7,3> 3839
T4M <5,4> 2047       T4MI <7,4> 3583
T5M <5,5> 4094      T5MI <7,5> 3071
T6M <5,6> 4093      T6MI <7,6> 2047
T7M <5,7> 4091      T7MI <7,7> 4094
T8M <5,8> 4087      T8MI <7,8> 4093
T9M <5,9> 4079      T9MI <7,9> 4091
T10M <5,10> 4063      T10MI <7,10> 4087
T11M <5,11> 4031      T11MI <7,11> 4079

The transformations that map this set to itself are: T0, T10I, T4M, 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 2045 Katogyllian Scale 2043 Lythyllian Scale 2039 Danyllian Scale 2031 Gadyllian Scale 2015 Messiaen Mode 7 Rotation 4 Scale 1983 Soryllian Scale 1919 Rocryllian Scale 1791 Aerygyllian Scale 1535 Mixodyllian Scale 1023 Chromatic Decamode Scale 3071 Chromatic Undecamode 2 Scale 4095 Chromatic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.