The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3003: "Genus Chromaticum"

Scale 3003: Genus Chromaticum, 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

Ancient Greek
Genus Chromaticum
Named After Composers
Tcherepnin Nonatonic Mode 1
Western Altered
Augmented Nine
Zeitler
Zydygic

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

{0,1,3,4,5,7,8,9,11}

Forte Number

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

9-12

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

[4, 8]

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.

[0, 2, 4]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

yes

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.

6 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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.

2

Prime Form

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

no
prime: 1911

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 1, 1, 2, 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.

<6, 6, 6, 9, 6, 3>

Interval Spectrum

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

p6m9n6s6d6t3

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

Spectra Variation

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

0.667

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.

2.799

Polygon Perimeter

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

6.106

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.

yes

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.

[0,4,8]

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".

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, 27, 108)

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}442.11
C♯{1,5,8}352.56
E{4,8,11}442.11
F{5,9,0}352.56
G♯{8,0,3}442.11
A{9,1,4}352.56
Minor Triadscm{0,3,7}352.56
c♯m{1,4,8}442.11
em{4,7,11}352.56
fm{5,8,0}442.11
g♯m{8,11,3}352.56
am{9,0,4}442.11
Augmented TriadsC+{0,4,8}631.67
C♯+{1,5,9}363
D♯+{3,7,11}363
Diminished Triadsc♯°{1,4,7}242.56
{5,8,11}242.56
{9,0,3}242.56
Parsimonious Voice Leading Between Common Triads of Scale 3003. Created by Ian Ring ©2019 cm cm 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# am am C+->am c#°->c#m C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F C#+->A D#+->em g#m g#m D#+->g#m em->E E->f° E->g#m f°->fm fm->F F->am g#m->G# G#->a° a°->am am->A

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 VerticesC+
Peripheral VerticesC♯+, D♯+

Modes

Modes are the rotational transformation of this scale. Scale 3003 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 3549
Scale 3549: Messiaen Mode 3 Inverse, Ian Ring Music TheoryMessiaen Mode 3 Inverse
3rd mode:
Scale 1911
Scale 1911: Messiaen Mode 3, Ian Ring Music TheoryMessiaen Mode 3This is the prime mode

Prime

The prime form of this scale is Scale 1911

Scale 1911Scale 1911: Messiaen Mode 3, Ian Ring Music TheoryMessiaen Mode 3

Complement

The enneatonic modal family [3003, 3549, 1911] (Forte: 9-12) is the complement of the tritonic modal family [273] (Forte: 3-12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3003 is itself, because it is a palindromic scale!

Scale 3003Scale 3003: Genus Chromaticum, Ian Ring Music TheoryGenus Chromaticum

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> 3003       T0I <11,0> 3003
T1 <1,1> 1911      T1I <11,1> 1911
T2 <1,2> 3822      T2I <11,2> 3822
T3 <1,3> 3549      T3I <11,3> 3549
T4 <1,4> 3003       T4I <11,4> 3003
T5 <1,5> 1911      T5I <11,5> 1911
T6 <1,6> 3822      T6I <11,6> 3822
T7 <1,7> 3549      T7I <11,7> 3549
T8 <1,8> 3003       T8I <11,8> 3003
T9 <1,9> 1911      T9I <11,9> 1911
T10 <1,10> 3822      T10I <11,10> 3822
T11 <1,11> 3549      T11I <11,11> 3549
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3003       T0MI <7,0> 3003
T1M <5,1> 1911      T1MI <7,1> 1911
T2M <5,2> 3822      T2MI <7,2> 3822
T3M <5,3> 3549      T3MI <7,3> 3549
T4M <5,4> 3003       T4MI <7,4> 3003
T5M <5,5> 1911      T5MI <7,5> 1911
T6M <5,6> 3822      T6MI <7,6> 3822
T7M <5,7> 3549      T7MI <7,7> 3549
T8M <5,8> 3003       T8MI <7,8> 3003
T9M <5,9> 1911      T9MI <7,9> 1911
T10M <5,10> 3822      T10MI <7,10> 3822
T11M <5,11> 3549      T11MI <7,11> 3549

The transformations that map this set to itself are: T0, T4, T8, T0I, T4I, T8I, T0M, T4M, T8M, T0MI, T4MI, T8MI

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 3001Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
Scale 3005Scale 3005: Gycrygic, Ian Ring Music TheoryGycrygic
Scale 3007Scale 3007: Zyryllian, Ian Ring Music TheoryZyryllian
Scale 2995Scale 2995: Raga Saurashtra, Ian Ring Music TheoryRaga Saurashtra
Scale 2999Scale 2999: Diminishing Nonamode, Ian Ring Music TheoryDiminishing Nonamode
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 2971Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic
Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic
Scale 3067Scale 3067: Goptyllian, Ian Ring Music TheoryGoptyllian
Scale 2875Scale 2875: Ganyllic, Ian Ring Music TheoryGanyllic
Scale 2939Scale 2939: Goptygic, Ian Ring Music TheoryGoptygic
Scale 2747Scale 2747: Stythyllic, Ian Ring Music TheoryStythyllic
Scale 2491Scale 2491: Layllic, Ian Ring Music TheoryLayllic
Scale 3515Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian
Scale 4027Scale 4027: Ragyllian, Ian Ring Music TheoryRagyllian
Scale 955Scale 955: Ionogyllic, Ian Ring Music TheoryIonogyllic
Scale 1979Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic

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.