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Scale 2111: "Heptatonic Chromatic 2"

Scale 2111: Heptatonic 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
Heptatonic Chromatic 2

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,3,4,5,11}

Forte Number

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

7-1

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.

[2]

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.

6 (multihemitonic)

Cohemitonia

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

5 (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.

5

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.

6

Prime Form

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

no
prime: 127

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

yes

Interval Formula

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

[1, 1, 1, 1, 1, 6, 1] 9

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, 5, 4, 3, 2, 1>

Interval Spectrum

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

p2m3n4s5d6t

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

Spectra Variation

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

4.286

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.

1.5

Polygon Perimeter

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

5.106

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.

[4]

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

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
Diminished Triads{11,2,5}000

The following pitch classes are not present in any of the common triads: {0,1,3,4}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 2111 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3103
Scale 3103: Heptatonic Chromatic 3, Ian Ring Music TheoryHeptatonic Chromatic 3
3rd mode:
Scale 3599
Scale 3599: Heptatonic Chromatic 4, Ian Ring Music TheoryHeptatonic Chromatic 4
4th mode:
Scale 3847
Scale 3847: Heptatonic Chromatic 5, Ian Ring Music TheoryHeptatonic Chromatic 5
5th mode:
Scale 3971
Scale 3971: Heptatonic Chromatic 6, Ian Ring Music TheoryHeptatonic Chromatic 6
6th mode:
Scale 4033
Scale 4033: Heptatonic Chromatic Descending, Ian Ring Music TheoryHeptatonic Chromatic Descending
7th mode:
Scale 127
Scale 127: Heptatonic Chromatic, Ian Ring Music TheoryHeptatonic ChromaticThis is the prime mode

Prime

The prime form of this scale is Scale 127

Scale 127Scale 127: Heptatonic Chromatic, Ian Ring Music TheoryHeptatonic Chromatic

Complement

The heptatonic modal family [2111, 3103, 3599, 3847, 3971, 4033, 127] (Forte: 7-1) is the complement of the pentatonic modal family [31, 2063, 3079, 3587, 3841] (Forte: 5-1)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2111 is 3971

Scale 3971Scale 3971: Heptatonic Chromatic 6, Ian Ring Music TheoryHeptatonic Chromatic 6

Transformations:

T0 2111  T0I 3971
T1 127  T1I 3847
T2 254  T2I 3599
T3 508  T3I 3103
T4 1016  T4I 2111
T5 2032  T5I 127
T6 4064  T6I 254
T7 4033  T7I 508
T8 3971  T8I 1016
T9 3847  T9I 2032
T10 3599  T10I 4064
T11 3103  T11I 4033

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 2109Scale 2109, Ian Ring Music Theory
Scale 2107Scale 2107, Ian Ring Music Theory
Scale 2103Scale 2103, Ian Ring Music Theory
Scale 2095Scale 2095, Ian Ring Music Theory
Scale 2079Scale 2079: Hexatonic Chromatic 4, Ian Ring Music TheoryHexatonic Chromatic 4
Scale 2143Scale 2143, Ian Ring Music Theory
Scale 2175Scale 2175: Octatonic Chromatic 2, Ian Ring Music TheoryOctatonic Chromatic 2
Scale 2239Scale 2239: Dacryllic, Ian Ring Music TheoryDacryllic
Scale 2367Scale 2367: Laryllic, Ian Ring Music TheoryLaryllic
Scale 2623Scale 2623: Aerylyllic, Ian Ring Music TheoryAerylyllic
Scale 3135Scale 3135: Octatonic Chromatic 3, Ian Ring Music TheoryOctatonic Chromatic 3
Scale 63Scale 63: Hexatonic Chromatic, Ian Ring Music TheoryHexatonic Chromatic
Scale 1087Scale 1087, Ian Ring Music Theory

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. 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.