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Scale 1113: "Locrian Pentatonic 2"

Scale 1113: Locrian Pentatonic 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

41161837294116105072918310504116183
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
Locrian Pentatonic 2
Zeitler
Aeronitonic

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,3,4,6,10}

Forte Number

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

5-28

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.

none

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.

yes
enantiomorph: 837

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

4

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.

4

Prime Form

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

no
prime: 333

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.

[3, 1, 2, 4, 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.

<1, 2, 2, 2, 1, 2>

Interval Spectrum

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

pm2n2s2dt2

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

Spectra Variation

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

2.4

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.

2.049

Polygon Perimeter

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

5.664

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.

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.

none

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
Minor Triadsd♯m{3,6,10}110.5
Diminished Triads{0,3,6}110.5

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

Parsimonious Voice Leading Between Common Triads of Scale 1113. Created by Ian Ring ©2019 d#m d#m c°->d#m

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1113 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 651
Scale 651: Golitonic, Ian Ring Music TheoryGolitonic
3rd mode:
Scale 2373
Scale 2373: Dyptitonic, Ian Ring Music TheoryDyptitonic
4th mode:
Scale 1617
Scale 1617: Phronitonic, Ian Ring Music TheoryPhronitonic
5th mode:
Scale 357
Scale 357: Banitonic, Ian Ring Music TheoryBanitonic

Prime

The prime form of this scale is Scale 333

Scale 333Scale 333: Bogitonic, Ian Ring Music TheoryBogitonic

Complement

The pentatonic modal family [1113, 651, 2373, 1617, 357] (Forte: 5-28) is the complement of the heptatonic modal family [747, 1431, 1629, 1881, 2421, 2763, 3429] (Forte: 7-28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1113 is 837

Scale 837Scale 837: Epaditonic, Ian Ring Music TheoryEpaditonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1113 is chiral, and its enantiomorph is scale 837

Scale 837Scale 837: Epaditonic, Ian Ring Music TheoryEpaditonic

Transformations:

T0 1113  T0I 837
T1 2226  T1I 1674
T2 357  T2I 3348
T3 714  T3I 2601
T4 1428  T4I 1107
T5 2856  T5I 2214
T6 1617  T6I 333
T7 3234  T7I 666
T8 2373  T8I 1332
T9 651  T9I 2664
T10 1302  T10I 1233
T11 2604  T11I 2466

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 1115Scale 1115: Superlocrian Hexamirror, Ian Ring Music TheorySuperlocrian Hexamirror
Scale 1117Scale 1117: Raptimic, Ian Ring Music TheoryRaptimic
Scale 1105Scale 1105: Messiaen Truncated Mode 6 Inverse, Ian Ring Music TheoryMessiaen Truncated Mode 6 Inverse
Scale 1109Scale 1109: Kataditonic, Ian Ring Music TheoryKataditonic
Scale 1097Scale 1097: Aeraphic, Ian Ring Music TheoryAeraphic
Scale 1129Scale 1129: Raga Jayakauns, Ian Ring Music TheoryRaga Jayakauns
Scale 1145Scale 1145: Zygimic, Ian Ring Music TheoryZygimic
Scale 1049Scale 1049, Ian Ring Music Theory
Scale 1081Scale 1081, Ian Ring Music Theory
Scale 1177Scale 1177: Garitonic, Ian Ring Music TheoryGaritonic
Scale 1241Scale 1241: Pygimic, Ian Ring Music TheoryPygimic
Scale 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic
Scale 1625Scale 1625: Lythimic, Ian Ring Music TheoryLythimic
Scale 89Scale 89, Ian Ring Music Theory
Scale 601Scale 601: Bycritonic, Ian Ring Music TheoryBycritonic
Scale 2137Scale 2137, Ian Ring Music Theory
Scale 3161Scale 3161: Kodimic, Ian Ring Music TheoryKodimic

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.