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Scale 1387: "Locrian"

Scale 1387: Locrian, 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
Locrian
Thang Klang
Southeast Asia
Khmer Hepatatonic 3
Thailand
Thang luk
Ancient Greek
Greek Mixolydian
Greek Hyperdorian
Greek Medieval Hyperaeolian
Medieval
Medieval Hypophrygian
Medieval Locrian
Unknown / Unsorted
Rut Biscale Descending
Pien chih
Jewish
Yishtabach
Zeitler
Locrian
Dozenal
LORian

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

{0,1,3,5,6,8,10}

Forte Number

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

7-35

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]

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.

2 (dihemitonic)

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.

1

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 Starr/Rahn algorithm.

yes

Generator

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

generator: 5
origin: 0

Deep Scale

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

yes

Interval Structure

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

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

<2, 5, 4, 3, 6, 1>

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.

<0, 0.75, 0.5, 0, 1, 0>

Interval Spectrum

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

p6m3n4s5d2t

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

Spectra Variation

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

0.857

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.

2.665

Polygon Perimeter

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

6.035

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

[6]

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, 1, 56)

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.

0.993

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.

0.556

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

Generator

This scale has a generator of 5, originating on 0.

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♯{1,5,8}231.71
F♯{6,10,1}231.71
G♯{8,0,3}231.71
Minor Triadsd♯m{3,6,10}231.71
fm{5,8,0}231.71
a♯m{10,1,5}231.71
Diminished Triads{0,3,6}231.71
Parsimonious Voice Leading Between Common Triads of Scale 1387. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C# C# fm fm C#->fm a#m a#m C#->a#m F# F# d#m->F# fm->G# F#->a#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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1387 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2741
Scale 2741: Major, Ian Ring Music TheoryMajor
3rd mode:
Scale 1709
Scale 1709: Dorian, Ian Ring Music TheoryDorian
4th mode:
Scale 1451
Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian
5th mode:
Scale 2773
Scale 2773: Lydian, Ian Ring Music TheoryLydian
6th mode:
Scale 1717
Scale 1717: Mixolydian, Ian Ring Music TheoryMixolydian
7th mode:
Scale 1453
Scale 1453: Aeolian, Ian Ring Music TheoryAeolian

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [1387, 2741, 1709, 1451, 2773, 1717, 1453] (Forte: 7-35) is the complement of the pentatonic modal family [661, 677, 1189, 1193, 1321] (Forte: 5-35)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1387 is 2773

Scale 2773Scale 2773: Lydian, Ian Ring Music TheoryLydian

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> 1387       T0I <11,0> 2773
T1 <1,1> 2774      T1I <11,1> 1451
T2 <1,2> 1453      T2I <11,2> 2902
T3 <1,3> 2906      T3I <11,3> 1709
T4 <1,4> 1717      T4I <11,4> 3418
T5 <1,5> 3434      T5I <11,5> 2741
T6 <1,6> 2773      T6I <11,6> 1387
T7 <1,7> 1451      T7I <11,7> 2774
T8 <1,8> 2902      T8I <11,8> 1453
T9 <1,9> 1709      T9I <11,9> 2906
T10 <1,10> 3418      T10I <11,10> 1717
T11 <1,11> 2741      T11I <11,11> 3434
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 127      T0MI <7,0> 4033
T1M <5,1> 254      T1MI <7,1> 3971
T2M <5,2> 508      T2MI <7,2> 3847
T3M <5,3> 1016      T3MI <7,3> 3599
T4M <5,4> 2032      T4MI <7,4> 3103
T5M <5,5> 4064      T5MI <7,5> 2111
T6M <5,6> 4033      T6MI <7,6> 127
T7M <5,7> 3971      T7MI <7,7> 254
T8M <5,8> 3847      T8MI <7,8> 508
T9M <5,9> 3599      T9MI <7,9> 1016
T10M <5,10> 3103      T10MI <7,10> 2032
T11M <5,11> 2111      T11MI <7,11> 4064

The transformations that map this set to itself are: T0, T6I

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 1385Scale 1385: Phracrimic, Ian Ring Music TheoryPhracrimic
Scale 1389Scale 1389: Minor Locrian, Ian Ring Music TheoryMinor Locrian
Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic
Scale 1379Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic
Scale 1383Scale 1383: Pynian, Ian Ring Music TheoryPynian
Scale 1395Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
Scale 1403Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
Scale 1355Scale 1355: Aeolorimic, Ian Ring Music TheoryAeolorimic
Scale 1371Scale 1371: Superlocrian, Ian Ring Music TheorySuperlocrian
Scale 1323Scale 1323: Ritsu, Ian Ring Music TheoryRitsu
Scale 1451Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian
Scale 1515Scale 1515: Phrygian/Locrian Mixed, Ian Ring Music TheoryPhrygian/Locrian Mixed
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian
Scale 1643Scale 1643: Locrian Natural 6, Ian Ring Music TheoryLocrian Natural 6
Scale 1899Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic
Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 875Scale 875: Locrian Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 7
Scale 2411Scale 2411: Aeolorian, Ian Ring Music TheoryAeolorian
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev

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. (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.