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Scale 1371: "Superlocrian"

Scale 1371: Superlocrian, 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
Superlocrian
Diminished Whole-tone
Altered
Western Altered
Altered Scale
Altered Dominant
Locrian Flat 4
Named After Composers
Pomeroy
Ravel
Western Mixed
Dominant Whole-tone Combo
Zeitler
Ionadian
Dozenal
Altian

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

{0,1,3,4,6,8,10}

Forte Number

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

7-34

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.

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.

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.

6

Prime Form

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

yes

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 Structure

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

[1, 2, 1, 2, 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, 4, 4, 2>

Interval Spectrum

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

p4m4n4s5d2t2

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> = {4,5,6}
<4> = {6,7,8}
<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.

1.143

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.

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 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.

[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".

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, 10, 72)

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.

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 TriadsF♯{6,10,1}231.71
G♯{8,0,3}231.71
Minor Triadsc♯m{1,4,8}231.71
d♯m{3,6,10}231.71
Augmented TriadsC+{0,4,8}231.71
Diminished Triads{0,3,6}231.71
a♯°{10,1,4}231.71
Parsimonious Voice Leading Between Common Triads of Scale 1371. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ c#m c#m C+->c#m C+->G# a#° a#° c#m->a#° F# F# d#m->F# F#->a#°

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 1371 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2733
Scale 2733: Melodic Minor Ascending, Ian Ring Music TheoryMelodic Minor Ascending
3rd mode:
Scale 1707
Scale 1707: Dorian Flat 2, Ian Ring Music TheoryDorian Flat 2
4th mode:
Scale 2901
Scale 2901: Lydian Augmented, Ian Ring Music TheoryLydian Augmented
5th mode:
Scale 1749
Scale 1749: Acoustic, Ian Ring Music TheoryAcoustic
6th mode:
Scale 1461
Scale 1461: Major-Minor, Ian Ring Music TheoryMajor-Minor
7th mode:
Scale 1389
Scale 1389: Minor Locrian, Ian Ring Music TheoryMinor Locrian

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [1371, 2733, 1707, 2901, 1749, 1461, 1389] (Forte: 7-34) is the complement of the pentatonic modal family [597, 681, 1173, 1317, 1353] (Forte: 5-34)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1371 is 2901

Scale 2901Scale 2901: Lydian Augmented, Ian Ring Music TheoryLydian Augmented

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> 1371       T0I <11,0> 2901
T1 <1,1> 2742      T1I <11,1> 1707
T2 <1,2> 1389      T2I <11,2> 3414
T3 <1,3> 2778      T3I <11,3> 2733
T4 <1,4> 1461      T4I <11,4> 1371
T5 <1,5> 2922      T5I <11,5> 2742
T6 <1,6> 1749      T6I <11,6> 1389
T7 <1,7> 3498      T7I <11,7> 2778
T8 <1,8> 2901      T8I <11,8> 1461
T9 <1,9> 1707      T9I <11,9> 2922
T10 <1,10> 3414      T10I <11,10> 1749
T11 <1,11> 2733      T11I <11,11> 3498
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 381      T0MI <7,0> 2001
T1M <5,1> 762      T1MI <7,1> 4002
T2M <5,2> 1524      T2MI <7,2> 3909
T3M <5,3> 3048      T3MI <7,3> 3723
T4M <5,4> 2001      T4MI <7,4> 3351
T5M <5,5> 4002      T5MI <7,5> 2607
T6M <5,6> 3909      T6MI <7,6> 1119
T7M <5,7> 3723      T7MI <7,7> 2238
T8M <5,8> 3351      T8MI <7,8> 381
T9M <5,9> 2607      T9MI <7,9> 762
T10M <5,10> 1119      T10MI <7,10> 1524
T11M <5,11> 2238      T11MI <7,11> 3048

The transformations that map this set to itself are: T0, T4I

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 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic
Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian
Scale 1375Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllic
Scale 1363Scale 1363: Gygimic, Ian Ring Music TheoryGygimic
Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse
Scale 1355Scale 1355: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 1403Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
Scale 1307Scale 1307: Katorimic, Ian Ring Music TheoryKatorimic
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 1435Scale 1435: Makam Huzzam, Ian Ring Music TheoryMakam Huzzam
Scale 1499Scale 1499: Bebop Locrian, Ian Ring Music TheoryBebop Locrian
Scale 1115Scale 1115: Superlocrian Hexamirror, Ian Ring Music TheorySuperlocrian Hexamirror
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian
Scale 1627Scale 1627: Zyptian, Ian Ring Music TheoryZyptian
Scale 1883Scale 1883: Lomian, Ian Ring Music TheoryLomian
Scale 347Scale 347: Barimic, Ian Ring Music TheoryBarimic
Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 2395Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
Scale 3419Scale 3419: Magen Abot 1, Ian Ring Music TheoryMagen Abot 1

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, and George Howlett for assistance with the Carnatic ragas.