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Scale 1115: "Superlocrian Hexamirror"

Scale 1115: Superlocrian Hexamirror, 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 Hexamirror
Zeitler
Locrimic
Dozenal
GUSian

Analysis

Cardinality

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

 6 (hexatonic)

Pitch Class Set

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

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

Forte Number

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

 6-Z23

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.

 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.

 5

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 365

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, an indicator of maximum hierarchization.

 no

Interval Structure

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

 [1, 2, 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.

 <2, 3, 4, 2, 2, 2>

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.

<0.4, 0.5, 0.8, 0, 0.4, 0.667>

Interval Spectrum

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

 p2m2n4s3d2t2

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

Spectra Variation

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

 2.667

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.

2.232

Polygon Perimeter

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

5.767

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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".

Improper

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.

(20, 6, 51)

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.

0.6

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.

0.32

Generator

This scale has no generator.

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}221
Minor Triadsd♯m{3,6,10}221
Diminished Triadsc°{0,3,6}131.5
a♯°{10,1,4}131.5
Parsimonious Voice Leading Between Common Triads of Scale 1115. Created by Ian Ring ©2019 c° c° d#m d#m c°->d#m F# F# d#m->F# a#° a#° 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
Radius2
Self-Centeredno
Central Verticesd♯m, F♯
Peripheral Verticesc°, a♯°

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Diminished: {0, 3, 6}
Diminished: {10, 1, 4}

Modes

Modes are the rotational transformation of this scale. Scale 1115 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2605		Scale 2605: Rylimic, Ian Ring Music TheoryRylimic
3rd mode:
Scale 1675		Scale 1675: Raga Salagavarali, Ian Ring Music TheoryRaga Salagavarali
4th mode:
Scale 2885		Scale 2885: Byrimic, Ian Ring Music TheoryByrimic
5th mode:
Scale 1745		Scale 1745: Raga Vutari, Ian Ring Music TheoryRaga Vutari
6th mode:
Scale 365		Scale 365: Marimic, Ian Ring Music TheoryMarimicThis is the prime mode

Prime

The prime form of this scale is Scale 365

Scale 365Scale 365: Marimic, Ian Ring Music TheoryMarimic

Complement

The hexatonic modal family [1115, 2605, 1675, 2885, 1745, 365] (Forte: 6-Z23) is the complement of the hexatonic modal family [605, 745, 1175, 1865, 2635, 3365] (Forte: 6-Z45)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1115 is 2885

Scale 2885Scale 2885: Byrimic, Ian Ring Music TheoryByrimic

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> 1115       T0I <11,0> 2885
T1 <1,1> 2230      T1I <11,1> 1675
T2 <1,2> 365      T2I <11,2> 3350
T3 <1,3> 730      T3I <11,3> 2605
T4 <1,4> 1460      T4I <11,4> 1115
T5 <1,5> 2920      T5I <11,5> 2230
T6 <1,6> 1745      T6I <11,6> 365
T7 <1,7> 3490      T7I <11,7> 730
T8 <1,8> 2885      T8I <11,8> 1460
T9 <1,9> 1675      T9I <11,9> 2920
T10 <1,10> 3350      T10I <11,10> 1745
T11 <1,11> 2605      T11I <11,11> 3490
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 365      T0MI <7,0> 1745
T1M <5,1> 730      T1MI <7,1> 3490
T2M <5,2> 1460      T2MI <7,2> 2885
T3M <5,3> 2920      T3MI <7,3> 1675
T4M <5,4> 1745      T4MI <7,4> 3350
T5M <5,5> 3490      T5MI <7,5> 2605
T6M <5,6> 2885      T6MI <7,6> 1115
T7M <5,7> 1675      T7MI <7,7> 2230
T8M <5,8> 3350      T8MI <7,8> 365
T9M <5,9> 2605      T9MI <7,9> 730
T10M <5,10> 1115       T10MI <7,10> 1460
T11M <5,11> 2230      T11MI <7,11> 2920

The transformations that map this set to itself are: T0, T4I, T10M, T6MI

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 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2
Scale 1117Scale 1117: Raptimic, Ian Ring Music TheoryRaptimic
Scale 1119Scale 1119: Rarian, Ian Ring Music TheoryRarian
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
Scale 1111Scale 1111: Sycrimic, Ian Ring Music TheorySycrimic
Scale 1099Scale 1099: Dyritonic, Ian Ring Music TheoryDyritonic
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian
Scale 1051Scale 1051: GIFian, Ian Ring Music TheoryGIFian
Scale 1083Scale 1083: GOYian, Ian Ring Music TheoryGOYian
Scale 1179Scale 1179: Sonimic, Ian Ring Music TheorySonimic
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian
Scale 1371Scale 1371: Superlocrian, Ian Ring Music TheorySuperlocrian
Scale 1627Scale 1627: Hungarian Major 4th Mode, Ian Ring Music TheoryHungarian Major 4th Mode
Scale 91Scale 91: ANOian, Ian Ring Music TheoryANOian
Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic
Scale 2139Scale 2139: NAMian, Ian Ring Music TheoryNAMian
Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian

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.