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Scale 1115: "Superlocrian Hexamirror"

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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.	6
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
Centre of Gravity Distance When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.	0.288675
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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Major Triads	F♯	{6,10,1}	2	2	1
Minor Triads	d♯m	{3,6,10}	2	2	1
Diminished Triads	c°	{0,3,6}	1	3	1.5
Diminished Triads	a♯°	{10,1,4}	1	3	1.5

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.

Diameter	3
Radius	2
Self-Centered	no
Central Vertices	d♯m, F♯
Peripheral Vertices	c°, 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	Rylimic
3rd mode: Scale 1675	Raga Salagavarali
4th mode: Scale 2885	Byrimic
5th mode: Scale 1745	Raga Vutari
6th mode: Scale 365	Marimic	This is the prime mode

Prime

The prime form of this scale is Scale 365

Scale 365

Marimic

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 2885

Byrimic

Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

Contradictions (20)

Ambiguities(6)

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

k	Hierarchizability	Breakdown Pattern
1	1	`110110100010`
2	1	`110110100010`
3	3	`([1]0[1])([1]0[1])000([1]0[1])`
4	3	`([1]0[1])([1]0[1])000([1]0[1])`
5	3	`([1]0[1])([1]0[1])000([1]0[1])`

Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center	(0.25, -0.144338)
Distance from Center	0.288675
Angle in degrees measured clockwise starting from the root.	60
Angle in cents 100 cents = 1 semitone.	200

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

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 1113				Locrian Pentatonic 2
Scale 1117				Raptimic
Scale 1119				Rarian
Scale 1107				Mogitonic
Scale 1111				Sycrimic
Scale 1099				Dyritonic
Scale 1131				Honchoshi Plagal Form
Scale 1147				Epynian
Scale 1051				GIFian
Scale 1083				GOYian
Scale 1179				Sonimic
Scale 1243				Epylian
Scale 1371				Superlocrian
Scale 1627				Hungarian Major 4th Mode
Scale 91				ANOian
Scale 603				Aeolygimic
Scale 2139				NAMian
Scale 3163				Rogian

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. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.