The Exciting Universe Of Music Theory

presents

more than you ever wanted to know about...

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

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 Rahn/Ring formula.	no prime: 365
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.	[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>
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 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".	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 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

Transformations:

T₀	1115	T₀I	2885
T₁	2230	T₁I	1675
T₂	365	T₂I	3350
T₃	730	T₃I	2605
T₄	1460	T₄I	1115
T₅	2920	T₅I	2230
T₆	1745	T₆I	365
T₇	3490	T₇I	730
T₈	2885	T₈I	1460
T₉	1675	T₉I	2920
T₁₀	3350	T₁₀I	1745
T₁₁	2605	T₁₁I	3490

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
Scale 1083
Scale 1179				Sonimic
Scale 1243				Epylian
Scale 1371				Superlocrian
Scale 1627				Zyptian
Scale 91
Scale 603				Aeolygimic
Scale 2139
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, 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.