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Scale 2115

Scale 2115, 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).

Analysis

Cardinality

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

4 (tetratonic)

Pitch Class Set

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

{0,1,6,11}

Forte Number

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

4-6

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.

[0]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

yes

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.

1 (uncohemitonic)

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.

2

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.

3

Prime Form

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

no
prime: 135

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, 5, 5, 1]

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, 1, 0, 0, 2, 1>

Interval Spectrum

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

p2sd2t

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

Spectra Variation

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

4

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.

1

Polygon Perimeter

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

4.899

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.

[0]

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

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 2115 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 3105
Scale 3105, Ian Ring Music Theory
3rd mode:
Scale 225
Scale 225, Ian Ring Music Theory
4th mode:
Scale 135
Scale 135, Ian Ring Music TheoryThis is the prime mode

Prime

The prime form of this scale is Scale 135

Scale 135Scale 135, Ian Ring Music Theory

Complement

The tetratonic modal family [2115, 3105, 225, 135] (Forte: 4-6) is the complement of the octatonic modal family [495, 1935, 2295, 3015, 3195, 3555, 3645, 3825] (Forte: 8-6)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2115 is itself, because it is a palindromic scale!

Scale 2115Scale 2115, Ian Ring Music Theory

Transformations:

T0 2115  T0I 2115
T1 135  T1I 135
T2 270  T2I 270
T3 540  T3I 540
T4 1080  T4I 1080
T5 2160  T5I 2160
T6 225  T6I 225
T7 450  T7I 450
T8 900  T8I 900
T9 1800  T9I 1800
T10 3600  T10I 3600
T11 3105  T11I 3105

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 2113Scale 2113, Ian Ring Music Theory
Scale 2117Scale 2117: Raga Sumukam, Ian Ring Music TheoryRaga Sumukam
Scale 2119Scale 2119, Ian Ring Music Theory
Scale 2123Scale 2123, Ian Ring Music Theory
Scale 2131Scale 2131, Ian Ring Music Theory
Scale 2147Scale 2147, Ian Ring Music Theory
Scale 2051Scale 2051: Tritonic Chromatic 2, Ian Ring Music TheoryTritonic Chromatic 2
Scale 2083Scale 2083, Ian Ring Music Theory
Scale 2179Scale 2179, Ian Ring Music Theory
Scale 2243Scale 2243, Ian Ring Music Theory
Scale 2371Scale 2371, Ian Ring Music Theory
Scale 2627Scale 2627, Ian Ring Music Theory
Scale 3139Scale 3139, Ian Ring Music Theory
Scale 67Scale 67, Ian Ring Music Theory
Scale 1091Scale 1091, Ian Ring Music Theory

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.