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Scale 2281: "Rathimic"

Scale 2281: Rathimic, 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

41161837294116105072918310504116183
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

Zeitler
Rathimic

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

{0,3,5,6,7,11}

Forte Number

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

6-Z17

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.

none

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.

yes
enantiomorph: 739

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

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.

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.

5

Prime Form

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

no
prime: 407

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.

[3, 2, 1, 1, 4, 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.

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

Interval Spectrum

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

p3m3n2s2d3t2

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

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.

2.116

Polygon Perimeter

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

5.699

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.

none

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 TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsB{11,3,6}221
Minor Triadscm{0,3,7}221
Augmented TriadsD♯+{3,7,11}221
Diminished Triads{0,3,6}221

The following pitch classes are not present in any of the common triads: {5}

Parsimonious Voice Leading Between Common Triads of Scale 2281. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ D#+->B

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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2281 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 797
Scale 797: Katocrimic, Ian Ring Music TheoryKatocrimic
3rd mode:
Scale 1223
Scale 1223: Phryptimic, Ian Ring Music TheoryPhryptimic
4th mode:
Scale 2659
Scale 2659: Katynimic, Ian Ring Music TheoryKatynimic
5th mode:
Scale 3377
Scale 3377: Phralimic, Ian Ring Music TheoryPhralimic
6th mode:
Scale 467
Scale 467: Raga Dhavalangam, Ian Ring Music TheoryRaga Dhavalangam

Prime

The prime form of this scale is Scale 407

Scale 407Scale 407: Zylimic, Ian Ring Music TheoryZylimic

Complement

The hexatonic modal family [2281, 797, 1223, 2659, 3377, 467] (Forte: 6-Z17) is the complement of the hexatonic modal family [359, 907, 1649, 2227, 2501, 3161] (Forte: 6-Z43)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2281 is 739

Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2281 is chiral, and its enantiomorph is scale 739

Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic

Transformations:

T0 2281  T0I 739
T1 467  T1I 1478
T2 934  T2I 2956
T3 1868  T3I 1817
T4 3736  T4I 3634
T5 3377  T5I 3173
T6 2659  T6I 2251
T7 1223  T7I 407
T8 2446  T8I 814
T9 797  T9I 1628
T10 1594  T10I 3256
T11 3188  T11I 2417

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 2283Scale 2283: Aeolyptian, Ian Ring Music TheoryAeolyptian
Scale 2285Scale 2285: Aerogian, Ian Ring Music TheoryAerogian
Scale 2273Scale 2273, Ian Ring Music Theory
Scale 2277Scale 2277: Kagimic, Ian Ring Music TheoryKagimic
Scale 2289Scale 2289: Mocrimic, Ian Ring Music TheoryMocrimic
Scale 2297Scale 2297: Thylian, Ian Ring Music TheoryThylian
Scale 2249Scale 2249: Raga Multani, Ian Ring Music TheoryRaga Multani
Scale 2265Scale 2265: Raga Rasamanjari, Ian Ring Music TheoryRaga Rasamanjari
Scale 2217Scale 2217: Kagitonic, Ian Ring Music TheoryKagitonic
Scale 2153Scale 2153, Ian Ring Music Theory
Scale 2409Scale 2409: Zacrimic, Ian Ring Music TheoryZacrimic
Scale 2537Scale 2537: Laptian, Ian Ring Music TheoryLaptian
Scale 2793Scale 2793: Eporian, Ian Ring Music TheoryEporian
Scale 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian
Scale 233Scale 233, Ian Ring Music Theory
Scale 1257Scale 1257: Blues Scale, Ian Ring Music TheoryBlues Scale

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.