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Scale 2821: "Rukian"

Scale 2821: Rukian, 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

Dozenal
Rukian

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,2,8,9,11}

Forte Number

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

5-10

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: 1051

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.

4

Prime Form

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

no
prime: 91

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.

no

Interval Structure

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

[2, 6, 1, 2, 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, 2, 3, 1, 1, 1>

Interval Spectrum

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

pmn3s2d2t

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

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.

1.366

Polygon Perimeter

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

5.035

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

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.

(14, 1, 30)

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
Diminished Triadsg♯°{8,11,2}000

The following pitch classes are not present in any of the common triads: {0,9}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 2821 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1729
Scale 1729: Kowian, Ian Ring Music TheoryKowian
3rd mode:
Scale 91
Scale 91: Anoian, Ian Ring Music TheoryAnoianThis is the prime mode
4th mode:
Scale 2093
Scale 2093: Mulian, Ian Ring Music TheoryMulian
5th mode:
Scale 1547
Scale 1547: Jopian, Ian Ring Music TheoryJopian

Prime

The prime form of this scale is Scale 91

Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian

Complement

The pentatonic modal family [2821, 1729, 91, 2093, 1547] (Forte: 5-10) is the complement of the heptatonic modal family [607, 761, 1993, 2351, 3223, 3659, 3877] (Forte: 7-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2821 is 1051

Scale 1051Scale 1051: Gifian, Ian Ring Music TheoryGifian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2821 is chiral, and its enantiomorph is scale 1051

Scale 1051Scale 1051: Gifian, Ian Ring Music TheoryGifian

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> 2821       T0I <11,0> 1051
T1 <1,1> 1547      T1I <11,1> 2102
T2 <1,2> 3094      T2I <11,2> 109
T3 <1,3> 2093      T3I <11,3> 218
T4 <1,4> 91      T4I <11,4> 436
T5 <1,5> 182      T5I <11,5> 872
T6 <1,6> 364      T6I <11,6> 1744
T7 <1,7> 728      T7I <11,7> 3488
T8 <1,8> 1456      T8I <11,8> 2881
T9 <1,9> 2912      T9I <11,9> 1667
T10 <1,10> 1729      T10I <11,10> 3334
T11 <1,11> 3458      T11I <11,11> 2573
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1681      T0MI <7,0> 301
T1M <5,1> 3362      T1MI <7,1> 602
T2M <5,2> 2629      T2MI <7,2> 1204
T3M <5,3> 1163      T3MI <7,3> 2408
T4M <5,4> 2326      T4MI <7,4> 721
T5M <5,5> 557      T5MI <7,5> 1442
T6M <5,6> 1114      T6MI <7,6> 2884
T7M <5,7> 2228      T7MI <7,7> 1673
T8M <5,8> 361      T8MI <7,8> 3346
T9M <5,9> 722      T9MI <7,9> 2597
T10M <5,10> 1444      T10MI <7,10> 1099
T11M <5,11> 2888      T11MI <7,11> 2198

The transformations that map this set to itself are: T0

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 2823Scale 2823: Rulian, Ian Ring Music TheoryRulian
Scale 2817Scale 2817, Ian Ring Music Theory
Scale 2819Scale 2819: Rujian, Ian Ring Music TheoryRujian
Scale 2825Scale 2825: Rumian, Ian Ring Music TheoryRumian
Scale 2829Scale 2829: Rupian, Ian Ring Music TheoryRupian
Scale 2837Scale 2837: Aelothimic, Ian Ring Music TheoryAelothimic
Scale 2853Scale 2853: Baptimic, Ian Ring Music TheoryBaptimic
Scale 2885Scale 2885: Byrimic, Ian Ring Music TheoryByrimic
Scale 2949Scale 2949: Sikian, Ian Ring Music TheorySikian
Scale 2565Scale 2565: Pogian, Ian Ring Music TheoryPogian
Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian
Scale 2309Scale 2309: Ocuian, Ian Ring Music TheoryOcuian
Scale 3333Scale 3333: Vacian, Ian Ring Music TheoryVacian
Scale 3845Scale 3845: Yihian, Ian Ring Music TheoryYihian
Scale 773Scale 773: Esuian, Ian Ring Music TheoryEsuian
Scale 1797Scale 1797: Lalian, Ian Ring Music TheoryLalian

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. 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.