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Scale 2163: "Nebian"

Scale 2163: Nebian, 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
Nebian

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

{0,1,4,5,6,11}

Forte Number

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

6-Z6

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.

[2.5]

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.

4 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

2 (dicohemitonic)

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.

5

Prime Form

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

no
prime: 231

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.

[1, 3, 1, 1, 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.

<4, 2, 1, 2, 4, 2>

Interval Spectrum

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

p4m2ns2d4t2

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

Spectra Variation

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

3

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.

1.75

Polygon Perimeter

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

5.417

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.

[5]

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.

(24, 10, 51)

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 2163 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3129
Scale 3129: Toqian, Ian Ring Music TheoryToqian
3rd mode:
Scale 903
Scale 903: Fosian, Ian Ring Music TheoryFosian
4th mode:
Scale 2499
Scale 2499: Pirian, Ian Ring Music TheoryPirian
5th mode:
Scale 3297
Scale 3297: Ullian, Ian Ring Music TheoryUllian
6th mode:
Scale 231
Scale 231: Bifian, Ian Ring Music TheoryBifianThis is the prime mode

Prime

The prime form of this scale is Scale 231

Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian

Complement

The hexatonic modal family [2163, 3129, 903, 2499, 3297, 231] (Forte: 6-Z6) is the complement of the hexatonic modal family [399, 483, 2247, 2289, 3171, 3633] (Forte: 6-Z38)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2163 is 2499

Scale 2499Scale 2499: Pirian, Ian Ring Music TheoryPirian

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> 2163       T0I <11,0> 2499
T1 <1,1> 231      T1I <11,1> 903
T2 <1,2> 462      T2I <11,2> 1806
T3 <1,3> 924      T3I <11,3> 3612
T4 <1,4> 1848      T4I <11,4> 3129
T5 <1,5> 3696      T5I <11,5> 2163
T6 <1,6> 3297      T6I <11,6> 231
T7 <1,7> 2499      T7I <11,7> 462
T8 <1,8> 903      T8I <11,8> 924
T9 <1,9> 1806      T9I <11,9> 1848
T10 <1,10> 3612      T10I <11,10> 3696
T11 <1,11> 3129      T11I <11,11> 3297
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 483      T0MI <7,0> 2289
T1M <5,1> 966      T1MI <7,1> 483
T2M <5,2> 1932      T2MI <7,2> 966
T3M <5,3> 3864      T3MI <7,3> 1932
T4M <5,4> 3633      T4MI <7,4> 3864
T5M <5,5> 3171      T5MI <7,5> 3633
T6M <5,6> 2247      T6MI <7,6> 3171
T7M <5,7> 399      T7MI <7,7> 2247
T8M <5,8> 798      T8MI <7,8> 399
T9M <5,9> 1596      T9MI <7,9> 798
T10M <5,10> 3192      T10MI <7,10> 1596
T11M <5,11> 2289      T11MI <7,11> 3192

The transformations that map this set to itself are: T0, T5I

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 2161Scale 2161: Nezian, Ian Ring Music TheoryNezian
Scale 2165Scale 2165: Necian, Ian Ring Music TheoryNecian
Scale 2167Scale 2167: Nedian, Ian Ring Music TheoryNedian
Scale 2171Scale 2171: Negian, Ian Ring Music TheoryNegian
Scale 2147Scale 2147: Narian, Ian Ring Music TheoryNarian
Scale 2155Scale 2155: Newian, Ian Ring Music TheoryNewian
Scale 2131Scale 2131: Nahian, Ian Ring Music TheoryNahian
Scale 2099Scale 2099: Raga Megharanji, Ian Ring Music TheoryRaga Megharanji
Scale 2227Scale 2227: Raga Gaula, Ian Ring Music TheoryRaga Gaula
Scale 2291Scale 2291: Zydian, Ian Ring Music TheoryZydian
Scale 2419Scale 2419: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 2675Scale 2675: Chromatic Lydian, Ian Ring Music TheoryChromatic Lydian
Scale 3187Scale 3187: Koptian, Ian Ring Music TheoryKoptian
Scale 115Scale 115: Ashian, Ian Ring Music TheoryAshian
Scale 1139Scale 1139: Aerygimic, Ian Ring Music TheoryAerygimic

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.