The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2169: "Nefian"

Scale 2169: Nefian, 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
Nefian

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

{0,3,4,5,6,11}

Forte Number

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

6-5

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: 963

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.

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: 207

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.

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

Interval Spectrum

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

p3m2n2s2d4t2

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> = {3,5,7,9}
<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.

3.667

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.

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.

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.

(30, 10, 55)

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}110.5
Diminished Triads{0,3,6}110.5

The following pitch classes are not present in any of the common triads: {4,5}

Parsimonious Voice Leading Between Common Triads of Scale 2169. Created by Ian Ring ©2019 B B c°->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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2169 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 783
Scale 783: Etuian, Ian Ring Music TheoryEtuian
3rd mode:
Scale 2439
Scale 2439: Pagian, Ian Ring Music TheoryPagian
4th mode:
Scale 3267
Scale 3267: Urfian, Ian Ring Music TheoryUrfian
5th mode:
Scale 3681
Scale 3681: Xahian, Ian Ring Music TheoryXahian
6th mode:
Scale 243
Scale 243: Bomian, Ian Ring Music TheoryBomian

Prime

The prime form of this scale is Scale 207

Scale 207Scale 207: Beqian, Ian Ring Music TheoryBeqian

Complement

The hexatonic modal family [2169, 783, 2439, 3267, 3681, 243] (Forte: 6-5) is the complement of the hexatonic modal family [207, 963, 2151, 2529, 3123, 3609] (Forte: 6-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2169 is 963

Scale 963Scale 963: Gacian, Ian Ring Music TheoryGacian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2169 is chiral, and its enantiomorph is scale 963

Scale 963Scale 963: Gacian, Ian Ring Music TheoryGacian

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> 2169       T0I <11,0> 963
T1 <1,1> 243      T1I <11,1> 1926
T2 <1,2> 486      T2I <11,2> 3852
T3 <1,3> 972      T3I <11,3> 3609
T4 <1,4> 1944      T4I <11,4> 3123
T5 <1,5> 3888      T5I <11,5> 2151
T6 <1,6> 3681      T6I <11,6> 207
T7 <1,7> 3267      T7I <11,7> 414
T8 <1,8> 2439      T8I <11,8> 828
T9 <1,9> 783      T9I <11,9> 1656
T10 <1,10> 1566      T10I <11,10> 3312
T11 <1,11> 3132      T11I <11,11> 2529
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 459      T0MI <7,0> 2673
T1M <5,1> 918      T1MI <7,1> 1251
T2M <5,2> 1836      T2MI <7,2> 2502
T3M <5,3> 3672      T3MI <7,3> 909
T4M <5,4> 3249      T4MI <7,4> 1818
T5M <5,5> 2403      T5MI <7,5> 3636
T6M <5,6> 711      T6MI <7,6> 3177
T7M <5,7> 1422      T7MI <7,7> 2259
T8M <5,8> 2844      T8MI <7,8> 423
T9M <5,9> 1593      T9MI <7,9> 846
T10M <5,10> 3186      T10MI <7,10> 1692
T11M <5,11> 2277      T11MI <7,11> 3384

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 2171Scale 2171: Negian, Ian Ring Music TheoryNegian
Scale 2173Scale 2173: Nehian, Ian Ring Music TheoryNehian
Scale 2161Scale 2161: Nezian, Ian Ring Music TheoryNezian
Scale 2165Scale 2165: Necian, Ian Ring Music TheoryNecian
Scale 2153Scale 2153: Navian, Ian Ring Music TheoryNavian
Scale 2137Scale 2137: Nalian, Ian Ring Music TheoryNalian
Scale 2105Scale 2105: Rigian, Ian Ring Music TheoryRigian
Scale 2233Scale 2233: Donimic, Ian Ring Music TheoryDonimic
Scale 2297Scale 2297: Thylian, Ian Ring Music TheoryThylian
Scale 2425Scale 2425: Rorian, Ian Ring Music TheoryRorian
Scale 2681Scale 2681: Aerycrian, Ian Ring Music TheoryAerycrian
Scale 3193Scale 3193: Zathian, Ian Ring Music TheoryZathian
Scale 121Scale 121: Asoian, Ian Ring Music TheoryAsoian
Scale 1145Scale 1145: Zygimic, Ian Ring Music TheoryZygimic

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.