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Scale 2165: "Necian"

Scale 2165: Necian, 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
Necian

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

{0,2,4,5,6,11}

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-Z12

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: 1475

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: 215

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, 2, 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.

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

Interval Spectrum

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

p3m2n2s3d3t2

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

Spectra Variation

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

3.333

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.

1.866

Polygon Perimeter

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

5.485

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.

(22, 14, 61)

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
Minor Triadsbm{11,2,6}110.5
Diminished Triads{11,2,5}110.5

The following pitch classes are not present in any of the common triads: {0,4}

Parsimonious Voice Leading Between Common Triads of Scale 2165. Created by Ian Ring ©2019 bm bm b°->bm

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 2165 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1565
Scale 1565: Jozian, Ian Ring Music TheoryJozian
3rd mode:
Scale 1415
Scale 1415: Impian, Ian Ring Music TheoryImpian
4th mode:
Scale 2755
Scale 2755: Rivian, Ian Ring Music TheoryRivian
5th mode:
Scale 3425
Scale 3425: Vihian, Ian Ring Music TheoryVihian
6th mode:
Scale 235
Scale 235: Bihian, Ian Ring Music TheoryBihian

Prime

The prime form of this scale is Scale 215

Scale 215Scale 215: Bivian, Ian Ring Music TheoryBivian

Complement

The hexatonic modal family [2165, 1565, 1415, 2755, 3425, 235] (Forte: 6-Z12) is the complement of the hexatonic modal family [335, 965, 1265, 2215, 3155, 3625] (Forte: 6-Z41)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2165 is 1475

Scale 1475Scale 1475: Uffian, Ian Ring Music TheoryUffian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2165 is chiral, and its enantiomorph is scale 1475

Scale 1475Scale 1475: Uffian, Ian Ring Music TheoryUffian

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> 2165       T0I <11,0> 1475
T1 <1,1> 235      T1I <11,1> 2950
T2 <1,2> 470      T2I <11,2> 1805
T3 <1,3> 940      T3I <11,3> 3610
T4 <1,4> 1880      T4I <11,4> 3125
T5 <1,5> 3760      T5I <11,5> 2155
T6 <1,6> 3425      T6I <11,6> 215
T7 <1,7> 2755      T7I <11,7> 430
T8 <1,8> 1415      T8I <11,8> 860
T9 <1,9> 2830      T9I <11,9> 1720
T10 <1,10> 1565      T10I <11,10> 3440
T11 <1,11> 3130      T11I <11,11> 2785
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1475      T0MI <7,0> 2165
T1M <5,1> 2950      T1MI <7,1> 235
T2M <5,2> 1805      T2MI <7,2> 470
T3M <5,3> 3610      T3MI <7,3> 940
T4M <5,4> 3125      T4MI <7,4> 1880
T5M <5,5> 2155      T5MI <7,5> 3760
T6M <5,6> 215      T6MI <7,6> 3425
T7M <5,7> 430      T7MI <7,7> 2755
T8M <5,8> 860      T8MI <7,8> 1415
T9M <5,9> 1720      T9MI <7,9> 2830
T10M <5,10> 3440      T10MI <7,10> 1565
T11M <5,11> 2785      T11MI <7,11> 3130

The transformations that map this set to itself are: T0, T0MI

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 2167Scale 2167: Nedian, Ian Ring Music TheoryNedian
Scale 2161Scale 2161: Nezian, Ian Ring Music TheoryNezian
Scale 2163Scale 2163: Nebian, Ian Ring Music TheoryNebian
Scale 2169Scale 2169: Nefian, Ian Ring Music TheoryNefian
Scale 2173Scale 2173: Nehian, Ian Ring Music TheoryNehian
Scale 2149Scale 2149: Nasian, Ian Ring Music TheoryNasian
Scale 2157Scale 2157: Nexian, Ian Ring Music TheoryNexian
Scale 2133Scale 2133: Raga Kumurdaki, Ian Ring Music TheoryRaga Kumurdaki
Scale 2101Scale 2101: Muqian, Ian Ring Music TheoryMuqian
Scale 2229Scale 2229: Raga Nalinakanti, Ian Ring Music TheoryRaga Nalinakanti
Scale 2293Scale 2293: Gorian, Ian Ring Music TheoryGorian
Scale 2421Scale 2421: Malian, Ian Ring Music TheoryMalian
Scale 2677Scale 2677: Thodian, Ian Ring Music TheoryThodian
Scale 3189Scale 3189: Aeolonian, Ian Ring Music TheoryAeolonian
Scale 117Scale 117: Anbian, Ian Ring Music TheoryAnbian
Scale 1141Scale 1141: Rynimic, Ian Ring Music TheoryRynimic

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.