The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2143: "Napian"

Scale 2143: Napian, 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
Napian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,3,4,6,11}

Forte Number

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

7-2

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: 3907

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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.

6

Prime Form

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

no
prime: 191

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, 1, 1, 1, 2, 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.

<5, 5, 4, 3, 3, 1>

Interval Spectrum

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

p3m3n4s5d5t

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

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.

1.933

Polygon Perimeter

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

5.52

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.

(62, 25, 86)

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}210.67
Minor Triadsbm{11,2,6}121
Diminished Triads{0,3,6}121

The following pitch classes are not present in any of the common triads: {1,4}

Parsimonious Voice Leading Between Common Triads of Scale 2143. Created by Ian Ring ©2019 B B c°->B bm bm bm->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
Radius1
Self-Centeredno
Central VerticesB
Peripheral Verticesc°, bm

Modes

Modes are the rotational transformation of this scale. Scale 2143 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3119
Scale 3119: Tikian, Ian Ring Music TheoryTikian
3rd mode:
Scale 3607
Scale 3607: Wopian, Ian Ring Music TheoryWopian
4th mode:
Scale 3851
Scale 3851: Yilian, Ian Ring Music TheoryYilian
5th mode:
Scale 3973
Scale 3973: Zehian, Ian Ring Music TheoryZehian
6th mode:
Scale 2017
Scale 2017: Meqian, Ian Ring Music TheoryMeqian
7th mode:
Scale 191
Scale 191: Begian, Ian Ring Music TheoryBegianThis is the prime mode

Prime

The prime form of this scale is Scale 191

Scale 191Scale 191: Begian, Ian Ring Music TheoryBegian

Complement

The heptatonic modal family [2143, 3119, 3607, 3851, 3973, 2017, 191] (Forte: 7-2) is the complement of the pentatonic modal family [47, 1921, 2071, 3083, 3589] (Forte: 5-2)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2143 is 3907

Scale 3907Scale 3907, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2143 is chiral, and its enantiomorph is scale 3907

Scale 3907Scale 3907, Ian Ring Music Theory

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> 2143       T0I <11,0> 3907
T1 <1,1> 191      T1I <11,1> 3719
T2 <1,2> 382      T2I <11,2> 3343
T3 <1,3> 764      T3I <11,3> 2591
T4 <1,4> 1528      T4I <11,4> 1087
T5 <1,5> 3056      T5I <11,5> 2174
T6 <1,6> 2017      T6I <11,6> 253
T7 <1,7> 4034      T7I <11,7> 506
T8 <1,8> 3973      T8I <11,8> 1012
T9 <1,9> 3851      T9I <11,9> 2024
T10 <1,10> 3607      T10I <11,10> 4048
T11 <1,11> 3119      T11I <11,11> 4001
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1513      T0MI <7,0> 757
T1M <5,1> 3026      T1MI <7,1> 1514
T2M <5,2> 1957      T2MI <7,2> 3028
T3M <5,3> 3914      T3MI <7,3> 1961
T4M <5,4> 3733      T4MI <7,4> 3922
T5M <5,5> 3371      T5MI <7,5> 3749
T6M <5,6> 2647      T6MI <7,6> 3403
T7M <5,7> 1199      T7MI <7,7> 2711
T8M <5,8> 2398      T8MI <7,8> 1327
T9M <5,9> 701      T9MI <7,9> 2654
T10M <5,10> 1402      T10MI <7,10> 1213
T11M <5,11> 2804      T11MI <7,11> 2426

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 2141Scale 2141: Nanian, Ian Ring Music TheoryNanian
Scale 2139Scale 2139: Namian, Ian Ring Music TheoryNamian
Scale 2135Scale 2135: Nakian, Ian Ring Music TheoryNakian
Scale 2127Scale 2127: Nafian, Ian Ring Music TheoryNafian
Scale 2159Scale 2159: Neyian, Ian Ring Music TheoryNeyian
Scale 2175Scale 2175: Octatonic Chromatic 2, Ian Ring Music TheoryOctatonic Chromatic 2
Scale 2079Scale 2079: Hexatonic Chromatic 4, Ian Ring Music TheoryHexatonic Chromatic 4
Scale 2111Scale 2111: Heptatonic Chromatic 2, Ian Ring Music TheoryHeptatonic Chromatic 2
Scale 2207Scale 2207: Mygian, Ian Ring Music TheoryMygian
Scale 2271Scale 2271: Poptyllic, Ian Ring Music TheoryPoptyllic
Scale 2399Scale 2399: Zanyllic, Ian Ring Music TheoryZanyllic
Scale 2655Scale 2655: Qojian, Ian Ring Music TheoryQojian
Scale 3167Scale 3167: Thynyllic, Ian Ring Music TheoryThynyllic
Scale 95Scale 95: Arkian, Ian Ring Music TheoryArkian
Scale 1119Scale 1119: Rarian, Ian Ring Music TheoryRarian

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.