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Scale 2147: "NARian"

Scale 2147: NARian, 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).



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


Forte Number

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 2243


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

3 (trihemitonic)


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

1 (uncohemitonic)


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.



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.


Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

prime: 199


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.


Interval Structure

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

[1, 4, 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, 1, 0, 1, 3, 2>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.75, 0.25, 0, 0, 0.75, 1>

Interval Spectrum

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


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

Spectra Variation

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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(4, 8, 30)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.


Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.



This scale has no generator.

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 2147 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3121
Scale 3121: TILian, Ian Ring Music TheoryTILian
3rd mode:
Scale 451
Scale 451: Raga Saugandhini, Ian Ring Music TheoryRaga Saugandhini
4th mode:
Scale 2273
Scale 2273: NURian, Ian Ring Music TheoryNURian
5th mode:
Scale 199
Scale 199: Raga Nabhomani, Ian Ring Music TheoryRaga NabhomaniThis is the prime mode


The prime form of this scale is Scale 199

Scale 199Scale 199: Raga Nabhomani, Ian Ring Music TheoryRaga Nabhomani


The pentatonic modal family [2147, 3121, 451, 2273, 199] (Forte: 5-7) is the complement of the heptatonic modal family [463, 967, 2279, 2531, 3187, 3313, 3641] (Forte: 7-7)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2147 is 2243

Scale 2243Scale 2243: NOYian, Ian Ring Music TheoryNOYian


Only scales that are chiral will have an enantiomorph. Scale 2147 is chiral, and its enantiomorph is scale 2243

Scale 2243Scale 2243: NOYian, Ian Ring Music TheoryNOYian


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> 2147       T0I <11,0> 2243
T1 <1,1> 199      T1I <11,1> 391
T2 <1,2> 398      T2I <11,2> 782
T3 <1,3> 796      T3I <11,3> 1564
T4 <1,4> 1592      T4I <11,4> 3128
T5 <1,5> 3184      T5I <11,5> 2161
T6 <1,6> 2273      T6I <11,6> 227
T7 <1,7> 451      T7I <11,7> 454
T8 <1,8> 902      T8I <11,8> 908
T9 <1,9> 1804      T9I <11,9> 1816
T10 <1,10> 3608      T10I <11,10> 3632
T11 <1,11> 3121      T11I <11,11> 3169
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 227      T0MI <7,0> 2273
T1M <5,1> 454      T1MI <7,1> 451
T2M <5,2> 908      T2MI <7,2> 902
T3M <5,3> 1816      T3MI <7,3> 1804
T4M <5,4> 3632      T4MI <7,4> 3608
T5M <5,5> 3169      T5MI <7,5> 3121
T6M <5,6> 2243      T6MI <7,6> 2147
T7M <5,7> 391      T7MI <7,7> 199
T8M <5,8> 782      T8MI <7,8> 398
T9M <5,9> 1564      T9MI <7,9> 796
T10M <5,10> 3128      T10MI <7,10> 1592
T11M <5,11> 2161      T11MI <7,11> 3184

The transformations that map this set to itself are: T0, T6MI

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 2145Scale 2145: Messiaen Mode 5 Truncation 2, Ian Ring Music TheoryMessiaen Mode 5 Truncation 2
Scale 2149Scale 2149: NASian, Ian Ring Music TheoryNASian
Scale 2151Scale 2151: NATian, Ian Ring Music TheoryNATian
Scale 2155Scale 2155: NEWian, Ian Ring Music TheoryNEWian
Scale 2163Scale 2163: NEBian, Ian Ring Music TheoryNEBian
Scale 2115Scale 2115: MUYian, Ian Ring Music TheoryMUYian
Scale 2131Scale 2131: NAHian, Ian Ring Music TheoryNAHian
Scale 2083Scale 2083: MOFian, Ian Ring Music TheoryMOFian
Scale 2211Scale 2211: Raga Gauri, Ian Ring Music TheoryRaga Gauri
Scale 2275Scale 2275: Messiaen Mode 5, Ian Ring Music TheoryMessiaen Mode 5
Scale 2403Scale 2403: Lycrimic, Ian Ring Music TheoryLycrimic
Scale 2659Scale 2659: Katynimic, Ian Ring Music TheoryKatynimic
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 99Scale 99: IPRian, Ian Ring Music TheoryIPRian
Scale 1123Scale 1123: Iwato, Ian Ring Music TheoryIwato

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler ( 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.