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Scale 2691: "Rahian"

Scale 2691: Rahian, 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




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: 2091


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

2 (dihemitonic)


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 Rahn/Ring formula.

prime: 87


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.


Interval Structure

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

[1, 6, 2, 2, 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.

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

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 Interval Spectra has 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.

(14, 7, 36)

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 2691 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3393
Scale 3393: Venian, Ian Ring Music TheoryVenian
3rd mode:
Scale 117
Scale 117: Anbian, Ian Ring Music TheoryAnbian
4th mode:
Scale 1053
Scale 1053: Gigian, Ian Ring Music TheoryGigian
5th mode:
Scale 1287
Scale 1287: Hutian, Ian Ring Music TheoryHutian


The prime form of this scale is Scale 87

Scale 87Scale 87: Asrian, Ian Ring Music TheoryAsrian


The pentatonic modal family [2691, 3393, 117, 1053, 1287] (Forte: 5-9) is the complement of the heptatonic modal family [351, 1521, 1989, 2223, 3159, 3627, 3861] (Forte: 7-9)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2691 is 2091

Scale 2091Scale 2091: Mukian, Ian Ring Music TheoryMukian


Only scales that are chiral will have an enantiomorph. Scale 2691 is chiral, and its enantiomorph is scale 2091

Scale 2091Scale 2091: Mukian, Ian Ring Music TheoryMukian


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> 2691       T0I <11,0> 2091
T1 <1,1> 1287      T1I <11,1> 87
T2 <1,2> 2574      T2I <11,2> 174
T3 <1,3> 1053      T3I <11,3> 348
T4 <1,4> 2106      T4I <11,4> 696
T5 <1,5> 117      T5I <11,5> 1392
T6 <1,6> 234      T6I <11,6> 2784
T7 <1,7> 468      T7I <11,7> 1473
T8 <1,8> 936      T8I <11,8> 2946
T9 <1,9> 1872      T9I <11,9> 1797
T10 <1,10> 3744      T10I <11,10> 3594
T11 <1,11> 3393      T11I <11,11> 3093
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2721      T0MI <7,0> 171
T1M <5,1> 1347      T1MI <7,1> 342
T2M <5,2> 2694      T2MI <7,2> 684
T3M <5,3> 1293      T3MI <7,3> 1368
T4M <5,4> 2586      T4MI <7,4> 2736
T5M <5,5> 1077      T5MI <7,5> 1377
T6M <5,6> 2154      T6MI <7,6> 2754
T7M <5,7> 213      T7MI <7,7> 1413
T8M <5,8> 426      T8MI <7,8> 2826
T9M <5,9> 852      T9MI <7,9> 1557
T10M <5,10> 1704      T10MI <7,10> 3114
T11M <5,11> 3408      T11MI <7,11> 2133

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 2689Scale 2689: Ragian, Ian Ring Music TheoryRagian
Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian
Scale 2695Scale 2695: Rakian, Ian Ring Music TheoryRakian
Scale 2699Scale 2699: Sythimic, Ian Ring Music TheorySythimic
Scale 2707Scale 2707: Banimic, Ian Ring Music TheoryBanimic
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 2755Scale 2755: Rivian, Ian Ring Music TheoryRivian
Scale 2563Scale 2563: Pofian, Ian Ring Music TheoryPofian
Scale 2627Scale 2627: Qerian, Ian Ring Music TheoryQerian
Scale 2819Scale 2819: Rujian, Ian Ring Music TheoryRujian
Scale 2947Scale 2947: Sijian, Ian Ring Music TheorySijian
Scale 2179Scale 2179, Ian Ring Music Theory
Scale 2435Scale 2435: Raga Deshgaur, Ian Ring Music TheoryRaga Deshgaur
Scale 3203Scale 3203: Etrian, Ian Ring Music TheoryEtrian
Scale 3715Scale 3715: Xician, Ian Ring Music TheoryXician
Scale 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 1667Scale 1667: Kekian, Ian Ring Music TheoryKekian

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 ( 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.