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Scale 641: "Duwian"

Scale 641: Duwian, 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.

3 (tritonic)

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: 41


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

0 (anhemitonic)


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

0 (ancohemitonic)


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: 37


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.

[7, 2, 3]

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.

<0, 1, 1, 0, 1, 0>

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> = {2,3,7}
<2> = {5,9,10}

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.

(1, 0, 6)

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 641 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 37
Scale 37: Afoian, Ian Ring Music TheoryAfoianThis is the prime mode
3rd mode:
Scale 1033
Scale 1033: Allian, Ian Ring Music TheoryAllian


The prime form of this scale is Scale 37

Scale 37Scale 37: Afoian, Ian Ring Music TheoryAfoian


The tritonic modal family [641, 37, 1033] (Forte: 3-7) is the complement of the enneatonic modal family [1471, 1789, 2027, 2783, 3061, 3439, 3767, 3931, 4013] (Forte: 9-7)


The inverse of a scale is a reflection using the root as its axis. The inverse of 641 is 41

Scale 41Scale 41: Vietnamese Tritonic, Ian Ring Music TheoryVietnamese Tritonic


Only scales that are chiral will have an enantiomorph. Scale 641 is chiral, and its enantiomorph is scale 41

Scale 41Scale 41: Vietnamese Tritonic, Ian Ring Music TheoryVietnamese Tritonic


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> 641       T0I <11,0> 41
T1 <1,1> 1282      T1I <11,1> 82
T2 <1,2> 2564      T2I <11,2> 164
T3 <1,3> 1033      T3I <11,3> 328
T4 <1,4> 2066      T4I <11,4> 656
T5 <1,5> 37      T5I <11,5> 1312
T6 <1,6> 74      T6I <11,6> 2624
T7 <1,7> 148      T7I <11,7> 1153
T8 <1,8> 296      T8I <11,8> 2306
T9 <1,9> 592      T9I <11,9> 517
T10 <1,10> 1184      T10I <11,10> 1034
T11 <1,11> 2368      T11I <11,11> 2068
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2561      T0MI <7,0> 11
T1M <5,1> 1027      T1MI <7,1> 22
T2M <5,2> 2054      T2MI <7,2> 44
T3M <5,3> 13      T3MI <7,3> 88
T4M <5,4> 26      T4MI <7,4> 176
T5M <5,5> 52      T5MI <7,5> 352
T6M <5,6> 104      T6MI <7,6> 704
T7M <5,7> 208      T7MI <7,7> 1408
T8M <5,8> 416      T8MI <7,8> 2816
T9M <5,9> 832      T9MI <7,9> 1537
T10M <5,10> 1664      T10MI <7,10> 3074
T11M <5,11> 3328      T11MI <7,11> 2053

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 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 645Scale 645: Duyian, Ian Ring Music TheoryDuyian
Scale 649Scale 649: Byptic, Ian Ring Music TheoryByptic
Scale 657Scale 657: Epathic, Ian Ring Music TheoryEpathic
Scale 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 705Scale 705: Edrian, Ian Ring Music TheoryEdrian
Scale 513Scale 513: Major Sixth Ditone, Ian Ring Music TheoryMajor Sixth Ditone
Scale 577Scale 577: Illian, Ian Ring Music TheoryIllian
Scale 769Scale 769: Enbian, Ian Ring Music TheoryEnbian
Scale 897Scale 897: Fopian, Ian Ring Music TheoryFopian
Scale 129Scale 129: Niagari, Ian Ring Music TheoryNiagari
Scale 385Scale 385: Civian, Ian Ring Music TheoryCivian
Scale 1153Scale 1153: Choian, Ian Ring Music TheoryChoian
Scale 1665Scale 1665: Kejian, Ian Ring Music TheoryKejian
Scale 2689Scale 2689: Ragian, Ian Ring Music TheoryRagian

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.