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Scale 3649: "Wupian"

Scale 3649: Wupian, 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: 79


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.

2 (dicohemitonic)


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: 79


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.

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

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.

(20, 3, 32)

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
Diminished Triadsf♯°{6,9,0}000

The following pitch classes are not present in any of the common triads: {10,11}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 3649 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 121
Scale 121: Asoian, Ian Ring Music TheoryAsoian
3rd mode:
Scale 527
Scale 527: Dedian, Ian Ring Music TheoryDedian
4th mode:
Scale 2311
Scale 2311: Raga Kumarapriya, Ian Ring Music TheoryRaga Kumarapriya
5th mode:
Scale 3203
Scale 3203: Etrian, Ian Ring Music TheoryEtrian


The prime form of this scale is Scale 79

Scale 79Scale 79: Appian, Ian Ring Music TheoryAppian


The pentatonic modal family [3649, 121, 527, 2311, 3203] (Forte: 5-4) is the complement of the heptatonic modal family [223, 1987, 2159, 3041, 3127, 3611, 3853] (Forte: 7-4)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3649 is 79

Scale 79Scale 79: Appian, Ian Ring Music TheoryAppian


Only scales that are chiral will have an enantiomorph. Scale 3649 is chiral, and its enantiomorph is scale 79

Scale 79Scale 79: Appian, Ian Ring Music TheoryAppian


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> 3649       T0I <11,0> 79
T1 <1,1> 3203      T1I <11,1> 158
T2 <1,2> 2311      T2I <11,2> 316
T3 <1,3> 527      T3I <11,3> 632
T4 <1,4> 1054      T4I <11,4> 1264
T5 <1,5> 2108      T5I <11,5> 2528
T6 <1,6> 121      T6I <11,6> 961
T7 <1,7> 242      T7I <11,7> 1922
T8 <1,8> 484      T8I <11,8> 3844
T9 <1,9> 968      T9I <11,9> 3593
T10 <1,10> 1936      T10I <11,10> 3091
T11 <1,11> 3872      T11I <11,11> 2087
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 709      T0MI <7,0> 1129
T1M <5,1> 1418      T1MI <7,1> 2258
T2M <5,2> 2836      T2MI <7,2> 421
T3M <5,3> 1577      T3MI <7,3> 842
T4M <5,4> 3154      T4MI <7,4> 1684
T5M <5,5> 2213      T5MI <7,5> 3368
T6M <5,6> 331      T6MI <7,6> 2641
T7M <5,7> 662      T7MI <7,7> 1187
T8M <5,8> 1324      T8MI <7,8> 2374
T9M <5,9> 2648      T9MI <7,9> 653
T10M <5,10> 1201      T10MI <7,10> 1306
T11M <5,11> 2402      T11MI <7,11> 2612

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 3651Scale 3651: Wuqian, Ian Ring Music TheoryWuqian
Scale 3653Scale 3653: Sathimic, Ian Ring Music TheorySathimic
Scale 3657Scale 3657: Epynimic, Ian Ring Music TheoryEpynimic
Scale 3665Scale 3665: Stalimic, Ian Ring Music TheoryStalimic
Scale 3681Scale 3681: Xahian, Ian Ring Music TheoryXahian
Scale 3585Scale 3585: Tetratonic Chromatic Descending, Ian Ring Music TheoryTetratonic Chromatic Descending
Scale 3617Scale 3617: Wovian, Ian Ring Music TheoryWovian
Scale 3713Scale 3713: Xibian, Ian Ring Music TheoryXibian
Scale 3777Scale 3777: Yarian, Ian Ring Music TheoryYarian
Scale 3905Scale 3905: Yusian, Ian Ring Music TheoryYusian
Scale 3137Scale 3137, Ian Ring Music Theory
Scale 3393Scale 3393: Venian, Ian Ring Music TheoryVenian
Scale 2625Scale 2625, Ian Ring Music Theory
Scale 1601Scale 1601: Juwian, Ian Ring Music TheoryJuwian

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.