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Scale 3489: "Vuvian"

Scale 3489: Vuvian, 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.

6 (hexatonic)

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: 183


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 Rahn/Ring formula.

prime: 183


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.

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

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.

(27, 11, 59)

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
Minor Triadsfm{5,8,0}110.5
Diminished Triads{5,8,11}110.5

The following pitch classes are not present in any of the common triads: {7,10}

Parsimonious Voice Leading Between Common Triads of Scale 3489. Created by Ian Ring ©2019 fm fm f°->fm

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.



Modes are the rotational transformation of this scale. Scale 3489 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 237
Scale 237: Bijian, Ian Ring Music TheoryBijian
3rd mode:
Scale 1083
Scale 1083: Goyian, Ian Ring Music TheoryGoyian
4th mode:
Scale 2589
Scale 2589: Puvian, Ian Ring Music TheoryPuvian
5th mode:
Scale 1671
Scale 1671: Kemian, Ian Ring Music TheoryKemian
6th mode:
Scale 2883
Scale 2883: Savian, Ian Ring Music TheorySavian


The prime form of this scale is Scale 183

Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian


The hexatonic modal family [3489, 237, 1083, 2589, 1671, 2883] (Forte: 6-Z11) is the complement of the hexatonic modal family [303, 753, 1929, 2199, 3147, 3621] (Forte: 6-Z40)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3489 is 183

Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian


Only scales that are chiral will have an enantiomorph. Scale 3489 is chiral, and its enantiomorph is scale 183

Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian


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> 3489       T0I <11,0> 183
T1 <1,1> 2883      T1I <11,1> 366
T2 <1,2> 1671      T2I <11,2> 732
T3 <1,3> 3342      T3I <11,3> 1464
T4 <1,4> 2589      T4I <11,4> 2928
T5 <1,5> 1083      T5I <11,5> 1761
T6 <1,6> 2166      T6I <11,6> 3522
T7 <1,7> 237      T7I <11,7> 2949
T8 <1,8> 474      T8I <11,8> 1803
T9 <1,9> 948      T9I <11,9> 3606
T10 <1,10> 1896      T10I <11,10> 3117
T11 <1,11> 3792      T11I <11,11> 2139
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2199      T0MI <7,0> 3363
T1M <5,1> 303      T1MI <7,1> 2631
T2M <5,2> 606      T2MI <7,2> 1167
T3M <5,3> 1212      T3MI <7,3> 2334
T4M <5,4> 2424      T4MI <7,4> 573
T5M <5,5> 753      T5MI <7,5> 1146
T6M <5,6> 1506      T6MI <7,6> 2292
T7M <5,7> 3012      T7MI <7,7> 489
T8M <5,8> 1929      T8MI <7,8> 978
T9M <5,9> 3858      T9MI <7,9> 1956
T10M <5,10> 3621      T10MI <7,10> 3912
T11M <5,11> 3147      T11MI <7,11> 3729

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 3491Scale 3491: Tharian, Ian Ring Music TheoryTharian
Scale 3493Scale 3493: Rathian, Ian Ring Music TheoryRathian
Scale 3497Scale 3497: Phrolian, Ian Ring Music TheoryPhrolian
Scale 3505Scale 3505: Stygian, Ian Ring Music TheoryStygian
Scale 3457Scale 3457: Vobian, Ian Ring Music TheoryVobian
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 3521Scale 3521: Wanian, Ian Ring Music TheoryWanian
Scale 3553Scale 3553: Wehian, Ian Ring Music TheoryWehian
Scale 3361Scale 3361: Vatian, Ian Ring Music TheoryVatian
Scale 3425Scale 3425: Vihian, Ian Ring Music TheoryVihian
Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian
Scale 3745Scale 3745: Xuvian, Ian Ring Music TheoryXuvian
Scale 4001Scale 4001: Ziyian, Ian Ring Music TheoryZiyian
Scale 2465Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani
Scale 2977Scale 2977: Sobian, Ian Ring Music TheorySobian
Scale 1441Scale 1441: Jabian, Ian Ring Music TheoryJabian

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.