The Exciting Universe Of Music Theory

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Scale 3607: "Wopian"

Scale 3607: Wopian, 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.

7 (heptatonic)

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: 3343


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

5 (multihemitonic)


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

4 (multicohemitonic)


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: 191


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, 1, 2, 5, 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.

<5, 5, 4, 3, 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> = {3,4,7,8}
<4> = {4,5,8,9}
<5> = {5,6,9,10}
<6> = {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.

(62, 25, 86)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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
Major TriadsA{9,1,4}210.67
Minor Triadsam{9,0,4}121
Diminished Triadsa♯°{10,1,4}121

The following pitch classes are not present in any of the common triads: {2,11}

Parsimonious Voice Leading Between Common Triads of Scale 3607. Created by Ian Ring ©2019 am am A A am->A a#° a#° A->a#°

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.

Central VerticesA
Peripheral Verticesam, a♯°


Modes are the rotational transformation of this scale. Scale 3607 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3851
Scale 3851: Yilian, Ian Ring Music TheoryYilian
3rd mode:
Scale 3973
Scale 3973: Zehian, Ian Ring Music TheoryZehian
4th mode:
Scale 2017
Scale 2017: Meqian, Ian Ring Music TheoryMeqian
5th mode:
Scale 191
Scale 191: Begian, Ian Ring Music TheoryBegianThis is the prime mode
6th mode:
Scale 2143
Scale 2143: Napian, Ian Ring Music TheoryNapian
7th mode:
Scale 3119
Scale 3119: Tikian, Ian Ring Music TheoryTikian


The prime form of this scale is Scale 191

Scale 191Scale 191: Begian, Ian Ring Music TheoryBegian


The heptatonic modal family [3607, 3851, 3973, 2017, 191, 2143, 3119] (Forte: 7-2) is the complement of the pentatonic modal family [47, 1921, 2071, 3083, 3589] (Forte: 5-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3607 is 3343

Scale 3343Scale 3343: Vajian, Ian Ring Music TheoryVajian


Only scales that are chiral will have an enantiomorph. Scale 3607 is chiral, and its enantiomorph is scale 3343

Scale 3343Scale 3343: Vajian, Ian Ring Music TheoryVajian


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> 3607       T0I <11,0> 3343
T1 <1,1> 3119      T1I <11,1> 2591
T2 <1,2> 2143      T2I <11,2> 1087
T3 <1,3> 191      T3I <11,3> 2174
T4 <1,4> 382      T4I <11,4> 253
T5 <1,5> 764      T5I <11,5> 506
T6 <1,6> 1528      T6I <11,6> 1012
T7 <1,7> 3056      T7I <11,7> 2024
T8 <1,8> 2017      T8I <11,8> 4048
T9 <1,9> 4034      T9I <11,9> 4001
T10 <1,10> 3973      T10I <11,10> 3907
T11 <1,11> 3851      T11I <11,11> 3719
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1957      T0MI <7,0> 1213
T1M <5,1> 3914      T1MI <7,1> 2426
T2M <5,2> 3733      T2MI <7,2> 757
T3M <5,3> 3371      T3MI <7,3> 1514
T4M <5,4> 2647      T4MI <7,4> 3028
T5M <5,5> 1199      T5MI <7,5> 1961
T6M <5,6> 2398      T6MI <7,6> 3922
T7M <5,7> 701      T7MI <7,7> 3749
T8M <5,8> 1402      T8MI <7,8> 3403
T9M <5,9> 2804      T9MI <7,9> 2711
T10M <5,10> 1513      T10MI <7,10> 1327
T11M <5,11> 3026      T11MI <7,11> 2654

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 3605Scale 3605: Olkian, Ian Ring Music TheoryOlkian
Scale 3603Scale 3603: Womian, Ian Ring Music TheoryWomian
Scale 3611Scale 3611: Worian, Ian Ring Music TheoryWorian
Scale 3615Scale 3615: Octatonic Chromatic 4, Ian Ring Music TheoryOctatonic Chromatic 4
Scale 3591Scale 3591: Wifian, Ian Ring Music TheoryWifian
Scale 3599Scale 3599: Heptatonic Chromatic 4, Ian Ring Music TheoryHeptatonic Chromatic 4
Scale 3623Scale 3623: Aerocrian, Ian Ring Music TheoryAerocrian
Scale 3639Scale 3639: Paptyllic, Ian Ring Music TheoryPaptyllic
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 3735Scale 3735: Xupian, Ian Ring Music TheoryXupian
Scale 3863Scale 3863: Eparyllic, Ian Ring Music TheoryEparyllic
Scale 3095Scale 3095: Tivian, Ian Ring Music TheoryTivian
Scale 3351Scale 3351: Crater Scale, Ian Ring Music TheoryCrater Scale
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 1559Scale 1559: Jowian, Ian Ring Music TheoryJowian

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.