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Scale 387: "Ciwian"

Scale 387: Ciwian, 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.

4 (tetratonic)

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.



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.

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: 99


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, 1, 4]

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, 0, 0, 1, 2, 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> = {1,4,6}
<2> = {5,7}
<3> = {6,8,11}

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.

(4, 1, 14)

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 387 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 2241
Scale 2241: Noxian, Ian Ring Music TheoryNoxian
3rd mode:
Scale 99
Scale 99: Iprian, Ian Ring Music TheoryIprianThis is the prime mode
4th mode:
Scale 2097
Scale 2097: Munian, Ian Ring Music TheoryMunian


The prime form of this scale is Scale 99

Scale 99Scale 99: Iprian, Ian Ring Music TheoryIprian


The tetratonic modal family [387, 2241, 99, 2097] (Forte: 4-8) is the complement of the octatonic modal family [927, 999, 2511, 2547, 3303, 3321, 3699, 3897] (Forte: 8-8)


The inverse of a scale is a reflection using the root as its axis. The inverse of 387 is 2097

Scale 2097Scale 2097: Munian, Ian Ring Music TheoryMunian


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> 387       T0I <11,0> 2097
T1 <1,1> 774      T1I <11,1> 99
T2 <1,2> 1548      T2I <11,2> 198
T3 <1,3> 3096      T3I <11,3> 396
T4 <1,4> 2097      T4I <11,4> 792
T5 <1,5> 99      T5I <11,5> 1584
T6 <1,6> 198      T6I <11,6> 3168
T7 <1,7> 396      T7I <11,7> 2241
T8 <1,8> 792      T8I <11,8> 387
T9 <1,9> 1584      T9I <11,9> 774
T10 <1,10> 3168      T10I <11,10> 1548
T11 <1,11> 2241      T11I <11,11> 3096
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2097      T0MI <7,0> 387
T1M <5,1> 99      T1MI <7,1> 774
T2M <5,2> 198      T2MI <7,2> 1548
T3M <5,3> 396      T3MI <7,3> 3096
T4M <5,4> 792      T4MI <7,4> 2097
T5M <5,5> 1584      T5MI <7,5> 99
T6M <5,6> 3168      T6MI <7,6> 198
T7M <5,7> 2241      T7MI <7,7> 396
T8M <5,8> 387       T8MI <7,8> 792
T9M <5,9> 774      T9MI <7,9> 1584
T10M <5,10> 1548      T10MI <7,10> 3168
T11M <5,11> 3096      T11MI <7,11> 2241

The transformations that map this set to itself are: T0, T8I, T8M, T0MI

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 385Scale 385: Civian, Ian Ring Music TheoryCivian
Scale 389Scale 389: Cixian, Ian Ring Music TheoryCixian
Scale 391Scale 391: Ciyian, Ian Ring Music TheoryCiyian
Scale 395Scale 395: Phrygian Pentatonic, Ian Ring Music TheoryPhrygian Pentatonic
Scale 403Scale 403: Raga Reva, Ian Ring Music TheoryRaga Reva
Scale 419Scale 419: Hon-kumoi-joshi, Ian Ring Music TheoryHon-kumoi-joshi
Scale 451Scale 451: Raga Saugandhini, Ian Ring Music TheoryRaga Saugandhini
Scale 259Scale 259: Gijian, Ian Ring Music TheoryGijian
Scale 323Scale 323: Cajian, Ian Ring Music TheoryCajian
Scale 131Scale 131: Atoian, Ian Ring Music TheoryAtoian
Scale 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 899Scale 899: Foqian, Ian Ring Music TheoryFoqian
Scale 1411Scale 1411: Iroian, Ian Ring Music TheoryIroian
Scale 2435Scale 2435: Raga Deshgaur, Ian Ring Music TheoryRaga Deshgaur

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.