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Scale 1665: "Kejian"

Scale 1665: Kejian, 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.

1 (unhemitonic)


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: 45


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, 1, 2]

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.

<1, 2, 2, 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> = {1,2,7}
<2> = {3,9}
<3> = {5,10,11}

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.

(5, 0, 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 1665 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 45
Scale 45: Aprian, Ian Ring Music TheoryAprianThis is the prime mode
3rd mode:
Scale 1035
Scale 1035: Givian, Ian Ring Music TheoryGivian
4th mode:
Scale 2565
Scale 2565: Pogian, Ian Ring Music TheoryPogian


The prime form of this scale is Scale 45

Scale 45Scale 45: Aprian, Ian Ring Music TheoryAprian


The tetratonic modal family [1665, 45, 1035, 2565] (Forte: 4-10) is the complement of the octatonic modal family [765, 1215, 2025, 2655, 3375, 3735, 3915, 4005] (Forte: 8-10)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1665 is 45

Scale 45Scale 45: Aprian, Ian Ring Music TheoryAprian


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> 1665       T0I <11,0> 45
T1 <1,1> 3330      T1I <11,1> 90
T2 <1,2> 2565      T2I <11,2> 180
T3 <1,3> 1035      T3I <11,3> 360
T4 <1,4> 2070      T4I <11,4> 720
T5 <1,5> 45      T5I <11,5> 1440
T6 <1,6> 90      T6I <11,6> 2880
T7 <1,7> 180      T7I <11,7> 1665
T8 <1,8> 360      T8I <11,8> 3330
T9 <1,9> 720      T9I <11,9> 2565
T10 <1,10> 1440      T10I <11,10> 1035
T11 <1,11> 2880      T11I <11,11> 2070
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2565      T0MI <7,0> 1035
T1M <5,1> 1035      T1MI <7,1> 2070
T2M <5,2> 2070      T2MI <7,2> 45
T3M <5,3> 45      T3MI <7,3> 90
T4M <5,4> 90      T4MI <7,4> 180
T5M <5,5> 180      T5MI <7,5> 360
T6M <5,6> 360      T6MI <7,6> 720
T7M <5,7> 720      T7MI <7,7> 1440
T8M <5,8> 1440      T8MI <7,8> 2880
T9M <5,9> 2880      T9MI <7,9> 1665
T10M <5,10> 1665       T10MI <7,10> 3330
T11M <5,11> 3330      T11MI <7,11> 2565

The transformations that map this set to itself are: T0, T7I, T10M, T9MI

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 1667Scale 1667: Kekian, Ian Ring Music TheoryKekian
Scale 1669Scale 1669: Raga Matha Kokila, Ian Ring Music TheoryRaga Matha Kokila
Scale 1673Scale 1673: Thocritonic, Ian Ring Music TheoryThocritonic
Scale 1681Scale 1681: Raga Valaji, Ian Ring Music TheoryRaga Valaji
Scale 1697Scale 1697: Raga Kuntvarali, Ian Ring Music TheoryRaga Kuntvarali
Scale 1729Scale 1729: Kowian, Ian Ring Music TheoryKowian
Scale 1537Scale 1537: Jijian, Ian Ring Music TheoryJijian
Scale 1601Scale 1601: Juwian, Ian Ring Music TheoryJuwian
Scale 1793Scale 1793: Lajian, Ian Ring Music TheoryLajian
Scale 1921Scale 1921: Lukian, Ian Ring Music TheoryLukian
Scale 1153Scale 1153: Choian, Ian Ring Music TheoryChoian
Scale 1409Scale 1409: Imsian, Ian Ring Music TheoryImsian
Scale 641Scale 641: Duwian, Ian Ring Music TheoryDuwian
Scale 2689Scale 2689: Ragian, Ian Ring Music TheoryRagian
Scale 3713Scale 3713: Xibian, Ian Ring Music TheoryXibian

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.