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Scale 449: "Cujian"

Scale 449: Cujian, 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.

enantiomorph: 113


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.

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: 71


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, 1, 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, 1, 0, 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> = {1,4,6}
<2> = {2,5,7,10}
<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.

(6, 1, 16)

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

2nd mode:
Scale 71
Scale 71: Aloian, Ian Ring Music TheoryAloianThis is the prime mode
3rd mode:
Scale 2083
Scale 2083: Mofian, Ian Ring Music TheoryMofian
4th mode:
Scale 3089
Scale 3089: Tirian, Ian Ring Music TheoryTirian


The prime form of this scale is Scale 71

Scale 71Scale 71: Aloian, Ian Ring Music TheoryAloian


The tetratonic modal family [449, 71, 2083, 3089] (Forte: 4-5) is the complement of the octatonic modal family [479, 1991, 2287, 3043, 3191, 3569, 3643, 3869] (Forte: 8-5)


The inverse of a scale is a reflection using the root as its axis. The inverse of 449 is 113

Scale 113Scale 113, Ian Ring Music Theory


Only scales that are chiral will have an enantiomorph. Scale 449 is chiral, and its enantiomorph is scale 113

Scale 113Scale 113, Ian Ring Music Theory


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> 449       T0I <11,0> 113
T1 <1,1> 898      T1I <11,1> 226
T2 <1,2> 1796      T2I <11,2> 452
T3 <1,3> 3592      T3I <11,3> 904
T4 <1,4> 3089      T4I <11,4> 1808
T5 <1,5> 2083      T5I <11,5> 3616
T6 <1,6> 71      T6I <11,6> 3137
T7 <1,7> 142      T7I <11,7> 2179
T8 <1,8> 284      T8I <11,8> 263
T9 <1,9> 568      T9I <11,9> 526
T10 <1,10> 1136      T10I <11,10> 1052
T11 <1,11> 2272      T11I <11,11> 2104
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2129      T0MI <7,0> 323
T1M <5,1> 163      T1MI <7,1> 646
T2M <5,2> 326      T2MI <7,2> 1292
T3M <5,3> 652      T3MI <7,3> 2584
T4M <5,4> 1304      T4MI <7,4> 1073
T5M <5,5> 2608      T5MI <7,5> 2146
T6M <5,6> 1121      T6MI <7,6> 197
T7M <5,7> 2242      T7MI <7,7> 394
T8M <5,8> 389      T8MI <7,8> 788
T9M <5,9> 778      T9MI <7,9> 1576
T10M <5,10> 1556      T10MI <7,10> 3152
T11M <5,11> 3112      T11MI <7,11> 2209

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 451Scale 451: Raga Saugandhini, Ian Ring Music TheoryRaga Saugandhini
Scale 453Scale 453: Raditonic, Ian Ring Music TheoryRaditonic
Scale 457Scale 457: Staptitonic, Ian Ring Music TheoryStaptitonic
Scale 465Scale 465: Zoditonic, Ian Ring Music TheoryZoditonic
Scale 481Scale 481: Dabian, Ian Ring Music TheoryDabian
Scale 385Scale 385: Civian, Ian Ring Music TheoryCivian
Scale 417Scale 417: Copian, Ian Ring Music TheoryCopian
Scale 321Scale 321: Cahian, Ian Ring Music TheoryCahian
Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari
Scale 705Scale 705: Edrian, Ian Ring Music TheoryEdrian
Scale 961Scale 961: Gabian, Ian Ring Music TheoryGabian
Scale 1473Scale 1473: Javian, Ian Ring Music TheoryJavian
Scale 2497Scale 2497: Peqian, Ian Ring Music TheoryPeqian

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.