The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 321: "Cahian"

Scale 321: Cahian, 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.

3 (tritonic)

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: 81


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

0 (anhemitonic)


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: 69


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, 2, 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.

<0, 1, 0, 1, 0, 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> = {2,4,6}
<2> = {6,8,10}

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.

(0, 1, 6)

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 321 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 69
Scale 69: Dezian, Ian Ring Music TheoryDezianThis is the prime mode
3rd mode:
Scale 1041
Scale 1041: Hitian, Ian Ring Music TheoryHitian


The prime form of this scale is Scale 69

Scale 69Scale 69: Dezian, Ian Ring Music TheoryDezian


The tritonic modal family [321, 69, 1041] (Forte: 3-8) is the complement of the enneatonic modal family [1503, 1917, 2007, 2799, 3051, 3447, 3573, 3771, 3933] (Forte: 9-8)


The inverse of a scale is a reflection using the root as its axis. The inverse of 321 is 81

Scale 81Scale 81: Disian, Ian Ring Music TheoryDisian


Only scales that are chiral will have an enantiomorph. Scale 321 is chiral, and its enantiomorph is scale 81

Scale 81Scale 81: Disian, Ian Ring Music TheoryDisian


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> 321       T0I <11,0> 81
T1 <1,1> 642      T1I <11,1> 162
T2 <1,2> 1284      T2I <11,2> 324
T3 <1,3> 2568      T3I <11,3> 648
T4 <1,4> 1041      T4I <11,4> 1296
T5 <1,5> 2082      T5I <11,5> 2592
T6 <1,6> 69      T6I <11,6> 1089
T7 <1,7> 138      T7I <11,7> 2178
T8 <1,8> 276      T8I <11,8> 261
T9 <1,9> 552      T9I <11,9> 522
T10 <1,10> 1104      T10I <11,10> 1044
T11 <1,11> 2208      T11I <11,11> 2088
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 81      T0MI <7,0> 321
T1M <5,1> 162      T1MI <7,1> 642
T2M <5,2> 324      T2MI <7,2> 1284
T3M <5,3> 648      T3MI <7,3> 2568
T4M <5,4> 1296      T4MI <7,4> 1041
T5M <5,5> 2592      T5MI <7,5> 2082
T6M <5,6> 1089      T6MI <7,6> 69
T7M <5,7> 2178      T7MI <7,7> 138
T8M <5,8> 261      T8MI <7,8> 276
T9M <5,9> 522      T9MI <7,9> 552
T10M <5,10> 1044      T10MI <7,10> 1104
T11M <5,11> 2088      T11MI <7,11> 2208

The transformations that map this set to itself are: T0, 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 323Scale 323: Cajian, Ian Ring Music TheoryCajian
Scale 325Scale 325: Messiaen Truncated Mode 6, Ian Ring Music TheoryMessiaen Truncated Mode 6
Scale 329Scale 329: Mynic 2, Ian Ring Music TheoryMynic 2
Scale 337Scale 337: Koptic, Ian Ring Music TheoryKoptic
Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian
Scale 257Scale 257: Minor Sixth Ditone, Ian Ring Music TheoryMinor Sixth Ditone
Scale 289Scale 289: Valian, Ian Ring Music TheoryValian
Scale 385Scale 385: Civian, Ian Ring Music TheoryCivian
Scale 449Scale 449: Cujian, Ian Ring Music TheoryCujian
Scale 65Scale 65: Tritone, Ian Ring Music TheoryTritone
Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari
Scale 577Scale 577: Illian, Ian Ring Music TheoryIllian
Scale 833Scale 833: Febian, Ian Ring Music TheoryFebian
Scale 1345Scale 1345: Iskian, Ian Ring Music TheoryIskian
Scale 2369Scale 2369: Offian, Ian Ring Music TheoryOffian

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.