The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 2309: "Ocuian"

Scale 2309: Ocuian, 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: 1043


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: 77


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.

[2, 6, 3, 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.

<1, 1, 2, 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> = {1,2,3,6}
<2> = {3,4,8,9}
<3> = {6,9,10,11}

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.

(4, 3, 18)

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
Diminished Triadsg♯°{8,11,2}000

The following pitch classes are not present in any of the common triads: {0}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 2309 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 1601
Scale 1601: Juwian, Ian Ring Music TheoryJuwian
3rd mode:
Scale 89
Scale 89: Aggian, Ian Ring Music TheoryAggian
4th mode:
Scale 523
Scale 523: Debian, Ian Ring Music TheoryDebian


The prime form of this scale is Scale 77

Scale 77Scale 77: Alvian, Ian Ring Music TheoryAlvian


The tetratonic modal family [2309, 1601, 89, 523] (Forte: 4-12) is the complement of the octatonic modal family [763, 1631, 2009, 2429, 2863, 3479, 3787, 3941] (Forte: 8-12)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2309 is 1043

Scale 1043Scale 1043: Gizian, Ian Ring Music TheoryGizian


Only scales that are chiral will have an enantiomorph. Scale 2309 is chiral, and its enantiomorph is scale 1043

Scale 1043Scale 1043: Gizian, Ian Ring Music TheoryGizian


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> 2309       T0I <11,0> 1043
T1 <1,1> 523      T1I <11,1> 2086
T2 <1,2> 1046      T2I <11,2> 77
T3 <1,3> 2092      T3I <11,3> 154
T4 <1,4> 89      T4I <11,4> 308
T5 <1,5> 178      T5I <11,5> 616
T6 <1,6> 356      T6I <11,6> 1232
T7 <1,7> 712      T7I <11,7> 2464
T8 <1,8> 1424      T8I <11,8> 833
T9 <1,9> 2848      T9I <11,9> 1666
T10 <1,10> 1601      T10I <11,10> 3332
T11 <1,11> 3202      T11I <11,11> 2569
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1169      T0MI <7,0> 293
T1M <5,1> 2338      T1MI <7,1> 586
T2M <5,2> 581      T2MI <7,2> 1172
T3M <5,3> 1162      T3MI <7,3> 2344
T4M <5,4> 2324      T4MI <7,4> 593
T5M <5,5> 553      T5MI <7,5> 1186
T6M <5,6> 1106      T6MI <7,6> 2372
T7M <5,7> 2212      T7MI <7,7> 649
T8M <5,8> 329      T8MI <7,8> 1298
T9M <5,9> 658      T9MI <7,9> 2596
T10M <5,10> 1316      T10MI <7,10> 1097
T11M <5,11> 2632      T11MI <7,11> 2194

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 2311Scale 2311: Raga Kumarapriya, Ian Ring Music TheoryRaga Kumarapriya
Scale 2305Scale 2305: Ollian, Ian Ring Music TheoryOllian
Scale 2307Scale 2307: Ocoian, Ian Ring Music TheoryOcoian
Scale 2313Scale 2313: Osrian, Ian Ring Music TheoryOsrian
Scale 2317Scale 2317: Odoian, Ian Ring Music TheoryOdoian
Scale 2325Scale 2325: Pynitonic, Ian Ring Music TheoryPynitonic
Scale 2341Scale 2341: Raga Priyadharshini, Ian Ring Music TheoryRaga Priyadharshini
Scale 2373Scale 2373: Dyptitonic, Ian Ring Music TheoryDyptitonic
Scale 2437Scale 2437: Pafian, Ian Ring Music TheoryPafian
Scale 2053Scale 2053: Powian, Ian Ring Music TheoryPowian
Scale 2181Scale 2181: Nemian, Ian Ring Music TheoryNemian
Scale 2565Scale 2565: Pogian, Ian Ring Music TheoryPogian
Scale 2821Scale 2821: Rukian, Ian Ring Music TheoryRukian
Scale 3333Scale 3333: Vacian, Ian Ring Music TheoryVacian
Scale 261Scale 261: Bozian, Ian Ring Music TheoryBozian
Scale 1285Scale 1285: Husian, Ian Ring Music TheoryHusian

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.