The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 2401: "OROian"

Scale 2401: OROian, 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).



Cardinality is the count of how many pitches are in the scale.

5 (pentatonic)

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: 211


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 Starr/Rahn algorithm.

prime: 203


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, an indicator of maximum hierarchization.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[5, 1, 2, 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.

<2, 1, 2, 1, 2, 2>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.5, 0.25, 0.5, 0, 0.5, 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> = {1,2,3,5}
<2> = {3,4,5,6}
<3> = {6,7,8,9}
<4> = {7,9,10,11}

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 Distribution Spectra have 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, 8, 36)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.


Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.



This scale has no generator.

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
Minor Triadsfm{5,8,0}110.5
Diminished Triads{5,8,11}110.5

The following pitch classes are not present in any of the common triads: {6}

Parsimonious Voice Leading Between Common Triads of Scale 2401. Created by Ian Ring ©2019 fm fm f°->fm

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.



Modes are the rotational transformation of this scale. Scale 2401 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 203
Scale 203: MiaoYao 5 Tone Type 3, Ian Ring Music TheoryMiaoYao 5 Tone Type 3This is the prime mode
3rd mode:
Scale 2149
Scale 2149: NASian, Ian Ring Music TheoryNASian
4th mode:
Scale 1561
Scale 1561: JOXian, Ian Ring Music TheoryJOXian
5th mode:
Scale 707
Scale 707: EHOian, Ian Ring Music TheoryEHOian


The prime form of this scale is Scale 203

Scale 203Scale 203: MiaoYao 5 Tone Type 3, Ian Ring Music TheoryMiaoYao 5 Tone Type 3


The pentatonic modal family [2401, 203, 2149, 1561, 707] (Forte: 5-19) is the complement of the heptatonic modal family [719, 971, 1657, 2407, 2533, 3251, 3673] (Forte: 7-19)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2401 is 211

Scale 211Scale 211: BISian, Ian Ring Music TheoryBISian


Only scales that are chiral will have an enantiomorph. Scale 2401 is chiral, and its enantiomorph is scale 211

Scale 211Scale 211: BISian, Ian Ring Music TheoryBISian


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> 2401       T0I <11,0> 211
T1 <1,1> 707      T1I <11,1> 422
T2 <1,2> 1414      T2I <11,2> 844
T3 <1,3> 2828      T3I <11,3> 1688
T4 <1,4> 1561      T4I <11,4> 3376
T5 <1,5> 3122      T5I <11,5> 2657
T6 <1,6> 2149      T6I <11,6> 1219
T7 <1,7> 203      T7I <11,7> 2438
T8 <1,8> 406      T8I <11,8> 781
T9 <1,9> 812      T9I <11,9> 1562
T10 <1,10> 1624      T10I <11,10> 3124
T11 <1,11> 3248      T11I <11,11> 2153
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 211      T0MI <7,0> 2401
T1M <5,1> 422      T1MI <7,1> 707
T2M <5,2> 844      T2MI <7,2> 1414
T3M <5,3> 1688      T3MI <7,3> 2828
T4M <5,4> 3376      T4MI <7,4> 1561
T5M <5,5> 2657      T5MI <7,5> 3122
T6M <5,6> 1219      T6MI <7,6> 2149
T7M <5,7> 2438      T7MI <7,7> 203
T8M <5,8> 781      T8MI <7,8> 406
T9M <5,9> 1562      T9MI <7,9> 812
T10M <5,10> 3124      T10MI <7,10> 1624
T11M <5,11> 2153      T11MI <7,11> 3248

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 2403Scale 2403: Lycrimic, Ian Ring Music TheoryLycrimic
Scale 2405Scale 2405: T4 First Rotation, Ian Ring Music TheoryT4 First Rotation
Scale 2409Scale 2409: Zacrimic, Ian Ring Music TheoryZacrimic
Scale 2417Scale 2417: Kanimic, Ian Ring Music TheoryKanimic
Scale 2369Scale 2369: OFFian, Ian Ring Music TheoryOFFian
Scale 2385Scale 2385: Karen 5tone Type 2, Ian Ring Music TheoryKaren 5tone Type 2
Scale 2337Scale 2337: OGOian, Ian Ring Music TheoryOGOian
Scale 2465Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani
Scale 2529Scale 2529: PIKian, Ian Ring Music TheoryPIKian
Scale 2145Scale 2145: Messiaen Mode 5 Truncation 2, Ian Ring Music TheoryMessiaen Mode 5 Truncation 2
Scale 2273Scale 2273: NURian, Ian Ring Music TheoryNURian
Scale 2657Scale 2657: QOKian, Ian Ring Music TheoryQOKian
Scale 2913Scale 2913: SENian, Ian Ring Music TheorySENian
Scale 3425Scale 3425: VIHian, Ian Ring Music TheoryVIHian
Scale 353Scale 353: CEBian, Ian Ring Music TheoryCEBian
Scale 1377Scale 1377: INSian, Ian Ring Music TheoryINSian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.