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Scale 903: "Fosian"

Scale 903: Fosian, 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.

6 (hexatonic)

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.

4 (multihemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

2 (dicohemitonic)


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: 231


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.

[1, 1, 5, 1, 1, 3]

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.

<4, 2, 1, 2, 4, 2>

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

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.

(24, 10, 51)

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 903 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2499
Scale 2499: Pirian, Ian Ring Music TheoryPirian
3rd mode:
Scale 3297
Scale 3297: Ullian, Ian Ring Music TheoryUllian
4th mode:
Scale 231
Scale 231: Bifian, Ian Ring Music TheoryBifianThis is the prime mode
5th mode:
Scale 2163
Scale 2163: Nebian, Ian Ring Music TheoryNebian
6th mode:
Scale 3129
Scale 3129: Toqian, Ian Ring Music TheoryToqian


The prime form of this scale is Scale 231

Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian


The hexatonic modal family [903, 2499, 3297, 231, 2163, 3129] (Forte: 6-Z6) is the complement of the hexatonic modal family [399, 483, 2247, 2289, 3171, 3633] (Forte: 6-Z38)


The inverse of a scale is a reflection using the root as its axis. The inverse of 903 is 3129

Scale 3129Scale 3129: Toqian, Ian Ring Music TheoryToqian


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> 903       T0I <11,0> 3129
T1 <1,1> 1806      T1I <11,1> 2163
T2 <1,2> 3612      T2I <11,2> 231
T3 <1,3> 3129      T3I <11,3> 462
T4 <1,4> 2163      T4I <11,4> 924
T5 <1,5> 231      T5I <11,5> 1848
T6 <1,6> 462      T6I <11,6> 3696
T7 <1,7> 924      T7I <11,7> 3297
T8 <1,8> 1848      T8I <11,8> 2499
T9 <1,9> 3696      T9I <11,9> 903
T10 <1,10> 3297      T10I <11,10> 1806
T11 <1,11> 2499      T11I <11,11> 3612
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3633      T0MI <7,0> 399
T1M <5,1> 3171      T1MI <7,1> 798
T2M <5,2> 2247      T2MI <7,2> 1596
T3M <5,3> 399      T3MI <7,3> 3192
T4M <5,4> 798      T4MI <7,4> 2289
T5M <5,5> 1596      T5MI <7,5> 483
T6M <5,6> 3192      T6MI <7,6> 966
T7M <5,7> 2289      T7MI <7,7> 1932
T8M <5,8> 483      T8MI <7,8> 3864
T9M <5,9> 966      T9MI <7,9> 3633
T10M <5,10> 1932      T10MI <7,10> 3171
T11M <5,11> 3864      T11MI <7,11> 2247

The transformations that map this set to itself are: T0, T9I

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 901Scale 901: Bofian, Ian Ring Music TheoryBofian
Scale 899Scale 899: Foqian, Ian Ring Music TheoryFoqian
Scale 907Scale 907: Tholimic, Ian Ring Music TheoryTholimic
Scale 911Scale 911: Radian, Ian Ring Music TheoryRadian
Scale 919Scale 919: Chromatic Phrygian Inverse, Ian Ring Music TheoryChromatic Phrygian Inverse
Scale 935Scale 935: Chromatic Dorian, Ian Ring Music TheoryChromatic Dorian
Scale 967Scale 967: Mela Salaga, Ian Ring Music TheoryMela Salaga
Scale 775Scale 775: Raga Putrika, Ian Ring Music TheoryRaga Putrika
Scale 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 647Scale 647: Duzian, Ian Ring Music TheoryDuzian
Scale 391Scale 391: Ciyian, Ian Ring Music TheoryCiyian
Scale 1415Scale 1415: Impian, Ian Ring Music TheoryImpian
Scale 1927Scale 1927: Lunian, Ian Ring Music TheoryLunian
Scale 2951Scale 2951: Silian, Ian Ring Music TheorySilian

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.