The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 1081: "Goxian"

Scale 1081: Goxian, 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.

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: 901


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: 167


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.

[3, 1, 1, 5, 2]

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

(9, 9, 38)

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 1081 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 647
Scale 647: Duzian, Ian Ring Music TheoryDuzian
3rd mode:
Scale 2371
Scale 2371: Omoian, Ian Ring Music TheoryOmoian
4th mode:
Scale 3233
Scale 3233: Unbian, Ian Ring Music TheoryUnbian
5th mode:
Scale 229
Scale 229: Bidian, Ian Ring Music TheoryBidian


The prime form of this scale is Scale 167

Scale 167Scale 167: Barian, Ian Ring Music TheoryBarian


The pentatonic modal family [1081, 647, 2371, 3233, 229] (Forte: 5-14) is the complement of the heptatonic modal family [431, 1507, 1933, 2263, 2801, 3179, 3637] (Forte: 7-14)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1081 is 901

Scale 901Scale 901: Bofian, Ian Ring Music TheoryBofian


Only scales that are chiral will have an enantiomorph. Scale 1081 is chiral, and its enantiomorph is scale 901

Scale 901Scale 901: Bofian, Ian Ring Music TheoryBofian


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> 1081       T0I <11,0> 901
T1 <1,1> 2162      T1I <11,1> 1802
T2 <1,2> 229      T2I <11,2> 3604
T3 <1,3> 458      T3I <11,3> 3113
T4 <1,4> 916      T4I <11,4> 2131
T5 <1,5> 1832      T5I <11,5> 167
T6 <1,6> 3664      T6I <11,6> 334
T7 <1,7> 3233      T7I <11,7> 668
T8 <1,8> 2371      T8I <11,8> 1336
T9 <1,9> 647      T9I <11,9> 2672
T10 <1,10> 1294      T10I <11,10> 1249
T11 <1,11> 2588      T11I <11,11> 2498
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 271      T0MI <7,0> 3601
T1M <5,1> 542      T1MI <7,1> 3107
T2M <5,2> 1084      T2MI <7,2> 2119
T3M <5,3> 2168      T3MI <7,3> 143
T4M <5,4> 241      T4MI <7,4> 286
T5M <5,5> 482      T5MI <7,5> 572
T6M <5,6> 964      T6MI <7,6> 1144
T7M <5,7> 1928      T7MI <7,7> 2288
T8M <5,8> 3856      T8MI <7,8> 481
T9M <5,9> 3617      T9MI <7,9> 962
T10M <5,10> 3139      T10MI <7,10> 1924
T11M <5,11> 2183      T11MI <7,11> 3848

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 1083Scale 1083: Goyian, Ian Ring Music TheoryGoyian
Scale 1085Scale 1085: Gozian, Ian Ring Music TheoryGozian
Scale 1073Scale 1073: Gosian, Ian Ring Music TheoryGosian
Scale 1077Scale 1077: Govian, Ian Ring Music TheoryGovian
Scale 1065Scale 1065: Gonian, Ian Ring Music TheoryGonian
Scale 1049Scale 1049: Gidian, Ian Ring Music TheoryGidian
Scale 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2
Scale 1145Scale 1145: Zygimic, Ian Ring Music TheoryZygimic
Scale 1209Scale 1209: Raga Bhanumanjari, Ian Ring Music TheoryRaga Bhanumanjari
Scale 1337Scale 1337: Epogimic, Ian Ring Music TheoryEpogimic
Scale 1593Scale 1593: Zogimic, Ian Ring Music TheoryZogimic
Scale 57Scale 57: Ahoian, Ian Ring Music TheoryAhoian
Scale 569Scale 569: Mothitonic, Ian Ring Music TheoryMothitonic
Scale 2105Scale 2105: Rigian, Ian Ring Music TheoryRigian
Scale 3129Scale 3129: Toqian, Ian Ring Music TheoryToqian

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.