The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 963: "Gacian"

Scale 963: Gacian, 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.

enantiomorph: 2169


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: 207


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, 5, 1, 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, 2, 2, 3, 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> = {3,5,7,9}
<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.

(30, 10, 55)

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 Triadsf♯m{6,9,1}110.5
Diminished Triadsf♯°{6,9,0}110.5

The following pitch classes are not present in any of the common triads: {7,8}

Parsimonious Voice Leading Between Common Triads of Scale 963. Created by Ian Ring ©2019 f#° f#° f#m f#m f#°->f#m

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

2nd mode:
Scale 2529
Scale 2529: Pikian, Ian Ring Music TheoryPikian
3rd mode:
Scale 207
Scale 207: Beqian, Ian Ring Music TheoryBeqianThis is the prime mode
4th mode:
Scale 2151
Scale 2151: Natian, Ian Ring Music TheoryNatian
5th mode:
Scale 3123
Scale 3123: Tomian, Ian Ring Music TheoryTomian
6th mode:
Scale 3609
Scale 3609: Woqian, Ian Ring Music TheoryWoqian


The prime form of this scale is Scale 207

Scale 207Scale 207: Beqian, Ian Ring Music TheoryBeqian


The hexatonic modal family [963, 2529, 207, 2151, 3123, 3609] (Forte: 6-5) is the complement of the hexatonic modal family [207, 963, 2151, 2529, 3123, 3609] (Forte: 6-5)


The inverse of a scale is a reflection using the root as its axis. The inverse of 963 is 2169

Scale 2169Scale 2169: Nefian, Ian Ring Music TheoryNefian


Only scales that are chiral will have an enantiomorph. Scale 963 is chiral, and its enantiomorph is scale 2169

Scale 2169Scale 2169: Nefian, Ian Ring Music TheoryNefian


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> 963       T0I <11,0> 2169
T1 <1,1> 1926      T1I <11,1> 243
T2 <1,2> 3852      T2I <11,2> 486
T3 <1,3> 3609      T3I <11,3> 972
T4 <1,4> 3123      T4I <11,4> 1944
T5 <1,5> 2151      T5I <11,5> 3888
T6 <1,6> 207      T6I <11,6> 3681
T7 <1,7> 414      T7I <11,7> 3267
T8 <1,8> 828      T8I <11,8> 2439
T9 <1,9> 1656      T9I <11,9> 783
T10 <1,10> 3312      T10I <11,10> 1566
T11 <1,11> 2529      T11I <11,11> 3132
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2673      T0MI <7,0> 459
T1M <5,1> 1251      T1MI <7,1> 918
T2M <5,2> 2502      T2MI <7,2> 1836
T3M <5,3> 909      T3MI <7,3> 3672
T4M <5,4> 1818      T4MI <7,4> 3249
T5M <5,5> 3636      T5MI <7,5> 2403
T6M <5,6> 3177      T6MI <7,6> 711
T7M <5,7> 2259      T7MI <7,7> 1422
T8M <5,8> 423      T8MI <7,8> 2844
T9M <5,9> 846      T9MI <7,9> 1593
T10M <5,10> 1692      T10MI <7,10> 3186
T11M <5,11> 3384      T11MI <7,11> 2277

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 961Scale 961: Gabian, Ian Ring Music TheoryGabian
Scale 965Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic
Scale 967Scale 967: Mela Salaga, Ian Ring Music TheoryMela Salaga
Scale 971Scale 971: Mela Gavambodhi, Ian Ring Music TheoryMela Gavambodhi
Scale 979Scale 979: Mela Dhavalambari, Ian Ring Music TheoryMela Dhavalambari
Scale 995Scale 995: Phrathian, Ian Ring Music TheoryPhrathian
Scale 899Scale 899: Foqian, Ian Ring Music TheoryFoqian
Scale 931Scale 931: Raga Kalakanthi, Ian Ring Music TheoryRaga Kalakanthi
Scale 835Scale 835: Fecian, Ian Ring Music TheoryFecian
Scale 707Scale 707: Ehoian, Ian Ring Music TheoryEhoian
Scale 451Scale 451: Raga Saugandhini, Ian Ring Music TheoryRaga Saugandhini
Scale 1475Scale 1475: Uffian, Ian Ring Music TheoryUffian
Scale 1987Scale 1987: Mexian, Ian Ring Music TheoryMexian
Scale 3011Scale 3011, Ian Ring Music Theory

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.