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Scale 1293: "HUXian"

Scale 1293: HUXian, 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: 1557


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 Starr/Rahn algorithm.

prime: 171


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.

[2, 1, 5, 2, 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.

<1, 3, 1, 2, 2, 1>

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.

<0.25, 0.75, 0.25, 0.333, 0.5, 0.5>

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

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.

(9, 3, 32)

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
Major TriadsG♯{8,0,3}000

The following pitch classes are not present in any of the common triads: {2,10}

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 1293 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1347
Scale 1347: IGOian, Ian Ring Music TheoryIGOian
3rd mode:
Scale 2721
Scale 2721: Raga Puruhutika, Ian Ring Music TheoryRaga Puruhutika
4th mode:
Scale 213
Scale 213: BITian, Ian Ring Music TheoryBITian
5th mode:
Scale 1077
Scale 1077: GOVian, Ian Ring Music TheoryGOVian


The prime form of this scale is Scale 171

Scale 171Scale 171: PRUian, Ian Ring Music TheoryPRUian


The pentatonic modal family [1293, 1347, 2721, 213, 1077] (Forte: 5-24) is the complement of the heptatonic modal family [687, 1401, 1509, 1941, 2391, 3243, 3669] (Forte: 7-24)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1293 is 1557

Scale 1557Scale 1557: JOVian, Ian Ring Music TheoryJOVian


Only scales that are chiral will have an enantiomorph. Scale 1293 is chiral, and its enantiomorph is scale 1557

Scale 1557Scale 1557: JOVian, Ian Ring Music TheoryJOVian


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> 1293       T0I <11,0> 1557
T1 <1,1> 2586      T1I <11,1> 3114
T2 <1,2> 1077      T2I <11,2> 2133
T3 <1,3> 2154      T3I <11,3> 171
T4 <1,4> 213      T4I <11,4> 342
T5 <1,5> 426      T5I <11,5> 684
T6 <1,6> 852      T6I <11,6> 1368
T7 <1,7> 1704      T7I <11,7> 2736
T8 <1,8> 3408      T8I <11,8> 1377
T9 <1,9> 2721      T9I <11,9> 2754
T10 <1,10> 1347      T10I <11,10> 1413
T11 <1,11> 2694      T11I <11,11> 2826
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1053      T0MI <7,0> 1797
T1M <5,1> 2106      T1MI <7,1> 3594
T2M <5,2> 117      T2MI <7,2> 3093
T3M <5,3> 234      T3MI <7,3> 2091
T4M <5,4> 468      T4MI <7,4> 87
T5M <5,5> 936      T5MI <7,5> 174
T6M <5,6> 1872      T6MI <7,6> 348
T7M <5,7> 3744      T7MI <7,7> 696
T8M <5,8> 3393      T8MI <7,8> 1392
T9M <5,9> 2691      T9MI <7,9> 2784
T10M <5,10> 1287      T10MI <7,10> 1473
T11M <5,11> 2574      T11MI <7,11> 2946

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 1295Scale 1295: HUYian, Ian Ring Music TheoryHUYian
Scale 1289Scale 1289: HUVian, Ian Ring Music TheoryHUVian
Scale 1291Scale 1291: HUWian, Ian Ring Music TheoryHUWian
Scale 1285Scale 1285: HUSian, Ian Ring Music TheoryHUSian
Scale 1301Scale 1301: Koditonic, Ian Ring Music TheoryKoditonic
Scale 1309Scale 1309: Pogimic, Ian Ring Music TheoryPogimic
Scale 1325Scale 1325: Phradimic, Ian Ring Music TheoryPhradimic
Scale 1357Scale 1357: Takemitsu Linea Mode 2, Ian Ring Music TheoryTakemitsu Linea Mode 2
Scale 1421Scale 1421: Raga Trimurti, Ian Ring Music TheoryRaga Trimurti
Scale 1037Scale 1037: Warao Tetratonic, Ian Ring Music TheoryWarao Tetratonic
Scale 1165Scale 1165: Gycritonic, Ian Ring Music TheoryGycritonic
Scale 1549Scale 1549: JOQian, Ian Ring Music TheoryJOQian
Scale 1805Scale 1805: LAQian, Ian Ring Music TheoryLAQian
Scale 269Scale 269: BOCian, Ian Ring Music TheoryBOCian
Scale 781Scale 781: ETOian, Ian Ring Music TheoryETOian
Scale 2317Scale 2317: ODOian, Ian Ring Music TheoryODOian
Scale 3341Scale 3341: VAHian, Ian Ring Music TheoryVAHian

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.