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Scale 1859: "Lixian"

Scale 1859: Lixian, 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: 2141


A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)


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: 187


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, 2, 1, 1, 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.

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

(29, 7, 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
Major TriadsF♯{6,10,1}121
Minor Triadsf♯m{6,9,1}210.67
Diminished Triadsf♯°{6,9,0}121

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

Parsimonious Voice Leading Between Common Triads of Scale 1859. Created by Ian Ring ©2019 f#° f#° f#m f#m f#°->f#m F# F# f#m->F#

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.

Central Verticesf♯m
Peripheral Verticesf♯°, F♯


Modes are the rotational transformation of this scale. Scale 1859 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2977
Scale 2977: Sobian, Ian Ring Music TheorySobian
3rd mode:
Scale 221
Scale 221: Biyian, Ian Ring Music TheoryBiyian
4th mode:
Scale 1079
Scale 1079: Gowian, Ian Ring Music TheoryGowian
5th mode:
Scale 2587
Scale 2587: Putian, Ian Ring Music TheoryPutian
6th mode:
Scale 3341
Scale 3341: Vahian, Ian Ring Music TheoryVahian


The prime form of this scale is Scale 187

Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian


The hexatonic modal family [1859, 2977, 221, 1079, 2587, 3341] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1859 is 2141

Scale 2141Scale 2141: Nanian, Ian Ring Music TheoryNanian


Only scales that are chiral will have an enantiomorph. Scale 1859 is chiral, and its enantiomorph is scale 2141

Scale 2141Scale 2141: Nanian, Ian Ring Music TheoryNanian


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> 1859       T0I <11,0> 2141
T1 <1,1> 3718      T1I <11,1> 187
T2 <1,2> 3341      T2I <11,2> 374
T3 <1,3> 2587      T3I <11,3> 748
T4 <1,4> 1079      T4I <11,4> 1496
T5 <1,5> 2158      T5I <11,5> 2992
T6 <1,6> 221      T6I <11,6> 1889
T7 <1,7> 442      T7I <11,7> 3778
T8 <1,8> 884      T8I <11,8> 3461
T9 <1,9> 1768      T9I <11,9> 2827
T10 <1,10> 3536      T10I <11,10> 1559
T11 <1,11> 2977      T11I <11,11> 3118
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 629      T0MI <7,0> 1481
T1M <5,1> 1258      T1MI <7,1> 2962
T2M <5,2> 2516      T2MI <7,2> 1829
T3M <5,3> 937      T3MI <7,3> 3658
T4M <5,4> 1874      T4MI <7,4> 3221
T5M <5,5> 3748      T5MI <7,5> 2347
T6M <5,6> 3401      T6MI <7,6> 599
T7M <5,7> 2707      T7MI <7,7> 1198
T8M <5,8> 1319      T8MI <7,8> 2396
T9M <5,9> 2638      T9MI <7,9> 697
T10M <5,10> 1181      T10MI <7,10> 1394
T11M <5,11> 2362      T11MI <7,11> 2788

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 1857Scale 1857: Liwian, Ian Ring Music TheoryLiwian
Scale 1861Scale 1861: Phrygimic, Ian Ring Music TheoryPhrygimic
Scale 1863Scale 1863: Pycrian, Ian Ring Music TheoryPycrian
Scale 1867Scale 1867: Solian, Ian Ring Music TheorySolian
Scale 1875Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale
Scale 1891Scale 1891: Thalian, Ian Ring Music TheoryThalian
Scale 1795Scale 1795: Lakian, Ian Ring Music TheoryLakian
Scale 1827Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic
Scale 1923Scale 1923: Lulian, Ian Ring Music TheoryLulian
Scale 1987Scale 1987: Mexian, Ian Ring Music TheoryMexian
Scale 1603Scale 1603: Juxian, Ian Ring Music TheoryJuxian
Scale 1731Scale 1731: Koxian, Ian Ring Music TheoryKoxian
Scale 1347Scale 1347: Igoian, Ian Ring Music TheoryIgoian
Scale 835Scale 835: Fecian, Ian Ring Music TheoryFecian
Scale 2883Scale 2883: Savian, Ian Ring Music TheorySavian
Scale 3907Scale 3907, 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.