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Scale 3851: "Yilian"

Scale 3851: Yilian, 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.

7 (heptatonic)

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: 2591


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

5 (multihemitonic)


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

4 (multicohemitonic)


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: 191


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, 2, 5, 1, 1, 1, 1]

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.

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

(62, 25, 86)

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}210.67
Minor Triadsg♯m{8,11,3}121
Diminished Triads{9,0,3}121

The following pitch classes are not present in any of the common triads: {1,10}

Parsimonious Voice Leading Between Common Triads of Scale 3851. Created by Ian Ring ©2019 g#m g#m G# G# g#m->G# G#->a°

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 VerticesG♯
Peripheral Verticesg♯m, a°


Modes are the rotational transformation of this scale. Scale 3851 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3973
Scale 3973: Zehian, Ian Ring Music TheoryZehian
3rd mode:
Scale 2017
Scale 2017: Meqian, Ian Ring Music TheoryMeqian
4th mode:
Scale 191
Scale 191: Begian, Ian Ring Music TheoryBegianThis is the prime mode
5th mode:
Scale 2143
Scale 2143: Napian, Ian Ring Music TheoryNapian
6th mode:
Scale 3119
Scale 3119: Tikian, Ian Ring Music TheoryTikian
7th mode:
Scale 3607
Scale 3607: Wopian, Ian Ring Music TheoryWopian


The prime form of this scale is Scale 191

Scale 191Scale 191: Begian, Ian Ring Music TheoryBegian


The heptatonic modal family [3851, 3973, 2017, 191, 2143, 3119, 3607] (Forte: 7-2) is the complement of the pentatonic modal family [47, 1921, 2071, 3083, 3589] (Forte: 5-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3851 is 2591

Scale 2591Scale 2591: Puwian, Ian Ring Music TheoryPuwian


Only scales that are chiral will have an enantiomorph. Scale 3851 is chiral, and its enantiomorph is scale 2591

Scale 2591Scale 2591: Puwian, Ian Ring Music TheoryPuwian


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> 3851       T0I <11,0> 2591
T1 <1,1> 3607      T1I <11,1> 1087
T2 <1,2> 3119      T2I <11,2> 2174
T3 <1,3> 2143      T3I <11,3> 253
T4 <1,4> 191      T4I <11,4> 506
T5 <1,5> 382      T5I <11,5> 1012
T6 <1,6> 764      T6I <11,6> 2024
T7 <1,7> 1528      T7I <11,7> 4048
T8 <1,8> 3056      T8I <11,8> 4001
T9 <1,9> 2017      T9I <11,9> 3907
T10 <1,10> 4034      T10I <11,10> 3719
T11 <1,11> 3973      T11I <11,11> 3343
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 701      T0MI <7,0> 1961
T1M <5,1> 1402      T1MI <7,1> 3922
T2M <5,2> 2804      T2MI <7,2> 3749
T3M <5,3> 1513      T3MI <7,3> 3403
T4M <5,4> 3026      T4MI <7,4> 2711
T5M <5,5> 1957      T5MI <7,5> 1327
T6M <5,6> 3914      T6MI <7,6> 2654
T7M <5,7> 3733      T7MI <7,7> 1213
T8M <5,8> 3371      T8MI <7,8> 2426
T9M <5,9> 2647      T9MI <7,9> 757
T10M <5,10> 1199      T10MI <7,10> 1514
T11M <5,11> 2398      T11MI <7,11> 3028

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 3849Scale 3849: Yikian, Ian Ring Music TheoryYikian
Scale 3853Scale 3853: Yomian, Ian Ring Music TheoryYomian
Scale 3855Scale 3855: Octatonic Chromatic 5, Ian Ring Music TheoryOctatonic Chromatic 5
Scale 3843Scale 3843: Hexatonic Chromatic 5, Ian Ring Music TheoryHexatonic Chromatic 5
Scale 3847Scale 3847: Heptatonic Chromatic 5, Ian Ring Music TheoryHeptatonic Chromatic 5
Scale 3859Scale 3859: Aeolarian, Ian Ring Music TheoryAeolarian
Scale 3867Scale 3867: Storyllic, Ian Ring Music TheoryStoryllic
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 3915Scale 3915: Yuyian, Ian Ring Music TheoryYuyian
Scale 3979Scale 3979: Dynyllic, Ian Ring Music TheoryDynyllic
Scale 3595Scale 3595: Wihian, Ian Ring Music TheoryWihian
Scale 3723Scale 3723: Myptian, Ian Ring Music TheoryMyptian
Scale 3339Scale 3339: Smuian, Ian Ring Music TheorySmuian
Scale 2827Scale 2827: Runian, Ian Ring Music TheoryRunian
Scale 1803Scale 1803: Lapian, Ian Ring Music TheoryLapian

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.