The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 221: "BIYian"

Scale 221: BIYian, 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.

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: 1889


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 Starr/Rahn algorithm.

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, an indicator of maximum hierarchization.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

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

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>

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.

<0.6, 0.5, 0.6, 0.25, 0.4, 0.333>

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 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.

(29, 7, 55)

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 TriadsC{0,4,7}121
Minor Triadscm{0,3,7}210.67
Diminished Triads{0,3,6}121

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

Parsimonious Voice Leading Between Common Triads of Scale 221. Created by Ian Ring ©2019 cm cm c°->cm C C cm->C

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 Verticescm
Peripheral Verticesc°, C


Modes are the rotational transformation of this scale. Scale 221 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1079
Scale 1079: GOWian, Ian Ring Music TheoryGOWian
3rd mode:
Scale 2587
Scale 2587: PUTian, Ian Ring Music TheoryPUTian
4th mode:
Scale 3341
Scale 3341: VAHian, Ian Ring Music TheoryVAHian
5th mode:
Scale 1859
Scale 1859: LIXian, Ian Ring Music TheoryLIXian
6th mode:
Scale 2977
Scale 2977: SOBian, Ian Ring Music TheorySOBian


The prime form of this scale is Scale 187

Scale 187Scale 187: BEDian, Ian Ring Music TheoryBEDian


The hexatonic modal family [221, 1079, 2587, 3341, 1859, 2977] (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 221 is 1889

Scale 1889Scale 1889: LOQian, Ian Ring Music TheoryLOQian


Only scales that are chiral will have an enantiomorph. Scale 221 is chiral, and its enantiomorph is scale 1889

Scale 1889Scale 1889: LOQian, Ian Ring Music TheoryLOQian


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

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 223Scale 223: BIZian, Ian Ring Music TheoryBIZian
Scale 217Scale 217: BIWian, Ian Ring Music TheoryBIWian
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 213Scale 213: BITian, Ian Ring Music TheoryBITian
Scale 205Scale 205: BEPian, Ian Ring Music TheoryBEPian
Scale 237Scale 237: BIJian, Ian Ring Music TheoryBIJian
Scale 253Scale 253: BOSian, Ian Ring Music TheoryBOSian
Scale 157Scale 157: BALian, Ian Ring Music TheoryBALian
Scale 189Scale 189: BEFian, Ian Ring Music TheoryBEFian
Scale 93Scale 93: ANUian, Ian Ring Music TheoryANUian
Scale 349Scale 349: Borimic, Ian Ring Music TheoryBorimic
Scale 477Scale 477: Stacrian, Ian Ring Music TheoryStacrian
Scale 733Scale 733: Donian, Ian Ring Music TheoryDonian
Scale 1245Scale 1245: Lathian, Ian Ring Music TheoryLathian
Scale 2269Scale 2269: Pygian, Ian Ring Music TheoryPygian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.