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Scale 217: "Biwian"

Scale 217: Biwian, 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.

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: 865


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

2 (dihemitonic)


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 Rahn/Ring formula.

prime: 155


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.

[3, 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.

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

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.

(11, 7, 36)

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
Parsimonious Voice Leading Between Common Triads of Scale 217. 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 217 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 539
Scale 539: Delian, Ian Ring Music TheoryDelian
3rd mode:
Scale 2317
Scale 2317: Odoian, Ian Ring Music TheoryOdoian
4th mode:
Scale 1603
Scale 1603: Juxian, Ian Ring Music TheoryJuxian
5th mode:
Scale 2849
Scale 2849: Rubian, Ian Ring Music TheoryRubian


The prime form of this scale is Scale 155

Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian


The pentatonic modal family [217, 539, 2317, 1603, 2849] (Forte: 5-16) is the complement of the heptatonic modal family [623, 889, 1939, 2359, 3017, 3227, 3661] (Forte: 7-16)


The inverse of a scale is a reflection using the root as its axis. The inverse of 217 is 865

Scale 865Scale 865: Jahian, Ian Ring Music TheoryJahian


Only scales that are chiral will have an enantiomorph. Scale 217 is chiral, and its enantiomorph is scale 865

Scale 865Scale 865: Jahian, Ian Ring Music TheoryJahian


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> 217       T0I <11,0> 865
T1 <1,1> 434      T1I <11,1> 1730
T2 <1,2> 868      T2I <11,2> 3460
T3 <1,3> 1736      T3I <11,3> 2825
T4 <1,4> 3472      T4I <11,4> 1555
T5 <1,5> 2849      T5I <11,5> 3110
T6 <1,6> 1603      T6I <11,6> 2125
T7 <1,7> 3206      T7I <11,7> 155
T8 <1,8> 2317      T8I <11,8> 310
T9 <1,9> 539      T9I <11,9> 620
T10 <1,10> 1078      T10I <11,10> 1240
T11 <1,11> 2156      T11I <11,11> 2480
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2377      T0MI <7,0> 595
T1M <5,1> 659      T1MI <7,1> 1190
T2M <5,2> 1318      T2MI <7,2> 2380
T3M <5,3> 2636      T3MI <7,3> 665
T4M <5,4> 1177      T4MI <7,4> 1330
T5M <5,5> 2354      T5MI <7,5> 2660
T6M <5,6> 613      T6MI <7,6> 1225
T7M <5,7> 1226      T7MI <7,7> 2450
T8M <5,8> 2452      T8MI <7,8> 805
T9M <5,9> 809      T9MI <7,9> 1610
T10M <5,10> 1618      T10MI <7,10> 3220
T11M <5,11> 3236      T11MI <7,11> 2345

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 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 221Scale 221: Biyian, Ian Ring Music TheoryBiyian
Scale 209Scale 209: Birian, Ian Ring Music TheoryBirian
Scale 213Scale 213: Bitian, Ian Ring Music TheoryBitian
Scale 201Scale 201: Bemian, Ian Ring Music TheoryBemian
Scale 233Scale 233: Bigian, Ian Ring Music TheoryBigian
Scale 249Scale 249: Boqian, Ian Ring Music TheoryBoqian
Scale 153Scale 153: Bajian, Ian Ring Music TheoryBajian
Scale 185Scale 185: Becian, Ian Ring Music TheoryBecian
Scale 89Scale 89: Aggian, Ian Ring Music TheoryAggian
Scale 345Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic
Scale 473Scale 473: Aeralimic, Ian Ring Music TheoryAeralimic
Scale 729Scale 729: Stygimic, Ian Ring Music TheoryStygimic
Scale 1241Scale 1241: Pygimic, Ian Ring Music TheoryPygimic
Scale 2265Scale 2265: Raga Rasamanjari, Ian Ring Music TheoryRaga Rasamanjari

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.