The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 211: "Bisian"

Scale 211: Bisian, 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

Dozenal
Bisian

Analysis

Cardinality

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

{0,1,4,6,7}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

5-19

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

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.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 2401

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

0 (ancohemitonic)

Imperfections

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.

3

Modes

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.

4

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 203

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

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

[1, 3, 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, 2, 1, 2, 2>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p2mn2sd2t2

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

2.8

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

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.

no

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.

1.683

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.381

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

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.

no

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.

none

Propriety

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".

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.

(4, 8, 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}110.5
Diminished Triadsc♯°{1,4,7}110.5

The following pitch classes are not present in any of the common triads: {6}

Parsimonious Voice Leading Between Common Triads of Scale 211. Created by Ian Ring ©2019 C C c#° c#° C->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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 211 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2153
Scale 2153: Navian, Ian Ring Music TheoryNavian
3rd mode:
Scale 781
Scale 781: Etoian, Ian Ring Music TheoryEtoian
4th mode:
Scale 1219
Scale 1219: Hidian, Ian Ring Music TheoryHidian
5th mode:
Scale 2657
Scale 2657: Qokian, Ian Ring Music TheoryQokian

Prime

The prime form of this scale is Scale 203

Scale 203Scale 203: Ichian, Ian Ring Music TheoryIchian

Complement

The pentatonic modal family [211, 2153, 781, 1219, 2657] (Forte: 5-19) is the complement of the heptatonic modal family [719, 971, 1657, 2407, 2533, 3251, 3673] (Forte: 7-19)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 211 is 2401

Scale 2401Scale 2401: Oroian, Ian Ring Music TheoryOroian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 211 is chiral, and its enantiomorph is scale 2401

Scale 2401Scale 2401: Oroian, Ian Ring Music TheoryOroian

Transformations:

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> 211       T0I <11,0> 2401
T1 <1,1> 422      T1I <11,1> 707
T2 <1,2> 844      T2I <11,2> 1414
T3 <1,3> 1688      T3I <11,3> 2828
T4 <1,4> 3376      T4I <11,4> 1561
T5 <1,5> 2657      T5I <11,5> 3122
T6 <1,6> 1219      T6I <11,6> 2149
T7 <1,7> 2438      T7I <11,7> 203
T8 <1,8> 781      T8I <11,8> 406
T9 <1,9> 1562      T9I <11,9> 812
T10 <1,10> 3124      T10I <11,10> 1624
T11 <1,11> 2153      T11I <11,11> 3248
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2401      T0MI <7,0> 211
T1M <5,1> 707      T1MI <7,1> 422
T2M <5,2> 1414      T2MI <7,2> 844
T3M <5,3> 2828      T3MI <7,3> 1688
T4M <5,4> 1561      T4MI <7,4> 3376
T5M <5,5> 3122      T5MI <7,5> 2657
T6M <5,6> 2149      T6MI <7,6> 1219
T7M <5,7> 203      T7MI <7,7> 2438
T8M <5,8> 406      T8MI <7,8> 781
T9M <5,9> 812      T9MI <7,9> 1562
T10M <5,10> 1624      T10MI <7,10> 3124
T11M <5,11> 3248      T11MI <7,11> 2153

The transformations that map this set to itself are: T0, T0MI

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 209Scale 209: Birian, Ian Ring Music TheoryBirian
Scale 213Scale 213: Bitian, Ian Ring Music TheoryBitian
Scale 215Scale 215: Bivian, Ian Ring Music TheoryBivian
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 195Scale 195: Messiaen Truncated Mode 5, Ian Ring Music TheoryMessiaen Truncated Mode 5
Scale 203Scale 203: Ichian, Ian Ring Music TheoryIchian
Scale 227Scale 227: Bician, Ian Ring Music TheoryBician
Scale 243Scale 243: Bomian, Ian Ring Music TheoryBomian
Scale 147Scale 147: Bafian, Ian Ring Music TheoryBafian
Scale 179Scale 179: Beyian, Ian Ring Music TheoryBeyian
Scale 83Scale 83: Amuian, Ian Ring Music TheoryAmuian
Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic
Scale 467Scale 467: Raga Dhavalangam, Ian Ring Music TheoryRaga Dhavalangam
Scale 723Scale 723: Ionadimic, Ian Ring Music TheoryIonadimic
Scale 1235Scale 1235: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2
Scale 2259Scale 2259: Raga Mandari, Ian Ring Music TheoryRaga Mandari

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 (http://allthescales.org) 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.