The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 215: "Bivian"

Scale 215: Bivian, 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
Bivian

Analysis

Cardinality

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

{0,1,2,4,6,7}

Forte Number

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

6-Z12

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: 3425

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

5

Prime Form

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

yes

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

Interval Spectrum

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

p3m2n2s3d3t2

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

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

3.333

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.

1.866

Polygon Perimeter

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

5.485

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.

(22, 14, 61)

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: {2,6}

Parsimonious Voice Leading Between Common Triads of Scale 215. 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 215 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2155
Scale 2155: Newian, Ian Ring Music TheoryNewian
3rd mode:
Scale 3125
Scale 3125: Tonian, Ian Ring Music TheoryTonian
4th mode:
Scale 1805
Scale 1805: Laqian, Ian Ring Music TheoryLaqian
5th mode:
Scale 1475
Scale 1475: Uffian, Ian Ring Music TheoryUffian
6th mode:
Scale 2785
Scale 2785: Ronian, Ian Ring Music TheoryRonian

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [215, 2155, 3125, 1805, 1475, 2785] (Forte: 6-Z12) is the complement of the hexatonic modal family [335, 965, 1265, 2215, 3155, 3625] (Forte: 6-Z41)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 215 is 3425

Scale 3425Scale 3425: Vihian, Ian Ring Music TheoryVihian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 215 is chiral, and its enantiomorph is scale 3425

Scale 3425Scale 3425: Vihian, Ian Ring Music TheoryVihian

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> 215       T0I <11,0> 3425
T1 <1,1> 430      T1I <11,1> 2755
T2 <1,2> 860      T2I <11,2> 1415
T3 <1,3> 1720      T3I <11,3> 2830
T4 <1,4> 3440      T4I <11,4> 1565
T5 <1,5> 2785      T5I <11,5> 3130
T6 <1,6> 1475      T6I <11,6> 2165
T7 <1,7> 2950      T7I <11,7> 235
T8 <1,8> 1805      T8I <11,8> 470
T9 <1,9> 3610      T9I <11,9> 940
T10 <1,10> 3125      T10I <11,10> 1880
T11 <1,11> 2155      T11I <11,11> 3760
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3425      T0MI <7,0> 215
T1M <5,1> 2755      T1MI <7,1> 430
T2M <5,2> 1415      T2MI <7,2> 860
T3M <5,3> 2830      T3MI <7,3> 1720
T4M <5,4> 1565      T4MI <7,4> 3440
T5M <5,5> 3130      T5MI <7,5> 2785
T6M <5,6> 2165      T6MI <7,6> 1475
T7M <5,7> 235      T7MI <7,7> 2950
T8M <5,8> 470      T8MI <7,8> 1805
T9M <5,9> 940      T9MI <7,9> 3610
T10M <5,10> 1880      T10MI <7,10> 3125
T11M <5,11> 3760      T11MI <7,11> 2155

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 213Scale 213: Bitian, Ian Ring Music TheoryBitian
Scale 211Scale 211: Bisian, Ian Ring Music TheoryBisian
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 223Scale 223: Bizian, Ian Ring Music TheoryBizian
Scale 199Scale 199: Raga Nabhomani, Ian Ring Music TheoryRaga Nabhomani
Scale 207Scale 207: Beqian, Ian Ring Music TheoryBeqian
Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian
Scale 247Scale 247: Bopian, Ian Ring Music TheoryBopian
Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian
Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian
Scale 87Scale 87: Asrian, Ian Ring Music TheoryAsrian
Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic
Scale 471Scale 471: Dodian, Ian Ring Music TheoryDodian
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 1239Scale 1239: Epaptian, Ian Ring Music TheoryEpaptian
Scale 2263Scale 2263: Lycrian, Ian Ring Music TheoryLycrian

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.