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Scale 209: "Birian"

Scale 209: Birian, 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
Birian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,4,6,7}

Forte Number

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

4-Z29

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: 353

Hemitonia

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

1 (unhemitonic)

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.

3

Prime Form

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

no
prime: 139

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.

[4, 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.

<1, 1, 1, 1, 1, 1>

Interval Spectrum

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

pmnsdt

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

Spectra Variation

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

3.5

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.

1.366

Polygon Perimeter

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

5.182

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, 0, 17)

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}000

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

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 209 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 269
Scale 269: Bocian, Ian Ring Music TheoryBocian
3rd mode:
Scale 1091
Scale 1091: Pedian, Ian Ring Music TheoryPedian
4th mode:
Scale 2593
Scale 2593: Puxian, Ian Ring Music TheoryPuxian

Prime

The prime form of this scale is Scale 139

Scale 139Scale 139: Ayoian, Ian Ring Music TheoryAyoian

Complement

The tetratonic modal family [209, 269, 1091, 2593] (Forte: 4-Z29) is the complement of the octatonic modal family [751, 1913, 1943, 2423, 3019, 3259, 3557, 3677] (Forte: 8-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 209 is 353

Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 209 is chiral, and its enantiomorph is scale 353

Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian

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> 209       T0I <11,0> 353
T1 <1,1> 418      T1I <11,1> 706
T2 <1,2> 836      T2I <11,2> 1412
T3 <1,3> 1672      T3I <11,3> 2824
T4 <1,4> 3344      T4I <11,4> 1553
T5 <1,5> 2593      T5I <11,5> 3106
T6 <1,6> 1091      T6I <11,6> 2117
T7 <1,7> 2182      T7I <11,7> 139
T8 <1,8> 269      T8I <11,8> 278
T9 <1,9> 538      T9I <11,9> 556
T10 <1,10> 1076      T10I <11,10> 1112
T11 <1,11> 2152      T11I <11,11> 2224
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2369      T0MI <7,0> 83
T1M <5,1> 643      T1MI <7,1> 166
T2M <5,2> 1286      T2MI <7,2> 332
T3M <5,3> 2572      T3MI <7,3> 664
T4M <5,4> 1049      T4MI <7,4> 1328
T5M <5,5> 2098      T5MI <7,5> 2656
T6M <5,6> 101      T6MI <7,6> 1217
T7M <5,7> 202      T7MI <7,7> 2434
T8M <5,8> 404      T8MI <7,8> 773
T9M <5,9> 808      T9MI <7,9> 1546
T10M <5,10> 1616      T10MI <7,10> 3092
T11M <5,11> 3232      T11MI <7,11> 2089

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 211Scale 211: Bisian, Ian Ring Music TheoryBisian
Scale 213Scale 213: Bitian, Ian Ring Music TheoryBitian
Scale 217Scale 217: Biwian, Ian Ring Music TheoryBiwian
Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari
Scale 201Scale 201: Bemian, Ian Ring Music TheoryBemian
Scale 225Scale 225: Bibian, Ian Ring Music TheoryBibian
Scale 241Scale 241: Bilian, Ian Ring Music TheoryBilian
Scale 145Scale 145: Raga Malasri, Ian Ring Music TheoryRaga Malasri
Scale 177Scale 177: Bexian, Ian Ring Music TheoryBexian
Scale 81Scale 81: Disian, Ian Ring Music TheoryDisian
Scale 337Scale 337: Koptic, Ian Ring Music TheoryKoptic
Scale 465Scale 465: Zoditonic, Ian Ring Music TheoryZoditonic
Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri
Scale 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic
Scale 2257Scale 2257: Lydian Pentatonic, Ian Ring Music TheoryLydian Pentatonic

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.