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Scale 157: "Balian"


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
Balian

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

{0,2,3,4,7}

Forte Number

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

5-11

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: 1825

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.

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.

4

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

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, an indicator of maximum hierarchization.

no

Interval Structure

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

[2, 1, 1, 3, 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, 2, 2, 2, 2, 0>

Interval Spectrum

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

p2m2n2s2d2

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

4

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 Distribution Spectra have 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.

(14, 8, 38)

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
Minor Triadscm{0,3,7}110.5

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

Parsimonious Voice Leading Between Common Triads of Scale 157. Created by Ian Ring ©2019 cm 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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 157 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1063
Scale 1063: Gomian, Ian Ring Music TheoryGomian
3rd mode:
Scale 2579
Scale 2579: Pupian, Ian Ring Music TheoryPupian
4th mode:
Scale 3337
Scale 3337: Vafian, Ian Ring Music TheoryVafian
5th mode:
Scale 929
Scale 929: Fujian, Ian Ring Music TheoryFujian

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [157, 1063, 2579, 3337, 929] (Forte: 5-11) is the complement of the heptatonic modal family [379, 1583, 1969, 2237, 2839, 3467, 3781] (Forte: 7-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 157 is 1825

Scale 1825Scale 1825: Lecian, Ian Ring Music TheoryLecian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 157 is chiral, and its enantiomorph is scale 1825

Scale 1825Scale 1825: Lecian, Ian Ring Music TheoryLecian

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> 157       T0I <11,0> 1825
T1 <1,1> 314      T1I <11,1> 3650
T2 <1,2> 628      T2I <11,2> 3205
T3 <1,3> 1256      T3I <11,3> 2315
T4 <1,4> 2512      T4I <11,4> 535
T5 <1,5> 929      T5I <11,5> 1070
T6 <1,6> 1858      T6I <11,6> 2140
T7 <1,7> 3716      T7I <11,7> 185
T8 <1,8> 3337      T8I <11,8> 370
T9 <1,9> 2579      T9I <11,9> 740
T10 <1,10> 1063      T10I <11,10> 1480
T11 <1,11> 2126      T11I <11,11> 2960
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3337      T0MI <7,0> 535
T1M <5,1> 2579      T1MI <7,1> 1070
T2M <5,2> 1063      T2MI <7,2> 2140
T3M <5,3> 2126      T3MI <7,3> 185
T4M <5,4> 157       T4MI <7,4> 370
T5M <5,5> 314      T5MI <7,5> 740
T6M <5,6> 628      T6MI <7,6> 1480
T7M <5,7> 1256      T7MI <7,7> 2960
T8M <5,8> 2512      T8MI <7,8> 1825
T9M <5,9> 929      T9MI <7,9> 3650
T10M <5,10> 1858      T10MI <7,10> 3205
T11M <5,11> 3716      T11MI <7,11> 2315

The transformations that map this set to itself are: T0, T4M

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 159Scale 159: Bamian, Ian Ring Music TheoryBamian
Scale 153Scale 153: Bajian, Ian Ring Music TheoryBajian
Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian
Scale 149Scale 149: Eskimo Tetratonic, Ian Ring Music TheoryEskimo Tetratonic
Scale 141Scale 141: Babian, Ian Ring Music TheoryBabian
Scale 173Scale 173: Raga Purnalalita, Ian Ring Music TheoryRaga Purnalalita
Scale 189Scale 189: Befian, Ian Ring Music TheoryBefian
Scale 221Scale 221: Biyian, Ian Ring Music TheoryBiyian
Scale 29Scale 29: Aduian, Ian Ring Music TheoryAduian
Scale 93Scale 93: Anuian, Ian Ring Music TheoryAnuian
Scale 285Scale 285: Zaritonic, Ian Ring Music TheoryZaritonic
Scale 413Scale 413: Ganimic, Ian Ring Music TheoryGanimic
Scale 669Scale 669: Gycrimic, Ian Ring Music TheoryGycrimic
Scale 1181Scale 1181: Katagimic, Ian Ring Music TheoryKatagimic
Scale 2205Scale 2205: Ionocrimic, Ian Ring Music TheoryIonocrimic

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.