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Scale 107: "Ansian"

Scale 107: Ansian, 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
Ansian

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

{0,1,3,5,6}

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-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.

[3]

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.

no

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.

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, 2, 2, 1, 6]

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, 1, 2, 1>

Interval Spectrum

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

p2mn2s2d2t

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

Spectra Variation

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

3.6

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.

5.035

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.

[6]

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, 1, 32)

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
Diminished Triads{0,3,6}000

The following pitch classes are not present in any of the common triads: {1,5}

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 107 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2101
Scale 2101: Muqian, Ian Ring Music TheoryMuqian
3rd mode:
Scale 1549
Scale 1549: Joqian, Ian Ring Music TheoryJoqian
4th mode:
Scale 1411
Scale 1411: Iroian, Ian Ring Music TheoryIroian
5th mode:
Scale 2753
Scale 2753: Ritian, Ian Ring Music TheoryRitian

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [107, 2101, 1549, 1411, 2753] (Forte: 5-Z12) is the complement of the heptatonic modal family [671, 997, 1273, 2383, 3239, 3667, 3881] (Forte: 7-Z12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 107 is 2753

Scale 2753Scale 2753: Ritian, Ian Ring Music TheoryRitian

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> 107       T0I <11,0> 2753
T1 <1,1> 214      T1I <11,1> 1411
T2 <1,2> 428      T2I <11,2> 2822
T3 <1,3> 856      T3I <11,3> 1549
T4 <1,4> 1712      T4I <11,4> 3098
T5 <1,5> 3424      T5I <11,5> 2101
T6 <1,6> 2753      T6I <11,6> 107
T7 <1,7> 1411      T7I <11,7> 214
T8 <1,8> 2822      T8I <11,8> 428
T9 <1,9> 1549      T9I <11,9> 856
T10 <1,10> 3098      T10I <11,10> 1712
T11 <1,11> 2101      T11I <11,11> 3424
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 107       T0MI <7,0> 2753
T1M <5,1> 214      T1MI <7,1> 1411
T2M <5,2> 428      T2MI <7,2> 2822
T3M <5,3> 856      T3MI <7,3> 1549
T4M <5,4> 1712      T4MI <7,4> 3098
T5M <5,5> 3424      T5MI <7,5> 2101
T6M <5,6> 2753      T6MI <7,6> 107
T7M <5,7> 1411      T7MI <7,7> 214
T8M <5,8> 2822      T8MI <7,8> 428
T9M <5,9> 1549      T9MI <7,9> 856
T10M <5,10> 3098      T10MI <7,10> 1712
T11M <5,11> 2101      T11MI <7,11> 3424

The transformations that map this set to itself are: T0, T6I, T0M, T6MI

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 105Scale 105, Ian Ring Music Theory
Scale 109Scale 109: Amsian, Ian Ring Music TheoryAmsian
Scale 111Scale 111: Aroian, Ian Ring Music TheoryAroian
Scale 99Scale 99: Iprian, Ian Ring Music TheoryIprian
Scale 103Scale 103: Apuian, Ian Ring Music TheoryApuian
Scale 115Scale 115: Ashian, Ian Ring Music TheoryAshian
Scale 123Scale 123: Asuian, Ian Ring Music TheoryAsuian
Scale 75Scale 75: Iloian, Ian Ring Music TheoryIloian
Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian
Scale 43Scale 43: Alfian, Ian Ring Music TheoryAlfian
Scale 171Scale 171: Pruian, Ian Ring Music TheoryPruian
Scale 235Scale 235: Bihian, Ian Ring Music TheoryBihian
Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 2155Scale 2155: Newian, Ian Ring Music TheoryNewian

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.