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Scale 87: "Asrian"

Scale 87: Asrian, 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
Asrian

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

{0,1,2,4,6}

Forte Number

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

5-9

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: 3393

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.

4

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

Interval Spectrum

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

pm2ns3d2t

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

Spectra Variation

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

4.4

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.

1.366

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.

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, 7, 36)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 87 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2091
Scale 2091: Mukian, Ian Ring Music TheoryMukian
3rd mode:
Scale 3093
Scale 3093: Buqian, Ian Ring Music TheoryBuqian
4th mode:
Scale 1797
Scale 1797: Lalian, Ian Ring Music TheoryLalian
5th mode:
Scale 1473
Scale 1473: Javian, Ian Ring Music TheoryJavian

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [87, 2091, 3093, 1797, 1473] (Forte: 5-9) is the complement of the heptatonic modal family [351, 1521, 1989, 2223, 3159, 3627, 3861] (Forte: 7-9)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 87 is 3393

Scale 3393Scale 3393: Venian, Ian Ring Music TheoryVenian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 87 is chiral, and its enantiomorph is scale 3393

Scale 3393Scale 3393: Venian, Ian Ring Music TheoryVenian

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> 87       T0I <11,0> 3393
T1 <1,1> 174      T1I <11,1> 2691
T2 <1,2> 348      T2I <11,2> 1287
T3 <1,3> 696      T3I <11,3> 2574
T4 <1,4> 1392      T4I <11,4> 1053
T5 <1,5> 2784      T5I <11,5> 2106
T6 <1,6> 1473      T6I <11,6> 117
T7 <1,7> 2946      T7I <11,7> 234
T8 <1,8> 1797      T8I <11,8> 468
T9 <1,9> 3594      T9I <11,9> 936
T10 <1,10> 3093      T10I <11,10> 1872
T11 <1,11> 2091      T11I <11,11> 3744
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1377      T0MI <7,0> 213
T1M <5,1> 2754      T1MI <7,1> 426
T2M <5,2> 1413      T2MI <7,2> 852
T3M <5,3> 2826      T3MI <7,3> 1704
T4M <5,4> 1557      T4MI <7,4> 3408
T5M <5,5> 3114      T5MI <7,5> 2721
T6M <5,6> 2133      T6MI <7,6> 1347
T7M <5,7> 171      T7MI <7,7> 2694
T8M <5,8> 342      T8MI <7,8> 1293
T9M <5,9> 684      T9MI <7,9> 2586
T10M <5,10> 1368      T10MI <7,10> 1077
T11M <5,11> 2736      T11MI <7,11> 2154

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 85Scale 85: Segian, Ian Ring Music TheorySegian
Scale 83Scale 83: Amuian, Ian Ring Music TheoryAmuian
Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian
Scale 95Scale 95: Arkian, Ian Ring Music TheoryArkian
Scale 71Scale 71: Aloian, Ian Ring Music TheoryAloian
Scale 79Scale 79: Appian, Ian Ring Music TheoryAppian
Scale 103Scale 103: Apuian, Ian Ring Music TheoryApuian
Scale 119Scale 119: Smoian, Ian Ring Music TheorySmoian
Scale 23Scale 23: Aphian, Ian Ring Music TheoryAphian
Scale 55Scale 55: Aspian, Ian Ring Music TheoryAspian
Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian
Scale 215Scale 215: Bivian, Ian Ring Music TheoryBivian
Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic
Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic
Scale 1111Scale 1111: Sycrimic, Ian Ring Music TheorySycrimic
Scale 2135Scale 2135: Nakian, Ian Ring Music TheoryNakian

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.