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Scale 83: "Amuian"


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
Amuian

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

{0,1,4,6}

Forte Number

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

4-Z15

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: 2369

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.

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, 3, 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.

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

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.

1.183

Polygon Perimeter

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

4.932

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, 1, 18)

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 83 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 2089
Scale 2089: Mujian, Ian Ring Music TheoryMujian
3rd mode:
Scale 773
Scale 773: Esuian, Ian Ring Music TheoryEsuian
4th mode:
Scale 1217
Scale 1217: Hician, Ian Ring Music TheoryHician

Prime

This is the prime form of this scale.

Complement

The tetratonic modal family [83, 2089, 773, 1217] (Forte: 4-Z15) is the complement of the octatonic modal family [863, 1523, 1997, 2479, 2809, 3287, 3691, 3893] (Forte: 8-Z15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 83 is 2369

Scale 2369Scale 2369: Offian, Ian Ring Music TheoryOffian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 83 is chiral, and its enantiomorph is scale 2369

Scale 2369Scale 2369: Offian, Ian Ring Music TheoryOffian

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

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 81Scale 81: Disian, Ian Ring Music TheoryDisian
Scale 85Scale 85: Segian, Ian Ring Music TheorySegian
Scale 87Scale 87: Asrian, Ian Ring Music TheoryAsrian
Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian
Scale 67Scale 67: Abrian, Ian Ring Music TheoryAbrian
Scale 75Scale 75: Iloian, Ian Ring Music TheoryIloian
Scale 99Scale 99: Iprian, Ian Ring Music TheoryIprian
Scale 115Scale 115: Ashian, Ian Ring Music TheoryAshian
Scale 19Scale 19: Acuian, Ian Ring Music TheoryAcuian
Scale 51Scale 51: Arfian, Ian Ring Music TheoryArfian
Scale 147Scale 147: Bafian, Ian Ring Music TheoryBafian
Scale 211Scale 211: Bisian, Ian Ring Music TheoryBisian
Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic
Scale 595Scale 595: Sogitonic, Ian Ring Music TheorySogitonic
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
Scale 2131Scale 2131: Nahian, Ian Ring Music TheoryNahian

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.