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Scale 1033: "Allian"

Scale 1033: Allian, 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
Allian

Analysis

Cardinality

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

3 (tritonic)

Pitch Class Set

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

{0,3,10}

Forte Number

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

3-7

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: 517

Hemitonia

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

0 (anhemitonic)

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.

2

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.

2

Prime Form

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

no
prime: 37

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.

[3, 7, 2]

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.

<0, 1, 1, 0, 1, 0>

Interval Spectrum

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

pns

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

Spectra Variation

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

3.333

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.

0.683

Polygon Perimeter

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

4.346

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.

(1, 0, 6)

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 1033 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 641
Scale 641: Duwian, Ian Ring Music TheoryDuwian
3rd mode:
Scale 37
Scale 37: Afoian, Ian Ring Music TheoryAfoianThis is the prime mode

Prime

The prime form of this scale is Scale 37

Scale 37Scale 37: Afoian, Ian Ring Music TheoryAfoian

Complement

The tritonic modal family [1033, 641, 37] (Forte: 3-7) is the complement of the enneatonic modal family [1471, 1789, 2027, 2783, 3061, 3439, 3767, 3931, 4013] (Forte: 9-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1033 is 517

Scale 517Scale 517: Aluian, Ian Ring Music TheoryAluian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1033 is chiral, and its enantiomorph is scale 517

Scale 517Scale 517: Aluian, Ian Ring Music TheoryAluian

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> 1033       T0I <11,0> 517
T1 <1,1> 2066      T1I <11,1> 1034
T2 <1,2> 37      T2I <11,2> 2068
T3 <1,3> 74      T3I <11,3> 41
T4 <1,4> 148      T4I <11,4> 82
T5 <1,5> 296      T5I <11,5> 164
T6 <1,6> 592      T6I <11,6> 328
T7 <1,7> 1184      T7I <11,7> 656
T8 <1,8> 2368      T8I <11,8> 1312
T9 <1,9> 641      T9I <11,9> 2624
T10 <1,10> 1282      T10I <11,10> 1153
T11 <1,11> 2564      T11I <11,11> 2306
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 13      T0MI <7,0> 1537
T1M <5,1> 26      T1MI <7,1> 3074
T2M <5,2> 52      T2MI <7,2> 2053
T3M <5,3> 104      T3MI <7,3> 11
T4M <5,4> 208      T4MI <7,4> 22
T5M <5,5> 416      T5MI <7,5> 44
T6M <5,6> 832      T6MI <7,6> 88
T7M <5,7> 1664      T7MI <7,7> 176
T8M <5,8> 3328      T8MI <7,8> 352
T9M <5,9> 2561      T9MI <7,9> 704
T10M <5,10> 1027      T10MI <7,10> 1408
T11M <5,11> 2054      T11MI <7,11> 2816

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 1035Scale 1035: Givian, Ian Ring Music TheoryGivian
Scale 1037Scale 1037: Warao Tetratonic, Ian Ring Music TheoryWarao Tetratonic
Scale 1025Scale 1025: Warao Ditonic, Ian Ring Music TheoryWarao Ditonic
Scale 1029Scale 1029: Ampian, Ian Ring Music TheoryAmpian
Scale 1041Scale 1041: Hitian, Ian Ring Music TheoryHitian
Scale 1049Scale 1049: Gidian, Ian Ring Music TheoryGidian
Scale 1065Scale 1065: Gonian, Ian Ring Music TheoryGonian
Scale 1097Scale 1097: Aeraphic, Ian Ring Music TheoryAeraphic
Scale 1161Scale 1161: Bi Yu, Ian Ring Music TheoryBi Yu
Scale 1289Scale 1289: Huvian, Ian Ring Music TheoryHuvian
Scale 1545Scale 1545: Jonian, Ian Ring Music TheoryJonian
Scale 9Scale 9: Minor Third Ditone, Ian Ring Music TheoryMinor Third Ditone
Scale 521Scale 521: Astian, Ian Ring Music TheoryAstian
Scale 2057Scale 2057: Mopian, Ian Ring Music TheoryMopian
Scale 3081Scale 3081: Temian, Ian Ring Music TheoryTemian

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.