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Scale 355: "Aeoloritonic"

Scale 355: Aeoloritonic, 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

Zeitler
Aeoloritonic
Dozenal
Cecian

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

{0,1,5,6,8}

Forte Number

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

5-20

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: 2257

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.

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.

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, 4, 1, 2, 4]

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, 1, 1, 2, 3, 1>

Interval Spectrum

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

p3m2nsd2t

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

Spectra Variation

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

2.4

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.

1.799

Polygon Perimeter

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

5.499

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, 30)

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♯{1,5,8}110.5
Minor Triadsfm{5,8,0}110.5

The following pitch classes are not present in any of the common triads: {6}

Parsimonious Voice Leading Between Common Triads of Scale 355. Created by Ian Ring ©2019 C# C# fm fm C#->fm

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

2nd mode:
Scale 2225
Scale 2225: Ionian Pentatonic, Ian Ring Music TheoryIonian Pentatonic
3rd mode:
Scale 395
Scale 395: Phrygian Pentatonic, Ian Ring Music TheoryPhrygian Pentatonic
4th mode:
Scale 2245
Scale 2245: Raga Vaijayanti, Ian Ring Music TheoryRaga Vaijayanti
5th mode:
Scale 1585
Scale 1585: Raga Khamaji Durga, Ian Ring Music TheoryRaga Khamaji Durga

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [355, 2225, 395, 2245, 1585] (Forte: 5-20) is the complement of the heptatonic modal family [743, 919, 1849, 2419, 2507, 3257, 3301] (Forte: 7-20)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 355 is 2257

Scale 2257Scale 2257: Lydian Pentatonic, Ian Ring Music TheoryLydian Pentatonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 355 is chiral, and its enantiomorph is scale 2257

Scale 2257Scale 2257: Lydian Pentatonic, Ian Ring Music TheoryLydian Pentatonic

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> 355       T0I <11,0> 2257
T1 <1,1> 710      T1I <11,1> 419
T2 <1,2> 1420      T2I <11,2> 838
T3 <1,3> 2840      T3I <11,3> 1676
T4 <1,4> 1585      T4I <11,4> 3352
T5 <1,5> 3170      T5I <11,5> 2609
T6 <1,6> 2245      T6I <11,6> 1123
T7 <1,7> 395      T7I <11,7> 2246
T8 <1,8> 790      T8I <11,8> 397
T9 <1,9> 1580      T9I <11,9> 794
T10 <1,10> 3160      T10I <11,10> 1588
T11 <1,11> 2225      T11I <11,11> 3176
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 115      T0MI <7,0> 2497
T1M <5,1> 230      T1MI <7,1> 899
T2M <5,2> 460      T2MI <7,2> 1798
T3M <5,3> 920      T3MI <7,3> 3596
T4M <5,4> 1840      T4MI <7,4> 3097
T5M <5,5> 3680      T5MI <7,5> 2099
T6M <5,6> 3265      T6MI <7,6> 103
T7M <5,7> 2435      T7MI <7,7> 206
T8M <5,8> 775      T8MI <7,8> 412
T9M <5,9> 1550      T9MI <7,9> 824
T10M <5,10> 3100      T10MI <7,10> 1648
T11M <5,11> 2105      T11MI <7,11> 3296

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 353Scale 353: Cebian, Ian Ring Music TheoryCebian
Scale 357Scale 357: Banitonic, Ian Ring Music TheoryBanitonic
Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic
Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 371Scale 371: Rythimic, Ian Ring Music TheoryRythimic
Scale 323Scale 323: Cajian, Ian Ring Music TheoryCajian
Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic
Scale 291Scale 291: Raga Lavangi, Ian Ring Music TheoryRaga Lavangi
Scale 419Scale 419: Hon-kumoi-joshi, Ian Ring Music TheoryHon-kumoi-joshi
Scale 483Scale 483: Kygimic, Ian Ring Music TheoryKygimic
Scale 99Scale 99: Iprian, Ian Ring Music TheoryIprian
Scale 227Scale 227: Bician, Ian Ring Music TheoryBician
Scale 611Scale 611: Anchihoye, Ian Ring Music TheoryAnchihoye
Scale 867Scale 867: Phrocrimic, Ian Ring Music TheoryPhrocrimic
Scale 1379Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic
Scale 2403Scale 2403: Lycrimic, Ian Ring Music TheoryLycrimic

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.