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Scale 555: "Aeolycritonic"

Scale 555: Aeolycritonic, 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
Aeolycritonic

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

{0,1,3,5,9}

Forte Number

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

5-26

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: 2697

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.

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.

no
prime: 309

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 2, 4, 3]

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, 2, 2, 3, 1, 1>

Interval Spectrum

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

pm3n2s2dt

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

Spectra Variation

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

2.8

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.

2.049

Polygon Perimeter

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

5.664

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.

(5, 7, 36)

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 TriadsF{5,9,0}210.67
Augmented TriadsC♯+{1,5,9}121
Diminished Triads{9,0,3}121
Parsimonious Voice Leading Between Common Triads of Scale 555. Created by Ian Ring ©2019 C#+ C#+ F F C#+->F F->a°

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesF
Peripheral VerticesC♯+, a°

Modes

Modes are the rotational transformation of this scale. Scale 555 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2325
Scale 2325: Pynitonic, Ian Ring Music TheoryPynitonic
3rd mode:
Scale 1605
Scale 1605: Zanitonic, Ian Ring Music TheoryZanitonic
4th mode:
Scale 1425
Scale 1425: Ryphitonic, Ian Ring Music TheoryRyphitonic
5th mode:
Scale 345
Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic

Prime

The prime form of this scale is Scale 309

Scale 309Scale 309: Palitonic, Ian Ring Music TheoryPalitonic

Complement

The pentatonic modal family [555, 2325, 1605, 1425, 345] (Forte: 5-26) is the complement of the heptatonic modal family [699, 1497, 1623, 1893, 2397, 2859, 3477] (Forte: 7-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 555 is 2697

Scale 2697Scale 2697: Katagitonic, Ian Ring Music TheoryKatagitonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 555 is chiral, and its enantiomorph is scale 2697

Scale 2697Scale 2697: Katagitonic, Ian Ring Music TheoryKatagitonic

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> 555       T0I <11,0> 2697
T1 <1,1> 1110      T1I <11,1> 1299
T2 <1,2> 2220      T2I <11,2> 2598
T3 <1,3> 345      T3I <11,3> 1101
T4 <1,4> 690      T4I <11,4> 2202
T5 <1,5> 1380      T5I <11,5> 309
T6 <1,6> 2760      T6I <11,6> 618
T7 <1,7> 1425      T7I <11,7> 1236
T8 <1,8> 2850      T8I <11,8> 2472
T9 <1,9> 1605      T9I <11,9> 849
T10 <1,10> 3210      T10I <11,10> 1698
T11 <1,11> 2325      T11I <11,11> 3396
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 555       T0MI <7,0> 2697
T1M <5,1> 1110      T1MI <7,1> 1299
T2M <5,2> 2220      T2MI <7,2> 2598
T3M <5,3> 345      T3MI <7,3> 1101
T4M <5,4> 690      T4MI <7,4> 2202
T5M <5,5> 1380      T5MI <7,5> 309
T6M <5,6> 2760      T6MI <7,6> 618
T7M <5,7> 1425      T7MI <7,7> 1236
T8M <5,8> 2850      T8MI <7,8> 2472
T9M <5,9> 1605      T9MI <7,9> 849
T10M <5,10> 3210      T10MI <7,10> 1698
T11M <5,11> 2325      T11MI <7,11> 3396

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

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 553Scale 553: Rothic 2, Ian Ring Music TheoryRothic 2
Scale 557Scale 557: Raga Abhogi, Ian Ring Music TheoryRaga Abhogi
Scale 559Scale 559: Lylimic, Ian Ring Music TheoryLylimic
Scale 547Scale 547: Pyrric, Ian Ring Music TheoryPyrric
Scale 551Scale 551: Aeoloditonic, Ian Ring Music TheoryAeoloditonic
Scale 563Scale 563: Thacritonic, Ian Ring Music TheoryThacritonic
Scale 571Scale 571: Kynimic, Ian Ring Music TheoryKynimic
Scale 523Scale 523, Ian Ring Music Theory
Scale 539Scale 539, Ian Ring Music Theory
Scale 587Scale 587: Pathitonic, Ian Ring Music TheoryPathitonic
Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic
Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic
Scale 811Scale 811: Radimic, Ian Ring Music TheoryRadimic
Scale 43Scale 43, Ian Ring Music Theory
Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini
Scale 1067Scale 1067, Ian Ring Music Theory
Scale 1579Scale 1579: Sagimic, Ian Ring Music TheorySagimic
Scale 2603Scale 2603: Gadimic, Ian Ring Music TheoryGadimic

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.