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Scale 2649: "Aeolythimic"

Scale 2649: Aeolythimic, 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
Aeolythimic
Dozenal
Qofian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,3,4,6,9,11}

Forte Number

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

6-Z29

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.

[1.5]

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.

no

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.

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.

5

Prime Form

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

no
prime: 717

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, 1, 2, 3, 2, 1]

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, 2, 4, 2, 3, 2>

Interval Spectrum

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

p3m2n4s2d2t2

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

Spectra Variation

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

1.667

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.

2.366

Polygon Perimeter

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

5.864

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.

[3]

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".

Proper

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.

(0, 12, 57)

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 TriadsB{11,3,6}231.5
Minor Triadsam{9,0,4}231.5
Diminished Triads{0,3,6}231.5
d♯°{3,6,9}231.5
f♯°{6,9,0}231.5
{9,0,3}231.5
Parsimonious Voice Leading Between Common Triads of Scale 2649. Created by Ian Ring ©2019 c°->a° B B c°->B d#° d#° f#° f#° d#°->f#° d#°->B am am f#°->am a°->am

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.

Diameter3
Radius3
Self-Centeredyes

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Minor: {9, 0, 4}
Major: {11, 3, 6}

Modes

Modes are the rotational transformation of this scale. Scale 2649 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 843
Scale 843: Molimic, Ian Ring Music TheoryMolimic
3rd mode:
Scale 2469
Scale 2469: Raga Bhinna Pancama, Ian Ring Music TheoryRaga Bhinna Pancama
4th mode:
Scale 1641
Scale 1641: Bocrimic, Ian Ring Music TheoryBocrimic
5th mode:
Scale 717
Scale 717: Raga Vijayanagari, Ian Ring Music TheoryRaga VijayanagariThis is the prime mode
6th mode:
Scale 1203
Scale 1203: Pagimic, Ian Ring Music TheoryPagimic

Prime

The prime form of this scale is Scale 717

Scale 717Scale 717: Raga Vijayanagari, Ian Ring Music TheoryRaga Vijayanagari

Complement

The hexatonic modal family [2649, 843, 2469, 1641, 717, 1203] (Forte: 6-Z29) is the complement of the hexatonic modal family [723, 813, 1227, 1689, 2409, 2661] (Forte: 6-Z50)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2649 is 843

Scale 843Scale 843: Molimic, Ian Ring Music TheoryMolimic

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> 2649       T0I <11,0> 843
T1 <1,1> 1203      T1I <11,1> 1686
T2 <1,2> 2406      T2I <11,2> 3372
T3 <1,3> 717      T3I <11,3> 2649
T4 <1,4> 1434      T4I <11,4> 1203
T5 <1,5> 2868      T5I <11,5> 2406
T6 <1,6> 1641      T6I <11,6> 717
T7 <1,7> 3282      T7I <11,7> 1434
T8 <1,8> 2469      T8I <11,8> 2868
T9 <1,9> 843      T9I <11,9> 1641
T10 <1,10> 1686      T10I <11,10> 3282
T11 <1,11> 3372      T11I <11,11> 2469
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 969      T0MI <7,0> 633
T1M <5,1> 1938      T1MI <7,1> 1266
T2M <5,2> 3876      T2MI <7,2> 2532
T3M <5,3> 3657      T3MI <7,3> 969
T4M <5,4> 3219      T4MI <7,4> 1938
T5M <5,5> 2343      T5MI <7,5> 3876
T6M <5,6> 591      T6MI <7,6> 3657
T7M <5,7> 1182      T7MI <7,7> 3219
T8M <5,8> 2364      T8MI <7,8> 2343
T9M <5,9> 633      T9MI <7,9> 591
T10M <5,10> 1266      T10MI <7,10> 1182
T11M <5,11> 2532      T11MI <7,11> 2364

The transformations that map this set to itself are: T0, T3I

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 2651Scale 2651: Panian, Ian Ring Music TheoryPanian
Scale 2653Scale 2653: Sygian, Ian Ring Music TheorySygian
Scale 2641Scale 2641: Raga Hindol, Ian Ring Music TheoryRaga Hindol
Scale 2645Scale 2645: Raga Mruganandana, Ian Ring Music TheoryRaga Mruganandana
Scale 2633Scale 2633: Bartók Beta Chord, Ian Ring Music TheoryBartók Beta Chord
Scale 2665Scale 2665: Aeradimic, Ian Ring Music TheoryAeradimic
Scale 2681Scale 2681: Aerycrian, Ian Ring Music TheoryAerycrian
Scale 2585Scale 2585: Otlian, Ian Ring Music TheoryOtlian
Scale 2617Scale 2617: Pylimic, Ian Ring Music TheoryPylimic
Scale 2713Scale 2713: Porimic, Ian Ring Music TheoryPorimic
Scale 2777Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1
Scale 2137Scale 2137: Nalian, Ian Ring Music TheoryNalian
Scale 2393Scale 2393: Zathimic, Ian Ring Music TheoryZathimic
Scale 3161Scale 3161: Kodimic, Ian Ring Music TheoryKodimic
Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian
Scale 601Scale 601: Bycritonic, Ian Ring Music TheoryBycritonic
Scale 1625Scale 1625: Lythimic, Ian Ring Music TheoryLythimic

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.