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Scale 3369: "Mixolimic"

Scale 3369: Mixolimic, 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
Mixolimic
Dozenal
Veyian

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

{0,3,5,8,10,11}

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-Z47

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: 663

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.

1 (uncohemitonic)

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.

5

Prime Form

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

no
prime: 663

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, 2, 3, 2, 1, 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, 3, 3, 2, 4, 1>

Interval Spectrum

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

p4m2n3s3d2t

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

Spectra Variation

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

2.333

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.

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.

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.

(10, 16, 62)

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 TriadsG♯{8,0,3}221
Minor Triadsfm{5,8,0}221
g♯m{8,11,3}221
Diminished Triads{5,8,11}221

The following pitch classes are not present in any of the common triads: {10}

Parsimonious Voice Leading Between Common Triads of Scale 3369. Created by Ian Ring ©2019 fm fm f°->fm g#m g#m f°->g#m G# G# fm->G# g#m->G#

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
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3369 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 933
Scale 933: Dadimic, Ian Ring Music TheoryDadimic
3rd mode:
Scale 1257
Scale 1257: Blues Scale, Ian Ring Music TheoryBlues Scale
4th mode:
Scale 669
Scale 669: Gycrimic, Ian Ring Music TheoryGycrimic
5th mode:
Scale 1191
Scale 1191: Pyrimic, Ian Ring Music TheoryPyrimic
6th mode:
Scale 2643
Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi

Prime

The prime form of this scale is Scale 663

Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic

Complement

The hexatonic modal family [3369, 933, 1257, 669, 1191, 2643] (Forte: 6-Z47) is the complement of the hexatonic modal family [363, 1419, 1581, 1713, 2229, 2757] (Forte: 6-Z25)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3369 is 663

Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3369 is chiral, and its enantiomorph is scale 663

Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic

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> 3369       T0I <11,0> 663
T1 <1,1> 2643      T1I <11,1> 1326
T2 <1,2> 1191      T2I <11,2> 2652
T3 <1,3> 2382      T3I <11,3> 1209
T4 <1,4> 669      T4I <11,4> 2418
T5 <1,5> 1338      T5I <11,5> 741
T6 <1,6> 2676      T6I <11,6> 1482
T7 <1,7> 1257      T7I <11,7> 2964
T8 <1,8> 2514      T8I <11,8> 1833
T9 <1,9> 933      T9I <11,9> 3666
T10 <1,10> 1866      T10I <11,10> 3237
T11 <1,11> 3732      T11I <11,11> 2379
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 159      T0MI <7,0> 3873
T1M <5,1> 318      T1MI <7,1> 3651
T2M <5,2> 636      T2MI <7,2> 3207
T3M <5,3> 1272      T3MI <7,3> 2319
T4M <5,4> 2544      T4MI <7,4> 543
T5M <5,5> 993      T5MI <7,5> 1086
T6M <5,6> 1986      T6MI <7,6> 2172
T7M <5,7> 3972      T7MI <7,7> 249
T8M <5,8> 3849      T8MI <7,8> 498
T9M <5,9> 3603      T9MI <7,9> 996
T10M <5,10> 3111      T10MI <7,10> 1992
T11M <5,11> 2127      T11MI <7,11> 3984

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 3371Scale 3371: Aeolylian, Ian Ring Music TheoryAeolylian
Scale 3373Scale 3373: Lodian, Ian Ring Music TheoryLodian
Scale 3361Scale 3361: Vatian, Ian Ring Music TheoryVatian
Scale 3365Scale 3365: Katolimic, Ian Ring Music TheoryKatolimic
Scale 3377Scale 3377: Phralimic, Ian Ring Music TheoryPhralimic
Scale 3385Scale 3385: Chromatic Phrygian, Ian Ring Music TheoryChromatic Phrygian
Scale 3337Scale 3337: Vafian, Ian Ring Music TheoryVafian
Scale 3353Scale 3353: Phraptimic, Ian Ring Music TheoryPhraptimic
Scale 3401Scale 3401: Palimic, Ian Ring Music TheoryPalimic
Scale 3433Scale 3433: Thonian, Ian Ring Music TheoryThonian
Scale 3497Scale 3497: Phrolian, Ian Ring Music TheoryPhrolian
Scale 3113Scale 3113: Tigian, Ian Ring Music TheoryTigian
Scale 3241Scale 3241: Dalimic, Ian Ring Music TheoryDalimic
Scale 3625Scale 3625: Podimic, Ian Ring Music TheoryPodimic
Scale 3881Scale 3881: Morian, Ian Ring Music TheoryMorian
Scale 2345Scale 2345: Raga Chandrakauns, Ian Ring Music TheoryRaga Chandrakauns
Scale 2857Scale 2857: Stythimic, Ian Ring Music TheoryStythimic
Scale 1321Scale 1321: Blues Minor, Ian Ring Music TheoryBlues Minor

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.