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Scale 3661: "Mixodorian"

Scale 3661: Mixodorian, 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
Mixodorian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,2,3,6,9,10,11}

Forte Number

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

7-16

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: 1615

Hemitonia

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

4 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

2 (dicohemitonic)

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.

6

Prime Form

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

no
prime: 623

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.

[2, 1, 3, 3, 1, 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.

<4, 3, 5, 4, 3, 2>

Interval Spectrum

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

p3m4n5s3d4t2

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

Spectra Variation

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

2.857

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.

2.433

Polygon Perimeter

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

5.899

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.

(32, 38, 102)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsD{2,6,9}331.67
B{11,3,6}331.67
Minor Triadsd♯m{3,6,10}331.67
bm{11,2,6}231.89
Augmented TriadsD+{2,6,10}331.67
Diminished Triads{0,3,6}231.89
d♯°{3,6,9}231.89
f♯°{6,9,0}231.89
{9,0,3}232
Parsimonious Voice Leading Between Common Triads of Scale 3661. Created by Ian Ring ©2019 c°->a° B B c°->B D D D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° d#m d#m D+->d#m bm bm D+->bm d#°->d#m d#m->B f#°->a° bm->B

view full size

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

Modes

Modes are the rotational transformation of this scale. Scale 3661 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1939
Scale 1939: Dathian, Ian Ring Music TheoryDathian
3rd mode:
Scale 3017
Scale 3017: Gacrian, Ian Ring Music TheoryGacrian
4th mode:
Scale 889
Scale 889: Borian, Ian Ring Music TheoryBorian
5th mode:
Scale 623
Scale 623: Sycrian, Ian Ring Music TheorySycrianThis is the prime mode
6th mode:
Scale 2359
Scale 2359: Gadian, Ian Ring Music TheoryGadian
7th mode:
Scale 3227
Scale 3227: Aeolocrian, Ian Ring Music TheoryAeolocrian

Prime

The prime form of this scale is Scale 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

Complement

The heptatonic modal family [3661, 1939, 3017, 889, 623, 2359, 3227] (Forte: 7-16) is the complement of the pentatonic modal family [155, 865, 1555, 2125, 2825] (Forte: 5-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3661 is 1615

Scale 1615Scale 1615: Sydian, Ian Ring Music TheorySydian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3661 is chiral, and its enantiomorph is scale 1615

Scale 1615Scale 1615: Sydian, Ian Ring Music TheorySydian

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> 3661       T0I <11,0> 1615
T1 <1,1> 3227      T1I <11,1> 3230
T2 <1,2> 2359      T2I <11,2> 2365
T3 <1,3> 623      T3I <11,3> 635
T4 <1,4> 1246      T4I <11,4> 1270
T5 <1,5> 2492      T5I <11,5> 2540
T6 <1,6> 889      T6I <11,6> 985
T7 <1,7> 1778      T7I <11,7> 1970
T8 <1,8> 3556      T8I <11,8> 3940
T9 <1,9> 3017      T9I <11,9> 3785
T10 <1,10> 1939      T10I <11,10> 3475
T11 <1,11> 3878      T11I <11,11> 2855
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1741      T0MI <7,0> 1645
T1M <5,1> 3482      T1MI <7,1> 3290
T2M <5,2> 2869      T2MI <7,2> 2485
T3M <5,3> 1643      T3MI <7,3> 875
T4M <5,4> 3286      T4MI <7,4> 1750
T5M <5,5> 2477      T5MI <7,5> 3500
T6M <5,6> 859      T6MI <7,6> 2905
T7M <5,7> 1718      T7MI <7,7> 1715
T8M <5,8> 3436      T8MI <7,8> 3430
T9M <5,9> 2777      T9MI <7,9> 2765
T10M <5,10> 1459      T10MI <7,10> 1435
T11M <5,11> 2918      T11MI <7,11> 2870

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 3663Scale 3663: Sonyllic, Ian Ring Music TheorySonyllic
Scale 3657Scale 3657: Epynimic, Ian Ring Music TheoryEpynimic
Scale 3659Scale 3659: Polian, Ian Ring Music TheoryPolian
Scale 3653Scale 3653: Sathimic, Ian Ring Music TheorySathimic
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3677Scale 3677: Xafian, Ian Ring Music TheoryXafian
Scale 3693Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic
Scale 3597Scale 3597: Wijian, Ian Ring Music TheoryWijian
Scale 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 3725Scale 3725: Kyrian, Ian Ring Music TheoryKyrian
Scale 3789Scale 3789: Eporyllic, Ian Ring Music TheoryEporyllic
Scale 3917Scale 3917: Katoptyllic, Ian Ring Music TheoryKatoptyllic
Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
Scale 3405Scale 3405: Stynian, Ian Ring Music TheoryStynian
Scale 2637Scale 2637: Raga Ranjani, Ian Ring Music TheoryRaga Ranjani
Scale 1613Scale 1613: Thylimic, Ian Ring Music TheoryThylimic

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.