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Scale 3207

Scale 3207, 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).

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

{0,1,2,7,10,11}

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-Z36

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: 3111

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.

3 (tricohemitonic)

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.

5

Prime Form

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

no
prime: 159

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, 1, 5, 3, 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, 3, 2, 2, 1>

Interval Spectrum

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

p2m2n3s3d4t

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

Spectra Variation

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

4.333

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.

1.75

Polygon Perimeter

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

5.417

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

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{7,11,2}121
Minor Triadsgm{7,10,2}210.67
Diminished Triads{7,10,1}121

The following pitch classes are not present in any of the common triads: {0}

Parsimonious Voice Leading Between Common Triads of Scale 3207. Created by Ian Ring ©2019 gm gm g°->gm Parsimonious Voice Leading Between Common Triads of Scale 3207. Created by Ian Ring ©2019 G gm->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
Radius1
Self-Centeredno
Central Verticesgm
Peripheral Verticesg°, G

Modes

Modes are the rotational transformation of this scale. Scale 3207 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3651
Scale 3651, Ian Ring Music Theory
3rd mode:
Scale 3873
Scale 3873, Ian Ring Music Theory
4th mode:
Scale 249
Scale 249, Ian Ring Music Theory
5th mode:
Scale 543
Scale 543, Ian Ring Music Theory
6th mode:
Scale 2319
Scale 2319, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 159

Scale 159Scale 159, Ian Ring Music Theory

Complement

The hexatonic modal family [3207, 3651, 3873, 249, 543, 2319] (Forte: 6-Z36) is the complement of the hexatonic modal family [111, 1923, 2103, 3009, 3099, 3597] (Forte: 6-Z3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3207 is 3111

Scale 3111Scale 3111, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3207 is chiral, and its enantiomorph is scale 3111

Scale 3111Scale 3111, Ian Ring Music Theory

Transformations:

T0 3207  T0I 3111
T1 2319  T1I 2127
T2 543  T2I 159
T3 1086  T3I 318
T4 2172  T4I 636
T5 249  T5I 1272
T6 498  T6I 2544
T7 996  T7I 993
T8 1992  T8I 1986
T9 3984  T9I 3972
T10 3873  T10I 3849
T11 3651  T11I 3603

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 3205Scale 3205, Ian Ring Music Theory
Scale 3203Scale 3203, Ian Ring Music Theory
Scale 3211Scale 3211: Epacrimic, Ian Ring Music TheoryEpacrimic
Scale 3215Scale 3215: Katydian, Ian Ring Music TheoryKatydian
Scale 3223Scale 3223: Thyphian, Ian Ring Music TheoryThyphian
Scale 3239Scale 3239: Mela Tanarupi, Ian Ring Music TheoryMela Tanarupi
Scale 3271Scale 3271: Mela Raghupriya, Ian Ring Music TheoryMela Raghupriya
Scale 3079Scale 3079: Pentatonic Chromatic 3, Ian Ring Music TheoryPentatonic Chromatic 3
Scale 3143Scale 3143: Polimic, Ian Ring Music TheoryPolimic
Scale 3335Scale 3335, Ian Ring Music Theory
Scale 3463Scale 3463, Ian Ring Music Theory
Scale 3719Scale 3719, Ian Ring Music Theory
Scale 2183Scale 2183, Ian Ring Music Theory
Scale 2695Scale 2695, Ian Ring Music Theory
Scale 1159Scale 1159, Ian Ring Music Theory

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. 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.