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Scale 209

Scale 209, 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.

4 (tetratonic)

Pitch Class Set

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

{0,4,6,7}

Forte Number

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

4-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.

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: 353

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.

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.

3

Prime Form

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

no
prime: 139

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.

[4, 2, 1, 5]

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, 1, 1, 1, 1, 1>

Interval Spectrum

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

pmnsdt

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

Spectra Variation

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

3.5

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.

1.366

Polygon Perimeter

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

5.182

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 TriadsC{0,4,7}000

The following pitch classes are not present in any of the common triads: {6}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 209 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 269
Scale 269, Ian Ring Music Theory
3rd mode:
Scale 1091
Scale 1091, Ian Ring Music Theory
4th mode:
Scale 2593
Scale 2593, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 139

Scale 139Scale 139, Ian Ring Music Theory

Complement

The tetratonic modal family [209, 269, 1091, 2593] (Forte: 4-Z29) is the complement of the octatonic modal family [751, 1913, 1943, 2423, 3019, 3259, 3557, 3677] (Forte: 8-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 209 is 353

Scale 353Scale 353, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 209 is chiral, and its enantiomorph is scale 353

Scale 353Scale 353, Ian Ring Music Theory

Transformations:

T0 209  T0I 353
T1 418  T1I 706
T2 836  T2I 1412
T3 1672  T3I 2824
T4 3344  T4I 1553
T5 2593  T5I 3106
T6 1091  T6I 2117
T7 2182  T7I 139
T8 269  T8I 278
T9 538  T9I 556
T10 1076  T10I 1112
T11 2152  T11I 2224

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 211Scale 211, Ian Ring Music Theory
Scale 213Scale 213, Ian Ring Music Theory
Scale 217Scale 217, Ian Ring Music Theory
Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari
Scale 201Scale 201, Ian Ring Music Theory
Scale 225Scale 225, Ian Ring Music Theory
Scale 241Scale 241, Ian Ring Music Theory
Scale 145Scale 145: Raga Malasri, Ian Ring Music TheoryRaga Malasri
Scale 177Scale 177, Ian Ring Music Theory
Scale 81Scale 81, Ian Ring Music Theory
Scale 337Scale 337: Koptic, Ian Ring Music TheoryKoptic
Scale 465Scale 465: Zoditonic, Ian Ring Music TheoryZoditonic
Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri
Scale 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic
Scale 2257Scale 2257: Lydian Pentatonic, Ian Ring Music TheoryLydian Pentatonic

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.