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Scale 609: "Docian"

Scale 609: Docian, 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

Dozenal
Docian

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

{0,5,6,9}

Forte Number

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

4-18

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: 201

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: 147

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.

[5, 1, 3, 3]

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, 0, 2, 1, 1, 1>

Interval Spectrum

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

pmn2dt

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

Spectra Variation

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

3

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.

1.5

Polygon Perimeter

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

5.278

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.

(2, 0, 15)

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 TriadsF{5,9,0}110.5
Diminished Triadsf♯°{6,9,0}110.5
Parsimonious Voice Leading Between Common Triads of Scale 609. Created by Ian Ring ©2019 F F f#° f#° F->f#°

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 609 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 147
Scale 147: Bafian, Ian Ring Music TheoryBafianThis is the prime mode
3rd mode:
Scale 2121
Scale 2121: Nabian, Ian Ring Music TheoryNabian
4th mode:
Scale 777
Scale 777: Empian, Ian Ring Music TheoryEmpian

Prime

The prime form of this scale is Scale 147

Scale 147Scale 147: Bafian, Ian Ring Music TheoryBafian

Complement

The tetratonic modal family [609, 147, 2121, 777] (Forte: 4-18) is the complement of the octatonic modal family [879, 1779, 1947, 2487, 2937, 3021, 3291, 3693] (Forte: 8-18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 609 is 201

Scale 201Scale 201: Bemian, Ian Ring Music TheoryBemian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 609 is chiral, and its enantiomorph is scale 201

Scale 201Scale 201: Bemian, Ian Ring Music TheoryBemian

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> 609       T0I <11,0> 201
T1 <1,1> 1218      T1I <11,1> 402
T2 <1,2> 2436      T2I <11,2> 804
T3 <1,3> 777      T3I <11,3> 1608
T4 <1,4> 1554      T4I <11,4> 3216
T5 <1,5> 3108      T5I <11,5> 2337
T6 <1,6> 2121      T6I <11,6> 579
T7 <1,7> 147      T7I <11,7> 1158
T8 <1,8> 294      T8I <11,8> 2316
T9 <1,9> 588      T9I <11,9> 537
T10 <1,10> 1176      T10I <11,10> 1074
T11 <1,11> 2352      T11I <11,11> 2148
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 579      T0MI <7,0> 2121
T1M <5,1> 1158      T1MI <7,1> 147
T2M <5,2> 2316      T2MI <7,2> 294
T3M <5,3> 537      T3MI <7,3> 588
T4M <5,4> 1074      T4MI <7,4> 1176
T5M <5,5> 2148      T5MI <7,5> 2352
T6M <5,6> 201      T6MI <7,6> 609
T7M <5,7> 402      T7MI <7,7> 1218
T8M <5,8> 804      T8MI <7,8> 2436
T9M <5,9> 1608      T9MI <7,9> 777
T10M <5,10> 3216      T10MI <7,10> 1554
T11M <5,11> 2337      T11MI <7,11> 3108

The transformations that map this set to itself are: T0, T6MI

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 611Scale 611: Anchihoye, Ian Ring Music TheoryAnchihoye
Scale 613Scale 613: Phralitonic, Ian Ring Music TheoryPhralitonic
Scale 617Scale 617: Katycritonic, Ian Ring Music TheoryKatycritonic
Scale 625Scale 625: Ionyptitonic, Ian Ring Music TheoryIonyptitonic
Scale 577Scale 577: Illian, Ian Ring Music TheoryIllian
Scale 593Scale 593: Saric, Ian Ring Music TheorySaric
Scale 545Scale 545: Dewian, Ian Ring Music TheoryDewian
Scale 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 737Scale 737: Truian, Ian Ring Music TheoryTruian
Scale 865Scale 865: Jahian, Ian Ring Music TheoryJahian
Scale 97Scale 97: Athian, Ian Ring Music TheoryAthian
Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian
Scale 1121Scale 1121: Guwian, Ian Ring Music TheoryGuwian
Scale 1633Scale 1633: Kapian, Ian Ring Music TheoryKapian
Scale 2657Scale 2657: Qokian, Ian Ring Music TheoryQokian

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.