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Scale 2647: "Dadian"

Scale 2647: Dadian, 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

41161837294116105072918310504116183
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
Dadian

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

{0,1,2,4,6,9,11}

Forte Number

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

7-23

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: 3403

Hemitonia

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

3 (trihemitonic)

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.

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.

6

Prime Form

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

no
prime: 701

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, 2, 2, 3, 2, 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.

<3, 5, 4, 3, 5, 1>

Interval Spectrum

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

p5m3n4s5d3t

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

Spectra Variation

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

2.571

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.

2.549

Polygon Perimeter

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

5.967

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

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}231.5
A{9,1,4}231.5
Minor Triadsf♯m{6,9,1}321.17
am{9,0,4}241.83
bm{11,2,6}142.17
Diminished Triadsf♯°{6,9,0}231.5
Parsimonious Voice Leading Between Common Triads of Scale 2647. Created by Ian Ring ©2019 D D f#m f#m D->f#m bm bm D->bm f#° f#° f#°->f#m am am f#°->am A A f#m->A am->A

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.

Diameter4
Radius2
Self-Centeredno
Central Verticesf♯m
Peripheral Verticesam, bm

Modes

Modes are the rotational transformation of this scale. Scale 2647 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3371
Scale 3371: Aeolylian, Ian Ring Music TheoryAeolylian
3rd mode:
Scale 3733
Scale 3733: Gycrian, Ian Ring Music TheoryGycrian
4th mode:
Scale 1957
Scale 1957: Pyrian, Ian Ring Music TheoryPyrian
5th mode:
Scale 1513
Scale 1513: Stathian, Ian Ring Music TheoryStathian
6th mode:
Scale 701
Scale 701: Mixonyphian, Ian Ring Music TheoryMixonyphianThis is the prime mode
7th mode:
Scale 1199
Scale 1199: Magian, Ian Ring Music TheoryMagian

Prime

The prime form of this scale is Scale 701

Scale 701Scale 701: Mixonyphian, Ian Ring Music TheoryMixonyphian

Complement

The heptatonic modal family [2647, 3371, 3733, 1957, 1513, 701, 1199] (Forte: 7-23) is the complement of the pentatonic modal family [173, 1067, 1441, 1669, 2581] (Forte: 5-23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2647 is 3403

Scale 3403Scale 3403: Bylian, Ian Ring Music TheoryBylian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2647 is chiral, and its enantiomorph is scale 3403

Scale 3403Scale 3403: Bylian, Ian Ring Music TheoryBylian

Transformations:

T0 2647  T0I 3403
T1 1199  T1I 2711
T2 2398  T2I 1327
T3 701  T3I 2654
T4 1402  T4I 1213
T5 2804  T5I 2426
T6 1513  T6I 757
T7 3026  T7I 1514
T8 1957  T8I 3028
T9 3914  T9I 1961
T10 3733  T10I 3922
T11 3371  T11I 3749

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 2645Scale 2645: Raga Mruganandana, Ian Ring Music TheoryRaga Mruganandana
Scale 2643Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi
Scale 2651Scale 2651: Panian, Ian Ring Music TheoryPanian
Scale 2655Scale 2655, Ian Ring Music Theory
Scale 2631Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
Scale 2639Scale 2639: Dothian, Ian Ring Music TheoryDothian
Scale 2663Scale 2663: Lalian, Ian Ring Music TheoryLalian
Scale 2679Scale 2679: Rathyllic, Ian Ring Music TheoryRathyllic
Scale 2583Scale 2583, Ian Ring Music Theory
Scale 2615Scale 2615: Thoptian, Ian Ring Music TheoryThoptian
Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian
Scale 2775Scale 2775: Godyllic, Ian Ring Music TheoryGodyllic
Scale 2903Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic
Scale 2135Scale 2135, Ian Ring Music Theory
Scale 2391Scale 2391: Molian, Ian Ring Music TheoryMolian
Scale 3159Scale 3159: Stocrian, Ian Ring Music TheoryStocrian
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic
Scale 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian

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.