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Scale 3999: "Dydyllian"

Scale 3999: Dydyllian, 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
Dydyllian

Analysis

Cardinality

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

10 (decatonic)

Pitch Class Set

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

{0,1,2,3,4,7,8,9,10,11}

Forte Number

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

10-1

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.

[5.5]

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.

no

Hemitonia

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

9 (multihemitonic)

Cohemitonia

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

8 (multicohemitonic)

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.

9

Prime Form

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

no
prime: 1023

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, 1, 1, 3, 1, 1, 1, 1, 1] 9

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.

<9, 8, 8, 8, 8, 4>

Interval Spectrum

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

p8m8n8s8d9t4

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

Spectra Variation

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

1.8

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.

2.75

Polygon Perimeter

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

6.073

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

yes

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.

[11]

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}452.4
D♯{3,7,10}352.7
E{4,8,11}352.4
G{7,11,2}352.7
G♯{8,0,3}452.4
A{9,1,4}352.9
Minor Triadscm{0,3,7}352.4
c♯m{1,4,8}352.7
em{4,7,11}452.4
gm{7,10,2}352.9
g♯m{8,11,3}452.4
am{9,0,4}352.7
Augmented TriadsC+{0,4,8}552.3
D♯+{3,7,11}552.3
Diminished Triadsc♯°{1,4,7}252.9
{4,7,10}252.9
{7,10,1}253.1
g♯°{8,11,2}252.9
{9,0,3}252.9
a♯°{10,1,4}253.1
Parsimonious Voice Leading Between Common Triads of Scale 3999. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° em em C->em c#m c#m C+->c#m E E C+->E C+->G# am am C+->am c#°->c#m A A c#m->A D# D# D#->D#+ D#->e° gm gm D#->gm D#+->em Parsimonious Voice Leading Between Common Triads of Scale 3999. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m e°->em em->E E->g#m g°->gm a#° a#° g°->a#° gm->G g#° g#° G->g#° g#°->g#m g#m->G# G#->a° a°->am am->A A->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.

Diameter5
Radius5
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3999 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 4047
Scale 4047: Thogyllian, Ian Ring Music TheoryThogyllian
3rd mode:
Scale 4071
Scale 4071: Rygyllian, Ian Ring Music TheoryRygyllian
4th mode:
Scale 4083
Scale 4083: Bathyllian, Ian Ring Music TheoryBathyllian
5th mode:
Scale 4089
Scale 4089: Katoryllian, Ian Ring Music TheoryKatoryllian
6th mode:
Scale 1023
Scale 1023: Chromatic Decamode, Ian Ring Music TheoryChromatic DecamodeThis is the prime mode
7th mode:
Scale 2559
Scale 2559: Zogyllian, Ian Ring Music TheoryZogyllian
8th mode:
Scale 3327
Scale 3327: Madyllian, Ian Ring Music TheoryMadyllian
9th mode:
Scale 3711
Scale 3711: Dycryllian, Ian Ring Music TheoryDycryllian
10th mode:
Scale 3903
Scale 3903: Aeogyllian, Ian Ring Music TheoryAeogyllian

Prime

The prime form of this scale is Scale 1023

Scale 1023Scale 1023: Chromatic Decamode, Ian Ring Music TheoryChromatic Decamode

Complement

The decatonic modal family [3999, 4047, 4071, 4083, 4089, 1023, 2559, 3327, 3711, 3903] (Forte: 10-1) is the complement of the ditonic modal family [3, 2049] (Forte: 2-1)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3999 is 3903

Scale 3903Scale 3903: Aeogyllian, Ian Ring Music TheoryAeogyllian

Transformations:

T0 3999  T0I 3903
T1 3903  T1I 3711
T2 3711  T2I 3327
T3 3327  T3I 2559
T4 2559  T4I 1023
T5 1023  T5I 2046
T6 2046  T6I 4092
T7 4092  T7I 4089
T8 4089  T8I 4083
T9 4083  T9I 4071
T10 4071  T10I 4047
T11 4047  T11I 3999

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 3997Scale 3997: Dogygic, Ian Ring Music TheoryDogygic
Scale 3995Scale 3995: Ionygic, Ian Ring Music TheoryIonygic
Scale 3991Scale 3991: Badygic, Ian Ring Music TheoryBadygic
Scale 3983Scale 3983: Thyptygic, Ian Ring Music TheoryThyptygic
Scale 4015Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
Scale 4031Scale 4031: Chromatic Undecamode 6, Ian Ring Music TheoryChromatic Undecamode 6
Scale 4063Scale 4063: Chromatic Undecamode 7, Ian Ring Music TheoryChromatic Undecamode 7
Scale 3871Scale 3871: Aerynygic, Ian Ring Music TheoryAerynygic
Scale 3935Scale 3935: Kataphyllian, Ian Ring Music TheoryKataphyllian
Scale 3743Scale 3743: Thadygic, Ian Ring Music TheoryThadygic
Scale 3487Scale 3487: Byptygic, Ian Ring Music TheoryByptygic
Scale 2975Scale 2975: Aeroptygic, Ian Ring Music TheoryAeroptygic
Scale 1951Scale 1951: Marygic, Ian Ring Music TheoryMarygic

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.