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Scale 999: "Ionodyllic"

Scale 999: Ionodyllic, 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

Zeitler
Ionodyllic
Dozenal
Geyian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,1,2,5,6,7,8,9}

Forte Number

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

8-8

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.

[1]

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.

6 (multihemitonic)

Cohemitonia

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

4 (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.

7

Prime Form

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

no
prime: 927

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.

[1, 1, 3, 1, 1, 1, 1, 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.

<6, 4, 4, 5, 6, 3>

Interval Spectrum

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

p6m5n4s4d6t3

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

Spectra Variation

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

2

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.

2.5

Polygon Perimeter

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

5.934

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.

[2]

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.

(44, 16, 93)

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♯{1,5,8}331.67
D{2,6,9}242
F{5,9,0}331.67
Minor Triadsdm{2,5,9}331.67
fm{5,8,0}242
f♯m{6,9,1}331.67
Augmented TriadsC♯+{1,5,9}421.33
Diminished Triads{2,5,8}242
f♯°{6,9,0}242

The following pitch classes are not present in any of the common triads: {7}

Parsimonious Voice Leading Between Common Triads of Scale 999. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m d°->dm D D dm->D D->f#m fm->F f#° f#° F->f#° f#°->f#m

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 VerticesC♯+
Peripheral Verticesd°, D, fm, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 999 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2547
Scale 2547: Raga Ramkali, Ian Ring Music TheoryRaga Ramkali
3rd mode:
Scale 3321
Scale 3321: Epagyllic, Ian Ring Music TheoryEpagyllic
4th mode:
Scale 927
Scale 927: Gaptyllic, Ian Ring Music TheoryGaptyllicThis is the prime mode
5th mode:
Scale 2511
Scale 2511: Aeroptyllic, Ian Ring Music TheoryAeroptyllic
6th mode:
Scale 3303
Scale 3303: Mylyllic, Ian Ring Music TheoryMylyllic
7th mode:
Scale 3699
Scale 3699: Galyllic, Ian Ring Music TheoryGalyllic
8th mode:
Scale 3897
Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic

Prime

The prime form of this scale is Scale 927

Scale 927Scale 927: Gaptyllic, Ian Ring Music TheoryGaptyllic

Complement

The octatonic modal family [999, 2547, 3321, 927, 2511, 3303, 3699, 3897] (Forte: 8-8) is the complement of the tetratonic modal family [99, 387, 2097, 2241] (Forte: 4-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 999 is 3321

Scale 3321Scale 3321: Epagyllic, Ian Ring Music TheoryEpagyllic

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> 999       T0I <11,0> 3321
T1 <1,1> 1998      T1I <11,1> 2547
T2 <1,2> 3996      T2I <11,2> 999
T3 <1,3> 3897      T3I <11,3> 1998
T4 <1,4> 3699      T4I <11,4> 3996
T5 <1,5> 3303      T5I <11,5> 3897
T6 <1,6> 2511      T6I <11,6> 3699
T7 <1,7> 927      T7I <11,7> 3303
T8 <1,8> 1854      T8I <11,8> 2511
T9 <1,9> 3708      T9I <11,9> 927
T10 <1,10> 3321      T10I <11,10> 1854
T11 <1,11> 2547      T11I <11,11> 3708
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3699      T0MI <7,0> 2511
T1M <5,1> 3303      T1MI <7,1> 927
T2M <5,2> 2511      T2MI <7,2> 1854
T3M <5,3> 927      T3MI <7,3> 3708
T4M <5,4> 1854      T4MI <7,4> 3321
T5M <5,5> 3708      T5MI <7,5> 2547
T6M <5,6> 3321      T6MI <7,6> 999
T7M <5,7> 2547      T7MI <7,7> 1998
T8M <5,8> 999       T8MI <7,8> 3996
T9M <5,9> 1998      T9MI <7,9> 3897
T10M <5,10> 3996      T10MI <7,10> 3699
T11M <5,11> 3897      T11MI <7,11> 3303

The transformations that map this set to itself are: T0, T2I, T8M, 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 997Scale 997: Rycrian, Ian Ring Music TheoryRycrian
Scale 995Scale 995: Phrathian, Ian Ring Music TheoryPhrathian
Scale 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 1007Scale 1007: Epitygic, Ian Ring Music TheoryEpitygic
Scale 1015Scale 1015: Ionodygic, Ian Ring Music TheoryIonodygic
Scale 967Scale 967: Mela Salaga, Ian Ring Music TheoryMela Salaga
Scale 983Scale 983: Thocryllic, Ian Ring Music TheoryThocryllic
Scale 935Scale 935: Chromatic Dorian, Ian Ring Music TheoryChromatic Dorian
Scale 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7
Scale 743Scale 743: Chromatic Hypophrygian Inverse, Ian Ring Music TheoryChromatic Hypophrygian Inverse
Scale 487Scale 487: Dynian, Ian Ring Music TheoryDynian
Scale 1511Scale 1511: Styptyllic, Ian Ring Music TheoryStyptyllic
Scale 2023Scale 2023: Zodygic, Ian Ring Music TheoryZodygic
Scale 3047Scale 3047: Panygic, Ian Ring Music TheoryPanygic

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.