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Scale 3687: "Zonyllic"

Scale 3687: Zonyllic, 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
Zonyllic
Dozenal
Xelian

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

{0,1,2,5,6,9,10,11}

Forte Number

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

8-7

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.

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.

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.

7

Prime Form

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

no
prime: 831

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, 3, 1, 1, 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.

<6, 4, 5, 6, 5, 2>

Interval Spectrum

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

p5m6n5s4d6t2

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> = {3,5,7}
<4> = {4,6,8}
<5> = {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.

2.5

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.

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.

[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

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.

(69, 36, 114)

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}331.83
F{5,9,0}252.5
F♯{6,10,1}331.83
A♯{10,2,5}441.83
Minor Triadsdm{2,5,9}331.83
f♯m{6,9,1}441.83
a♯m{10,1,5}331.83
bm{11,2,6}252.5
Augmented TriadsC♯+{1,5,9}441.83
D+{2,6,10}441.83
Diminished Triadsf♯°{6,9,0}252.5
{11,2,5}252.5
Parsimonious Voice Leading Between Common Triads of Scale 3687. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m a#m a#m C#+->a#m D D dm->D A# A# dm->A# D+ D+ D->D+ D->f#m F# F# D+->F# D+->A# bm bm D+->bm f#° f#° F->f#° f#°->f#m f#m->F# F#->a#m a#m->A# A#->b° b°->bm

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
Radius3
Self-Centeredno
Central Verticesdm, D, F♯, a♯m
Peripheral VerticesF, f♯°, b°, bm

Modes

Modes are the rotational transformation of this scale. Scale 3687 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3891
Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
3rd mode:
Scale 3993
Scale 3993: Ioniptyllic, Ian Ring Music TheoryIoniptyllic
4th mode:
Scale 1011
Scale 1011: Kycryllic, Ian Ring Music TheoryKycryllic
5th mode:
Scale 2553
Scale 2553: Aeolaptyllic, Ian Ring Music TheoryAeolaptyllic
6th mode:
Scale 831
Scale 831: Rodyllic, Ian Ring Music TheoryRodyllicThis is the prime mode
7th mode:
Scale 2463
Scale 2463: Ionathyllic, Ian Ring Music TheoryIonathyllic
8th mode:
Scale 3279
Scale 3279: Pythyllic, Ian Ring Music TheoryPythyllic

Prime

The prime form of this scale is Scale 831

Scale 831Scale 831: Rodyllic, Ian Ring Music TheoryRodyllic

Complement

The octatonic modal family [3687, 3891, 3993, 1011, 2553, 831, 2463, 3279] (Forte: 8-7) is the complement of the tetratonic modal family [51, 771, 2073, 2433] (Forte: 4-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3687 is 3279

Scale 3279Scale 3279: Pythyllic, Ian Ring Music TheoryPythyllic

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> 3687       T0I <11,0> 3279
T1 <1,1> 3279      T1I <11,1> 2463
T2 <1,2> 2463      T2I <11,2> 831
T3 <1,3> 831      T3I <11,3> 1662
T4 <1,4> 1662      T4I <11,4> 3324
T5 <1,5> 3324      T5I <11,5> 2553
T6 <1,6> 2553      T6I <11,6> 1011
T7 <1,7> 1011      T7I <11,7> 2022
T8 <1,8> 2022      T8I <11,8> 4044
T9 <1,9> 4044      T9I <11,9> 3993
T10 <1,10> 3993      T10I <11,10> 3891
T11 <1,11> 3891      T11I <11,11> 3687
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1767      T0MI <7,0> 3309
T1M <5,1> 3534      T1MI <7,1> 2523
T2M <5,2> 2973      T2MI <7,2> 951
T3M <5,3> 1851      T3MI <7,3> 1902
T4M <5,4> 3702      T4MI <7,4> 3804
T5M <5,5> 3309      T5MI <7,5> 3513
T6M <5,6> 2523      T6MI <7,6> 2931
T7M <5,7> 951      T7MI <7,7> 1767
T8M <5,8> 1902      T8MI <7,8> 3534
T9M <5,9> 3804      T9MI <7,9> 2973
T10M <5,10> 3513      T10MI <7,10> 1851
T11M <5,11> 2931      T11MI <7,11> 3702

The transformations that map this set to itself are: T0, T11I

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 3685Scale 3685: Kodian, Ian Ring Music TheoryKodian
Scale 3683Scale 3683: Dycrian, Ian Ring Music TheoryDycrian
Scale 3691Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic
Scale 3695Scale 3695: Kodygic, Ian Ring Music TheoryKodygic
Scale 3703Scale 3703: Katalygic, Ian Ring Music TheoryKatalygic
Scale 3655Scale 3655: Mathian, Ian Ring Music TheoryMathian
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 3623Scale 3623: Aerocrian, Ian Ring Music TheoryAerocrian
Scale 3751Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
Scale 3815Scale 3815: Galygic, Ian Ring Music TheoryGalygic
Scale 3943Scale 3943: Zynygic, Ian Ring Music TheoryZynygic
Scale 3175Scale 3175: Eponian, Ian Ring Music TheoryEponian
Scale 3431Scale 3431: Zyptyllic, Ian Ring Music TheoryZyptyllic
Scale 2663Scale 2663: Lalian, Ian Ring Music TheoryLalian
Scale 1639Scale 1639: Aeolothian, Ian Ring Music TheoryAeolothian

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.