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Scale 3579: "Zyphyllian"

Scale 3579: Zyphyllian, 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
Zyphyllian
Dozenal
Wapian
Wixian

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

{0,1,3,4,5,6,7,8,10,11}

Forte Number

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

10-5

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.

8 (multihemitonic)

Cohemitonia

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

6 (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.

1

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: 1983

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 5
origin: 7

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, 2, 1, 1, 1, 1, 1, 2, 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.

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

Interval Spectrum

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

p9m8n8s8d8t4

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

Spectra Variation

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

1

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.

yes

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.

2.866

Polygon Perimeter

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

6.141

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".

Proper

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.

(0, 96, 177)

Generator

This scale has a generator of 5, originating on 7.

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}452.41
C♯{1,5,8}352.86
D♯{3,7,10}452.68
E{4,8,11}452.5
F♯{6,10,1}352.95
G♯{8,0,3}352.59
B{11,3,6}352.77
Minor Triadscm{0,3,7}452.5
c♯m{1,4,8}452.68
d♯m{3,6,10}352.86
em{4,7,11}452.41
fm{5,8,0}352.77
g♯m{8,11,3}352.59
a♯m{10,1,5}352.95
Augmented TriadsC+{0,4,8}552.41
D♯+{3,7,11}552.41
Diminished Triads{0,3,6}253.05
c♯°{1,4,7}252.86
{4,7,10}252.86
{5,8,11}253.05
{7,10,1}253.05
a♯°{10,1,4}253.05
Parsimonious Voice Leading Between Common Triads of Scale 3579. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B 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 fm fm C+->fm C+->G# c#°->c#m C# C# c#m->C# a#° a#° c#m->a#° C#->fm a#m a#m C#->a#m d#m d#m D# D# d#m->D# F# F# d#m->F# d#m->B D#->D#+ D#->e° D#->g° D#+->em g#m g#m D#+->g#m D#+->B e°->em em->E E->f° E->g#m f°->fm F#->g° F#->a#m g#m->G# a#°->a#m

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 3579 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 3837
Scale 3837: Minor Pentatonic With Leading Tones, Ian Ring Music TheoryMinor Pentatonic With Leading Tones
3rd mode:
Scale 1983
Scale 1983: Soryllian, Ian Ring Music TheorySoryllianThis is the prime mode
4th mode:
Scale 3039
Scale 3039: Godyllian, Ian Ring Music TheoryGodyllian
5th mode:
Scale 3567
Scale 3567: Epityllian, Ian Ring Music TheoryEpityllian
6th mode:
Scale 3831
Scale 3831: Ionyllian, Ian Ring Music TheoryIonyllian
7th mode:
Scale 3963
Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian
8th mode:
Scale 4029
Scale 4029: Major/Minor Mixed, Ian Ring Music TheoryMajor/Minor Mixed
9th mode:
Scale 2031
Scale 2031: Gadyllian, Ian Ring Music TheoryGadyllian
10th mode:
Scale 3063
Scale 3063: Solyllian, Ian Ring Music TheorySolyllian

Prime

The prime form of this scale is Scale 1983

Scale 1983Scale 1983: Soryllian, Ian Ring Music TheorySoryllian

Complement

The decatonic modal family [3579, 3837, 1983, 3039, 3567, 3831, 3963, 4029, 2031, 3063] (Forte: 10-5) is the complement of the ditonic modal family [33, 129] (Forte: 2-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3579 is 3063

Scale 3063Scale 3063: Solyllian, Ian Ring Music TheorySolyllian

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> 3579       T0I <11,0> 3063
T1 <1,1> 3063      T1I <11,1> 2031
T2 <1,2> 2031      T2I <11,2> 4062
T3 <1,3> 4062      T3I <11,3> 4029
T4 <1,4> 4029      T4I <11,4> 3963
T5 <1,5> 3963      T5I <11,5> 3831
T6 <1,6> 3831      T6I <11,6> 3567
T7 <1,7> 3567      T7I <11,7> 3039
T8 <1,8> 3039      T8I <11,8> 1983
T9 <1,9> 1983      T9I <11,9> 3966
T10 <1,10> 3966      T10I <11,10> 3837
T11 <1,11> 3837      T11I <11,11> 3579
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2559      T0MI <7,0> 4083
T1M <5,1> 1023      T1MI <7,1> 4071
T2M <5,2> 2046      T2MI <7,2> 4047
T3M <5,3> 4092      T3MI <7,3> 3999
T4M <5,4> 4089      T4MI <7,4> 3903
T5M <5,5> 4083      T5MI <7,5> 3711
T6M <5,6> 4071      T6MI <7,6> 3327
T7M <5,7> 4047      T7MI <7,7> 2559
T8M <5,8> 3999      T8MI <7,8> 1023
T9M <5,9> 3903      T9MI <7,9> 2046
T10M <5,10> 3711      T10MI <7,10> 4092
T11M <5,11> 3327      T11MI <7,11> 4089

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 3577Scale 3577: Loptygic, Ian Ring Music TheoryLoptygic
Scale 3581Scale 3581: Epocryllian, Ian Ring Music TheoryEpocryllian
Scale 3583Scale 3583: Chromatic Undecamode 3, Ian Ring Music TheoryChromatic Undecamode 3
Scale 3571Scale 3571: Dyrygic, Ian Ring Music TheoryDyrygic
Scale 3575Scale 3575: Symmetrical Decatonic, Ian Ring Music TheorySymmetrical Decatonic
Scale 3563Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic
Scale 3547Scale 3547: Sadygic, Ian Ring Music TheorySadygic
Scale 3515Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 3323Scale 3323: Lacrygic, Ian Ring Music TheoryLacrygic
Scale 3835Scale 3835: Katodyllian, Ian Ring Music TheoryKatodyllian
Scale 4091Scale 4091: Chromatic Undecamode 10, Ian Ring Music TheoryChromatic Undecamode 10
Scale 2555Scale 2555: Bythygic, Ian Ring Music TheoryBythygic
Scale 3067Scale 3067: Goptyllian, Ian Ring Music TheoryGoptyllian
Scale 1531Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic

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.