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Scale 2813: "Zolygic"

Scale 2813: Zolygic, 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
Zolygic

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,2,3,4,5,6,7,9,11}

Forte Number

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

9-7

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.

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: 2027

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.

8

Prime Form

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

no
prime: 1471

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

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

<6, 7, 7, 6, 7, 3>

Interval Spectrum

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

p7m6n7s7d6t3

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

Spectra Variation

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

1.778

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.

2.799

Polygon Perimeter

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

6.106

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

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.

(25, 109, 196)

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}342.44
D{2,6,9}442.31
F{5,9,0}342.44
G{7,11,2}242.38
B{11,3,6}442.13
Minor Triadscm{0,3,7}442.31
dm{2,5,9}342.44
em{4,7,11}242.56
am{9,0,4}342.44
bm{11,2,6}442.19
Augmented TriadsD♯+{3,7,11}442.19
Diminished Triads{0,3,6}242.44
d♯°{3,6,9}242.44
f♯°{6,9,0}242.56
{9,0,3}242.56
{11,2,5}242.56
Parsimonious Voice Leading Between Common Triads of Scale 2813. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ cm->a° em em C->em am am C->am dm dm D D dm->D F F dm->F dm->b° d#° d#° D->d#° f#° f#° D->f#° bm bm D->bm d#°->B D#+->em Parsimonious Voice Leading Between Common Triads of Scale 2813. Created by Ian Ring ©2019 G D#+->G D#+->B F->f#° F->am G->bm a°->am b°->bm bm->B

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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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2813 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 1727
Scale 1727: Sydygic, Ian Ring Music TheorySydygic
3rd mode:
Scale 2911
Scale 2911: Katygic, Ian Ring Music TheoryKatygic
4th mode:
Scale 3503
Scale 3503: Zyphygic, Ian Ring Music TheoryZyphygic
5th mode:
Scale 3799
Scale 3799: Aeralygic, Ian Ring Music TheoryAeralygic
6th mode:
Scale 3947
Scale 3947: Ryptygic, Ian Ring Music TheoryRyptygic
7th mode:
Scale 4021
Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
8th mode:
Scale 2029
Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi
9th mode:
Scale 1531
Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic

Prime

The prime form of this scale is Scale 1471

Scale 1471Scale 1471: Radygic, Ian Ring Music TheoryRadygic

Complement

The enneatonic modal family [2813, 1727, 2911, 3503, 3799, 3947, 4021, 2029, 1531] (Forte: 9-7) is the complement of the tritonic modal family [37, 641, 1033] (Forte: 3-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2813 is 2027

Scale 2027Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2813 is chiral, and its enantiomorph is scale 2027

Scale 2027Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic

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> 2813       T0I <11,0> 2027
T1 <1,1> 1531      T1I <11,1> 4054
T2 <1,2> 3062      T2I <11,2> 4013
T3 <1,3> 2029      T3I <11,3> 3931
T4 <1,4> 4058      T4I <11,4> 3767
T5 <1,5> 4021      T5I <11,5> 3439
T6 <1,6> 3947      T6I <11,6> 2783
T7 <1,7> 3799      T7I <11,7> 1471
T8 <1,8> 3503      T8I <11,8> 2942
T9 <1,9> 2911      T9I <11,9> 1789
T10 <1,10> 1727      T10I <11,10> 3578
T11 <1,11> 3454      T11I <11,11> 3061
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4043      T0MI <7,0> 2687
T1M <5,1> 3991      T1MI <7,1> 1279
T2M <5,2> 3887      T2MI <7,2> 2558
T3M <5,3> 3679      T3MI <7,3> 1021
T4M <5,4> 3263      T4MI <7,4> 2042
T5M <5,5> 2431      T5MI <7,5> 4084
T6M <5,6> 767      T6MI <7,6> 4073
T7M <5,7> 1534      T7MI <7,7> 4051
T8M <5,8> 3068      T8MI <7,8> 4007
T9M <5,9> 2041      T9MI <7,9> 3919
T10M <5,10> 4082      T10MI <7,10> 3743
T11M <5,11> 4069      T11MI <7,11> 3391

The transformations that map this set to itself are: T0

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 2815Scale 2815: Aeradyllian, Ian Ring Music TheoryAeradyllian
Scale 2809Scale 2809: Gythyllic, Ian Ring Music TheoryGythyllic
Scale 2811Scale 2811: Barygic, Ian Ring Music TheoryBarygic
Scale 2805Scale 2805: Ishikotsucho, Ian Ring Music TheoryIshikotsucho
Scale 2797Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
Scale 2781Scale 2781: Gycryllic, Ian Ring Music TheoryGycryllic
Scale 2749Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
Scale 2685Scale 2685: Ionoryllic, Ian Ring Music TheoryIonoryllic
Scale 2941Scale 2941: Laptygic, Ian Ring Music TheoryLaptygic
Scale 3069Scale 3069: Maqam Shawq Afza, Ian Ring Music TheoryMaqam Shawq Afza
Scale 2301Scale 2301: Bydyllic, Ian Ring Music TheoryBydyllic
Scale 2557Scale 2557: Dothygic, Ian Ring Music TheoryDothygic
Scale 3325Scale 3325: Mixolygic, Ian Ring Music TheoryMixolygic
Scale 3837Scale 3837: Minor Pentatonic With Leading Tones, Ian Ring Music TheoryMinor Pentatonic With Leading Tones
Scale 765Scale 765, Ian Ring Music Theory
Scale 1789Scale 1789: Blues Enneatonic II, Ian Ring Music TheoryBlues Enneatonic II

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.