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Scale 3053: "Zycrygic"

Scale 3053: Zycrygic, 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
Zycrygic
Dozenal
Tavian

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

{0,2,3,5,6,7,8,9,11}

Forte Number

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

9-10

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.

3 (tricohemitonic)

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.

8

Prime Form

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

no
prime: 1759

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.

[2, 1, 2, 1, 1, 1, 1, 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, 6, 8, 6, 6, 4>

Interval Spectrum

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

p6m6n8s6d6t4

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> = {3,4,5}
<4> = {4,5,6}
<5> = {6,7,8}
<6> = {7,8,9}
<7> = {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.

1.333

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.

[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.

(4, 84, 168)

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}442.37
F{5,9,0}442.42
G{7,11,2}342.47
G♯{8,0,3}442.32
B{11,3,6}442.37
Minor Triadscm{0,3,7}342.47
dm{2,5,9}442.42
fm{5,8,0}442.37
g♯m{8,11,3}442.37
bm{11,2,6}442.32
Augmented TriadsD♯+{3,7,11}442.32
Diminished Triads{0,3,6}242.74
{2,5,8}242.63
d♯°{3,6,9}242.63
{5,8,11}242.63
f♯°{6,9,0}242.63
g♯°{8,11,2}242.74
{9,0,3}242.58
{11,2,5}242.58
Parsimonious Voice Leading Between Common Triads of Scale 3053. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ G# G# cm->G# dm dm d°->dm fm fm d°->fm D D dm->D F F dm->F dm->b° d#° d#° D->d#° f#° f#° D->f#° bm bm D->bm d#°->B Parsimonious Voice Leading Between Common Triads of Scale 3053. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m D#+->B f°->fm f°->g#m fm->F fm->G# F->f#° F->a° g#° g#° G->g#° G->bm g#°->g#m g#m->G# G#->a° b°->bm bm->B

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3053 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 1787
Scale 1787: Mycrygic, Ian Ring Music TheoryMycrygic
3rd mode:
Scale 2941
Scale 2941: Laptygic, Ian Ring Music TheoryLaptygic
4th mode:
Scale 1759
Scale 1759: Pylygic, Ian Ring Music TheoryPylygicThis is the prime mode
5th mode:
Scale 2927
Scale 2927: Rodygic, Ian Ring Music TheoryRodygic
6th mode:
Scale 3511
Scale 3511: Epolygic, Ian Ring Music TheoryEpolygic
7th mode:
Scale 3803
Scale 3803: Epidygic, Ian Ring Music TheoryEpidygic
8th mode:
Scale 3949
Scale 3949: Koptygic, Ian Ring Music TheoryKoptygic
9th mode:
Scale 2011
Scale 2011: Raphygic, Ian Ring Music TheoryRaphygic

Prime

The prime form of this scale is Scale 1759

Scale 1759Scale 1759: Pylygic, Ian Ring Music TheoryPylygic

Complement

The enneatonic modal family [3053, 1787, 2941, 1759, 2927, 3511, 3803, 3949, 2011] (Forte: 9-10) is the complement of the tritonic modal family [73, 521, 577] (Forte: 3-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3053 is 1787

Scale 1787Scale 1787: Mycrygic, Ian Ring Music TheoryMycrygic

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> 3053       T0I <11,0> 1787
T1 <1,1> 2011      T1I <11,1> 3574
T2 <1,2> 4022      T2I <11,2> 3053
T3 <1,3> 3949      T3I <11,3> 2011
T4 <1,4> 3803      T4I <11,4> 4022
T5 <1,5> 3511      T5I <11,5> 3949
T6 <1,6> 2927      T6I <11,6> 3803
T7 <1,7> 1759      T7I <11,7> 3511
T8 <1,8> 3518      T8I <11,8> 2927
T9 <1,9> 2941      T9I <11,9> 1759
T10 <1,10> 1787      T10I <11,10> 3518
T11 <1,11> 3574      T11I <11,11> 2941
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3803      T0MI <7,0> 2927
T1M <5,1> 3511      T1MI <7,1> 1759
T2M <5,2> 2927      T2MI <7,2> 3518
T3M <5,3> 1759      T3MI <7,3> 2941
T4M <5,4> 3518      T4MI <7,4> 1787
T5M <5,5> 2941      T5MI <7,5> 3574
T6M <5,6> 1787      T6MI <7,6> 3053
T7M <5,7> 3574      T7MI <7,7> 2011
T8M <5,8> 3053       T8MI <7,8> 4022
T9M <5,9> 2011      T9MI <7,9> 3949
T10M <5,10> 4022      T10MI <7,10> 3803
T11M <5,11> 3949      T11MI <7,11> 3511

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 3055Scale 3055: Messiaen Mode 7, Ian Ring Music TheoryMessiaen Mode 7
Scale 3049Scale 3049: Phrydyllic, Ian Ring Music TheoryPhrydyllic
Scale 3051Scale 3051: Stalygic, Ian Ring Music TheoryStalygic
Scale 3045Scale 3045: Raptyllic, Ian Ring Music TheoryRaptyllic
Scale 3061Scale 3061: Apinygic, Ian Ring Music TheoryApinygic
Scale 3069Scale 3069: Maqam Shawq Afza, Ian Ring Music TheoryMaqam Shawq Afza
Scale 3021Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
Scale 3037Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale
Scale 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 2925Scale 2925: Diminished, Ian Ring Music TheoryDiminished
Scale 2797Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
Scale 2541Scale 2541: Algerian, Ian Ring Music TheoryAlgerian
Scale 3565Scale 3565: Aeolorygic, Ian Ring Music TheoryAeolorygic
Scale 4077Scale 4077: Gothyllian, Ian Ring Music TheoryGothyllian
Scale 1005Scale 1005: Radyllic, Ian Ring Music TheoryRadyllic
Scale 2029Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi

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.