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Scale 3961: "Zathygic"

Scale 3961: Zathygic, 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

41161837294116105072918310504116183
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
Zathygic

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

{0,3,4,5,6,8,9,10,11}

Forte Number

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

9-5

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.

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: 991

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

7 (multihemitonic)

Cohemitonia

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

5 (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: 991

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.

[3, 1, 1, 1, 2, 1, 1, 1, 1] 9

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.

<7, 6, 6, 6, 7, 4>

Interval Spectrum

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

p7m6n6s6d7t4

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> = {1,2,3}
<2> = {2,3,4}
<3> = {3,4,5}
<4> = {4,5,6}
<5> = {6,7,8}
<6> = {7,8,9}
<7> = {8,9,10}
<8> = {9,10,11}

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.

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.

2.683

Polygon Perimeter

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

6.038

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

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 TriadsE{4,8,11}342.21
F{5,9,0}342.29
G♯{8,0,3}442.07
B{11,3,6}342.36
Minor Triadsd♯m{3,6,10}242.64
fm{5,8,0}342.21
g♯m{8,11,3}342.14
am{9,0,4}342.21
Augmented TriadsC+{0,4,8}442
Diminished Triads{0,3,6}242.43
d♯°{3,6,9}242.71
{5,8,11}242.57
f♯°{6,9,0}242.57
{9,0,3}242.43
Parsimonious Voice Leading Between Common Triads of Scale 3961. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ E E C+->E fm fm C+->fm C+->G# am am C+->am d#° d#° d#m d#m d#°->d#m f#° f#° d#°->f#° d#m->B E->f° g#m g#m E->g#m f°->fm F F fm->F F->f#° F->am g#m->G# g#m->B G#->a° a°->am

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 3961 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 1007
Scale 1007: Epitygic, Ian Ring Music TheoryEpitygic
3rd mode:
Scale 2551
Scale 2551: Thocrygic, Ian Ring Music TheoryThocrygic
4th mode:
Scale 3323
Scale 3323: Lacrygic, Ian Ring Music TheoryLacrygic
5th mode:
Scale 3709
Scale 3709: Katynygic, Ian Ring Music TheoryKatynygic
6th mode:
Scale 1951
Scale 1951: Marygic, Ian Ring Music TheoryMarygic
7th mode:
Scale 3023
Scale 3023: Mothygic, Ian Ring Music TheoryMothygic
8th mode:
Scale 3559
Scale 3559: Thophygic, Ian Ring Music TheoryThophygic
9th mode:
Scale 3827
Scale 3827: Bodygic, Ian Ring Music TheoryBodygic

Prime

The prime form of this scale is Scale 991

Scale 991Scale 991: Aeolygic, Ian Ring Music TheoryAeolygic

Complement

The enneatonic modal family [3961, 1007, 2551, 3323, 3709, 1951, 3023, 3559, 3827] (Forte: 9-5) is the complement of the tritonic modal family [67, 193, 2081] (Forte: 3-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3961 is 991

Scale 991Scale 991: Aeolygic, Ian Ring Music TheoryAeolygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3961 is chiral, and its enantiomorph is scale 991

Scale 991Scale 991: Aeolygic, Ian Ring Music TheoryAeolygic

Transformations:

T0 3961  T0I 991
T1 3827  T1I 1982
T2 3559  T2I 3964
T3 3023  T3I 3833
T4 1951  T4I 3571
T5 3902  T5I 3047
T6 3709  T6I 1999
T7 3323  T7I 3998
T8 2551  T8I 3901
T9 1007  T9I 3707
T10 2014  T10I 3319
T11 4028  T11I 2543

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 3963Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian
Scale 3965Scale 3965: Messiaen Mode 7 Inverse, Ian Ring Music TheoryMessiaen Mode 7 Inverse
Scale 3953Scale 3953: Thagyllic, Ian Ring Music TheoryThagyllic
Scale 3957Scale 3957: Porygic, Ian Ring Music TheoryPorygic
Scale 3945Scale 3945: Lydyllic, Ian Ring Music TheoryLydyllic
Scale 3929Scale 3929: Aeolothyllic, Ian Ring Music TheoryAeolothyllic
Scale 3897Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic
Scale 4025Scale 4025: Kalygic, Ian Ring Music TheoryKalygic
Scale 4089Scale 4089: Katoryllian, Ian Ring Music TheoryKatoryllian
Scale 3705Scale 3705: Messiaen Mode 4 Inverse, Ian Ring Music TheoryMessiaen Mode 4 Inverse
Scale 3833Scale 3833: Dycrygic, Ian Ring Music TheoryDycrygic
Scale 3449Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
Scale 2937Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic
Scale 1913Scale 1913, Ian Ring Music Theory

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. 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.