The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2991: "Zanygic"

Scale 2991: Zanygic, 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
Zanygic
Dozenal
Sukian

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

{0,1,2,3,5,7,8,9,11}

Forte Number

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

9-8

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: 3771

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.

8

Prime Form

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

no
prime: 1503

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, 1, 2, 2, 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, 7, 6, 7, 6, 4>

Interval Spectrum

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

p6m7n6s7d6t4

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}
<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.

1.556

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.

(8, 94, 180)

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♯{1,5,8}342.33
F{5,9,0}342.33
G{7,11,2}342.47
G♯{8,0,3}442.07
Minor Triadscm{0,3,7}242.47
dm{2,5,9}342.47
fm{5,8,0}442.07
g♯m{8,11,3}442.2
Augmented TriadsC♯+{1,5,9}342.4
D♯+{3,7,11}342.4
Diminished Triads{2,5,8}242.67
{5,8,11}242.33
g♯°{8,11,2}242.53
{9,0,3}242.47
{11,2,5}242.53
Parsimonious Voice Leading Between Common Triads of Scale 2991. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F d°->dm dm->b° Parsimonious Voice Leading Between Common Triads of Scale 2991. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m f°->fm f°->g#m fm->F fm->G# F->a° g#° g#° G->g#° G->b° g#°->g#m g#m->G# G#->a°

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 2991 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3543
Scale 3543: Aeolonygic, Ian Ring Music TheoryAeolonygic
3rd mode:
Scale 3819
Scale 3819: Aeolanygic, Ian Ring Music TheoryAeolanygic
4th mode:
Scale 3957
Scale 3957: Porygic, Ian Ring Music TheoryPorygic
5th mode:
Scale 2013
Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic
6th mode:
Scale 1527
Scale 1527: Aeolyrigic, Ian Ring Music TheoryAeolyrigic
7th mode:
Scale 2811
Scale 2811: Barygic, Ian Ring Music TheoryBarygic
8th mode:
Scale 3453
Scale 3453: Katarygic, Ian Ring Music TheoryKatarygic
9th mode:
Scale 1887
Scale 1887: Aerocrygic, Ian Ring Music TheoryAerocrygic

Prime

The prime form of this scale is Scale 1503

Scale 1503Scale 1503: Padygic, Ian Ring Music TheoryPadygic

Complement

The enneatonic modal family [2991, 3543, 3819, 3957, 2013, 1527, 2811, 3453, 1887] (Forte: 9-8) is the complement of the tritonic modal family [69, 321, 1041] (Forte: 3-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2991 is 3771

Scale 3771Scale 3771: Stophygic, Ian Ring Music TheoryStophygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2991 is chiral, and its enantiomorph is scale 3771

Scale 3771Scale 3771: Stophygic, Ian Ring Music TheoryStophygic

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> 2991       T0I <11,0> 3771
T1 <1,1> 1887      T1I <11,1> 3447
T2 <1,2> 3774      T2I <11,2> 2799
T3 <1,3> 3453      T3I <11,3> 1503
T4 <1,4> 2811      T4I <11,4> 3006
T5 <1,5> 1527      T5I <11,5> 1917
T6 <1,6> 3054      T6I <11,6> 3834
T7 <1,7> 2013      T7I <11,7> 3573
T8 <1,8> 4026      T8I <11,8> 3051
T9 <1,9> 3957      T9I <11,9> 2007
T10 <1,10> 3819      T10I <11,10> 4014
T11 <1,11> 3543      T11I <11,11> 3933
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3771      T0MI <7,0> 2991
T1M <5,1> 3447      T1MI <7,1> 1887
T2M <5,2> 2799      T2MI <7,2> 3774
T3M <5,3> 1503      T3MI <7,3> 3453
T4M <5,4> 3006      T4MI <7,4> 2811
T5M <5,5> 1917      T5MI <7,5> 1527
T6M <5,6> 3834      T6MI <7,6> 3054
T7M <5,7> 3573      T7MI <7,7> 2013
T8M <5,8> 3051      T8MI <7,8> 4026
T9M <5,9> 2007      T9MI <7,9> 3957
T10M <5,10> 4014      T10MI <7,10> 3819
T11M <5,11> 3933      T11MI <7,11> 3543

The transformations that map this set to itself are: T0, T0MI

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 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 2983Scale 2983: Zythyllic, Ian Ring Music TheoryZythyllic
Scale 2999Scale 2999: Diminishing Nonamode, Ian Ring Music TheoryDiminishing Nonamode
Scale 3007Scale 3007: Zyryllian, Ian Ring Music TheoryZyryllian
Scale 2959Scale 2959: Dygyllic, Ian Ring Music TheoryDygyllic
Scale 2975Scale 2975: Aeroptygic, Ian Ring Music TheoryAeroptygic
Scale 3023Scale 3023: Mothygic, Ian Ring Music TheoryMothygic
Scale 3055Scale 3055: Messiaen Mode 7, Ian Ring Music TheoryMessiaen Mode 7
Scale 2863Scale 2863: Aerogyllic, Ian Ring Music TheoryAerogyllic
Scale 2927Scale 2927: Rodygic, Ian Ring Music TheoryRodygic
Scale 2735Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic
Scale 2479Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed
Scale 3503Scale 3503: Zyphygic, Ian Ring Music TheoryZyphygic
Scale 4015Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic
Scale 1967Scale 1967: Diatonic Dorian Mixed, Ian Ring Music TheoryDiatonic Dorian Mixed

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.