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Scale 3443: "Verdi's Scala Enigmatica"

Scale 3443: Verdi's Scala Enigmatica, 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

Named After Composers
Verdi's Scala Enigmatica

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Mugham Saränc
Mugham Saränc
Zeitler
Epathyllic
Dozenal
VOSIAN

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

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

Forte Number

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

8-16

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: 2519

Hemitonia

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

5 (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.

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.

7

Prime Form

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

no
prime: 943

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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.

[5, 5, 4, 5, 6, 3]

Interval Spectrum

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

p6m5n4s5d5t3

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

Spectra Variation

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

1.75

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.

2.616

Polygon Perimeter

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

6.002

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 TriadsC♯{1,5,8}331.56
E{4,8,11}252.33
F♯{6,10,1}152.67
Minor Triadsc♯m{1,4,8}331.67
fm{5,8,0}331.67
a♯m{10,1,5}341.89
Augmented TriadsC+{0,4,8}341.78
Diminished Triads{5,8,11}242.22
a♯°{10,1,4}242
Parsimonious Voice Leading Between Common Triads of Scale 3443. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E fm fm C+->fm C# C# c#m->C# a#° a#° c#m->a#° C#->fm a#m a#m C#->a#m E->f° f°->fm F# F# F#->a#m 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
Radius3
Self-Centeredno
Central Verticesc♯m, C♯, fm
Peripheral VerticesE, F♯

Modes

Modes are the rotational transformation of this scale. Scale 3443 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3769
Scale 3769: Aeracryllic, Ian Ring Music TheoryAeracryllic
3rd mode:
Scale 983
Scale 983: Epygyllic, Ian Ring Music TheoryEpygyllic
4th mode:
Scale 2539
Scale 2539: Half-Diminished Bebop, Ian Ring Music TheoryHalf-Diminished Bebop
5th mode:
Scale 3317
Scale 3317: Lanyllic, Ian Ring Music TheoryLanyllic
6th mode:
Scale 1853
Scale 1853: Phrynyllic, Ian Ring Music TheoryPhrynyllic
7th mode:
Scale 1487
Scale 1487: Lycryllic, Ian Ring Music TheoryLycryllic
8th mode:
Scale 2791
Scale 2791: Ionyptyllic, Ian Ring Music TheoryIonyptyllic

Prime

The prime form of this scale is Scale 943

Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic

Complement

The octatonic modal family [3443, 3769, 983, 2539, 3317, 1853, 1487, 2791] (Forte: 8-16) is the complement of the tetratonic modal family [163, 389, 1121, 2129] (Forte: 4-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3443 is 2519

Scale 2519Scale 2519: Dathyllic, Ian Ring Music TheoryDathyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3443 is chiral, and its enantiomorph is scale 2519

Scale 2519Scale 2519: Dathyllic, Ian Ring Music TheoryDathyllic

Transformations:

T0 3443  T0I 2519
T1 2791  T1I 943
T2 1487  T2I 1886
T3 2974  T3I 3772
T4 1853  T4I 3449
T5 3706  T5I 2803
T6 3317  T6I 1511
T7 2539  T7I 3022
T8 983  T8I 1949
T9 1966  T9I 3898
T10 3932  T10I 3701
T11 3769  T11I 3307

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 3441Scale 3441: Thacrian, Ian Ring Music TheoryThacrian
Scale 3445Scale 3445: Messiaen Mode 6 Inverse, Ian Ring Music TheoryMessiaen Mode 6 Inverse
Scale 3447Scale 3447: Kynygic, Ian Ring Music TheoryKynygic
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 3427Scale 3427: Zacrian, Ian Ring Music TheoryZacrian
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3379Scale 3379: Verdi's Scala Enigmatica Descending, Ian Ring Music TheoryVerdi's Scala Enigmatica Descending
Scale 3507Scale 3507: Maqam Hijaz, Ian Ring Music TheoryMaqam Hijaz
Scale 3571Scale 3571: Dyrygic, Ian Ring Music TheoryDyrygic
Scale 3187Scale 3187: Koptian, Ian Ring Music TheoryKoptian
Scale 3315Scale 3315: Tcherepnin Octatonic Mode 1, Ian Ring Music TheoryTcherepnin Octatonic Mode 1
Scale 3699Scale 3699: Aeolylyllic, Ian Ring Music TheoryAeolylyllic
Scale 3955Scale 3955: Galygic, Ian Ring Music TheoryGalygic
Scale 2419Scale 2419: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 2931Scale 2931: Zathyllic, Ian Ring Music TheoryZathyllic
Scale 1395Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.