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Scale 3379: "Verdi's Scala Enigmatica Descending"

Scale 3379: Verdi's Scala Enigmatica Descending, 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 Descending
Zeitler
Sothian
Dozenal
Vefian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

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

Forte Number

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

7-Z18

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: 2455

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

6

Prime Form

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

no
prime: 755

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, 3, 1, 3, 2, 1, 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.

<4, 3, 4, 4, 4, 2>

Interval Spectrum

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

p4m4n4s3d4t2

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

Spectra Variation

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

2.571

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.

2.433

Polygon Perimeter

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

5.899

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.

(23, 34, 100)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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}331.5
E{4,8,11}242
Minor Triadsc♯m{1,4,8}331.5
fm{5,8,0}331.5
a♯m{10,1,5}242
Augmented TriadsC+{0,4,8}331.5
Diminished Triads{5,8,11}242
a♯°{10,1,4}242
Parsimonious Voice Leading Between Common Triads of Scale 3379. 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 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.

Diameter4
Radius3
Self-Centeredno
Central VerticesC+, c♯m, C♯, fm
Peripheral VerticesE, f°, a♯°, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 3379 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3737
Scale 3737: Phrocrian, Ian Ring Music TheoryPhrocrian
3rd mode:
Scale 979
Scale 979: Mela Dhavalambari, Ian Ring Music TheoryMela Dhavalambari
4th mode:
Scale 2537
Scale 2537: Laptian, Ian Ring Music TheoryLaptian
5th mode:
Scale 829
Scale 829: Lygian, Ian Ring Music TheoryLygian
6th mode:
Scale 1231
Scale 1231: Logian, Ian Ring Music TheoryLogian
7th mode:
Scale 2663
Scale 2663: Lalian, Ian Ring Music TheoryLalian

Prime

The prime form of this scale is Scale 755

Scale 755Scale 755: Phrythian, Ian Ring Music TheoryPhrythian

Complement

The heptatonic modal family [3379, 3737, 979, 2537, 829, 1231, 2663] (Forte: 7-Z18) is the complement of the pentatonic modal family [179, 779, 1633, 2137, 2437] (Forte: 5-Z18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3379 is 2455

Scale 2455Scale 2455: Bothian, Ian Ring Music TheoryBothian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3379 is chiral, and its enantiomorph is scale 2455

Scale 2455Scale 2455: Bothian, Ian Ring Music TheoryBothian

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> 3379       T0I <11,0> 2455
T1 <1,1> 2663      T1I <11,1> 815
T2 <1,2> 1231      T2I <11,2> 1630
T3 <1,3> 2462      T3I <11,3> 3260
T4 <1,4> 829      T4I <11,4> 2425
T5 <1,5> 1658      T5I <11,5> 755
T6 <1,6> 3316      T6I <11,6> 1510
T7 <1,7> 2537      T7I <11,7> 3020
T8 <1,8> 979      T8I <11,8> 1945
T9 <1,9> 1958      T9I <11,9> 3890
T10 <1,10> 3916      T10I <11,10> 3685
T11 <1,11> 3737      T11I <11,11> 3275
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 439      T0MI <7,0> 3505
T1M <5,1> 878      T1MI <7,1> 2915
T2M <5,2> 1756      T2MI <7,2> 1735
T3M <5,3> 3512      T3MI <7,3> 3470
T4M <5,4> 2929      T4MI <7,4> 2845
T5M <5,5> 1763      T5MI <7,5> 1595
T6M <5,6> 3526      T6MI <7,6> 3190
T7M <5,7> 2957      T7MI <7,7> 2285
T8M <5,8> 1819      T8MI <7,8> 475
T9M <5,9> 3638      T9MI <7,9> 950
T10M <5,10> 3181      T10MI <7,10> 1900
T11M <5,11> 2267      T11MI <7,11> 3800

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 3377Scale 3377: Phralimic, Ian Ring Music TheoryPhralimic
Scale 3381Scale 3381: Katanian, Ian Ring Music TheoryKatanian
Scale 3383Scale 3383: Zoptyllic, Ian Ring Music TheoryZoptyllic
Scale 3387Scale 3387: Aeryptyllic, Ian Ring Music TheoryAeryptyllic
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic
Scale 3371Scale 3371: Aeolylian, Ian Ring Music TheoryAeolylian
Scale 3347Scale 3347: Synimic, Ian Ring Music TheorySynimic
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3443Scale 3443: Verdi's Scala Enigmatica, Ian Ring Music TheoryVerdi's Scala Enigmatica
Scale 3507Scale 3507: Maqam Hijaz, Ian Ring Music TheoryMaqam Hijaz
Scale 3123Scale 3123: Tomian, Ian Ring Music TheoryTomian
Scale 3251Scale 3251: Mela Hatakambari, Ian Ring Music TheoryMela Hatakambari
Scale 3635Scale 3635: Katygian, Ian Ring Music TheoryKatygian
Scale 3891Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
Scale 2355Scale 2355: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 2867Scale 2867: Socrian, Ian Ring Music TheorySocrian
Scale 1331Scale 1331: Raga Vasantabhairavi, Ian Ring Music TheoryRaga Vasantabhairavi

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.