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Scale 3411: "Enigmatic"

Scale 3411: Enigmatic, 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

Unknown / Unsorted
Enigmatic
Named After Composers
Verdi's Scala Enigmatica Ascending
Zeitler
Ionathian
Dozenal
Viyian

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

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

Forte Number

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

7-24

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: 2391

Hemitonia

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

3 (trihemitonic)

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: 687

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, 2, 2, 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.

<3, 5, 3, 4, 4, 2>

Interval Spectrum

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

p4m4n3s5d3t2

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.

2.549

Polygon Perimeter

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

5.967

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.

(19, 38, 102)

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 TriadsE{4,8,11}142
F♯{6,10,1}142
Minor Triadsc♯m{1,4,8}221.2
Augmented TriadsC+{0,4,8}231.4
Diminished Triadsa♯°{10,1,4}231.4
Parsimonious Voice Leading Between Common Triads of Scale 3411. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E a#° a#° c#m->a#° F# F# F#->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
Radius2
Self-Centeredno
Central Verticesc♯m
Peripheral VerticesE, F♯

Modes

Modes are the rotational transformation of this scale. Scale 3411 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3753
Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian
3rd mode:
Scale 981
Scale 981: Mela Kantamani, Ian Ring Music TheoryMela Kantamani
4th mode:
Scale 1269
Scale 1269: Katythian, Ian Ring Music TheoryKatythian
5th mode:
Scale 1341
Scale 1341: Madian, Ian Ring Music TheoryMadian
6th mode:
Scale 1359
Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
7th mode:
Scale 2727
Scale 2727: Mela Manavati, Ian Ring Music TheoryMela Manavati

Prime

The prime form of this scale is Scale 687

Scale 687Scale 687: Aeolythian, Ian Ring Music TheoryAeolythian

Complement

The heptatonic modal family [3411, 3753, 981, 1269, 1341, 1359, 2727] (Forte: 7-24) is the complement of the pentatonic modal family [171, 1377, 1413, 1557, 2133] (Forte: 5-24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3411 is 2391

Scale 2391Scale 2391: Molian, Ian Ring Music TheoryMolian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3411 is chiral, and its enantiomorph is scale 2391

Scale 2391Scale 2391: Molian, Ian Ring Music TheoryMolian

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> 3411       T0I <11,0> 2391
T1 <1,1> 2727      T1I <11,1> 687
T2 <1,2> 1359      T2I <11,2> 1374
T3 <1,3> 2718      T3I <11,3> 2748
T4 <1,4> 1341      T4I <11,4> 1401
T5 <1,5> 2682      T5I <11,5> 2802
T6 <1,6> 1269      T6I <11,6> 1509
T7 <1,7> 2538      T7I <11,7> 3018
T8 <1,8> 981      T8I <11,8> 1941
T9 <1,9> 1962      T9I <11,9> 3882
T10 <1,10> 3924      T10I <11,10> 3669
T11 <1,11> 3753      T11I <11,11> 3243
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 501      T0MI <7,0> 1521
T1M <5,1> 1002      T1MI <7,1> 3042
T2M <5,2> 2004      T2MI <7,2> 1989
T3M <5,3> 4008      T3MI <7,3> 3978
T4M <5,4> 3921      T4MI <7,4> 3861
T5M <5,5> 3747      T5MI <7,5> 3627
T6M <5,6> 3399      T6MI <7,6> 3159
T7M <5,7> 2703      T7MI <7,7> 2223
T8M <5,8> 1311      T8MI <7,8> 351
T9M <5,9> 2622      T9MI <7,9> 702
T10M <5,10> 1149      T10MI <7,10> 1404
T11M <5,11> 2298      T11MI <7,11> 2808

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 3409Scale 3409: Katanimic, Ian Ring Music TheoryKatanimic
Scale 3413Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone
Scale 3415Scale 3415: Ionaptyllic, Ian Ring Music TheoryIonaptyllic
Scale 3419Scale 3419: Magen Abot 1, Ian Ring Music TheoryMagen Abot 1
Scale 3395Scale 3395: Vepian, Ian Ring Music TheoryVepian
Scale 3403Scale 3403: Bylian, Ian Ring Music TheoryBylian
Scale 3427Scale 3427: Zacrian, Ian Ring Music TheoryZacrian
Scale 3443Scale 3443: Verdi's Scala Enigmatica, Ian Ring Music TheoryVerdi's Scala Enigmatica
Scale 3347Scale 3347: Synimic, Ian Ring Music TheorySynimic
Scale 3379Scale 3379: Verdi's Scala Enigmatica Descending, Ian Ring Music TheoryVerdi's Scala Enigmatica Descending
Scale 3475Scale 3475: Kylian, Ian Ring Music TheoryKylian
Scale 3539Scale 3539: Aeoryllic, Ian Ring Music TheoryAeoryllic
Scale 3155Scale 3155: Ladimic, Ian Ring Music TheoryLadimic
Scale 3283Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari
Scale 3667Scale 3667: Kaptian, Ian Ring Music TheoryKaptian
Scale 3923Scale 3923: Stoptyllic, Ian Ring Music TheoryStoptyllic
Scale 2387Scale 2387: Paptimic, Ian Ring Music TheoryPaptimic
Scale 2899Scale 2899: Kagian, Ian Ring Music TheoryKagian
Scale 1363Scale 1363: Gygimic, Ian Ring Music TheoryGygimic

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.