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Scale 3413: "Leading Whole-tone"

Scale 3413: Leading Whole-tone, 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

Western Modern
Leading Whole-tone
Major Locrian
Zeitler
Aeolynian

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

{0,2,4,6,8,10,11}

Forte Number

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

7-33

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.

[5]

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.

no

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

5

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: 1367

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.

[2, 6, 2, 6, 2, 3]

Interval Spectrum

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

p2m6n2s6d2t3

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

Spectra Variation

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

1.429

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.

2.665

Polygon Perimeter

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

6.035

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.

[10]

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".

Proper

Harmonic Chords

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}231.4
Minor Triadsbm{11,2,6}231.4
Augmented TriadsC+{0,4,8}142
D+{2,6,10}142
Diminished Triadsg♯°{8,11,2}221.2
Parsimonious Voice Leading Between Common Triads of Scale 3413. Created by Ian Ring ©2019 C+ C+ E E C+->E D+ D+ bm bm D+->bm g#° g#° E->g#° g#°->bm

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 Verticesg♯°
Peripheral VerticesC+, D+

Modes

Modes are the rotational transformation of this scale. Scale 3413 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1877
Scale 1877: Aeroptian, Ian Ring Music TheoryAeroptian
3rd mode:
Scale 1493
Scale 1493: Lydian Minor, Ian Ring Music TheoryLydian Minor
4th mode:
Scale 1397
Scale 1397: Major Locrian, Ian Ring Music TheoryMajor Locrian
5th mode:
Scale 1373
Scale 1373: Storian, Ian Ring Music TheoryStorian
6th mode:
Scale 1367
Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone InverseThis is the prime mode
7th mode:
Scale 2731
Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major

Prime

The prime form of this scale is Scale 1367

Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse

Complement

The heptatonic modal family [3413, 1877, 1493, 1397, 1373, 1367, 2731] (Forte: 7-33) is the complement of the pentatonic modal family [341, 1109, 1301, 1349, 1361] (Forte: 5-33)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3413 is 1367

Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse

Transformations:

T0 3413  T0I 1367
T1 2731  T1I 2734
T2 1367  T2I 1373
T3 2734  T3I 2746
T4 1373  T4I 1397
T5 2746  T5I 2794
T6 1397  T6I 1493
T7 2794  T7I 2986
T8 1493  T8I 1877
T9 2986  T9I 3754
T10 1877  T10I 3413
T11 3754  T11I 2731

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 3415Scale 3415: Ionaptyllic, Ian Ring Music TheoryIonaptyllic
Scale 3409Scale 3409: Katanimic, Ian Ring Music TheoryKatanimic
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3417Scale 3417: Golian, Ian Ring Music TheoryGolian
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 3397Scale 3397: Sydimic, Ian Ring Music TheorySydimic
Scale 3405Scale 3405: Stynian, Ian Ring Music TheoryStynian
Scale 3429Scale 3429: Marian, Ian Ring Music TheoryMarian
Scale 3445Scale 3445: Messiaen Mode 6 Inverse, Ian Ring Music TheoryMessiaen Mode 6 Inverse
Scale 3349Scale 3349: Aeolocrimic, Ian Ring Music TheoryAeolocrimic
Scale 3381Scale 3381: Katanian, Ian Ring Music TheoryKatanian
Scale 3477Scale 3477: Kyptian, Ian Ring Music TheoryKyptian
Scale 3541Scale 3541: Racryllic, Ian Ring Music TheoryRacryllic
Scale 3157Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic
Scale 3285Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3925Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic
Scale 2389Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
Scale 2901Scale 2901: Lydian Augmented, Ian Ring Music TheoryLydian Augmented
Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone

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.