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Scale 3557: "Wekian"

Scale 3557: Wekian, 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

Dozenal
Wekian

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

{0,2,5,6,7,8,10,11}

Forte Number

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

8-Z29

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: 1271

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.

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.

7

Prime Form

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

no
prime: 751

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.

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

<5, 5, 5, 5, 5, 3>

Interval Spectrum

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

p5m5n5s5d5t3

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

Spectra Variation

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

2.25

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

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.

(30, 60, 141)

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 TriadsG{7,11,2}341.9
A♯{10,2,5}341.9
Minor Triadsfm{5,8,0}242.3
gm{7,10,2}242.1
bm{11,2,6}341.9
Augmented TriadsD+{2,6,10}341.9
Diminished Triads{2,5,8}242.1
{5,8,11}242.3
g♯°{8,11,2}242.1
{11,2,5}242.1
Parsimonious Voice Leading Between Common Triads of Scale 3557. Created by Ian Ring ©2019 fm fm d°->fm A# A# d°->A# D+ D+ gm gm D+->gm D+->A# bm bm D+->bm f°->fm g#° g#° f°->g#° Parsimonious Voice Leading Between Common Triads of Scale 3557. Created by Ian Ring ©2019 G gm->G G->g#° G->bm A#->b° b°->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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3557 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1913
Scale 1913: Lofian, Ian Ring Music TheoryLofian
3rd mode:
Scale 751
Scale 751: Epoian, Ian Ring Music TheoryEpoianThis is the prime mode
4th mode:
Scale 2423
Scale 2423: Otuian, Ian Ring Music TheoryOtuian
5th mode:
Scale 3259
Scale 3259: Ulian, Ian Ring Music TheoryUlian
6th mode:
Scale 3677
Scale 3677: Xafian, Ian Ring Music TheoryXafian
7th mode:
Scale 1943
Scale 1943: Luxian, Ian Ring Music TheoryLuxian
8th mode:
Scale 3019
Scale 3019: Subian, Ian Ring Music TheorySubian

Prime

The prime form of this scale is Scale 751

Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian

Complement

The octatonic modal family [3557, 1913, 751, 2423, 3259, 3677, 1943, 3019] (Forte: 8-Z29) is the complement of the tetratonic modal family [139, 353, 1553, 2117] (Forte: 4-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3557 is 1271

Scale 1271Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3557 is chiral, and its enantiomorph is scale 1271

Scale 1271Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic

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> 3557       T0I <11,0> 1271
T1 <1,1> 3019      T1I <11,1> 2542
T2 <1,2> 1943      T2I <11,2> 989
T3 <1,3> 3886      T3I <11,3> 1978
T4 <1,4> 3677      T4I <11,4> 3956
T5 <1,5> 3259      T5I <11,5> 3817
T6 <1,6> 2423      T6I <11,6> 3539
T7 <1,7> 751      T7I <11,7> 2983
T8 <1,8> 1502      T8I <11,8> 1871
T9 <1,9> 3004      T9I <11,9> 3742
T10 <1,10> 1913      T10I <11,10> 3389
T11 <1,11> 3826      T11I <11,11> 2683
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3287      T0MI <7,0> 3431
T1M <5,1> 2479      T1MI <7,1> 2767
T2M <5,2> 863      T2MI <7,2> 1439
T3M <5,3> 1726      T3MI <7,3> 2878
T4M <5,4> 3452      T4MI <7,4> 1661
T5M <5,5> 2809      T5MI <7,5> 3322
T6M <5,6> 1523      T6MI <7,6> 2549
T7M <5,7> 3046      T7MI <7,7> 1003
T8M <5,8> 1997      T8MI <7,8> 2006
T9M <5,9> 3994      T9MI <7,9> 4012
T10M <5,10> 3893      T10MI <7,10> 3929
T11M <5,11> 3691      T11MI <7,11> 3763

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 3559Scale 3559: Thophygic, Ian Ring Music TheoryThophygic
Scale 3553Scale 3553: Wehian, Ian Ring Music TheoryWehian
Scale 3555Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
Scale 3561Scale 3561: Pothyllic, Ian Ring Music TheoryPothyllic
Scale 3565Scale 3565: Aeolorygic, Ian Ring Music TheoryAeolorygic
Scale 3573Scale 3573: Kaptygic, Ian Ring Music TheoryKaptygic
Scale 3525Scale 3525: Zocrian, Ian Ring Music TheoryZocrian
Scale 3541Scale 3541: Racryllic, Ian Ring Music TheoryRacryllic
Scale 3493Scale 3493: Rathian, Ian Ring Music TheoryRathian
Scale 3429Scale 3429: Marian, Ian Ring Music TheoryMarian
Scale 3301Scale 3301: Chromatic Mixolydian Inverse, Ian Ring Music TheoryChromatic Mixolydian Inverse
Scale 3813Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
Scale 4069Scale 4069: Starygic, Ian Ring Music TheoryStarygic
Scale 2533Scale 2533: Podian, Ian Ring Music TheoryPodian
Scale 3045Scale 3045: Raptyllic, Ian Ring Music TheoryRaptyllic
Scale 1509Scale 1509: Ragian, Ian Ring Music TheoryRagian

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.