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Scale 631: "Zygian"

Scale 631: Zygian, 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

Zeitler
Zygian
Dozenal
Duqian

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

{0,1,2,4,5,6,9}

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-Z17

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.

[3]

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.

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.

yes

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, 1, 2, 1, 1, 3, 3]

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, 5, 4, 1>

Interval Spectrum

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

p4m5n4s3d4t

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

[6]

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.

(26, 33, 96)

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 TriadsD{2,6,9}242
F{5,9,0}331.5
A{9,1,4}231.75
Minor Triadsdm{2,5,9}231.75
f♯m{6,9,1}331.5
am{9,0,4}242
Augmented TriadsC♯+{1,5,9}421.25
Diminished Triadsf♯°{6,9,0}231.75
Parsimonious Voice Leading Between Common Triads of Scale 631. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m A A C#+->A D D dm->D D->f#m f#° f#° F->f#° am am F->am f#°->f#m am->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♯+
Peripheral VerticesD, am

Modes

Modes are the rotational transformation of this scale. Scale 631 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2363
Scale 2363: Kataptian, Ian Ring Music TheoryKataptian
3rd mode:
Scale 3229
Scale 3229: Aeolaptian, Ian Ring Music TheoryAeolaptian
4th mode:
Scale 1831
Scale 1831: Pothian, Ian Ring Music TheoryPothian
5th mode:
Scale 2963
Scale 2963: Bygian, Ian Ring Music TheoryBygian
6th mode:
Scale 3529
Scale 3529: Stalian, Ian Ring Music TheoryStalian
7th mode:
Scale 953
Scale 953: Mela Yagapriya, Ian Ring Music TheoryMela Yagapriya

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [631, 2363, 3229, 1831, 2963, 3529, 953] (Forte: 7-Z17) is the complement of the pentatonic modal family [283, 433, 1571, 2189, 2833] (Forte: 5-Z17)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 631 is 3529

Scale 3529Scale 3529: Stalian, Ian Ring Music TheoryStalian

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> 631       T0I <11,0> 3529
T1 <1,1> 1262      T1I <11,1> 2963
T2 <1,2> 2524      T2I <11,2> 1831
T3 <1,3> 953      T3I <11,3> 3662
T4 <1,4> 1906      T4I <11,4> 3229
T5 <1,5> 3812      T5I <11,5> 2363
T6 <1,6> 3529      T6I <11,6> 631
T7 <1,7> 2963      T7I <11,7> 1262
T8 <1,8> 1831      T8I <11,8> 2524
T9 <1,9> 3662      T9I <11,9> 953
T10 <1,10> 3229      T10I <11,10> 1906
T11 <1,11> 2363      T11I <11,11> 3812
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1891      T0MI <7,0> 2269
T1M <5,1> 3782      T1MI <7,1> 443
T2M <5,2> 3469      T2MI <7,2> 886
T3M <5,3> 2843      T3MI <7,3> 1772
T4M <5,4> 1591      T4MI <7,4> 3544
T5M <5,5> 3182      T5MI <7,5> 2993
T6M <5,6> 2269      T6MI <7,6> 1891
T7M <5,7> 443      T7MI <7,7> 3782
T8M <5,8> 886      T8MI <7,8> 3469
T9M <5,9> 1772      T9MI <7,9> 2843
T10M <5,10> 3544      T10MI <7,10> 1591
T11M <5,11> 2993      T11MI <7,11> 3182

The transformations that map this set to itself are: T0, T6I

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 629Scale 629: Aeronimic, Ian Ring Music TheoryAeronimic
Scale 627Scale 627: Mogimic, Ian Ring Music TheoryMogimic
Scale 635Scale 635: Epolian, Ian Ring Music TheoryEpolian
Scale 639Scale 639: Ionaryllic, Ian Ring Music TheoryIonaryllic
Scale 615Scale 615: Schoenberg Hexachord, Ian Ring Music TheorySchoenberg Hexachord
Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian
Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic
Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
Scale 695Scale 695: Sarian, Ian Ring Music TheorySarian
Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic
Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic
Scale 119Scale 119: Smoian, Ian Ring Music TheorySmoian
Scale 375Scale 375: Sodian, Ian Ring Music TheorySodian
Scale 1143Scale 1143: Styrian, Ian Ring Music TheoryStyrian
Scale 1655Scale 1655: Katygyllic, Ian Ring Music TheoryKatygyllic
Scale 2679Scale 2679: Rathyllic, Ian Ring Music TheoryRathyllic

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.