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Scale 3013: "Thynian"

Scale 3013: Thynian, 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
Thynian
Dozenal
Suxian

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

{0,2,6,7,8,9,11}

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-Z36

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: 1147

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: 367

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, 4, 1, 1, 1, 2, 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, 4, 4, 3, 4, 2>

Interval Spectrum

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

p4m3n4s4d4t2

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> = {1,2,4}
<2> = {2,3,5,6}
<3> = {3,4,5,6,7}
<4> = {5,6,7,8,9}
<5> = {6,7,9,10}
<6> = {8,10,11}

Spectra Variation

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

3.143

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.

2.299

Polygon Perimeter

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

5.803

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.

(38, 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 TriadsD{2,6,9}231.4
G{7,11,2}231.4
Minor Triadsbm{11,2,6}221.2
Diminished Triadsf♯°{6,9,0}142
g♯°{8,11,2}142
Parsimonious Voice Leading Between Common Triads of Scale 3013. Created by Ian Ring ©2019 D D f#° f#° D->f#° bm bm D->bm Parsimonious Voice Leading Between Common Triads of Scale 3013. Created by Ian Ring ©2019 G g#° g#° G->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 Verticesbm
Peripheral Verticesf♯°, g♯°

Modes

Modes are the rotational transformation of this scale. Scale 3013 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1777
Scale 1777: Saptian, Ian Ring Music TheorySaptian
3rd mode:
Scale 367
Scale 367: Aerodian, Ian Ring Music TheoryAerodianThis is the prime mode
4th mode:
Scale 2231
Scale 2231: Macrian, Ian Ring Music TheoryMacrian
5th mode:
Scale 3163
Scale 3163: Rogian, Ian Ring Music TheoryRogian
6th mode:
Scale 3629
Scale 3629: Boptian, Ian Ring Music TheoryBoptian
7th mode:
Scale 1931
Scale 1931: Stogian, Ian Ring Music TheoryStogian

Prime

The prime form of this scale is Scale 367

Scale 367Scale 367: Aerodian, Ian Ring Music TheoryAerodian

Complement

The heptatonic modal family [3013, 1777, 367, 2231, 3163, 3629, 1931] (Forte: 7-Z36) is the complement of the pentatonic modal family [151, 737, 1801, 2123, 3109] (Forte: 5-Z36)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3013 is 1147

Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3013 is chiral, and its enantiomorph is scale 1147

Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian

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> 3013       T0I <11,0> 1147
T1 <1,1> 1931      T1I <11,1> 2294
T2 <1,2> 3862      T2I <11,2> 493
T3 <1,3> 3629      T3I <11,3> 986
T4 <1,4> 3163      T4I <11,4> 1972
T5 <1,5> 2231      T5I <11,5> 3944
T6 <1,6> 367      T6I <11,6> 3793
T7 <1,7> 734      T7I <11,7> 3491
T8 <1,8> 1468      T8I <11,8> 2887
T9 <1,9> 2936      T9I <11,9> 1679
T10 <1,10> 1777      T10I <11,10> 3358
T11 <1,11> 3554      T11I <11,11> 2621
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3793      T0MI <7,0> 367
T1M <5,1> 3491      T1MI <7,1> 734
T2M <5,2> 2887      T2MI <7,2> 1468
T3M <5,3> 1679      T3MI <7,3> 2936
T4M <5,4> 3358      T4MI <7,4> 1777
T5M <5,5> 2621      T5MI <7,5> 3554
T6M <5,6> 1147      T6MI <7,6> 3013
T7M <5,7> 2294      T7MI <7,7> 1931
T8M <5,8> 493      T8MI <7,8> 3862
T9M <5,9> 986      T9MI <7,9> 3629
T10M <5,10> 1972      T10MI <7,10> 3163
T11M <5,11> 3944      T11MI <7,11> 2231

The transformations that map this set to itself are: T0, T6MI

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 3015Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic
Scale 3009Scale 3009: Suvian, Ian Ring Music TheorySuvian
Scale 3011Scale 3011, Ian Ring Music Theory
Scale 3017Scale 3017: Gacrian, Ian Ring Music TheoryGacrian
Scale 3021Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
Scale 3045Scale 3045: Raptyllic, Ian Ring Music TheoryRaptyllic
Scale 2949Scale 2949: Sikian, Ian Ring Music TheorySikian
Scale 2981Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
Scale 2885Scale 2885: Byrimic, Ian Ring Music TheoryByrimic
Scale 2757Scale 2757: Raga Nishadi, Ian Ring Music TheoryRaga Nishadi
Scale 2501Scale 2501: Ralimic, Ian Ring Music TheoryRalimic
Scale 3525Scale 3525: Zocrian, Ian Ring Music TheoryZocrian
Scale 4037Scale 4037: Ionyllic, Ian Ring Music TheoryIonyllic
Scale 965Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic
Scale 1989Scale 1989: Dydian, Ian Ring Music TheoryDydian

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.