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Scale 2931: "Zathyllic"

Scale 2931: Zathyllic, 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
Zathyllic
Dozenal
Siyian

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

{0,1,4,5,6,8,9,11}

Forte Number

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

8-20

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.

[2.5]

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.

5 (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.

2

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: 951

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, 1, 1, 2, 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.

<5, 4, 5, 6, 6, 2>

Interval Spectrum

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

p6m6n5s4d5t2

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

1.5

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.

[5]

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.

(4, 48, 126)

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 TriadsC♯{1,5,8}331.83
E{4,8,11}252.5
F{5,9,0}431.67
A{9,1,4}342
Minor Triadsc♯m{1,4,8}342
fm{5,8,0}431.67
f♯m{6,9,1}252.5
am{9,0,4}331.83
Augmented TriadsC+{0,4,8}441.83
C♯+{1,5,9}441.83
Diminished Triads{5,8,11}242.33
f♯°{6,9,0}242.33
Parsimonious Voice Leading Between Common Triads of Scale 2931. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E fm fm C+->fm am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F f#m f#m C#+->f#m C#+->A E->f° f°->fm fm->F f#° f#° F->f#° 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.

Diameter5
Radius3
Self-Centeredno
Central VerticesC♯, fm, F, am
Peripheral VerticesE, f♯m

Modes

Modes are the rotational transformation of this scale. Scale 2931 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3513
Scale 3513: Dydyllic, Ian Ring Music TheoryDydyllic
3rd mode:
Scale 951
Scale 951: Thogyllic, Ian Ring Music TheoryThogyllicThis is the prime mode
4th mode:
Scale 2523
Scale 2523: Mirage Scale, Ian Ring Music TheoryMirage Scale
5th mode:
Scale 3309
Scale 3309: Bycryllic, Ian Ring Music TheoryBycryllic
6th mode:
Scale 1851
Scale 1851: Zacryllic, Ian Ring Music TheoryZacryllic
7th mode:
Scale 2973
Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
8th mode:
Scale 1767
Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic

Prime

The prime form of this scale is Scale 951

Scale 951Scale 951: Thogyllic, Ian Ring Music TheoryThogyllic

Complement

The octatonic modal family [2931, 3513, 951, 2523, 3309, 1851, 2973, 1767] (Forte: 8-20) is the complement of the tetratonic modal family [291, 393, 561, 2193] (Forte: 4-20)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2931 is 2523

Scale 2523Scale 2523: Mirage Scale, Ian Ring Music TheoryMirage Scale

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> 2931       T0I <11,0> 2523
T1 <1,1> 1767      T1I <11,1> 951
T2 <1,2> 3534      T2I <11,2> 1902
T3 <1,3> 2973      T3I <11,3> 3804
T4 <1,4> 1851      T4I <11,4> 3513
T5 <1,5> 3702      T5I <11,5> 2931
T6 <1,6> 3309      T6I <11,6> 1767
T7 <1,7> 2523      T7I <11,7> 3534
T8 <1,8> 951      T8I <11,8> 2973
T9 <1,9> 1902      T9I <11,9> 1851
T10 <1,10> 3804      T10I <11,10> 3702
T11 <1,11> 3513      T11I <11,11> 3309
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1011      T0MI <7,0> 2553
T1M <5,1> 2022      T1MI <7,1> 1011
T2M <5,2> 4044      T2MI <7,2> 2022
T3M <5,3> 3993      T3MI <7,3> 4044
T4M <5,4> 3891      T4MI <7,4> 3993
T5M <5,5> 3687      T5MI <7,5> 3891
T6M <5,6> 3279      T6MI <7,6> 3687
T7M <5,7> 2463      T7MI <7,7> 3279
T8M <5,8> 831      T8MI <7,8> 2463
T9M <5,9> 1662      T9MI <7,9> 831
T10M <5,10> 3324      T10MI <7,10> 1662
T11M <5,11> 2553      T11MI <7,11> 3324

The transformations that map this set to itself are: T0, T5I

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 2929Scale 2929: Aeolathian, Ian Ring Music TheoryAeolathian
Scale 2933Scale 2933: Sizian, Ian Ring Music TheorySizian
Scale 2935Scale 2935: Modygic, Ian Ring Music TheoryModygic
Scale 2939Scale 2939: Goptygic, Ian Ring Music TheoryGoptygic
Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2899Scale 2899: Kagian, Ian Ring Music TheoryKagian
Scale 2867Scale 2867: Socrian, Ian Ring Music TheorySocrian
Scale 2995Scale 2995: Raga Saurashtra, Ian Ring Music TheoryRaga Saurashtra
Scale 3059Scale 3059: Madygic, Ian Ring Music TheoryMadygic
Scale 2675Scale 2675: Chromatic Lydian, Ian Ring Music TheoryChromatic Lydian
Scale 2803Scale 2803: Raga Bhatiyar, Ian Ring Music TheoryRaga Bhatiyar
Scale 2419Scale 2419: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 3443Scale 3443: Verdi's Scala Enigmatica, Ian Ring Music TheoryVerdi's Scala Enigmatica
Scale 3955Scale 3955: Pothygic, Ian Ring Music TheoryPothygic
Scale 883Scale 883: Ralian, Ian Ring Music TheoryRalian
Scale 1907Scale 1907: Lynyllic, Ian Ring Music TheoryLynyllic

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.