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# Scale 4059: "Zolyllian"

### 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).

Zeitler
Zolyllian
Dozenal
ZUJian

## Analysis

#### Cardinality

Cardinality is the count of how many pitches are in the scale.

10 (decatonic)

#### Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,3,4,6,7,8,9,10,11}

#### Forte Number

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

10-3

#### 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.

[3.5]

#### 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.

8 (multihemitonic)

#### Cohemitonia

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

6 (multicohemitonic)

#### 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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

10

#### Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 1791

#### 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, an indicator of maximum hierarchization.

no

#### Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

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

<8, 8, 9, 8, 8, 4>

#### Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0, 0, 1, 0, 0, 0>

#### Interval Spectrum

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

p8m8n9s8d8t4

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

#### Spectra Variation

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

1.4

#### 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.

2.866

#### Polygon Perimeter

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

6.141

#### Myhill Property

A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.

no

#### Centre of Gravity Distance

When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.

0.141422

#### 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.

[7]

#### 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.

(20, 154, 235)

#### Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.768

#### Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.42

## Generator

This scale has no generator.

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

D♯{3,7,10}452.67
E{4,8,11}352.75
F♯{6,10,1}452.83
G♯{8,0,3}452.67
A{9,1,4}452.75
B{11,3,6}352.75
c♯m{1,4,8}352.75
d♯m{3,6,10}452.75
em{4,7,11}452.67
f♯m{6,9,1}452.83
g♯m{8,11,3}352.75
am{9,0,4}452.67
D♯+{3,7,11}552.5
c♯°{1,4,7}253
d♯°{3,6,9}253
{4,7,10}253
f♯°{6,9,0}253
{7,10,1}253
{9,0,3}253
a♯°{10,1,4}253

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.

Diameter 5 5 yes

## Modes

Modes are the rotational transformation of this scale. Scale 4059 can be rotated to make 9 other scales. The 1st mode is itself.

 2nd mode:Scale 4077 Gothyllian 3rd mode:Scale 2043 Lythyllian 4th mode:Scale 3069 Bacryllian 5th mode:Scale 1791 Aerygyllian This is the prime mode 6th mode:Scale 2943 Dathyllian 7th mode:Scale 3519 Raga Sindhi-Bhairavi 8th mode:Scale 3807 Bagyllian 9th mode:Scale 3951 Mathyllian 10th mode:Scale 4023 Styptyllian

## Prime

The prime form of this scale is Scale 1791

 Scale 1791 Aerygyllian

## Complement

The decatonic modal family [4059, 4077, 2043, 3069, 1791, 2943, 3519, 3807, 3951, 4023] (Forte: 10-3) is the complement of the ditonic modal family [9, 513] (Forte: 2-3)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4059 is 2943

 Scale 2943 Dathyllian

## Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

## Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
11110110111111
22(11)0(11)0(11)(11)(11)
32(11)0(11)0(11)(11)(11)
420110(1111)(1111)
520110(1111)(1111)

## Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center (-0.136603, -0.036603) 0.141422 285 950

## 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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 4059       T0I <11,0> 2943
T1 <1,1> 4023      T1I <11,1> 1791
T2 <1,2> 3951      T2I <11,2> 3582
T3 <1,3> 3807      T3I <11,3> 3069
T4 <1,4> 3519      T4I <11,4> 2043
T5 <1,5> 2943      T5I <11,5> 4086
T6 <1,6> 1791      T6I <11,6> 4077
T7 <1,7> 3582      T7I <11,7> 4059
T8 <1,8> 3069      T8I <11,8> 4023
T9 <1,9> 2043      T9I <11,9> 3951
T10 <1,10> 4086      T10I <11,10> 3807
T11 <1,11> 4077      T11I <11,11> 3519
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3069      T0MI <7,0> 2043
T1M <5,1> 2043      T1MI <7,1> 4086
T2M <5,2> 4086      T2MI <7,2> 4077
T3M <5,3> 4077      T3MI <7,3> 4059
T4M <5,4> 4059       T4MI <7,4> 4023
T5M <5,5> 4023      T5MI <7,5> 3951
T6M <5,6> 3951      T6MI <7,6> 3807
T7M <5,7> 3807      T7MI <7,7> 3519
T8M <5,8> 3519      T8MI <7,8> 2943
T9M <5,9> 2943      T9MI <7,9> 1791
T10M <5,10> 1791      T10MI <7,10> 3582
T11M <5,11> 3582      T11MI <7,11> 3069

The transformations that map this set to itself are: T0, T7I, T4M, T3MI

## 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 4057 Phrygic Scale 4061 Staptyllian Scale 4063 Chromatic Undecamode 7 Scale 4051 Ionilygic Scale 4055 Dagyllian Scale 4043 Phrocrygic Scale 4075 Katyllian Scale 4091 Chromatic Undecamode 10 Scale 3995 Ionygic Scale 4027 Ragyllian Scale 3931 Aerygic Scale 3803 Epidygic Scale 3547 Diminishing Nonamode 9th Rotation Scale 3035 Gocrygic Scale 2011 Raphygic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.