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Scale 2719: "Zocryllic"

Scale 2719: Zocryllic, 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
Zocryllic
Dozenal
REYian

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

 {0,1,2,3,4,7,9,11}

Forte Number

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

 8-11

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: 3883

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.

 4 (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.

 3

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.

 8

Prime Form

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

 no
prime: 703

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, 1, 1, 1, 3, 2, 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, 6, 5, 5, 5, 2>

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.

<0.333, 0.667, 0.25, 0.333, 0.333, 0>

Interval Spectrum

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

 p5m5n5s6d5t2

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

 2.75

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 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.

 0.279508

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.

(59, 69, 149)

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.

0.519

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.

0.24

Generator

This scale has no generator.

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{0,4,7}431.44
G{7,11,2}152.67
A{9,1,4}252.33
Minor Triadscm{0,3,7}331.56
em{4,7,11}231.78
am{9,0,4}341.78
Augmented TriadsD♯+{3,7,11}341.89
Diminished Triadsc♯°{1,4,7}242
a°{9,0,3}231.89
Parsimonious Voice Leading Between Common Triads of Scale 2719. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ a° a° cm->a° c#° c#° C->c#° em em C->em am am C->am A A c#°->A D#+->em Parsimonious Voice Leading Between Common Triads of Scale 2719. Created by Ian Ring ©2019 G D#+->G a°->am 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 Verticescm, C, em, a°
Peripheral VerticesG, A

Modes

Modes are the rotational transformation of this scale. Scale 2719 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3407		Scale 3407: Katocryllic, Ian Ring Music TheoryKatocryllic
3rd mode:
Scale 3751		Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
4th mode:
Scale 3923		Scale 3923: Stoptyllic, Ian Ring Music TheoryStoptyllic
5th mode:
Scale 4009		Scale 4009: Phranyllic, Ian Ring Music TheoryPhranyllic
6th mode:
Scale 1013		Scale 1013: Stydyllic, Ian Ring Music TheoryStydyllic
7th mode:
Scale 1277		Scale 1277: Zadyllic, Ian Ring Music TheoryZadyllic
8th mode:
Scale 1343		Scale 1343: Zalyllic, Ian Ring Music TheoryZalyllic

Prime

The prime form of this scale is Scale 703

Scale 703Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllic

Complement

The octatonic modal family [2719, 3407, 3751, 3923, 4009, 1013, 1277, 1343] (Forte: 8-11) is the complement of the tetratonic modal family [43, 1409, 1541, 2069] (Forte: 4-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2719 is 3883

Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic

Interval Matrix

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

Contradictions (59)

Ambiguities(69)

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
111111100101012719k = 1h = 1
211111100101012719k = 2h = 1
311111100101012719k = 3h = 1
42(1)(1)(1)(1)(1)00(1)0(1)0(1)2719k = 4h = 2
52(1)(1)(1)(1)(1)00(1)0(1)0(1)2719k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2719 is chiral, and its enantiomorph is scale 3883

Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic

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

(0.154006, -0.233253)
Distance from Center0.279508
Angle in degrees

measured clockwise starting from the root.

33.435
Angle in cents

100 cents = 1 semitone.

111.45

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> 2719       T0I <11,0> 3883
T1 <1,1> 1343      T1I <11,1> 3671
T2 <1,2> 2686      T2I <11,2> 3247
T3 <1,3> 1277      T3I <11,3> 2399
T4 <1,4> 2554      T4I <11,4> 703
T5 <1,5> 1013      T5I <11,5> 1406
T6 <1,6> 2026      T6I <11,6> 2812
T7 <1,7> 4052      T7I <11,7> 1529
T8 <1,8> 4009      T8I <11,8> 3058
T9 <1,9> 3923      T9I <11,9> 2021
T10 <1,10> 3751      T10I <11,10> 4042
T11 <1,11> 3407      T11I <11,11> 3989
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4009      T0MI <7,0> 703
T1M <5,1> 3923      T1MI <7,1> 1406
T2M <5,2> 3751      T2MI <7,2> 2812
T3M <5,3> 3407      T3MI <7,3> 1529
T4M <5,4> 2719       T4MI <7,4> 3058
T5M <5,5> 1343      T5MI <7,5> 2021
T6M <5,6> 2686      T6MI <7,6> 4042
T7M <5,7> 1277      T7MI <7,7> 3989
T8M <5,8> 2554      T8MI <7,8> 3883
T9M <5,9> 1013      T9MI <7,9> 3671
T10M <5,10> 2026      T10MI <7,10> 3247
T11M <5,11> 4052      T11MI <7,11> 2399

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

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 2717Scale 2717: Melodic ♭4, Ian Ring Music TheoryMelodic 4
Scale 2715Scale 2715: Neapolitan Major ♭4, Ian Ring Music TheoryNeapolitan Major 4
Scale 2711Scale 2711: Erratic, Ian Ring Music TheoryErratic
Scale 2703Scale 2703: Erratic minor, Ian Ring Music TheoryErratic minor
Scale 2735Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic
Scale 2751Scale 2751: Sylygic, Ian Ring Music TheorySylygic
Scale 2783Scale 2783: Gothygic, Ian Ring Music TheoryGothygic
Scale 2591Scale 2591: Alt ♮67𝄫34𝄫♭5, Ian Ring Music TheoryAlt 67345
Scale 2655Scale 2655: Thocryllic, Ian Ring Music TheoryThocryllic
Scale 2847Scale 2847: Phracryllic, Ian Ring Music TheoryPhracryllic
Scale 2975Scale 2975: Gaptygic, Ian Ring Music TheoryGaptygic
Scale 2207Scale 2207: Alt ♮7𝄫346𝄫♭5, Ian Ring Music TheoryAlt 73465
Scale 2463Scale 2463: Ionathyllic, Ian Ring Music TheoryIonathyllic
Scale 3231Scale 3231: Kataptyllic, Ian Ring Music TheoryKataptyllic
Scale 3743Scale 3743: Thadygic, Ian Ring Music TheoryThadygic
Scale 671Scale 671: Infra-Alt 5, Ian Ring Music TheoryInfra-Alt 5
Scale 1695Scale 1695: Phrodyllic, Ian Ring Music TheoryPhrodyllic

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, and ©2023 Robert Bedwell, 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.