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Scale 3639: "Paptyllic"

Scale 3639: Paptyllic, 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
Paptyllic
Dozenal
WUJian

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

 {0,1,2,4,5,9,10,11}

Forte Number

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

 8-4

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: 3471

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

 6 (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: 447

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, 2, 1, 4, 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.

 <6, 5, 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.

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

Interval Spectrum

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

 p5m5n5s5d6t2

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

Spectra Variation

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

 3

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.

2.366

Polygon Perimeter

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

5.838

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.

 0.299271

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.

(75, 56, 136)

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.

0.508

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.

0.306

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 TriadsF{5,9,0}242
A{9,1,4}341.78
A♯{10,2,5}341.89
Minor Triadsdm{2,5,9}231.78
am{9,0,4}252.33
a♯m{10,1,5}331.56
Augmented TriadsC♯+{1,5,9}431.44
Diminished Triadsa♯°{10,1,4}231.89
b°{11,2,5}152.67
Parsimonious Voice Leading Between Common Triads of Scale 3639. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F A A C#+->A a#m a#m C#+->a#m A# A# dm->A# am am F->am am->A a#° a#° A->a#° a#°->a#m a#m->A# b° b° A#->b°

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♯+, dm, a♯°, a♯m
Peripheral Verticesam, b°

Modes

Modes are the rotational transformation of this scale. Scale 3639 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3867		Scale 3867: Storyllic, Ian Ring Music TheoryStoryllic
3rd mode:
Scale 3981		Scale 3981: Phrycryllic, Ian Ring Music TheoryPhrycryllic
4th mode:
Scale 2019		Scale 2019: Palyllic, Ian Ring Music TheoryPalyllic
5th mode:
Scale 3057		Scale 3057: Phroryllic, Ian Ring Music TheoryPhroryllic
6th mode:
Scale 447		Scale 447: Thyphyllic, Ian Ring Music TheoryThyphyllicThis is the prime mode
7th mode:
Scale 2271		Scale 2271: Poptyllic, Ian Ring Music TheoryPoptyllic
8th mode:
Scale 3183		Scale 3183: Mixonyllic, Ian Ring Music TheoryMixonyllic

Prime

The prime form of this scale is Scale 447

Scale 447Scale 447: Thyphyllic, Ian Ring Music TheoryThyphyllic

Complement

The octatonic modal family [3639, 3867, 3981, 2019, 3057, 447, 2271, 3183] (Forte: 8-4) is the complement of the tetratonic modal family [39, 897, 2067, 3081] (Forte: 4-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3639 is 3471

Scale 3471Scale 3471: Gyryllic, Ian Ring Music TheoryGyryllic

Interval Matrix

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

Contradictions (75)

Ambiguities(56)

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
111110110001113639k = 1h = 1
211110110001113639k = 2h = 1
311110110001113639k = 3h = 1
42(11)0(11)000(11)(11)3639k = 4h = 2
52(11)0(11)000(11)(11)3639k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3639 is chiral, and its enantiomorph is scale 3471

Scale 3471Scale 3471: Gyryllic, Ian Ring Music TheoryGyryllic

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

(0.045753, -0.295753)
Distance from Center0.299271
Angle in degrees

measured clockwise starting from the root.

8.794
Angle in cents

100 cents = 1 semitone.

29.313333

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> 3639       T0I <11,0> 3471
T1 <1,1> 3183      T1I <11,1> 2847
T2 <1,2> 2271      T2I <11,2> 1599
T3 <1,3> 447      T3I <11,3> 3198
T4 <1,4> 894      T4I <11,4> 2301
T5 <1,5> 1788      T5I <11,5> 507
T6 <1,6> 3576      T6I <11,6> 1014
T7 <1,7> 3057      T7I <11,7> 2028
T8 <1,8> 2019      T8I <11,8> 4056
T9 <1,9> 4038      T9I <11,9> 4017
T10 <1,10> 3981      T10I <11,10> 3939
T11 <1,11> 3867      T11I <11,11> 3783
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1959      T0MI <7,0> 3261
T1M <5,1> 3918      T1MI <7,1> 2427
T2M <5,2> 3741      T2MI <7,2> 759
T3M <5,3> 3387      T3MI <7,3> 1518
T4M <5,4> 2679      T4MI <7,4> 3036
T5M <5,5> 1263      T5MI <7,5> 1977
T6M <5,6> 2526      T6MI <7,6> 3954
T7M <5,7> 957      T7MI <7,7> 3813
T8M <5,8> 1914      T8MI <7,8> 3531
T9M <5,9> 3828      T9MI <7,9> 2967
T10M <5,10> 3561      T10MI <7,10> 1839
T11M <5,11> 3027      T11MI <7,11> 3678

The transformations that map this set to itself are: T0

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 3637Scale 3637: Ionian ♯6𝄪5, Ian Ring Music TheoryIonian 65
Scale 3635Scale 3635: Ionian ♭2♯6𝄪5, Ian Ring Music TheoryIonian 265
Scale 3643Scale 3643: Kydyllic, Ian Ring Music TheoryKydyllic
Scale 3647Scale 3647: Nonatonic Chromatic 4, Ian Ring Music TheoryNonatonic Chromatic 4
Scale 3623Scale 3623: Ionian ♭2♯6𝄫3𝄪5, Ian Ring Music TheoryIonian 2635
Scale 3631Scale 3631: Gydyllic, Ian Ring Music TheoryGydyllic
Scale 3607Scale 3607: Alt ♮7♯6𝄪5𝄫3, Ian Ring Music TheoryAlt 7653
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 3703Scale 3703: Katalygic, Ian Ring Music TheoryKatalygic
Scale 3767Scale 3767: Chromatic Bebop, Ian Ring Music TheoryChromatic Bebop
Scale 3895Scale 3895: Eparygic, Ian Ring Music TheoryEparygic
Scale 3127Scale 3127: Alt ♮7♯6𝄫35, Ian Ring Music TheoryAlt 7635
Scale 3383Scale 3383: Daptyllic, Ian Ring Music TheoryDaptyllic
Scale 2615Scale 2615: Alt ♮67𝄫35, Ian Ring Music TheoryAlt 6735
Scale 1591Scale 1591: Alt ♮6𝄫35, Ian Ring Music TheoryAlt 635

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.