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Scale 3825: "Pynyllic"

Scale 3825: Pynyllic, 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
Pynyllic
Dozenal
YIVian

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

 {0,4,5,6,7,9,10,11}

Forte Number

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

 8-6

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]

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.

 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.

 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.

 8

Prime Form

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

 no
prime: 495

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.

 [4, 1, 1, 1, 2, 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, 4, 4, 6, 3>

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, 0, 0.667, 0.5>

Interval Spectrum

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

 p6m4n4s5d6t3

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}
<4> = {5,7}
<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.

 2.5

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.

2.366

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.

 0.216506

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.

 [4]

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.

(60, 47, 124)

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.

0.598

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.

0.367

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}231.5
F{5,9,0}241.83
Minor Triadsem{4,7,11}241.83
am{9,0,4}231.5
Diminished Triadse°{4,7,10}152.5
f♯°{6,9,0}152.5
Parsimonious Voice Leading Between Common Triads of Scale 3825. Created by Ian Ring ©2019 C C em em C->em am am C->am e° e° e°->em F F f#° f#° F->f#° F->am

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, am
Peripheral Verticese°, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 3825 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 495		Scale 495: Bocryllic, Ian Ring Music TheoryBocryllicThis is the prime mode
3rd mode:
Scale 2295		Scale 2295: Kogyllic, Ian Ring Music TheoryKogyllic
4th mode:
Scale 3195		Scale 3195: Raryllic, Ian Ring Music TheoryRaryllic
5th mode:
Scale 3645		Scale 3645: Zycryllic, Ian Ring Music TheoryZycryllic
6th mode:
Scale 1935		Scale 1935: Mycryllic, Ian Ring Music TheoryMycryllic
7th mode:
Scale 3015		Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic
8th mode:
Scale 3555		Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic

Prime

The prime form of this scale is Scale 495

Scale 495Scale 495: Bocryllic, Ian Ring Music TheoryBocryllic

Complement

The octatonic modal family [3825, 495, 2295, 3195, 3645, 1935, 3015, 3555] (Forte: 8-6) is the complement of the tetratonic modal family [135, 225, 2115, 3105] (Forte: 4-6)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3825 is 495

Scale 495Scale 495: Bocryllic, Ian Ring Music TheoryBocryllic

Interval Matrix

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

Contradictions (60)

Ambiguities(47)

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
111000111101113825k = 1h = 1
2300(0[11][11])(0[11][11])3825k = 2h = 3
3300(0[11][11])(0[11][11])3825k = 3h = 3
4300(0[11][11])(0[11][11])3825k = 4h = 3
5300(0[11][11])(0[11][11])3825k = 5h = 3

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

(-0.1875, 0.108253)
Distance from Center0.216506
Angle in degrees

measured clockwise starting from the root.

240
Angle in cents

100 cents = 1 semitone.

800

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> 3825       T0I <11,0> 495
T1 <1,1> 3555      T1I <11,1> 990
T2 <1,2> 3015      T2I <11,2> 1980
T3 <1,3> 1935      T3I <11,3> 3960
T4 <1,4> 3870      T4I <11,4> 3825
T5 <1,5> 3645      T5I <11,5> 3555
T6 <1,6> 3195      T6I <11,6> 3015
T7 <1,7> 2295      T7I <11,7> 1935
T8 <1,8> 495      T8I <11,8> 3870
T9 <1,9> 990      T9I <11,9> 3645
T10 <1,10> 1980      T10I <11,10> 3195
T11 <1,11> 3960      T11I <11,11> 2295
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3015      T0MI <7,0> 3195
T1M <5,1> 1935      T1MI <7,1> 2295
T2M <5,2> 3870      T2MI <7,2> 495
T3M <5,3> 3645      T3MI <7,3> 990
T4M <5,4> 3195      T4MI <7,4> 1980
T5M <5,5> 2295      T5MI <7,5> 3960
T6M <5,6> 495      T6MI <7,6> 3825
T7M <5,7> 990      T7MI <7,7> 3555
T8M <5,8> 1980      T8MI <7,8> 3015
T9M <5,9> 3960      T9MI <7,9> 1935
T10M <5,10> 3825       T10MI <7,10> 3870
T11M <5,11> 3555      T11MI <7,11> 3645

The transformations that map this set to itself are: T0, T4I, T10M, 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 3827Scale 3827: Dorygic, Ian Ring Music TheoryDorygic
Scale 3829Scale 3829: Taishikicho, Ian Ring Music TheoryTaishikicho
Scale 3833Scale 3833: Dycrygic, Ian Ring Music TheoryDycrygic
Scale 3809Scale 3809: YELian, Ian Ring Music TheoryYELian
Scale 3817Scale 3817: Zoryllic, Ian Ring Music TheoryZoryllic
Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
Scale 3761Scale 3761: Raga Madhuri, Ian Ring Music TheoryRaga Madhuri
Scale 3697Scale 3697: Ionarian, Ian Ring Music TheoryIonarian
Scale 3953Scale 3953: Thagyllic, Ian Ring Music TheoryThagyllic
Scale 4081Scale 4081: Nonatonic Chromatic Descending, Ian Ring Music TheoryNonatonic Chromatic Descending
Scale 3313Scale 3313: Aeolacrian, Ian Ring Music TheoryAeolacrian
Scale 3569Scale 3569: Aeoladyllic, Ian Ring Music TheoryAeoladyllic
Scale 2801Scale 2801: Zogian, Ian Ring Music TheoryZogian
Scale 1777Scale 1777: Saptian, Ian Ring Music TheorySaptian

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.