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Scale 3309: "Bycryllic"

Scale 3309: Bycryllic, 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
Bycryllic
Dozenal
UPPian

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

 {0,2,3,5,6,7,10,11}

Forte Number

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

 8-20

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 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: 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, an indicator of maximum hierarchization.

 no

Interval Structure

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

 [2, 1, 2, 1, 1, 3, 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.

 <5, 4, 5, 6, 6, 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, 0.25, 0.667, 0.667, 0>

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 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.

 0.064705

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)

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.

0.805

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.

0.357

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 TriadsD♯{3,7,10}342
G{7,11,2}331.83
A♯{10,2,5}252.5
B{11,3,6}431.67
Minor Triadscm{0,3,7}252.5
d♯m{3,6,10}331.83
gm{7,10,2}342
bm{11,2,6}431.67
Augmented TriadsD+{2,6,10}441.83
D♯+{3,7,11}441.83
Diminished Triadsc°{0,3,6}242.33
b°{11,2,5}242.33
Parsimonious Voice Leading Between Common Triads of Scale 3309. Created by Ian Ring ©2019 c° c° cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ D+ D+ d#m d#m D+->d#m gm gm D+->gm A# A# D+->A# bm bm D+->bm D# D# d#m->D# d#m->B D#->D#+ D#->gm Parsimonious Voice Leading Between Common Triads of Scale 3309. Created by Ian Ring ©2019 G D#+->G D#+->B gm->G G->bm b° b° A#->b° b°->bm bm->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 Verticesd♯m, G, bm, B
Peripheral Verticescm, A♯

Modes

Modes are the rotational transformation of this scale. Scale 3309 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1851		Scale 1851: Zacryllic, Ian Ring Music TheoryZacryllic
3rd mode:
Scale 2973		Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
4th mode:
Scale 1767		Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic
5th mode:
Scale 2931		Scale 2931: Zathyllic, Ian Ring Music TheoryZathyllic
6th mode:
Scale 3513		Scale 3513: Dydyllic, Ian Ring Music TheoryDydyllic
7th mode:
Scale 951		Scale 951: Thogyllic, Ian Ring Music TheoryThogyllicThis is the prime mode
8th mode:
Scale 2523		Scale 2523: Mirage Scale, Ian Ring Music TheoryMirage Scale

Prime

The prime form of this scale is Scale 951

Scale 951Scale 951: Thogyllic, Ian Ring Music TheoryThogyllic

Complement

The octatonic modal family [3309, 1851, 2973, 1767, 2931, 3513, 951, 2523] (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 3309 is 1767

Scale 1767Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic

Interval Matrix

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

Contradictions (4)

Ambiguities(48)

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
111011011100113309k = 1h = 1
2311(0[1][1][1]0)(0[1][1][1]0)3309k = 2h = 3
3311(0[1][1][1]0)(0[1][1][1]0)3309k = 3h = 3
4311(0[1][1][1]0)(0[1][1][1]0)3309k = 4h = 3
5311(0[1][1][1]0)(0[1][1][1]0)3309k = 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

(0.0625, -0.016747)
Distance from Center0.064705
Angle in degrees

measured clockwise starting from the root.

75
Angle in cents

100 cents = 1 semitone.

250

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

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 3311Scale 3311: Mixodygic, Ian Ring Music TheoryMixodygic
Scale 3305Scale 3305: Lydian ♯236, Ian Ring Music TheoryLydian 236
Scale 3307Scale 3307: Boptyllic, Ian Ring Music TheoryBoptyllic
Scale 3301Scale 3301: Lydian ♯36, Ian Ring Music TheoryLydian 36
Scale 3317Scale 3317: Lanyllic, Ian Ring Music TheoryLanyllic
Scale 3325Scale 3325: Epygic, Ian Ring Music TheoryEpygic
Scale 3277Scale 3277: Lydian ♭3♯6, Ian Ring Music TheoryLydian 36
Scale 3293Scale 3293: Saryllic, Ian Ring Music TheorySaryllic
Scale 3245Scale 3245: Melodic ♯6, Ian Ring Music TheoryMelodic 6
Scale 3181Scale 3181: Melodic ♭5♯6, Ian Ring Music TheoryMelodic 56
Scale 3437Scale 3437: Gathyllic, Ian Ring Music TheoryGathyllic
Scale 3565Scale 3565: Aeolorygic, Ian Ring Music TheoryAeolorygic
Scale 3821Scale 3821: Diminishing Nonamode 8th Rotation, Ian Ring Music TheoryDiminishing Nonamode 8th Rotation
Scale 2285Scale 2285: Supine, Ian Ring Music TheorySupine
Scale 2797Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
Scale 1261Scale 1261: Aeolian ♭5𝄫6, Ian Ring Music TheoryAeolian 56

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.