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Scale 495: "Bocryllic"

Scale 495: Bocryllic, 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
Bocryllic

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

{0,1,2,3,5,6,7,8}

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.

[4]

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 does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

7

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

yes

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.

no

Interval Structure

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

[1, 1, 1, 2, 1, 1, 1, 4]

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>

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 Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

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.

[8]

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)

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♯{1,5,8}241.83
G♯{8,0,3}231.5
Minor Triadscm{0,3,7}241.83
fm{5,8,0}231.5
Diminished Triads{0,3,6}152.5
{2,5,8}152.5
Parsimonious Voice Leading Between Common Triads of Scale 495. Created by Ian Ring ©2019 cm cm c°->cm G# G# cm->G# C# C# C#->d° fm fm C#->fm fm->G#

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 Verticesfm, G♯
Peripheral Verticesc°, d°

Modes

Modes are the rotational transformation of this scale. Scale 495 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2295
Scale 2295: Kogyllic, Ian Ring Music TheoryKogyllic
3rd mode:
Scale 3195
Scale 3195: Raryllic, Ian Ring Music TheoryRaryllic
4th mode:
Scale 3645
Scale 3645: Zycryllic, Ian Ring Music TheoryZycryllic
5th mode:
Scale 1935
Scale 1935: Mycryllic, Ian Ring Music TheoryMycryllic
6th mode:
Scale 3015
Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic
7th mode:
Scale 3555
Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
8th mode:
Scale 3825
Scale 3825: Pynyllic, Ian Ring Music TheoryPynyllic

Prime

This is the prime form of this scale.

Complement

The octatonic modal family [495, 2295, 3195, 3645, 1935, 3015, 3555, 3825] (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 495 is 3825

Scale 3825Scale 3825: Pynyllic, Ian Ring Music TheoryPynyllic

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

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 495       T0I <11,0> 3825
T1 <1,1> 990      T1I <11,1> 3555
T2 <1,2> 1980      T2I <11,2> 3015
T3 <1,3> 3960      T3I <11,3> 1935
T4 <1,4> 3825      T4I <11,4> 3870
T5 <1,5> 3555      T5I <11,5> 3645
T6 <1,6> 3015      T6I <11,6> 3195
T7 <1,7> 1935      T7I <11,7> 2295
T8 <1,8> 3870      T8I <11,8> 495
T9 <1,9> 3645      T9I <11,9> 990
T10 <1,10> 3195      T10I <11,10> 1980
T11 <1,11> 2295      T11I <11,11> 3960
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3195      T0MI <7,0> 3015
T1M <5,1> 2295      T1MI <7,1> 1935
T2M <5,2> 495       T2MI <7,2> 3870
T3M <5,3> 990      T3MI <7,3> 3645
T4M <5,4> 1980      T4MI <7,4> 3195
T5M <5,5> 3960      T5MI <7,5> 2295
T6M <5,6> 3825      T6MI <7,6> 495
T7M <5,7> 3555      T7MI <7,7> 990
T8M <5,8> 3015      T8MI <7,8> 1980
T9M <5,9> 1935      T9MI <7,9> 3960
T10M <5,10> 3870      T10MI <7,10> 3825
T11M <5,11> 3645      T11MI <7,11> 3555

The transformations that map this set to itself are: T0, T8I, T2M, 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 493Scale 493: Rygian, Ian Ring Music TheoryRygian
Scale 491Scale 491: Aeolyrian, Ian Ring Music TheoryAeolyrian
Scale 487Scale 487: Dynian, Ian Ring Music TheoryDynian
Scale 503Scale 503: Thoptyllic, Ian Ring Music TheoryThoptyllic
Scale 511Scale 511: Chromatic Nonamode, Ian Ring Music TheoryChromatic Nonamode
Scale 463Scale 463: Zythian, Ian Ring Music TheoryZythian
Scale 479Scale 479: Kocryllic, Ian Ring Music TheoryKocryllic
Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian
Scale 367Scale 367: Aerodian, Ian Ring Music TheoryAerodian
Scale 239Scale 239: Bikian, Ian Ring Music TheoryBikian
Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian
Scale 1007Scale 1007: Epitygic, Ian Ring Music TheoryEpitygic
Scale 1519Scale 1519: Locrian/Aeolian Mixed, Ian Ring Music TheoryLocrian/Aeolian Mixed
Scale 2543Scale 2543: Dydygic, Ian Ring Music TheoryDydygic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

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 the Carnatic ragas.