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Scale 3307: "Boptyllic"

Scale 3307: Boptyllic, 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
Boptyllic
Dozenal
Ulkian

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

{0,1,3,5,6,7,10,11}

Forte Number

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

8-16

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: 2791

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.

3 (tricohemitonic)

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.

no
prime: 943

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, 2, 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, 5, 4, 5, 6, 3>

Interval Spectrum

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

p6m5n4s5d5t3

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

1.75

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 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.

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.

(12, 44, 123)

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}331.67
F♯{6,10,1}341.89
B{11,3,6}331.67
Minor Triadscm{0,3,7}252.33
d♯m{3,6,10}331.56
a♯m{10,1,5}152.67
Augmented TriadsD♯+{3,7,11}341.78
Diminished Triads{0,3,6}242.22
{7,10,1}242
Parsimonious Voice Leading Between Common Triads of Scale 3307. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ d#m d#m D# D# d#m->D# F# F# d#m->F# d#m->B D#->D#+ D#->g° D#+->B F#->g° a#m a#m F#->a#m

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, D♯, B
Peripheral Verticescm, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 3307 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3701
Scale 3701: Bagyllic, Ian Ring Music TheoryBagyllic
3rd mode:
Scale 1949
Scale 1949: Mathyllic, Ian Ring Music TheoryMathyllic
4th mode:
Scale 1511
Scale 1511: Styptyllic, Ian Ring Music TheoryStyptyllic
5th mode:
Scale 2803
Scale 2803: Raga Bhatiyar, Ian Ring Music TheoryRaga Bhatiyar
6th mode:
Scale 3449
Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
7th mode:
Scale 943
Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllicThis is the prime mode
8th mode:
Scale 2519
Scale 2519: Dathyllic, Ian Ring Music TheoryDathyllic

Prime

The prime form of this scale is Scale 943

Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic

Complement

The octatonic modal family [3307, 3701, 1949, 1511, 2803, 3449, 943, 2519] (Forte: 8-16) is the complement of the tetratonic modal family [163, 389, 1121, 2129] (Forte: 4-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3307 is 2791

Scale 2791Scale 2791: Mixothyllic, Ian Ring Music TheoryMixothyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3307 is chiral, and its enantiomorph is scale 2791

Scale 2791Scale 2791: Mixothyllic, Ian Ring Music TheoryMixothyllic

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> 3307       T0I <11,0> 2791
T1 <1,1> 2519      T1I <11,1> 1487
T2 <1,2> 943      T2I <11,2> 2974
T3 <1,3> 1886      T3I <11,3> 1853
T4 <1,4> 3772      T4I <11,4> 3706
T5 <1,5> 3449      T5I <11,5> 3317
T6 <1,6> 2803      T6I <11,6> 2539
T7 <1,7> 1511      T7I <11,7> 983
T8 <1,8> 3022      T8I <11,8> 1966
T9 <1,9> 1949      T9I <11,9> 3932
T10 <1,10> 3898      T10I <11,10> 3769
T11 <1,11> 3701      T11I <11,11> 3443
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2287      T0MI <7,0> 3811
T1M <5,1> 479      T1MI <7,1> 3527
T2M <5,2> 958      T2MI <7,2> 2959
T3M <5,3> 1916      T3MI <7,3> 1823
T4M <5,4> 3832      T4MI <7,4> 3646
T5M <5,5> 3569      T5MI <7,5> 3197
T6M <5,6> 3043      T6MI <7,6> 2299
T7M <5,7> 1991      T7MI <7,7> 503
T8M <5,8> 3982      T8MI <7,8> 1006
T9M <5,9> 3869      T9MI <7,9> 2012
T10M <5,10> 3643      T10MI <7,10> 4024
T11M <5,11> 3191      T11MI <7,11> 3953

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 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian
Scale 3309Scale 3309: Bycryllic, Ian Ring Music TheoryBycryllic
Scale 3311Scale 3311: Mixodygic, Ian Ring Music TheoryMixodygic
Scale 3299Scale 3299: Syptian, Ian Ring Music TheorySyptian
Scale 3303Scale 3303: Mylyllic, Ian Ring Music TheoryMylyllic
Scale 3315Scale 3315: Tcherepnin Octatonic Mode 1, Ian Ring Music TheoryTcherepnin Octatonic Mode 1
Scale 3323Scale 3323: Lacrygic, Ian Ring Music TheoryLacrygic
Scale 3275Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani
Scale 3291Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic
Scale 3243Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
Scale 3179Scale 3179: Daptian, Ian Ring Music TheoryDaptian
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3563Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic
Scale 3819Scale 3819: Aeolanygic, Ian Ring Music TheoryAeolanygic
Scale 2283Scale 2283: Aeolyptian, Ian Ring Music TheoryAeolyptian
Scale 2795Scale 2795: Van der Horst Octatonic, Ian Ring Music TheoryVan der Horst Octatonic
Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian

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.