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Scale 361: "Bocritonic"

Scale 361: Bocritonic, 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
Bocritonic

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

5 (pentatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,3,5,6,8}

Forte Number

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

5-25

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: 721

Hemitonia

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

1 (unhemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

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 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.

4

Prime Form

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

no
prime: 301

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

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

[3, 2, 1, 2, 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.

<1, 2, 3, 1, 2, 1>

Interval Spectrum

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

p2mn3s2dt

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> = {1,2,3,4}
<2> = {3,5,6,7}
<3> = {5,6,7,9}
<4> = {8,9,10,11}

Spectra Variation

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

2.8

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.

2.049

Polygon Perimeter

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

5.664

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

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 TriadsG♯{8,0,3}210.67
Minor Triadsfm{5,8,0}121
Diminished Triads{0,3,6}121
Parsimonious Voice Leading Between Common Triads of Scale 361. Created by Ian Ring ©2019 G# G# c°->G# fm fm fm->G#

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesG♯
Peripheral Verticesc°, fm

Modes

Modes are the rotational transformation of this scale. Scale 361 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 557
Scale 557: Raga Abhogi, Ian Ring Music TheoryRaga Abhogi
3rd mode:
Scale 1163
Scale 1163: Raga Rukmangi, Ian Ring Music TheoryRaga Rukmangi
4th mode:
Scale 2629
Scale 2629: Raga Shubravarni, Ian Ring Music TheoryRaga Shubravarni
5th mode:
Scale 1681
Scale 1681: Raga Valaji, Ian Ring Music TheoryRaga Valaji

Prime

The prime form of this scale is Scale 301

Scale 301Scale 301: Raga Audav Tukhari, Ian Ring Music TheoryRaga Audav Tukhari

Complement

The pentatonic modal family [361, 557, 1163, 2629, 1681] (Forte: 5-25) is the complement of the heptatonic modal family [733, 1207, 1769, 1867, 2651, 2981, 3373] (Forte: 7-25)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 361 is 721

Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 361 is chiral, and its enantiomorph is scale 721

Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri

Transformations:

T0 361  T0I 721
T1 722  T1I 1442
T2 1444  T2I 2884
T3 2888  T3I 1673
T4 1681  T4I 3346
T5 3362  T5I 2597
T6 2629  T6I 1099
T7 1163  T7I 2198
T8 2326  T8I 301
T9 557  T9I 602
T10 1114  T10I 1204
T11 2228  T11I 2408

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 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 365Scale 365: Marimic, Ian Ring Music TheoryMarimic
Scale 353Scale 353, Ian Ring Music Theory
Scale 357Scale 357: Banitonic, Ian Ring Music TheoryBanitonic
Scale 369Scale 369: Laditonic, Ian Ring Music TheoryLaditonic
Scale 377Scale 377: Kathimic, Ian Ring Music TheoryKathimic
Scale 329Scale 329: Mynic 2, Ian Ring Music TheoryMynic 2
Scale 345Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic
Scale 297Scale 297: Mynic, Ian Ring Music TheoryMynic
Scale 425Scale 425: Raga Kokil Pancham, Ian Ring Music TheoryRaga Kokil Pancham
Scale 489Scale 489: Phrathimic, Ian Ring Music TheoryPhrathimic
Scale 105Scale 105, Ian Ring Music Theory
Scale 233Scale 233, Ian Ring Music Theory
Scale 617Scale 617: Katycritonic, Ian Ring Music TheoryKatycritonic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 1385Scale 1385: Phracrimic, Ian Ring Music TheoryPhracrimic
Scale 2409Scale 2409: Zacrimic, Ian Ring Music TheoryZacrimic

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. 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.