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Scale 247: "Bopian"

Scale 247: Bopian, 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

Dozenal
Bopian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,4,5,6,7}

Forte Number

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

7-5

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: 3553

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.

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.

6

Prime Form

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

no
prime: 239

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, 2, 1, 1, 1, 5]

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, 3, 3, 4, 2>

Interval Spectrum

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

p4m3n3s4d5t2

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

Spectra Variation

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

3.429

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.

1.933

Polygon Perimeter

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

5.52

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.

(55, 20, 84)

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}110.5
Diminished Triadsc♯°{1,4,7}110.5

The following pitch classes are not present in any of the common triads: {2,5,6}

Parsimonious Voice Leading Between Common Triads of Scale 247. Created by Ian Ring ©2019 C C c#° c#° C->c#°

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 247 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2171
Scale 2171: Negian, Ian Ring Music TheoryNegian
3rd mode:
Scale 3133
Scale 3133: Tosian, Ian Ring Music TheoryTosian
4th mode:
Scale 1807
Scale 1807: Larian, Ian Ring Music TheoryLarian
5th mode:
Scale 2951
Scale 2951: Silian, Ian Ring Music TheorySilian
6th mode:
Scale 3523
Scale 3523, Ian Ring Music Theory
7th mode:
Scale 3809
Scale 3809: Yelian, Ian Ring Music TheoryYelian

Prime

The prime form of this scale is Scale 239

Scale 239Scale 239: Bikian, Ian Ring Music TheoryBikian

Complement

The heptatonic modal family [247, 2171, 3133, 1807, 2951, 3523, 3809] (Forte: 7-5) is the complement of the pentatonic modal family [143, 481, 2119, 3107, 3601] (Forte: 5-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 247 is 3553

Scale 3553Scale 3553: Wehian, Ian Ring Music TheoryWehian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 247 is chiral, and its enantiomorph is scale 3553

Scale 3553Scale 3553: Wehian, Ian Ring Music TheoryWehian

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> 247       T0I <11,0> 3553
T1 <1,1> 494      T1I <11,1> 3011
T2 <1,2> 988      T2I <11,2> 1927
T3 <1,3> 1976      T3I <11,3> 3854
T4 <1,4> 3952      T4I <11,4> 3613
T5 <1,5> 3809      T5I <11,5> 3131
T6 <1,6> 3523      T6I <11,6> 2167
T7 <1,7> 2951      T7I <11,7> 239
T8 <1,8> 1807      T8I <11,8> 478
T9 <1,9> 3614      T9I <11,9> 956
T10 <1,10> 3133      T10I <11,10> 1912
T11 <1,11> 2171      T11I <11,11> 3824
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3427      T0MI <7,0> 2263
T1M <5,1> 2759      T1MI <7,1> 431
T2M <5,2> 1423      T2MI <7,2> 862
T3M <5,3> 2846      T3MI <7,3> 1724
T4M <5,4> 1597      T4MI <7,4> 3448
T5M <5,5> 3194      T5MI <7,5> 2801
T6M <5,6> 2293      T6MI <7,6> 1507
T7M <5,7> 491      T7MI <7,7> 3014
T8M <5,8> 982      T8MI <7,8> 1933
T9M <5,9> 1964      T9MI <7,9> 3866
T10M <5,10> 3928      T10MI <7,10> 3637
T11M <5,11> 3761      T11MI <7,11> 3179

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 245Scale 245: Raga Dipak, Ian Ring Music TheoryRaga Dipak
Scale 243Scale 243: Bomian, Ian Ring Music TheoryBomian
Scale 251Scale 251: Borian, Ian Ring Music TheoryBorian
Scale 255Scale 255: Chromatic Octamode, Ian Ring Music TheoryChromatic Octamode
Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian
Scale 239Scale 239: Bikian, Ian Ring Music TheoryBikian
Scale 215Scale 215: Bivian, Ian Ring Music TheoryBivian
Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian
Scale 119Scale 119: Smoian, Ian Ring Music TheorySmoian
Scale 375Scale 375: Sodian, Ian Ring Music TheorySodian
Scale 503Scale 503: Thoptyllic, Ian Ring Music TheoryThoptyllic
Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic
Scale 1271Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic
Scale 2295Scale 2295: Kogyllic, Ian Ring Music TheoryKogyllic

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.