The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2961: "Bygimic"

Scale 2961: Bygimic, 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
Bygimic
Dozenal
Sorian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,4,7,8,9,11}

Forte Number

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

6-14

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: 315

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

5

Prime Form

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

no
prime: 315

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.

[4, 3, 1, 1, 2, 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.

<3, 2, 3, 4, 3, 0>

Interval Spectrum

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

p3m4n3s2d3

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

Spectra Variation

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

3.333

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.

2.116

Polygon Perimeter

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

5.699

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.

(19, 22, 65)

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}221.2
E{4,8,11}221.2
Minor Triadsem{4,7,11}231.4
am{9,0,4}131.6
Augmented TriadsC+{0,4,8}321
Parsimonious Voice Leading Between Common Triads of Scale 2961. Created by Ian Ring ©2019 C C C+ C+ C->C+ em em C->em E E C+->E am am C+->am em->E

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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC, C+, E
Peripheral Verticesem, am

Modes

Modes are the rotational transformation of this scale. Scale 2961 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 441
Scale 441: Thycrimic, Ian Ring Music TheoryThycrimic
3rd mode:
Scale 567
Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
4th mode:
Scale 2331
Scale 2331: Dylimic, Ian Ring Music TheoryDylimic
5th mode:
Scale 3213
Scale 3213: Eponimic, Ian Ring Music TheoryEponimic
6th mode:
Scale 1827
Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic

Prime

The prime form of this scale is Scale 315

Scale 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic

Complement

The hexatonic modal family [2961, 441, 567, 2331, 3213, 1827] (Forte: 6-14) is the complement of the hexatonic modal family [315, 945, 1575, 2205, 2835, 3465] (Forte: 6-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2961 is 315

Scale 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2961 is chiral, and its enantiomorph is scale 315

Scale 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic

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> 2961       T0I <11,0> 315
T1 <1,1> 1827      T1I <11,1> 630
T2 <1,2> 3654      T2I <11,2> 1260
T3 <1,3> 3213      T3I <11,3> 2520
T4 <1,4> 2331      T4I <11,4> 945
T5 <1,5> 567      T5I <11,5> 1890
T6 <1,6> 1134      T6I <11,6> 3780
T7 <1,7> 2268      T7I <11,7> 3465
T8 <1,8> 441      T8I <11,8> 2835
T9 <1,9> 882      T9I <11,9> 1575
T10 <1,10> 1764      T10I <11,10> 3150
T11 <1,11> 3528      T11I <11,11> 2205
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2961       T0MI <7,0> 315
T1M <5,1> 1827      T1MI <7,1> 630
T2M <5,2> 3654      T2MI <7,2> 1260
T3M <5,3> 3213      T3MI <7,3> 2520
T4M <5,4> 2331      T4MI <7,4> 945
T5M <5,5> 567      T5MI <7,5> 1890
T6M <5,6> 1134      T6MI <7,6> 3780
T7M <5,7> 2268      T7MI <7,7> 3465
T8M <5,8> 441      T8MI <7,8> 2835
T9M <5,9> 882      T9MI <7,9> 1575
T10M <5,10> 1764      T10MI <7,10> 3150
T11M <5,11> 3528      T11MI <7,11> 2205

The transformations that map this set to itself are: T0, T0M

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 2963Scale 2963: Bygian, Ian Ring Music TheoryBygian
Scale 2965Scale 2965: Darian, Ian Ring Music TheoryDarian
Scale 2969Scale 2969: Tholian, Ian Ring Music TheoryTholian
Scale 2945Scale 2945: Sihian, Ian Ring Music TheorySihian
Scale 2953Scale 2953: Ionylimic, Ian Ring Music TheoryIonylimic
Scale 2977Scale 2977: Sobian, Ian Ring Music TheorySobian
Scale 2993Scale 2993: Stythian, Ian Ring Music TheoryStythian
Scale 3025Scale 3025: Epycrian, Ian Ring Music TheoryEpycrian
Scale 2833Scale 2833: Dolitonic, Ian Ring Music TheoryDolitonic
Scale 2897Scale 2897: Rycrimic, Ian Ring Music TheoryRycrimic
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 2449Scale 2449: Zacritonic, Ian Ring Music TheoryZacritonic
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 3985Scale 3985: Thadian, Ian Ring Music TheoryThadian
Scale 913Scale 913: Aeolyritonic, Ian Ring Music TheoryAeolyritonic
Scale 1937Scale 1937: Galimic, Ian Ring Music TheoryGalimic

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.