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Scale 2977: "SOBian"

Scale 2977: SOBian, 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).

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

 {0,5,7,8,9,11}

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-Z10

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: 187

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.

 4

Modes

Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 6

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 187

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, an indicator of maximum hierarchization.

 no

Interval Structure

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

 [5, 2, 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, 3, 3, 3, 2, 1>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.6, 0.5, 0.6, 0.25, 0.4, 0.333>

Interval Spectrum

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

 p2m3n3s3d3t

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

Spectra Variation

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

 3.667

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.

1.866

Polygon Perimeter

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

5.485

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(29, 7, 55)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.446

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.267

Generator

This scale has no generator.

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 TriadsF{5,9,0}121
Minor Triadsfm{5,8,0}210.67
Diminished Triadsf°{5,8,11}121

The following pitch classes are not present in any of the common triads: {7}

Parsimonious Voice Leading Between Common Triads of Scale 2977. Created by Ian Ring ©2019 f° f° fm fm f°->fm F F fm->F

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 Verticesfm
Peripheral Verticesf°, F

Modes

Modes are the rotational transformation of this scale. Scale 2977 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 221		Scale 221: BIYian, Ian Ring Music TheoryBIYian
3rd mode:
Scale 1079		Scale 1079: GOWian, Ian Ring Music TheoryGOWian
4th mode:
Scale 2587		Scale 2587: PUTian, Ian Ring Music TheoryPUTian
5th mode:
Scale 3341		Scale 3341: VAHian, Ian Ring Music TheoryVAHian
6th mode:
Scale 1859		Scale 1859: LIXian, Ian Ring Music TheoryLIXian

Prime

The prime form of this scale is Scale 187

Scale 187Scale 187: BEDian, Ian Ring Music TheoryBEDian

Complement

The hexatonic modal family [2977, 221, 1079, 2587, 3341, 1859] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2977 is 187

Scale 187Scale 187: BEDian, Ian Ring Music TheoryBEDian

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111000010111012977k = 1h = 1
211000010111012977k = 2h = 1
311000010111012977k = 3h = 1
430000([1]0[1][1])([1]0[1][1])2977k = 4h = 3
530000([1]0[1][1])([1]0[1][1])2977k = 5h = 3

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2977 is chiral, and its enantiomorph is scale 187

Scale 187Scale 187: BEDian, Ian Ring Music TheoryBEDian

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2977       T0I <11,0> 187
T1 <1,1> 1859      T1I <11,1> 374
T2 <1,2> 3718      T2I <11,2> 748
T3 <1,3> 3341      T3I <11,3> 1496
T4 <1,4> 2587      T4I <11,4> 2992
T5 <1,5> 1079      T5I <11,5> 1889
T6 <1,6> 2158      T6I <11,6> 3778
T7 <1,7> 221      T7I <11,7> 3461
T8 <1,8> 442      T8I <11,8> 2827
T9 <1,9> 884      T9I <11,9> 1559
T10 <1,10> 1768      T10I <11,10> 3118
T11 <1,11> 3536      T11I <11,11> 2141
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2707      T0MI <7,0> 2347
T1M <5,1> 1319      T1MI <7,1> 599
T2M <5,2> 2638      T2MI <7,2> 1198
T3M <5,3> 1181      T3MI <7,3> 2396
T4M <5,4> 2362      T4MI <7,4> 697
T5M <5,5> 629      T5MI <7,5> 1394
T6M <5,6> 1258      T6MI <7,6> 2788
T7M <5,7> 2516      T7MI <7,7> 1481
T8M <5,8> 937      T8MI <7,8> 2962
T9M <5,9> 1874      T9MI <7,9> 1829
T10M <5,10> 3748      T10MI <7,10> 3658
T11M <5,11> 3401      T11MI <7,11> 3221

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 2979Scale 2979: Gyptian, Ian Ring Music TheoryGyptian
Scale 2981Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
Scale 2985Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian
Scale 2993Scale 2993: Stythian, Ian Ring Music TheoryStythian
Scale 2945Scale 2945: SIHian, Ian Ring Music TheorySIHian
Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic
Scale 3009Scale 3009: SUVian, Ian Ring Music TheorySUVian
Scale 3041Scale 3041: TANian, Ian Ring Music TheoryTANian
Scale 2849Scale 2849: RUBian, Ian Ring Music TheoryRUBian
Scale 2913Scale 2913: SENian, Ian Ring Music TheorySENian
Scale 2721Scale 2721: Raga Puruhutika, Ian Ring Music TheoryRaga Puruhutika
Scale 2465Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani
Scale 3489Scale 3489: VUVian, Ian Ring Music TheoryVUVian
Scale 4001Scale 4001: ZIYian, Ian Ring Music TheoryZIYian
Scale 929Scale 929: FUJian, Ian Ring Music TheoryFUJian
Scale 1953Scale 1953: MACian, Ian Ring Music TheoryMACian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.