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Scale 2989: "Bebop Minor"

Scale 2989: Bebop Minor, 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

Jazz and Blues
Bebop Minor
Melodic Minor Bebop
Minor Bebop
Arabic
Zirafkend
Dozenal
Sujian

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

{0,2,3,5,7,8,9,11}

Forte Number

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

8-27

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: 1723

Hemitonia

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

4 (multihemitonic)

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.

7

Prime Form

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

no
prime: 1463

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.

[2, 1, 2, 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.

<4, 5, 6, 5, 5, 3>

Interval Spectrum

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

p5m5n6s5d4t3

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

Spectra Variation

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

1.25

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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".

Proper

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.

(0, 24, 103)

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}342.15
G{7,11,2}342.23
G♯{8,0,3}441.85
Minor Triadscm{0,3,7}242.23
dm{2,5,9}342.23
fm{5,8,0}441.92
g♯m{8,11,3}441.92
Augmented TriadsD♯+{3,7,11}342.15
Diminished Triads{2,5,8}242.31
{5,8,11}242.15
g♯°{8,11,2}242.31
{9,0,3}242.23
{11,2,5}242.31
Parsimonious Voice Leading Between Common Triads of Scale 2989. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# dm dm d°->dm fm fm d°->fm F F dm->F dm->b° Parsimonious Voice Leading Between Common Triads of Scale 2989. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m f°->fm f°->g#m fm->F fm->G# F->a° g#° g#° G->g#° G->b° g#°->g#m g#m->G# G#->a°

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2989 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1771
Scale 1771: Kuwian, Ian Ring Music TheoryKuwian
3rd mode:
Scale 2933
Scale 2933: Sizian, Ian Ring Music TheorySizian
4th mode:
Scale 1757
Scale 1757: Kunian, Ian Ring Music TheoryKunian
5th mode:
Scale 1463
Scale 1463: Ugrian, Ian Ring Music TheoryUgrianThis is the prime mode
6th mode:
Scale 2779
Scale 2779: Shostakovich, Ian Ring Music TheoryShostakovich
7th mode:
Scale 3437
Scale 3437: Vopian, Ian Ring Music TheoryVopian
8th mode:
Scale 1883
Scale 1883: Lomian, Ian Ring Music TheoryLomian

Prime

The prime form of this scale is Scale 1463

Scale 1463Scale 1463: Ugrian, Ian Ring Music TheoryUgrian

Complement

The octatonic modal family [2989, 1771, 2933, 1757, 1463, 2779, 3437, 1883] (Forte: 8-27) is the complement of the tetratonic modal family [293, 593, 649, 1097] (Forte: 4-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2989 is 1723

Scale 1723Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2989 is chiral, and its enantiomorph is scale 1723

Scale 1723Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic

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> 2989       T0I <11,0> 1723
T1 <1,1> 1883      T1I <11,1> 3446
T2 <1,2> 3766      T2I <11,2> 2797
T3 <1,3> 3437      T3I <11,3> 1499
T4 <1,4> 2779      T4I <11,4> 2998
T5 <1,5> 1463      T5I <11,5> 1901
T6 <1,6> 2926      T6I <11,6> 3802
T7 <1,7> 1757      T7I <11,7> 3509
T8 <1,8> 3514      T8I <11,8> 2923
T9 <1,9> 2933      T9I <11,9> 1751
T10 <1,10> 1771      T10I <11,10> 3502
T11 <1,11> 3542      T11I <11,11> 2909
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3739      T0MI <7,0> 2863
T1M <5,1> 3383      T1MI <7,1> 1631
T2M <5,2> 2671      T2MI <7,2> 3262
T3M <5,3> 1247      T3MI <7,3> 2429
T4M <5,4> 2494      T4MI <7,4> 763
T5M <5,5> 893      T5MI <7,5> 1526
T6M <5,6> 1786      T6MI <7,6> 3052
T7M <5,7> 3572      T7MI <7,7> 2009
T8M <5,8> 3049      T8MI <7,8> 4018
T9M <5,9> 2003      T9MI <7,9> 3941
T10M <5,10> 4006      T10MI <7,10> 3787
T11M <5,11> 3917      T11MI <7,11> 3479

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 2991Scale 2991: Zanygic, Ian Ring Music TheoryZanygic
Scale 2985Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 2981Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
Scale 2997Scale 2997: Major Bebop, Ian Ring Music TheoryMajor Bebop
Scale 3005Scale 3005: Gycrygic, Ian Ring Music TheoryGycrygic
Scale 2957Scale 2957: Thygian, Ian Ring Music TheoryThygian
Scale 2973Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
Scale 3021Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
Scale 3053Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic
Scale 2861Scale 2861: Katothian, Ian Ring Music TheoryKatothian
Scale 2925Scale 2925: Diminished, Ian Ring Music TheoryDiminished
Scale 2733Scale 2733: Melodic Minor Ascending, Ian Ring Music TheoryMelodic Minor Ascending
Scale 2477Scale 2477: Harmonic Minor, Ian Ring Music TheoryHarmonic Minor
Scale 3501Scale 3501: Maqam Nahawand, Ian Ring Music TheoryMaqam Nahawand
Scale 4013Scale 4013: Raga Pilu, Ian Ring Music TheoryRaga Pilu
Scale 941Scale 941: Mela Jhankaradhvani, Ian Ring Music TheoryMela Jhankaradhvani
Scale 1965Scale 1965: Raga Mukhari, Ian Ring Music TheoryRaga Mukhari

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.