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Scale 2997: "Major Bebop"

Scale 2997: Major Bebop, 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
Major Bebop
Bebop Major
Western Modern
Altered Mixolydian
Zeitler
Ionoptyllic

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

{0,2,4,5,7,8,9,11}

Forte Number

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

8-26

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.

[2]

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.

no

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.

2

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: 1467

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 Formula

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

[2, 2, 1, 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, 6, 2>

Interval Spectrum

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

p6m5n6s5d4t2

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.

[4]

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, 32, 110)

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}242.23
E{4,8,11}441.92
F{5,9,0}342.15
G{7,11,2}342.23
Minor Triadsdm{2,5,9}342.23
em{4,7,11}342.15
fm{5,8,0}441.92
am{9,0,4}242.23
Augmented TriadsC+{0,4,8}441.85
Diminished Triads{2,5,8}242.31
{5,8,11}242.15
g♯°{8,11,2}242.31
{11,2,5}242.31
Parsimonious Voice Leading Between Common Triads of Scale 2997. Created by Ian Ring ©2019 C C C+ C+ C->C+ em em C->em E E C+->E fm fm C+->fm am am C+->am dm dm d°->dm d°->fm F F dm->F dm->b° em->E Parsimonious Voice Leading Between Common Triads of Scale 2997. Created by Ian Ring ©2019 G em->G E->f° g#° g#° E->g#° f°->fm fm->F F->am G->g#° G->b°

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 2997 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1773
Scale 1773: Blues Scale II, Ian Ring Music TheoryBlues Scale II
3rd mode:
Scale 1467
Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish PhrygianThis is the prime mode
4th mode:
Scale 2781
Scale 2781: Gycryllic, Ian Ring Music TheoryGycryllic
5th mode:
Scale 1719
Scale 1719: Lyryllic, Ian Ring Music TheoryLyryllic
6th mode:
Scale 2907
Scale 2907: Magen Abot 2, Ian Ring Music TheoryMagen Abot 2
7th mode:
Scale 3501
Scale 3501: Maqam Nahawand, Ian Ring Music TheoryMaqam Nahawand
8th mode:
Scale 1899
Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic

Prime

The prime form of this scale is Scale 1467

Scale 1467Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish Phrygian

Complement

The octatonic modal family [2997, 1773, 1467, 2781, 1719, 2907, 3501, 1899] (Forte: 8-26) is the complement of the tetratonic modal family [297, 549, 657, 1161] (Forte: 4-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2997 is 1467

Scale 1467Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish Phrygian

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> 2997       T0I <11,0> 1467
T1 <1,1> 1899      T1I <11,1> 2934
T2 <1,2> 3798      T2I <11,2> 1773
T3 <1,3> 3501      T3I <11,3> 3546
T4 <1,4> 2907      T4I <11,4> 2997
T5 <1,5> 1719      T5I <11,5> 1899
T6 <1,6> 3438      T6I <11,6> 3798
T7 <1,7> 2781      T7I <11,7> 3501
T8 <1,8> 1467      T8I <11,8> 2907
T9 <1,9> 2934      T9I <11,9> 1719
T10 <1,10> 1773      T10I <11,10> 3438
T11 <1,11> 3546      T11I <11,11> 2781
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3987      T0MI <7,0> 2367
T1M <5,1> 3879      T1MI <7,1> 639
T2M <5,2> 3663      T2MI <7,2> 1278
T3M <5,3> 3231      T3MI <7,3> 2556
T4M <5,4> 2367      T4MI <7,4> 1017
T5M <5,5> 639      T5MI <7,5> 2034
T6M <5,6> 1278      T6MI <7,6> 4068
T7M <5,7> 2556      T7MI <7,7> 4041
T8M <5,8> 1017      T8MI <7,8> 3987
T9M <5,9> 2034      T9MI <7,9> 3879
T10M <5,10> 4068      T10MI <7,10> 3663
T11M <5,11> 4041      T11MI <7,11> 3231

The transformations that map this set to itself are: T0, T4I

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 2999Scale 2999: Diminishing Nonamode, Ian Ring Music TheoryDiminishing Nonamode
Scale 2993Scale 2993: Stythian, Ian Ring Music TheoryStythian
Scale 2995Scale 2995: Raga Saurashtra, Ian Ring Music TheoryRaga Saurashtra
Scale 3001Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
Scale 3005Scale 3005: Gycrygic, Ian Ring Music TheoryGycrygic
Scale 2981Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
Scale 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 2965Scale 2965: Darian, Ian Ring Music TheoryDarian
Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
Scale 3061Scale 3061: Apinygic, Ian Ring Music TheoryApinygic
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
Scale 2933Scale 2933, Ian Ring Music Theory
Scale 2741Scale 2741: Major, Ian Ring Music TheoryMajor
Scale 2485Scale 2485: Harmonic Major, Ian Ring Music TheoryHarmonic Major
Scale 3509Scale 3509: Stogyllic, Ian Ring Music TheoryStogyllic
Scale 4021Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
Scale 949Scale 949: Mela Mararanjani, Ian Ring Music TheoryMela Mararanjani
Scale 1973Scale 1973: Zyryllic, Ian Ring Music TheoryZyryllic

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.