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# Scale 2539: "Half-Diminished Bebop" ### 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
Half-Diminished Bebop
Western
Half Diminished Bebop
Zeitler
Thonyllic
Dozenal
POQian

## 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

{0,1,3,5,6,7,8,11}

#### Forte Number

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

8-16

#### 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: 2803

#### 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.

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 Starr/Rahn algorithm.

no
prime: 943

#### 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.

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

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

#### 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.

<0.333, 0.333, 0, 0.333, 0.667, 0.5>

#### Interval Spectrum

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

p6m5n4s5d5t3

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

#### Spectra Variation

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

1.75

#### 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.

2.616

#### Polygon Perimeter

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

6.002

#### 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.

(12, 44, 123)

#### 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.

0.789

#### 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.

0.372

## Generator

This scale has no generator.

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

G♯{8,0,3}331.56
B{11,3,6}252.33
fm{5,8,0}341.89
g♯m{8,11,3}331.67
{5,8,11}242

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.

Diameter 5 3 no cm, g♯m, G♯ C♯, B

## Modes

Modes are the rotational transformation of this scale. Scale 2539 can be rotated to make 7 other scales. The 1st mode is itself.

 2nd mode:Scale 3317 Lanyllic 3rd mode:Scale 1853 Phrynyllic 4th mode:Scale 1487 Lycryllic 5th mode:Scale 2791 Ionyptyllic 6th mode:Scale 3443 Verdi's Scala Enigmatica 7th mode:Scale 3769 Aeracryllic 8th mode:Scale 983 Epygyllic

## Prime

The prime form of this scale is Scale 943

 Scale 943 Aerygyllic

## Complement

The octatonic modal family [2539, 3317, 1853, 1487, 2791, 3443, 3769, 983] (Forte: 8-16) is the complement of the tetratonic modal family [163, 389, 1121, 2129] (Forte: 4-16)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2539 is 2803

 Scale 2803 Raga Bhatiyar

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2539 is chiral, and its enantiomorph is scale 2803

 Scale 2803 Raga Bhatiyar

## 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> 2539       T0I <11,0> 2803
T1 <1,1> 983      T1I <11,1> 1511
T2 <1,2> 1966      T2I <11,2> 3022
T3 <1,3> 3932      T3I <11,3> 1949
T4 <1,4> 3769      T4I <11,4> 3898
T5 <1,5> 3443      T5I <11,5> 3701
T6 <1,6> 2791      T6I <11,6> 3307
T7 <1,7> 1487      T7I <11,7> 2519
T8 <1,8> 2974      T8I <11,8> 943
T9 <1,9> 1853      T9I <11,9> 1886
T10 <1,10> 3706      T10I <11,10> 3772
T11 <1,11> 3317      T11I <11,11> 3449
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2299      T0MI <7,0> 3043
T1M <5,1> 503      T1MI <7,1> 1991
T2M <5,2> 1006      T2MI <7,2> 3982
T3M <5,3> 2012      T3MI <7,3> 3869
T4M <5,4> 4024      T4MI <7,4> 3643
T5M <5,5> 3953      T5MI <7,5> 3191
T6M <5,6> 3811      T6MI <7,6> 2287
T7M <5,7> 3527      T7MI <7,7> 479
T8M <5,8> 2959      T8MI <7,8> 958
T9M <5,9> 1823      T9MI <7,9> 1916
T10M <5,10> 3646      T10MI <7,10> 3832
T11M <5,11> 3197      T11MI <7,11> 3569

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 2537 Laptian Scale 2541 Algerian Scale 2543 Dydygic Scale 2531 Danian Scale 2535 Messiaen Mode 4 Scale 2547 Raga Ramkali Scale 2555 Bythygic Scale 2507 Todi That Scale 2523 Mirage Scale Scale 2475 Neapolitan Minor Scale 2411 Aeolorian Scale 2283 Aeolyptian Scale 2795 Van der Horst Octatonic Scale 3051 Stalygic Scale 3563 Ionoptygic Scale 491 Aeolyrian Scale 1515 Phrygian/Locrian Mixed

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.