The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3767: "Chromatic Bebop"

Scale 3767: Chromatic 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
Chromatic Bebop
Zeitler
Bacrygic
Dozenal
Yalian

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,1,2,4,5,7,9,10,11}

Forte Number

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

9-7

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: 3503

Hemitonia

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

6 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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.

8

Prime Form

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

no
prime: 1471

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.

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

<6, 7, 7, 6, 7, 3>

Interval Spectrum

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

p7m6n7s7d6t3

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

Spectra Variation

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

1.778

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.

2.799

Polygon Perimeter

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

6.106

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.

(25, 109, 196)

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}342.44
F{5,9,0}242.56
G{7,11,2}342.44
A{9,1,4}442.31
A♯{10,2,5}442.19
Minor Triadsdm{2,5,9}242.38
em{4,7,11}342.44
gm{7,10,2}442.31
am{9,0,4}342.44
a♯m{10,1,5}442.13
Augmented TriadsC♯+{1,5,9}442.19
Diminished Triadsc♯°{1,4,7}242.56
{4,7,10}242.56
{7,10,1}242.44
a♯°{10,1,4}242.44
{11,2,5}242.56
Parsimonious Voice Leading Between Common Triads of Scale 3767. Created by Ian Ring ©2019 C C c#° c#° C->c#° em em C->em am am C->am A A c#°->A C#+ C#+ dm dm C#+->dm F F C#+->F C#+->A a#m a#m C#+->a#m A# A# dm->A# e°->em gm gm e°->gm Parsimonious Voice Leading Between Common Triads of Scale 3767. Created by Ian Ring ©2019 G em->G F->am g°->gm g°->a#m gm->G gm->A# G->b° am->A a#° a#° A->a#° a#°->a#m a#m->A# A#->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 3767 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3931
Scale 3931: Aerygic, Ian Ring Music TheoryAerygic
3rd mode:
Scale 4013
Scale 4013: Raga Pilu, Ian Ring Music TheoryRaga Pilu
4th mode:
Scale 2027
Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic
5th mode:
Scale 3061
Scale 3061: Apinygic, Ian Ring Music TheoryApinygic
6th mode:
Scale 1789
Scale 1789: Blues Enneatonic II, Ian Ring Music TheoryBlues Enneatonic II
7th mode:
Scale 1471
Scale 1471: Radygic, Ian Ring Music TheoryRadygicThis is the prime mode
8th mode:
Scale 2783
Scale 2783: Gothygic, Ian Ring Music TheoryGothygic
9th mode:
Scale 3439
Scale 3439: Lythygic, Ian Ring Music TheoryLythygic

Prime

The prime form of this scale is Scale 1471

Scale 1471Scale 1471: Radygic, Ian Ring Music TheoryRadygic

Complement

The enneatonic modal family [3767, 3931, 4013, 2027, 3061, 1789, 1471, 2783, 3439] (Forte: 9-7) is the complement of the tritonic modal family [37, 641, 1033] (Forte: 3-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3767 is 3503

Scale 3503Scale 3503: Zyphygic, Ian Ring Music TheoryZyphygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3767 is chiral, and its enantiomorph is scale 3503

Scale 3503Scale 3503: Zyphygic, Ian Ring Music TheoryZyphygic

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> 3767       T0I <11,0> 3503
T1 <1,1> 3439      T1I <11,1> 2911
T2 <1,2> 2783      T2I <11,2> 1727
T3 <1,3> 1471      T3I <11,3> 3454
T4 <1,4> 2942      T4I <11,4> 2813
T5 <1,5> 1789      T5I <11,5> 1531
T6 <1,6> 3578      T6I <11,6> 3062
T7 <1,7> 3061      T7I <11,7> 2029
T8 <1,8> 2027      T8I <11,8> 4058
T9 <1,9> 4054      T9I <11,9> 4021
T10 <1,10> 4013      T10I <11,10> 3947
T11 <1,11> 3931      T11I <11,11> 3799
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4007      T0MI <7,0> 3263
T1M <5,1> 3919      T1MI <7,1> 2431
T2M <5,2> 3743      T2MI <7,2> 767
T3M <5,3> 3391      T3MI <7,3> 1534
T4M <5,4> 2687      T4MI <7,4> 3068
T5M <5,5> 1279      T5MI <7,5> 2041
T6M <5,6> 2558      T6MI <7,6> 4082
T7M <5,7> 1021      T7MI <7,7> 4069
T8M <5,8> 2042      T8MI <7,8> 4043
T9M <5,9> 4084      T9MI <7,9> 3991
T10M <5,10> 4073      T10MI <7,10> 3887
T11M <5,11> 4051      T11MI <7,11> 3679

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 3765Scale 3765: Dominant Bebop, Ian Ring Music TheoryDominant Bebop
Scale 3763Scale 3763: Modyllic, Ian Ring Music TheoryModyllic
Scale 3771Scale 3771: Stophygic, Ian Ring Music TheoryStophygic
Scale 3775Scale 3775: Loptyllian, Ian Ring Music TheoryLoptyllian
Scale 3751Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
Scale 3759Scale 3759: Darygic, Ian Ring Music TheoryDarygic
Scale 3735Scale 3735: Xupian, Ian Ring Music TheoryXupian
Scale 3799Scale 3799: Aeralygic, Ian Ring Music TheoryAeralygic
Scale 3831Scale 3831: Ionyllian, Ian Ring Music TheoryIonyllian
Scale 3639Scale 3639: Paptyllic, Ian Ring Music TheoryPaptyllic
Scale 3703Scale 3703: Katalygic, Ian Ring Music TheoryKatalygic
Scale 3895Scale 3895: Eparygic, Ian Ring Music TheoryEparygic
Scale 4023Scale 4023: Styptyllian, Ian Ring Music TheoryStyptyllian
Scale 3255Scale 3255: Daryllic, Ian Ring Music TheoryDaryllic
Scale 3511Scale 3511: Epolygic, Ian Ring Music TheoryEpolygic
Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic
Scale 1719Scale 1719: Lyryllic, Ian Ring Music TheoryLyryllic

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.