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Scale 3135: "Octatonic Chromatic 3"

Scale 3135: Octatonic Chromatic 3, 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

Western Modern
Octatonic Chromatic 3
Dozenal
Totian

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

{0,1,2,3,4,5,10,11}

Forte Number

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

8-1

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.

[1.5]

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.

7 (multihemitonic)

Cohemitonia

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

6 (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.

4

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: 255

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 1
origin: 10

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, 1, 1, 1, 5, 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.

<7, 6, 5, 4, 4, 2>

Interval Spectrum

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

p4m4n5s6d7t2

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> = {1,5}
<2> = {2,6}
<3> = {3,7}
<4> = {4,8}
<5> = {5,9}
<6> = {6,10}
<7> = {7,11}

Spectra Variation

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

3.5

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

Polygon Perimeter

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

5.555

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

yes

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.

[3]

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.

(111, 10, 84)

Generator

This scale has a generator of 1, originating on 10.

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 TriadsA♯{10,2,5}221
Minor Triadsa♯m{10,1,5}221
Diminished Triadsa♯°{10,1,4}131.5
{11,2,5}131.5

The following pitch classes are not present in any of the common triads: {0,3}

Parsimonious Voice Leading Between Common Triads of Scale 3135. Created by Ian Ring ©2019 a#° a#° a#m a#m a#°->a#m A# A# a#m->A# A#->b°

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesa♯m, A♯
Peripheral Verticesa♯°, b°

Modes

Modes are the rotational transformation of this scale. Scale 3135 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3615
Scale 3615: Octatonic Chromatic 4, Ian Ring Music TheoryOctatonic Chromatic 4
3rd mode:
Scale 3855
Scale 3855: Octatonic Chromatic 5, Ian Ring Music TheoryOctatonic Chromatic 5
4th mode:
Scale 3975
Scale 3975: Octatonic Chromatic 6, Ian Ring Music TheoryOctatonic Chromatic 6
5th mode:
Scale 4035
Scale 4035: Octatonic Chromatic 7, Ian Ring Music TheoryOctatonic Chromatic 7
6th mode:
Scale 4065
Scale 4065: Octatonic Chromatic Descending, Ian Ring Music TheoryOctatonic Chromatic Descending
7th mode:
Scale 255
Scale 255: Chromatic Octamode, Ian Ring Music TheoryChromatic OctamodeThis is the prime mode
8th mode:
Scale 2175
Scale 2175: Octatonic Chromatic 2, Ian Ring Music TheoryOctatonic Chromatic 2

Prime

The prime form of this scale is Scale 255

Scale 255Scale 255: Chromatic Octamode, Ian Ring Music TheoryChromatic Octamode

Complement

The octatonic modal family [3135, 3615, 3855, 3975, 4035, 4065, 255, 2175] (Forte: 8-1) is the complement of the tetratonic modal family [15, 2055, 3075, 3585] (Forte: 4-1)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3135 is 3975

Scale 3975Scale 3975: Octatonic Chromatic 6, Ian Ring Music TheoryOctatonic Chromatic 6

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> 3135       T0I <11,0> 3975
T1 <1,1> 2175      T1I <11,1> 3855
T2 <1,2> 255      T2I <11,2> 3615
T3 <1,3> 510      T3I <11,3> 3135
T4 <1,4> 1020      T4I <11,4> 2175
T5 <1,5> 2040      T5I <11,5> 255
T6 <1,6> 4080      T6I <11,6> 510
T7 <1,7> 4065      T7I <11,7> 1020
T8 <1,8> 4035      T8I <11,8> 2040
T9 <1,9> 3975      T9I <11,9> 4080
T10 <1,10> 3855      T10I <11,10> 4065
T11 <1,11> 3615      T11I <11,11> 4035
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1455      T0MI <7,0> 3765
T1M <5,1> 2910      T1MI <7,1> 3435
T2M <5,2> 1725      T2MI <7,2> 2775
T3M <5,3> 3450      T3MI <7,3> 1455
T4M <5,4> 2805      T4MI <7,4> 2910
T5M <5,5> 1515      T5MI <7,5> 1725
T6M <5,6> 3030      T6MI <7,6> 3450
T7M <5,7> 1965      T7MI <7,7> 2805
T8M <5,8> 3930      T8MI <7,8> 1515
T9M <5,9> 3765      T9MI <7,9> 3030
T10M <5,10> 3435      T10MI <7,10> 1965
T11M <5,11> 2775      T11MI <7,11> 3930

The transformations that map this set to itself are: T0, T3I

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 3133Scale 3133: Tosian, Ian Ring Music TheoryTosian
Scale 3131Scale 3131: Torian, Ian Ring Music TheoryTorian
Scale 3127Scale 3127: Topian, Ian Ring Music TheoryTopian
Scale 3119Scale 3119: Tikian, Ian Ring Music TheoryTikian
Scale 3103Scale 3103: Heptatonic Chromatic 3, Ian Ring Music TheoryHeptatonic Chromatic 3
Scale 3167Scale 3167: Thynyllic, Ian Ring Music TheoryThynyllic
Scale 3199Scale 3199: Nonatonic Chromatic 3, Ian Ring Music TheoryNonatonic Chromatic 3
Scale 3263Scale 3263: Pyrygic, Ian Ring Music TheoryPyrygic
Scale 3391Scale 3391: Aeolynygic, Ian Ring Music TheoryAeolynygic
Scale 3647Scale 3647: Nonatonic Chromatic 4, Ian Ring Music TheoryNonatonic Chromatic 4
Scale 2111Scale 2111: Heptatonic Chromatic 2, Ian Ring Music TheoryHeptatonic Chromatic 2
Scale 2623Scale 2623: Aerylyllic, Ian Ring Music TheoryAerylyllic
Scale 1087Scale 1087: Gobian, Ian Ring Music TheoryGobian

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.