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Scale 3705: "Messiaen Mode 4 Inverse"

Scale 3705: Messiaen Mode 4 Inverse, 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

41161837294116105072918310504116183
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

Messiaen
Messiaen Mode 4 Inverse
Named After Composers
Tcherepnin Octatonic Mode 4
Zeitler
Sydyllic

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

{0,3,4,5,6,9,10,11}

Forte Number

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

8-9

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

[6]

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, 4.5]

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.

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.

3

Prime Form

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

no
prime: 975

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.

[3, 1, 1, 1, 3, 1, 1, 1] 9

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, 4, 4, 4, 6, 4>

Interval Spectrum

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

p6m4n4s4d6t4

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

Spectra Variation

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

1.5

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.

2.5

Polygon Perimeter

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

5.934

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.

yes

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.

[3,9]

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

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}242
B{11,3,6}242
Minor Triadsd♯m{3,6,10}242
am{9,0,4}242
Diminished Triads{0,3,6}242
d♯°{3,6,9}242
f♯°{6,9,0}242
{9,0,3}242
Parsimonious Voice Leading Between Common Triads of Scale 3705. Created by Ian Ring ©2019 c°->a° B B c°->B d#° d#° d#m d#m d#°->d#m f#° f#° d#°->f#° d#m->B F F F->f#° am am F->am a°->am

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 3705 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 975
Scale 975: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4This is the prime mode
3rd mode:
Scale 2535
Scale 2535: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4
4th mode:
Scale 3315
Scale 3315: Tcherepnin Octatonic Mode 1, Ian Ring Music TheoryTcherepnin Octatonic Mode 1

Prime

The prime form of this scale is Scale 975

Scale 975Scale 975: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4

Complement

The octatonic modal family [3705, 975, 2535, 3315] (Forte: 8-9) is the complement of the tetratonic modal family [195, 2145] (Forte: 4-9)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3705 is 975

Scale 975Scale 975: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4

Transformations:

T0 3705  T0I 975
T1 3315  T1I 1950
T2 2535  T2I 3900
T3 975  T3I 3705
T4 1950  T4I 3315
T5 3900  T5I 2535
T6 3705  T6I 975
T7 3315  T7I 1950
T8 2535  T8I 3900
T9 975  T9I 3705
T10 1950  T10I 3315
T11 3900  T11I 2535

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 3707Scale 3707: Rynygic, Ian Ring Music TheoryRynygic
Scale 3709Scale 3709: Katynygic, Ian Ring Music TheoryKatynygic
Scale 3697Scale 3697: Ionarian, Ian Ring Music TheoryIonarian
Scale 3701Scale 3701: Bagyllic, Ian Ring Music TheoryBagyllic
Scale 3689Scale 3689: Katocrian, Ian Ring Music TheoryKatocrian
Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian
Scale 3641Scale 3641: Thocrian, Ian Ring Music TheoryThocrian
Scale 3769Scale 3769: Eponyllic, Ian Ring Music TheoryEponyllic
Scale 3833Scale 3833: Dycrygic, Ian Ring Music TheoryDycrygic
Scale 3961Scale 3961: Zathygic, Ian Ring Music TheoryZathygic
Scale 3193Scale 3193: Zathian, Ian Ring Music TheoryZathian
Scale 3449Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
Scale 2681Scale 2681: Aerycrian, Ian Ring Music TheoryAerycrian
Scale 1657Scale 1657: Ionothian, Ian Ring Music TheoryIonothian

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. 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.