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Scale 1235: "Messiaen Truncated Mode 2"

Scale 1235: Messiaen Truncated Mode 2, 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

Messiaen
Messiaen Truncated Mode 2
Dozenal
Honian
Carnatic
Raga Indupriya
Western Modern
Tritone Scale
Zeitler
Stylimic

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,1,4,6,7,10}

Forte Number

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

6-30

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.

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: 2405

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

2

Prime Form

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

no
prime: 715

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, 3, 2, 1, 3, 2]

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.

<2, 2, 4, 2, 2, 3>

Interval Spectrum

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

p2m2n4s2d2t3

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

Spectra Variation

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

1.333

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.

2.366

Polygon Perimeter

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

5.864

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.

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".

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, 8, 48)

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}231.5
F♯{6,10,1}231.5
Diminished Triadsc♯°{1,4,7}231.5
{4,7,10}231.5
{7,10,1}231.5
a♯°{10,1,4}231.5
Parsimonious Voice Leading Between Common Triads of Scale 1235. Created by Ian Ring ©2019 C C c#° c#° C->c#° C->e° a#° a#° c#°->a#° e°->g° F# F# F#->g° F#->a#°

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
Radius3
Self-Centeredyes

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Major: {0, 4, 7}
Major: {6, 10, 1}

Modes

Modes are the rotational transformation of this scale. Scale 1235 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 2665
Scale 2665: Aeradimic, Ian Ring Music TheoryAeradimic
3rd mode:
Scale 845
Scale 845: Raga Neelangi, Ian Ring Music TheoryRaga Neelangi

Prime

The prime form of this scale is Scale 715

Scale 715Scale 715: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2

Complement

The hexatonic modal family [1235, 2665, 845] (Forte: 6-30) is the complement of the hexatonic modal family [715, 1625, 2405] (Forte: 6-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1235 is 2405

Scale 2405Scale 2405: Katalimic, Ian Ring Music TheoryKatalimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1235 is chiral, and its enantiomorph is scale 2405

Scale 2405Scale 2405: Katalimic, Ian Ring Music TheoryKatalimic

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> 1235       T0I <11,0> 2405
T1 <1,1> 2470      T1I <11,1> 715
T2 <1,2> 845      T2I <11,2> 1430
T3 <1,3> 1690      T3I <11,3> 2860
T4 <1,4> 3380      T4I <11,4> 1625
T5 <1,5> 2665      T5I <11,5> 3250
T6 <1,6> 1235       T6I <11,6> 2405
T7 <1,7> 2470      T7I <11,7> 715
T8 <1,8> 845      T8I <11,8> 1430
T9 <1,9> 1690      T9I <11,9> 2860
T10 <1,10> 3380      T10I <11,10> 1625
T11 <1,11> 2665      T11I <11,11> 3250
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2405      T0MI <7,0> 1235
T1M <5,1> 715      T1MI <7,1> 2470
T2M <5,2> 1430      T2MI <7,2> 845
T3M <5,3> 2860      T3MI <7,3> 1690
T4M <5,4> 1625      T4MI <7,4> 3380
T5M <5,5> 3250      T5MI <7,5> 2665
T6M <5,6> 2405      T6MI <7,6> 1235
T7M <5,7> 715      T7MI <7,7> 2470
T8M <5,8> 1430      T8MI <7,8> 845
T9M <5,9> 2860      T9MI <7,9> 1690
T10M <5,10> 1625      T10MI <7,10> 3380
T11M <5,11> 3250      T11MI <7,11> 2665

The transformations that map this set to itself are: T0, T6, T0MI, T6MI

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 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic
Scale 1237Scale 1237: Salimic, Ian Ring Music TheorySalimic
Scale 1239Scale 1239: Epaptian, Ian Ring Music TheoryEpaptian
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian
Scale 1219Scale 1219: Hidian, Ian Ring Music TheoryHidian
Scale 1227Scale 1227: Thacrimic, Ian Ring Music TheoryThacrimic
Scale 1251Scale 1251: Sylimic, Ian Ring Music TheorySylimic
Scale 1267Scale 1267: Katynian, Ian Ring Music TheoryKatynian
Scale 1171Scale 1171: Raga Manaranjani I, Ian Ring Music TheoryRaga Manaranjani I
Scale 1203Scale 1203: Pagimic, Ian Ring Music TheoryPagimic
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
Scale 1363Scale 1363: Gygimic, Ian Ring Music TheoryGygimic
Scale 1491Scale 1491: Namanarayani, Ian Ring Music TheoryNamanarayani
Scale 1747Scale 1747: Mela Ramapriya, Ian Ring Music TheoryMela Ramapriya
Scale 211Scale 211: Bisian, Ian Ring Music TheoryBisian
Scale 723Scale 723: Ionadimic, Ian Ring Music TheoryIonadimic
Scale 2259Scale 2259: Raga Mandari, Ian Ring Music TheoryRaga Mandari
Scale 3283Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari

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.