Modes of Limited Transpositions: Quick Reference
Overview
The seven modes of limited transpositions are symmetrical pitch collections that cannot be transposed beyond a certain number of times without reproducing their original pitch content. This "impossibility" of unlimited transposition creates their distinctive harmonic character—existing "in the atmosphere of several tonalities at once, without polytonality."
Mode 1: Whole-Tone Scale
| Property | Value |
|---|---|
| Pitch count | 6 |
| Transpositions | 2 |
| Interval pattern | Tone–Tone–Tone–Tone–Tone–Tone |
| Set class | 6-35 [02468T] |
Character: Maximum symmetry; divides octave into six equal parts. Creates ambiguous, floating harmonies lacking perfect fifths.
Messiaen's assessment: "Exhausted" by Debussy and Dukas; Messiaen avoids except when concealed in polymodal superpositions.
Associated chord: Augmented triad
Mode 2: Octatonic Scale
| Property | Value |
|---|---|
| Pitch count | 8 |
| Transpositions | 3 |
| Interval pattern | Semitone–Tone–Semitone–Tone–Semitone–Tone–Semitone–Tone |
| Set class | 8-28 [0134679T] |
Three transpositions:
- C–C♯–D♯–E–F♯–G–A–B♭
- C♯–D–E–F–G–A♭–B♭–B
- D–E♭–F–F♯–A♭–A–B–C
Character: The most frequently used mode. Suggests four major/minor tonalities simultaneously (in transposition 1: C, E♭, F♯, A). Rich in tritones and minor thirds.
Associated chord: Perfect chord with added sixth and augmented fourth (the characteristic Messiaen sonority)
Tonal implications: Each transposition contains four diminished seventh chords, suggesting four major and four minor keys.
Mode 3
| Property | Value |
|---|---|
| Pitch count | 9 |
| Transpositions | 4 |
| Interval pattern | Tone–Semitone–Semitone (repeated 3×) |
| Set class | 9-12 [01245689T] |
Four transpositions:
- C–D–E♭–E–F♯–G–A♭–B♭–B
- C♯–D♯–E–F–G–A♭–A–B–C
- D–E–F–F♯–G♯–A–B♭–C–C♯
- E♭–F–F♯–G–A–B♭–B–C♯–D
Character: Approaches chromatic saturation (9 of 12 pitches) while maintaining modal identity through characteristic gaps.
Associated chord: Chord of resonance (derived from overtone series)
Mode 4
| Property | Value |
|---|---|
| Pitch count | 8 |
| Transpositions | 6 |
| Interval pattern | Semitone–Semitone–Minor 3rd–Semitone (repeated 2×) |
| Set class | 8-9 [01236789] |
Character: One of the "lesser interest" modes due to greater transposability. Contains characteristic minor third gaps.
Usage: Messiaen employs sparingly; appears in specific coloristic contexts.
Mode 5
| Property | Value |
|---|---|
| Pitch count | 6 |
| Transpositions | 6 |
| Interval pattern | Semitone–Major 3rd–Semitone (repeated 2×) |
| Set class | 6-7 [012678] |
Character: Distinctive major-third gaps create ambiguity between major and minor.
Associated chord: Chord in fourths (quartal harmony)
Associated melodic formula: Specific contour connected to this mode (discussed in Ch. X)
Mode 6
| Property | Value |
|---|---|
| Pitch count | 8 |
| Transpositions | 6 |
| Interval pattern | Tone–Tone–Semitone–Semitone (repeated 2×) |
| Set class | 8-25 [0124678T] |
Character: Contains two tritones; creates whole-tone segments interrupted by semitone clusters.
Usage: Appears in Les Bergers and La Vierge et l'Enfant from La Nativité du Seigneur.
Mode 7
| Property | Value |
|---|---|
| Pitch count | 10 |
| Transpositions | 6 |
| Interval pattern | Semitone–Semitone–Semitone–Tone–Semitone (repeated 2×) |
| Set class | 10-6 [012346789T] |
Character: Near-chromatic (10 of 12 pitches); most pitches of any mode. Minimal gaps create subtle coloring.
Usage: Used in specific passages; referenced in Ch. XIV examples.
Comparative Table
| Mode | Pitches | Transpositions | Intervals per group | Messiaen's "Interest" |
|---|---|---|---|---|
| 1 | 6 | 2 | 6 tones | Low (exhausted) |
| 2 | 8 | 3 | S–T–S–T | High (most used) |
| 3 | 9 | 4 | T–S–S | High |
| 4 | 8 | 6 | S–S–m3–S | Moderate |
| 5 | 6 | 6 | S–M3–S | Moderate |
| 6 | 8 | 6 | T–T–S–S | Moderate |
| 7 | 10 | 6 | S–S–S–T–S | Moderate |
S = Semitone, T = Tone, m3 = minor third, M3 = major third
The Symmetry Principle
All modes share the defining property: internal symmetry creates transpositional limitation.
- Modes divide the octave into equal segments
- Each segment contains the same interval pattern
- After a certain number of transpositions, the same pitch-classes recur
This connects to nonretrogradable rhythms (Ch. V):
- Modes: cannot transpose without repetition → "tonal ubiquity"
- Rhythms: cannot retrograde without identity → "unity of movement"
Both embody the "charm of impossibilities"—structures whose mathematical properties create perceptual effects of suspension, timelessness, and transcendence.
Practical Applications
Melodic Use
All pitches in a melodic line belong to the chosen mode. Characteristic intervals emerge naturally from modal content.
Harmonic Use
All chord tones belong to the mode. Characteristic sonorities:
- Mode 2 → Added sixth + augmented fourth chord
- Mode 3 → Chord of resonance
- Mode 5 → Chord in fourths
Modal Modulation (Ch. XVII)
- To same mode, different transposition: Shifts pitch content while maintaining modal character
- To different mode: Changes intervallic structure and color
- Mixing with tonality: Dominant seventh chords or tonic emphasis create tonal orientation within modal context
Polymodality (Ch. XIX)
Superposing different modes in different textural layers:
- Two-mode combinations: foundational polymodal texture
- Three-mode combinations: maximum harmonic density
- Polymodal modulation: large-scale transformation of modal combinations
Historical Note
Messiaen emphasizes these modes have nothing in common with:
- Indian modal systems (ragas)
- Chinese modal systems
- Greek modes
- Plainchant modes
Those systems are all transposable twelve times within equal temperament. Messiaen's modes are defined precisely by their limited transposability—a fundamentally different organizational principle based on symmetry rather than scalar tradition.