Theory of the Modes of Limited Transpositions
Definition: Pitch collections formed from symmetrical intervallic groups that, due to their internal structure, can only be transposed a limited number of times before reproducing the same pitch-class content. These modes exist within the twelve-tone equal-tempered system and are mathematically closed—no additional modes of this type can be discovered within the system.
Messiaen's Treatment: Messiaen explains that these modes consist of several symmetrical groups, with the last note of each group always held in common with the first note of the following group. After a certain number of chromatic transpositions (which varies with each mode), they cease to be transposable—the fourth transposition yields exactly the same notes as the first, the fifth the same as the second, and so forth. Messiaen emphasizes enharmonic equivalence: when he says "the same notes," he means enharmonically, within the tempered system where C-sharp equals D-flat. Three modes exhibit this property most strongly (modes 1, 2, and 3), while four other modes (4, 5, 6, 7) are transposable six times but present less interest due to their greater number of transpositions. All seven modes can be used melodically and especially harmonically, with melody and harmonies never leaving the notes of the mode.
The theoretical foundation connects to the "charm of impossibilities" discussed in Chapter I: their impossibility of transposition creates their distinctive character. They exist in the atmosphere of several tonalities simultaneously, without polytonality, allowing the composer to give predominance to one tonality or leave the tonal impression unsettled. Their series is closed—it is mathematically impossible to find others within the tempered system of twelve semitones. Messiaen notes that in quarter-tone systems (explored by composers like Haba and Wischnegradsky), corresponding series exist, though he cannot elaborate on this topic as it lies outside his work's scope.
Messiaen adds that the modes of limited transpositions have nothing in common with the three great modal systems of India, China, and ancient Greece, nor with plainchant modes (relatives of the Greek modes)—all these scales being transposable twelve times.
Modern Context: Contemporary pitch-class set theory recognizes these structures through the concept of transpositional symmetry. A pitch-class set exhibits limited transposability when it maps onto itself under transposition by some interval other than the octave. The degree of symmetry determines the number of distinct transpositions: the whole-tone scale (Mode 1) has maximal symmetry (transpositionally equivalent at six levels), the octatonic collection (Mode 2) has three distinct transpositions, and the augmented or hexatonic collection (Mode 3) has four distinct transpositions. Modes 4–7 have two distinct transpositions each. Allen Forte's set-class theory systematizes these relationships, assigning each symmetrical collection a Tn/TnI-type designation indicating its transpositional and inversional symmetry.
What distinguishes Messiaen's approach from abstract set theory is his insistence on practical compositional application—these are not merely mathematical curiosities but living harmonic resources with distinctive coloristic properties. His claim that the series is closed anticipates later theoretical work demonstrating that only certain interval-class vectors permit transpositional symmetry within twelve-tone equal temperament. The connection to non-Western modal systems reveals Messiaen's awareness of ethnomusicological research while maintaining that his modes represent a fundamentally different organizational principle based on symmetry rather than scalar tradition. The theological dimension—modes existing in "the atmosphere of several tonalities" without committing to any—suggests Messiaen heard these structures as embodying a kind of harmonic transcendence or suspension of earthly tonal gravity.
Examples: Theoretical exposition throughout; practical demonstrations in Examples 312–357
