Quarter-Tone Music

Definition: Acknowledgment of the theoretical possibility that modes of limited transpositions could exist in microtonal systems dividing the octave into units smaller than semitones, particularly quarter-tone systems, though Messiaen declines to explore this territory in detail.

Messiaen's Treatment: Messiaen briefly mentions (recalling Chapter XVI, article 1) that the tempered quarter-tone system, advocated by composers Haba and Wischnegradsky, offers a series of modes of limited transpositions that continues the series Messiaen has presented. He cites one example: a scale of eight sounds in three-quarters of a tone, where each degree forms with its neighbor an interval of three-quarters of a tone or a neuter second (smaller by a quarter tone than the major second). This system comprises twenty-four sounds, twenty-four intervals, twenty-four transpositions—entirely new melodies and chords requiring special notation and special instruments. Messiaen states he cannot extend himself further on this question, which alone would fill several treatises.

Modern Context: Quarter-tone composition remained relatively marginal in Messiaen's time and has continued as a specialized practice rather than mainstream development. Composers like Alois Hába, Ivan Wyschnegradsky, Julian Carrillo, and later Easley Blackwood explored systematic microtonal divisions of the octave. Messiaen's insight—that symmetrical pitch collections with limited transposability can exist in any equal-tempered system—demonstrates mathematical understanding of the underlying principles. Just as twelve-tone equal temperament yields seven modes of limited transpositions (the maximum number possible in that system), twenty-four-tone equal temperament would yield its own set of symmetrical collections with analogous properties.

Contemporary microtonal composers and theorists have explored these relationships systematically, discovering that any equal division of the octave generates characteristic symmetrical subsets. The number and types of these subsets depend on the divisor's factors—systems with more factors (like 24, which is divisible by 2, 3, 4, 6, 8, and 12) yield more symmetrical collections than systems with fewer factors. Messiaen's decision not to pursue quarter-tone composition reflects practical concerns—in 1944, quarter-tone instruments remained rare, notation unstandardized, and performance traditions undeveloped. His focus on twelve-tone equal temperament reflected the overwhelming dominance of that system in Western art music. The brief mention serves primarily to delimit his treatise's scope rather than to advocate for microtonal exploration. It also demonstrates Messiaen's awareness of contemporary experimental practices even when choosing not to adopt them in his own work.

Examples: Discussed conceptually only; no musical examples provided