One of the most influential of Messiaen’s innovation was extending the ideals of twelve-tone music to go beyond pitch. The twelve-tone method, as developed Arnold Schoenberg and his followers, involved arranging pitches in fixed orders so that each note would have equal prominence and the tendency to write music too-mired in the sounds of the past would be eliminated (or at least greatly reduced). While much of the musical world reacted (and still reacts) to Schoenberg by believing he went too far in alienating modern music from the past, Messiaen and a small group of (at the time) young composers believed that he did not go far enough. By not organizing rhythms and loudness (called dynamics or, in French, intensités) similarly to the way he had serialized pitch, Schoenberg’s music could still be rooted in the rhythms and swells of the past.
Messiaen’s 1949 piece “Mode de valeurs et d’intensités” from Quatre études de rythme (“Organization of durations and dynamics” from Four studies in rhythm) was the first European work of “total serialism.” (In America, the composer Milton Babbitt had already made independent but similar creative discoveries). In this piece for piano, three “rows” are used simultaneously. Each pitch in each row is then assigned a particular duration and dynamic that always appears with that pitch (and only with that pitch.) Within each row, the durations get longer as notes get lower.
The precise relationship between pitch and duration is often hard to imagine when looking at the score, but with music21 these relationships become obvious. This code runs a Finale version of the score for the middle row of the Messiaen piece through our graphing tools:
messiaen = converter.parse('d:/desktop/messiaen_valeurs.xml')
notes = messiaen.flat.stripTies()
g = graph.PlotScatterWeightedPitchSpaceQuarterLength(notes,
title='Messiaen, Mode de Valeurs, middle voice', xLog=False)
The 1:1 correspondence between pitch and duration is obvious: each pitch has exactly one quarter length associated with it, and the lengths get longer as the pitch descends. But the non-serially determined aspects of the score are equally obvious. Unlike Schoenberg’s ideal of using each pitch the same number of times, higher pitches appear much more often in the Messiaen piece. But the correspondence is not perfect. For instance, the pitch D 4 appears 11 times while the higher E-flat 4 appears only 10 times. For the most part, the amount of time that each pitch is sounded is roughly constant – at around 20 quarter-notes. But the two shortest notes (G5 and C5) appear as outliers to this theory, appearing only for about 9 and 13 quarters respectively.
What also jumps out in the graph is its shape: it is cubic, that is, it approximates the graph of f(x) = –x3. There are several functions in nature and human society that are modeled by cubic equations, such as magnetic strength, the twisting force of rubber bands (like in wind-up toy planes), and the costs involved with manufacturing. One example from one piece is of course nowhere near enough evidence to suggest that post-tonal pitch and duration relationships could be another place where cubic equations might guide composers. But it does give an idea for further research. As more serial and other post-tonal compositions are inputted into machine-readable formats, I’ll be checking back with whether this relationship holds often enough in Messiaen and other’s music to be significant.
The code for this example has been submitted to the International Symposium on Music Information Retrieval as part of a proposed paper on the music21 system.