(The longer original post gives more background on Messiaen and serialism. An extract appears below...)
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')
messiaen.show()
notes = messiaen.flat.stripTies()
g = graph.PlotScatterWeightedPitchSpaceQuarterLength(notes,
title='Messiaen, Mode de Valeurs, middle voice', xLog=False)
g.process()
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.
