Australian
composer Carl Vine's Piano Sonata from 1990 begins with the pianist silently
depressing A0, C1, E1, G1, and B1 and keeping them undamped with the sostenuto
pedal for the first 12 measures. In a very short preface, the composer tells
the performer that the tempo markings are "not suggestions but indications
of absolute speed." The first 103 measures of the work, before a
"poco allargando" in m. 104 and "Poco meno" tempo change
involving a factor of 11 in m. 105, establish various isochronous pulses with
wavelengths longer than 250 ms. These pulses group into exactly five
"tempo classes" by equivalence of 2:1 ratios. If the first
"tempo class" in the work is represented as 1, then the other four tempo
classes can be represented as 1.2, 1.5, 1.8, and 2.25. If the frequency
of the pitch class A is represented as 1, then the other four pitch-class
frequencies in the opening "silent" chord can be represented in just
intonation as 1.2, 1.5, 1.8, and 2.25. (Such an analogy could also be
demonstrated using the method David Lewin used in his 1987 book Generalized
Musical Intervals and Transformations to analyze a passage from
Carter's First String Quintet.)
