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Most composers who use mathematics to construct music do not intend it
to be heard. Tom Johnson does. When he performed his piano cycle "Counting Keys"
in Cologne in 1986 he began each piece by counting out its structure in
numbers so that the audience could follow the logic of the music.
A particularly appreciative member of that audience was John Cage. One
might wonder what the famous advocate of indeterminacy could find
interesting in this exactly determined, highly predictable music, until
one considers his favourite definition of art as the imitation of
nature in her manner of operation. Much of what we see in nature,
whether flowers, sea-shells, ice-crystals or patterns of growth and
decay, is determined by mathematical processes.
The first piece in "Counting Keys"
begins with a single high note. This is repeated followed by a 2-note
cluster played twice, the whole being repeated followed by a 3-note
cluster played three times, the process being repeated chromatically
down the keyboard ending in the lowest register with a 12-note cluster
played twelve times. The keys are literally counted. The effect is like
an avalanche, beginning with a snowflake and ending in a pile of rubble
at the foot of a mountain.
Over the years, the mathematics of Tom Johnson's music has become increasingly complex. "Organ and Silence",
completed in 2000, is a cycle of 28 pieces for organ answering the
challenge of composing music consisting more of silence than of sound.
It is also a compendium of the compositional techniques he had
developed up till then, from simple procedures like subtraction or
permutation to the more sophisticated operations he refers to as
automata and self-similar melodies. A consequence of the many long
pauses it contains is the slowing down of the mathematical processes,
making them much easier to hear. "Organ and Silence for Piano" is a
reworking of eight of these pieces for piano made by the composer and
myself in 2002.
"Tilework for Piano" is
one of a series of pieces for solo instruments utilising the
mathematics of arranging tiles in a single line along a wall to produce
interlocking repeated patterns. In musical terms, if each tile is a
note, motive or, as in "Tilework for Piano",
chord, this means creating counterpoint with one voice. Imagine a row
of 15 coloured tiles without gaps. There are 5 different colours; 3 of
each. The same-coloured tiles are arranged in 5 equally-spaced
interlocking triplets, each triplet having a different spacing. There
is only one way to do this and this provides the basic structure of "Tilework for Piano".
The triplets are first presented singly. Then every combination
of 2 triplets is presented twice, every combination of 3 triplets
three times and so on, until finally all 5 triplets together are
presented five times.
The mathematical term "block design"
comes from combination theory and refers to the distribution of a fixed
number of elements into blocks of a fixed size according to specific
combinatory preconditions. In "Block Design for Piano"
the blocks are a sequence of 330 different ascending 6-note arpeggios.
Their notes are taken from a fixed 12-note chord in such a way that
every combination of four particular notes occurs 10 times in 10
different arpeggios. The 330 arpeggios are presented in 30 groups of
eleven. Unlike most of Tom Johnson's compositions, the mathematical
logic is not easily discernable. In an introductory text he sent me
shortly after completing the piece, he commented on the difficulty of
predicting the course of the music and wrote: "As I listen, I can
imagine that I am hearing something derived from nature." Indeed, like
the waves washing up on a beach, no two them identical yet all
following the same unfathomable laws, this music has the inevitability
and timelessness of the processes of nature.
Tom Johnson has written extensively about his compositional techniques. His writings can be found at www.tom.johnson.org