as-
pect of sound, since,
of all the as-
pects of sound in-
cluding frequen-
cy, amplitude,
and timbre, dur-
ation, alone,
was also a
characteris-
tic of silence.
The structure, then,
was a divi-
sion of actu-
al time by con-
ventional met-
rical means, me-
ter taken as
simply the meas-
urement of quan-
tity. ¶In the
case of the So-natas and In-terludes (which I finished in nine- teen forty-eight), only structure was organized, quite roughly for the work as a whole, exactly, however, with- in each single piece. The method was that of con- sidered impro- visation (main- ly at the pi- ano, though i- deas came to me at some mo- ments away from the instrument.
The materi-
als, the pia-
no prepara-
tions, were chosen
as one chooses
shells while walking
along a beach.
The form was as
natural as
my taste permit-
ted: so that where,
as in all of
the Sonatas and two of the Interludes, parts were to be re- peated, the for- mal concern was to make the prog- ress from the end of a section to its begin- ning seem inev- itable. ¶The structure of one of the Sona-tas, the fourth, was one hundred meas- ures of two-two time, divided into ten u- nits of ten meas- ures each. These u- nits were combined in the propor- tion three, three, two, two, to give the piece large parts, and they were subdi- vided in the same proportion to give small parts to each unit. In contrast to a structure based on the frequen- cy aspect of sound, tonali- ty, that is, this rhythmic structure was as hospi- table to non- musical sounds, noises, as it was to those of the convention- al scales and in- struments. For noth- ing about the structure was de- termined by the materials which were to oc- cur in it; it was conceived, in fact, so that it could be as well expressed by the absence of these materials as by their pres- ence. ¶In terms of the oppo- sition of free- dom and law, a piece written ten years before the Sonatas andInterludes, Con-struction in Met-al, presents the same relation- ship, but reversed: structure, method, and materi- als were all of them subjected to organi- zation. The mor- phology of the continu- ity, form, a- lone was free. Draw- ing a straight line between this sit- uation and that presented
by the later
work, the deduc-
tion might be made
that there is a
tendency in
my composi-
tion means away
from ideas
of order towards
no ideas
of order. And
though when exam-
ined the histo-
ry would probab-
ly not read as
a straight line, re-
cent works, begin-
ning with the Mu-sic of Changes, support the ac- curacy of this deduction. ¶For, in the Mu-sic of Changes, the note-to-note procedure, the method, is the function of chance operations. And the structure, though planned precise- ly as those of the Sonatasand Interludes, and more thorough- ly since it en- compassed the whole span of the com- position, was only a se- ries of numbers, three, five, six and three quarters, six and three quarters, five, three and one eighth, which became, on the one hand, the number of units within each section, and, on the other, number of meas- ures of four-four within each u- nit. At each small structural di- vision in the Music of Chan-ges, at the be- ginning, for ex- ample, and a-
gain at the fourth
and ninth measures
and so on, chance
operations
determined sta-
bility or
change of tempo.
Thus, by intro-
ducing the ac-
tion of method
into the bod-
y of the struc-
ture, and these two
opposed in terms
of order and
freedom, that struc-
ture became in-
determinate:
it was not pos-
sible to know the
total time-length
of the piece un-
til the final
chance opera-
tion, the last toss
of coins af-
fecting the rate
of tempo, had
been made. Being
indetermi-
nate, though still pres-
ent, it became
apparent that
structure was not
necessary,
even though it had
certain uses.
¶One of these u-
ses was the de-
termination
of density,
the determi-
nation, that is,
of how many
of the poten-
tially present
eight lines, each com-
posed of sounds and
silences, were
actually
to be present
within a giv-
en small structur-
al part. ¶Anoth-
er use of the
structure affect-
ed the charts of
sounds and silen-
ces, amplitudes,
durations, po-
tentially ac-
tive in the con-
tinuity.
These twenty-four
charts, eight for sounds
and silences,
eight for ampli-
tudes, eight for du-
rations, were, through-
out the course of
a single struc-
tural unit, half
of them mobile
and half of them
immobile. Mo-
bile meant that once
any of the
elements