Step 2: Define vectors a,b M. Create a pattern C in M consisting of elements
B + na + mb, (m,n Z), modulo a preditermined integer p (in our case p was
Step 3: Define a set G of main 2-dimensional transformations g : 2(M) 2(M), where
2 =projection of M on T × P, T × L, T × D, P × L, P × D or L × D.
Step 4: Apply certain transformations g G to local parts (subsets) of C. Note that the
transformations are applied to any two-dimensional projection.
Step 5: Apply other freely chosen transformations h Gc to local parts (subsets) of
Step 6: Shape a detailed tempo curve.
Step 7: Define an instrumentation.
Steps 2–7 are applied repeatedly, and not necessarily in the given sequence.
A few remarks need to be made at this point, since at first the construction given above
may appear completely abstract and too mathematical:
- The composition was done using “musical intuition”, without conscious application
of mathematics. Only a vague global plan existed – the description above is
an a posteriori analysis that is much more precise than the original plan. This
corresponds to the usual compositional process. A global plan is used, consciously
or subconsciously, in most composed (and even in improvised) music. (Also, for
traditional music it is not unusual that an a posteriori analysis reveals structural
features the composer was not quite aware of.)
- An intuitive musical application of transforms was made possible by the graphical
MIDI-software PRESTO (Mazzola 1989–94). No mathematical knowledge was
required. Due to the help of the software, a vast variety musical possibilities could
be tried out by listening. This allowed “controlled improvisation” leading to an
enormously fast, intuitive and spontaneous way of composing.
- It should be emphasized that the software used here does not compose, The
computer is here “only” a tool that enables us to pursue new musical ideas in an
1.3. Basic principles
The following basic principles were used:
- Representation of basic aspects of the composition (excluding e.g. instrumentation,
microtuning etc.) as a “point cloud” in an algebraic module (linear space).
- Hierarchy (global structures, local structures).
- Exchangibility of musical parameters (time, pitch, loudness, duration).