Content uploaded by Örjan Sandred
Author content
All content in this area was uploaded by Örjan Sandred on Apr 06, 2016
Content may be subject to copyright.
1
Revisiting the Illiac Suite – a rule based approach
to stochastic processes
Örjan Sandred
University of Manitoba
Studio FLAT
Mikael Laurson
Sibelius Academy
CMT
Mika Kuuskankare
Sibelius Academy
CMT
ABSTRACT
This article will first discuss the use of probability distribution in L.A. Hiller and L.M.
Isaacson’s string quartet the Illiac Suite. After pointing out some limitations with the technique
used in the Illiac Suite, the use of stochastic rules in a constraint-based system will be introduced.
Finally two possible versions of the beginning of the last movement in the Illiac Suite will be
used to demonstrate the combination of stochastic and ordinary rules.
1. Introduction
Many early experiments in Computer Assisted Composition were inspired by the use of
indeterminacy in contemporary music. John Cage and the group that grew up around him used
chance to do away with the traditional control over the material. When composers began using
computers, they had a tool that could take random decisions very quickly.
The first examples of random processes in computer music composition can be found in
L.A. Hiller and L.M. Isaacson’s string quartet the Illiac Suite (Hiller and Isaacson 1957, 1958).
Hiller and Isaacson describe the Illiac Suite as a chronological record of experiments. The general
idea is to use screening rules to accept or reject randomly generated pitches and rhythms.
Probability distribution and Markov processes can also be found in the suite.
Around the time the Illiac Suite was composed, I. Xenakis established himself as the
pioneer who explored stochastic techniques (with or without the aid of a computer). His music
can serve as a catalogue of possible approaches (Xenakis 2001). Several composers have
continued Hiller’s and Xenakis’ early work: J. Tenney’s first instrumental work using Computer
Assisted Composition, Stochastic String Quartet from 1963 (Tenney 1988), was inspired by both
Xenakis and Hiller. A later example is T. DeLio’s use of Markov Chains in his Serenade from
1976. C. Ames, a student of L.A. Hiller, has contributed both as a composer and a writer (Ames
1987, 1990).
Stochastic techniques are good for formalizing overall tendencies in music. The weakness
of stochastic techniques is that the details are left to the randomness that characterizes the
process. Some composers have dealt with this weakness by using larger units than pitches or
durations, for example chords or phrases, within stochastic processes. However, stochastic
techniques remain inaccurate for managing structural details.
2
The objective of our work described in this article has been to combine stochastic processes
with the detailed control that is possible with constraint-based computing. Our intention has been
to leave the definition of tendencies and rules to the composer of the music. Our interest has not
been to research into machine learning techniques nor to recreate a musical style. We have used
the 4th movement in the Illiac suite as a case study for this article.
We have implemented our examples in the PatchWork Musical Constraint (PWMC)
system, a further development of the OpenMusic Rhythmical Constraint system (Sandred 2006).
PWMC is an extension to the PatchWorkGL visual programming language (Laurson and
Kuuskankare 2006) for Computer Assisted Composition. PWMC is a framework on top of the
PMC constraints solver (Laurson 1996) that is part of the PWGLConstraints system and
programmed in Lisp. PWMC was developed by Örjan Sandred at the University of Manitoba,
Canada, and PWGLConstraints was developed by Mikael Laurson at the Sibelius Academy,
Finland. There are currently several other approaches oriented towards musical search in addition
to our constraint-based systems, such as Situation (Rueda et al., 1998), OMClouds (Truchet et al.,
2001) and the recent Strasheela system based on the OZ programming language (Anders, 2006).
2. The 4th movement of the Illiac Suite
The pioneering work done in Computer Assisted Composition by L.A. Hiller and L.M.
Isaacson in the 1950s shows several examples of random processes.
2.1. The Illiac Suite and probability tables
In the 4th movement of their string quartet the Illiac Suite Hiller and Isaacson used
probability tables to control the distribution of melodic intervals in the four voices. The rhythm in
the movement was set to an ostinato eight-note pulse; the computer was only used to decide what
pitches to assign to the eight-notes.
The probability table with the distribution of melodic intervals was changed every second
measure. The first table set the probability for repeated pitches to 100% and any other melodic
interval to 0%. If the computer assigned the same pitch to two consecutive eight-notes they were
automatically slurred to constitute a longer note value. Thus the movement starts with a sustained
pitch for two measures (without any melodic movement) in all four voices.
Step-by-step, the vocabulary of melodic intervals expands to include octaves, fifths,
fourths, major thirds, minor sixths, minor thirds, major sixth, major seconds, minor sevenths,
minor seconds, major sevenths and tritons (in that order). In this way the linearity becomes
increasingly more complex. Hiller and Isaacson always setup each probability table so that
simpler intervals have higher probabilities (i.e. unison is more likely than an octave, and octave is
more likely than a fifth, etc).
In an article about the Illiac Suite Hiller briefly mentions that they used more complex
dependencies for the probabilities later on in the movement, such as letting the last choice of
interval have an impact on the next choice.
3
2.2. The Illiac Suite – a critical listening to the aural result
A consequence of using probability or Markov tables in music composition is that the
distribution can only be controlled in one dimension in a score. In Hiller’s case the horizontal
melodic contour was generated, but having a preference for what the vertical harmonies should
be is not “part of the game”. The opposite would have been equally possible.
Our critique of Hiller and Isaacson’s experiment is that the aural effect of the music is very
much colored by what is not defined in the probability tables. The lack of harmonic control is
soon obvious; already when the third probability table is used (that includes repeated pitches,
octaves and fifths as possible melodic intervals) in measure 5 – 6 it becomes clear that the
harmonies are there as a side-effect of melodic movements. After another few measures the four
voices walk over the full chromatic spectrum.
The movement begun without any consideration of the melodic context; the melodic
intervals are distributed independently of each other. This allows the melodies to “walk away” to
form very complex profiles. Hiller and Isaacson are conscious of this situation, and they
experiment with different approaches where choices partly depend on the opening note of the
passage, or on the preceding interval. It is not clear how this was done in detail, and judging from
the aural result and the score, the impact is not always obvious.
3. Probability distribution as stochastic rules
We propose to implement probability distribution as a stochastic rule in a constraint-based
system. Rules have the advantage of being modular – it is possible to add rules to restrict related
dimensions (i.e. both the horizontal melodic contour and the vertical harmonic structure can be
controlled by separate rules).
We have tried two different approaches to design probability rules. Both designs imitate the
behavior of a stochastic process. Instead of using a probability table in a random process, the rule
makes sure pitches (or rhythms) in a generated sequence are an acceptable representation of a
probability table. The constraint system can be set to generate the sequence by proposing pitches
(or rhythms) randomly, or systematically going through the domain of possible candidates. In the
first case, the process resembles the stochastic process. In the latter case, the process is
deterministic. In both cases the probability rule will filter out a sequence that follows the
probability distribution.
Our first design uses a count value as a threshold for how many times a given pitch may
exist in the sequence. For example if the constraint based system generates a sequence of 50
pitches, and the probability table states that 20% of these should be the pitch C4, then a
maximum of 10 pitches can be a C4. As a result of the combination of all count values, an
accurate representation of the probability table will exist when the sequence is complete (this is
only true if the pitches that are not part of the probability table have the count value 0). For more
detailed information see Laurson (1996).
4
Our second design checks that the probability table is an acceptable representation of the
generated sequence of pitches (or rhythms) every time a new element is added to the sequence.
Only pitches (or rhythms) that maintain an acceptable distribution at each step during the
calculation are allowed to be added. The maximum accuracy that can be expected depends on the
length of the sequence. For example if a sequence of pitches only contains two pitches, a pitch
can not exist, exist once or exist twice in the sequence. Possible probabilities are thus 0%, 50% or
100%. If the table states that a pitch should have a 40% probability to occur in the sequence, a
precise representation is not possible. Our implementation calculates the maximum deviation
from the given probability table that might occur. The rule will always accept this deviation.
Since the sequence gradually will become longer, the accuracy will increase during the
generation of the sequence. In the given example the highest guaranteed accuracy will be +/-
50%.
€
a=100
n
€
a
= highest guaranteed accuracy (percent)
€
n
= length of sequence
The disadvantage of the first design is that the probabilities will not be accurate until the
end of the sequence. The beginning of the sequence has little if any restrictions, while the end
will be forced to strictly fit into the count values. In combination with other rules, time
consuming backtracking might be necessary late in the search. The disadvantage of the second
design is that the sequence might follow the probability table too closely, without allowing any
fluctuations. The sequence will have the same distribution throughout its whole length. This
might not be desired; for example unexpected events and surprises (that can be of great value in
music) are if not impossible very unlikely.
In both designs, we add a tolerance factor that will allow additional deviations from the
given probability table. Especially when combining a probability rule with other rules, it might be
necessary to allow a certain deviation to find a possible solution. Deviations also open up for
unexpected events in the second design above. Pitches (or rhythms) that are not defined in the
probability table are by default not allowed in the sequence (the tolerance factor will only affect
the distribution of events that are defined in the probability table).
The text below will refer to our second design of the probability rule.
4. The implementation – PWMC
The experiments described in this article were done using the Patch Work Musical
Constraints system (PWMC). PWMC is a system for generating musical structures based on a
domain of pitches, rhythms and metric units. PWMC stores the generated score in a database.
The database is gradually filled with candidates from the domain that fulfill rules defined by the
user. In PWMC, pitch, rhythm and time signatures can all be unknown before the search is done,
but it is also possible to set the sequence of pitches, rhythms or time signatures to a predefined
sequence (as is the case with the sequence of rhythms in the example below). The rules typically
5
constrain how pitch and rhythm can interact in relationship to their metric positions within a
single voice as well as between two or more voices.
PWMC does not use the pattern-matching mechanism from PWGLConstraints. Instead
PWMC searches its database to find the appropriate instances when rules are checked.
PWGLConstraints has a strictly defined search order for how variables in the generated
sequence are assigned values. The search order in the PWMC system is more open. PWMC
searches for a solution sequentially within each voice (starting from the beginning of the
sequence). Beside this fundamental principal, the order for how voices are built is by default not
known. The order between searching for pitches and rhythms within a voice is also not known.
Typically the next assigned value will be in the voice with the shortest sequence. The values are
typically alternating between durations and pitches in the solution.
It is outside the scope of this article to fully describe the PWMC system. The full power of
PWMC was not used in the examples for this article: rhythm was only treated as a side effect of
repeated pitches.
Fig.1. The PWMC patch used to generate the examples described in this article. The stochastic
rule is called “rule-interval-dynamic-prob” in the patch, and the beginning of the probability
table can be seen in the text box above the rule.
6
5. The 4th movement of the Illiac Suite; two new possible versions
Our objective here is to demonstrate how it is possible to have control over harmony and
voice leading without violating Hiller and Isaacson’s probability tables for the 4th movement of
the Illiac Suite. We created a probability rule for melodic intervals based on the 2nd design
described above. We used Hiller and Isaacson’s original probability tables (changing the table
every second measure) for generating pitches in four voices, and we allowed maximum 2%
deviation from the table. The rhythm was set to a sequence of repeated eighth notes and the time
signature was 6/8.
To this we added a rule for controlling the harmonies made up of the four voices. The rule
would only allow certain chord structures between the voices. In our first experiment we did the
obvious; we allowed only major and minor triads (and all their possible inversions). We also
added a second rule for controlling voice leading. We did not allow parallel unisons, octaves or
fifths between neighboring voices, neither between the outer voices. Finally we added a heuristic
rule (i.e. a rule that is not strict, but gives the system a preference for certain types of solutions);
the first voice is preferred to be above the second voice. Since the two violins have the same
register, voice crossing would otherwise frequently occur for no reason. Despite this heuristic
rule, the computer found reasons to have quite a few voice crossings.
The result is as expected a far more tonal-sounding Illiac movement than Hiller and
Isaacson’s original composition. Knowing the amount of possible variations that can be generate
from the original probability tables alone, it is highly unlikely that Hiller and Isaacson would
have come up with this version. At the same time, it is easy to prove that Hiller and Isaacson’s
probability tables are respected. Our version is therefore just as likely as any other version of the
movement.
In a second experiment we changed the allowed chord structures to be based on either
fourths, fifths or augmented fourths; a set of four allowed chord structures was given to the
harmony rule (see fig.2). The rule would also allow all possible inversions of these chord
structures, however we made a restriction: minor seconds were not allowed between neighboring
voices (inversions of the last two chords in fig.2 could otherwise have created minor seconds in
the harmony). At the first beat in each measure we made the rule stricter; at these positions only
open fifths (and octaves) were allowed.
Fig. 2
We kept the voice-leading rule from our first experiment, but we added a link between the
top two voices as well as between the bottom two voices. Each time there is a repeated pitch in
one of the voices, the other voice in the pair must repeat its pitch as well. Since repeated pitches
are automatically slurred (this is how Hiller and Isaacson created sustained pitches in the original
version), the top two voices will have the same rhythm. The bottom two voices will have the
7
same rhythm as well. We also added the heuristic rule from our first experiment to avoid voice
crossing. Finally we restricted the maximum distance between the outer notes for two
consecutive melodic intervals; maximum an octave was allowed. The purpose of this rule is to
avoid melodies that walk over the whole register (for example this rule will force an octave to be
followed by a repeated pitch or a contrary motion).
Each probability table in the Illiac Suite is valid for two measures. Within two measures
each voice has 12 consecutive melodic intervals. The highest guaranteed accuracy will thus be
100 / 12 = 8.3%. With the tolerance set to 2% we can expect our score to show a probability
distribution that differs up to +/- 10.3% from the given probability table.
Our second experiment shows many aural structures. Just as it would have been unlikely
for Hiller and Isaacson to come up with anything that resembled our first experiment, our second
experiment is equally unlikely without the added rules. However our second experiment respects
Hiller and Isaacson’s probability tables and is therefore a valid version.
Fig.3. The beginning of the score for our second experiment.
8
6. Conclusion
The concept of probability distribution can be useful in music. Development over longer
time spans, as well as frequency of events, can be controlled. By implementing probability
distribution as a rule in a constraint based system it is possible to combine this concept with other
structural restrictions. The constraint solving systems of today (that were not available when the
Illiac Suite was composed in 1956) make complex problems with dependencies between linear
and vertical pitch dimensions possible to solve.
7. References
Ames, C. (1987). Automated Composition in Retrospect: 1956-1986. In Leonardo, Vol. 20, No.
2: 169-185.
Ames, C. (1990). Statistics and Compositional Balance. In Perspectives of New Music, Vol. 28,
No. 1: 80-111.
Anders, T. (2007). Composing Music by Composing Rules: Design and Usage of a Generic
Music Constraint System. PhD Dissertation, Queen's University, Belfast.
Hiller, L. and Isaacson, L. (1958). Musical composition with a high-speed digital computer. In
Journal of Audio Engineering Society 6: 154-60.
Hiller, L. and Isaacson, L. (1957). Illiac Suite. Score, Theodore Presser Co, New York, USA.
Laurson, M. (1996). PATCHWORK: A Visual Programming Language and Some Musical
Applications. Ph.D. Dissertation, Sibelius Academy, Helsinki, Finland.
Laurson, M. and Kuuskankare M. (2006). Recent trends in PWGL. In Proc. ICMC 2006, New
Orleans, USA.
Rueda C., Lindberg M., Laurson M., Bloch G. & Assayag G. (1998). Integrating Constraint
Programming in Visual Musical Composition Languages. ECAI 98 Workshop on Constraints for
Artistic Applications. Brighton, England.
Sandred, Ö. (2006). Kalejdoskop for clarinet, viola and piano. In The OM Composer's Book -
Vol.1, Editions Delatour France / IRCAM. 223-235.
Tenney, J. (1988). Stochastic String Quartet. Score, Smith Publications/Sonic Art Editions,
Baltimore, USA.
Truchet, C., Assayag, G. & Codognet, P. (2001). Visual and Adaptive Constraint Programming
in Music. Proceedings of the International Computer Music Conference. Havana, Cuba. 346-352.
Xenakis, Iannis. (2001). Formalized Music: Thought and Mathematics in Composition.
(Harmonologia Series No.6). Hillsdale, NY: Pendragon Press.
