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Extensions to higher orders as well as variable-orders (Ron et al.
1996
) do not
change substantially the basic principle of Markov generation. The Markov hypoth-
esis, in all its forms, can be seen as a concrete implementation of Longuet-Higgins's
memoryless assumption
(see Sect.
5.2
).
The Markovian aspects of musical sequences have long been acknowledged, see
e.g. Brooks et al. (
1957
). Many attempts to model musical style have therefore ex-
ploited Markov chains in various ways (Nierhaus
2009
), notably for sequence gen-
eration.
Many experiments in musical Markov models have shown that there is indeed
a strong Markovian dimension in musical surface in most genres of tonal music,
including jazz (see e.g. Nierhaus
2009
for a survey). The
Continuator system
(Pachet
2003
) was the first to propose a real-time improvisation generation system based on
Markov chains, producing sequences as continuations of input sequences played
by humans. This system was shown to deliver striking results, even passing 'jazz
Turing tests' (Van Veenendaal
2004
).
Most Markov generators are based on a
random walk
process, exploiting a prob-
abilistic model of the input phrases. The generation is performed step-by-step, in
our case, note by note, using a random draw scheme, which takes into account the
context, i.e. the phrase generated so far:
Iteration at step i:
next
=
Random
_
Draw(context
i
)
;
context
i
+
1
Concatenate(context
i
, next)
;
In practice, the context is limited to a certain maximal
order
. Random choice is
performed as a weighted random draw, using an efficient representation of all en-
countered suffixes computed from the training set, which yields a probability table.
The generated event is then concatenated to the context, and the process is iter-
ated.
It has been shown that this model enables the creation of realistic outputs in
many musical styles, with professional musicians (Pachet
2003
, Assayag and Dub-
nov
2004
) as well as children (Addessi and Pachet
2005
). Like previous approaches,
these systems use a general, agnostic algorithm, uniformly applied to music se-
quences of any style. Consequently, the qualities of its outputs are also indepen-
dent of the style of its inputs, and uniformly good . . . or bad. However, it should
be noted that these systems perform best in
musically unconstrained
contexts,
such as free-form improvisation. No convincing results were obtained when used
in a
bebop
setting with the constraints we have introduced in the preceding sec-
tion.
Random walk approaches have shown limitations when used for generating
com-
plete pieces
as this strategy does not always favour the most probable sequences in
the long term (Conklin
2003
). This is not an issue in our case, as we will see how
the generation can be controlled using higher-level
controls
that determine global
characteristics of the generated sequences, taking precedence over the details of the
basic generation algorithm. Indefinite memory length is the main claimed advantage
of the system proposed by Assayag and Dubnov (
2004
). In our context, this problem
is irrelevant, as our goal is not to reproduce similar pieces, but to use the training
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