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solvers . Virtuosity can be objectively measured, observed, and as such is, 'as a con-
cept, closer to the ground, than creativity' (Howard 2008 ).
Indeed, the capacity to effortlessly navigate in large search space in real-time
is not only a matter of physiological prowess. By transferring part of the decision
processes to the body, a virtuoso naturally compiles his knowledge in a remarkable
way that can teach us a lot about innovative problem-solving.
For instance, virtuosos in mental calculation invent and make extensive use of
so-called mathematical tricks . As an example, squaring any number ending in 5 can
be done easily using a simple formula (take the first digits except the last 5, multiply
it by itself plus 1, and then concatenate 25 at the end). Some of these tricks are well-
known, but many others are not, and probably ignored by their inventors: intense
training may result in exceptional performance, not necessarily in clear explana-
tions. In the following sections, we show how jazz virtuosos produce interesting
inventions, and how modelling this virtuosity leads to interesting insights about the
nature of invention and creativity.
5.2 The Case of Jazz
Much like language, the ability of humans to spontaneously improvise music in
real time is considered by many as an extraordinary skill, a sort of magic. Most of
this magic, again, comes from hard training. As Levine ( 1995 ) states in his intro-
duction: “ A great jazz solo consists of 1 % magic and 99 % stuff that is Explain-
able, Analyzable, Categorizeable [sic], Doable. This topic is mostly about the 99 %
stuff .” This chapter is about putting the 99 % stuff in a machine, and making the
remaining 1% explicit. In particular, our aim is to separate clearly what can be
reasonably automated—what virtuosos are able to do unconsciously—from what
emanates from artistic, conscious decision making.
Invented in the 1940s with Charlie Parkerand Dizzy Gillespie, bebop is an id-
iom of jazz where a strong emphasis is put on melodic and harmonic dimen-
sions.Virtually all instruments of the classical orchestra have been used by bebop
musicians. Nowadays, bebop musicians continue expanding the style. The case of
jazz bebop improvisation is particularly interesting because the specific constraints
of bebop are shared unambiguously and can be easily expressed using well-defined
languages: In some sense, jazz improvisation is a special form of computing.
Scientists have long tried to debunk the magic of jazz improvisation, starting
with the psychologist Philip Johnson-Laird. His work is not to be judged by the
musical quality of his algorithmic productions, but by the seminal nature of his ar-
guments. One of his main claims is that the ability to produce an improvisation
does not require any 'short-term memory' (Johnson-Laird 1991 ; 2002 ). He demon-
strated this idea by proposing memoryless automata that automatically generate
rhythmic and melodic material. Since then, more powerful algorithmic techniques
have been used to produce jazz improvisation (see Sect. 5.3 ), but it can be said that
the problem of modelling 'basic' bebop improvisation has been solved, notably by
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