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Fig. 1.
Final Solution of Sudoku Puzzle Obtained by HS
3 Music Composition
Music can be composed by metaheuristic algorithms. Horner and Goldberg [3] ap-
plied a GA model to bridge-music composition in the minimalist style. The fitness
function in their model was the degree of pattern match and duration. Ralley [4] pro-
posed another GA model to develop music melody. However, the melody developed
by GA could not be evaluated because there was no appropriate fitness function. Biles
[5] developed an interactive GA model, GenJam, to play jazz solos. The GenJam
model has been applied to many jazz tunes. This section shows another music compo-
sition example using HS.
3.1 Formulation of Medieval Music Composition
Gregorian chant is a monophonic unaccompanied song of the Roman Catholic Church
in the middle age, and organum is an early form of polyphonic music accompanying
the Gregorian chant.
The organum has the following composing characteristics: 1) the original harmony
line progresses in parallel since it starts on the same note; 2) for the parallel motion,
the interval of perfect fourth is frequently used while those of perfect fifth and unison
(or octave) are also preferred; 3) in order to distinguish the vox principalis (chant
melody) from vox organalis (harmony), the former should always be located above
the latter. The above-mentioned organum composition technique can be formulated as
the following optimization problem:
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