Information Technology Reference
In-Depth Information
N
N
∑
∑
Minimize
Rank
(
x
)
+
Penalty
(
x
),
i
=
1
2
"
,
N
(9)
i
i
i
=
1
i
=
1
s.t.
x
≤
m
,
i
=
1
2
"
,
N
(10)
i
i
|
x
−
x
|
≤
|
m
−
m
|,
i
=
2
3
"
,
N
(11)
i
i
−
1
i
i
−
1
x
=
m
(12)
Start
Start
x
=
m
(13)
End
End
{
}
+
x
∈
Do,
Re,
Mi,
Fa,
Sol,
La,
Si,
Do
(14)
i
where
x
is the
t
i
pitch in harmony line and
m
is the
t
i
pitch in original chant line.
Equation 9 represents the fitness function, which is the summation of the rank term and
the penalty term for each pitch in the harmony line (vox organalis). As the interval of a
perfect fourth between vox principalis and vox organalis is most preferred, it receives
the highest priority (rank = 1). The ranking for intervals is a prior knowledge and ex-
plicitly determined. Equation 10 represents the constraint that organum pitch should be
lower than or equal to corresponding chant pitch. Equation 11 represents the constraint
that the interval between two consecutive organum pitches should be less than or equal
to that in two consecutive chant pitches. Equations 12 and 13 represent boundary con-
ditions that constrain starting and ending pitches in organum.
3.2 Example Composition
Figure 2 shows a Gregorian chant 'Rex caeli Domine' as well as a corresponding or-
ganum composed by HS [6]. The upper line in the figure is the Gregorian chant mel-
ody and the lower line is the organum line. There are 28 pitches (number of decision
variables) accompanied by the organum line, which represents
28
25
com-
8
(
=
1
.
93
×
10
)
binatorial possibilities.
Fig. 2.
Organum Composed by the HS algorithm
The HS model composed the organum line by generating up to 3,000 improvisa-
tions within one second on Intel Celeron 1.8GHz CPU and obtained aesthetically
pleasing organum as shown in Figure 2. Also, a more complex organum piece, which
has 50 pitches, was also successfully composed using the same process.
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