Ashish Arya Ludmila Botelho Fabiola Cañete Dhruvi Kapadia Özlem Salehi
"...might compose elaborate and scientific pieces of music of any degree of complexity or extent"
Proposition: Find the minimum energy in a landscape
H(t)=(1−Tt)H0+ TtHP
HP=−i>j∑Jijsisj−i∑hisi,
H(t)=(1−Tt)H0+ TtHP
- Can be encoded in Binary Values
si∈{−1,+1}
si↔xi=21−si
xi∈{0,1}
Minimizing quadratic functions
y=xTQx
Penalty Method
min y=f(x)
subject to: x1+x2+x3=1
min y=f(x)+P(i=1∑3xi−1)2
Binary variables
Constants
4
1
rest
measure
Durations
2
Pitches
C
D
A
E
G
F
B
C
j∈P∑xi,j=1
How to define the binary variables?
1−j∈P∑xi,j2
xi,C+xi+1,D≤1
xi,C+xi+1,C+xi+2,C≤2
P={p1,p2,…,pk}
xi,j for i∈[n] and j∈P
P={C,D,E,G}
Optimization
−i∈[n−1] j,j′∈P∑Wj,j′xi,jxi+1,j′
weights
Exemple: Ode to Joy excerpt
WF#4,E4=2,
WF#4,F#4=2,
WF#4,G4=1
i∈[n]j∈D∑d(j)yi,j=L
j∈D∑yi,j=1
D={1,2,3,4}
j∈D
j∈P∑xi,j=3
(1−xi,pij)
G
E
D
G
B
B
G
Melody Harmonized melody