APPLICATIONS OF QUANTUM ANNEALING TO MUSIC THEORY
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences
Ludmila Botelho
"...might compose elaborate and scientific pieces of music of any degree of complexity or extent"
Ada Lovelace
Charles Babbage
Quantum annealing
Proposition: Find the minimum energy in a landscape
- Ground state = optimal solution
$$ H(t) = \left(1-\frac{t}{T}\right)H_0+ \frac{t}{T}H_P $$
- Different from gate based
$$ H_P = - \sum_{i > j} J_{ij}s_i s_j - \sum _i h_is_i, $$
$$ H(t) = \left(1-\frac{t}{T}\right)H_0+ \frac{t}{T}H_P $$
- Can be encoded in Binary Values
Ising model
- Mathematical model to describe magnetization
\( s_i \in \{-1,+1\} \)
NP-hard!
\( s_i \leftrightarrow x_i = \frac{1-s_i}{2} \)
\( x_i \in\{0,1\} \)
quadratic unconstrained binary optimization
-
Minimizing quadratic functions
$$ y=x^TQx $$
-
Penalty Method
$$ \text{min } y=f(x) $$
$$ \text{subject to: } x_1 +x_2 + x_3= 1 $$
$$\text{min }y=f(x)+P\left(\sum_{i=1}^3 x_i -1\right)^2 $$
Binary variables
Constants
And now for something completely different
Music Generation
4
1
rest
measure
Durations
2
Pitches
C
D
A
E
G
F
B
C
Music Generation
$$\sum_{j \in P} x_{i,j} = 1$$
How to define the binary variables?
$$\left ( 1- \sum_{j \in P} x_{i,j} \right )^2$$
- Pitches
- Rules about consecutive notes
$$x_{i,C} + x_{i+1,D} \leq 1$$
$$x_{i,C} + x_{i+1,C} + x_{i+2,C} \leq 2 $$
$$P=\{p_1,p_2,\dots,p_k \}$$
\(x_{i,j}\) for \(i \in [n]\) and \(j \in P\)
$$P=\{\mathtt{C}, \mathtt{D}, \mathtt{E}, \mathtt{G}\}$$
Melody Generation
Optimization
$$ -\sum_{ \substack{i\in [n-1] \\ j,j' \in P}} W_{j,j'} x_{i,j} x_{i+1,j'} $$
weights
Exemple: Ode to Joy excerpt
$$W_{F\#4,E4} = 2, $$
$$W_{F\#4,F\#4} = 2,$$
$$W_{F\#4,G4} = 1$$
melody Generation
$$\sum_{\substack{i \in [n] \\ j \in D}} d(j)y_{i,j} = L$$
- Duration of notes
Rhythm Generation
$$ \sum_{j \in D} y_{i,j} = 1$$
$$D = \{1,2,3,4\}$$
$$j \in D$$
Harmonization
$$\sum_{j \in P} x_{i,j} = 3$$
$$(1-x_{i,p_{i_j}})$$
G
E
D
G
B
B
G
Boléro (Ravel, Maurice)
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Monophonic
Polyphonic
Music is complex
-
Phrase Identification
But also adaptable!
Huang, Jiun-Long, Shih-Chuan Chiu, and Man-Kwan Shan. "Towards an automatic music arrangement framework using score reduction." ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM) 8.1 (2012): 1-23.
But also adaptable!
-
Phrase Identification
But also adaptable!
But also adaptable!
But also adaptable!
But also adaptable!
What is relevant?
QUBO Formulation
- Variables and objective function
- Entropy
- Pitch and rhythm
- Maximize information
$$x_{ij} =\begin{cases}1,& \text{$j$'th phrase of track $i$ is selected}\\ 0, & \text{otherwise.}\end{cases} $$
$$- \sum_{i=1}^N\sum_{j=1}^{P_i} e(p_{ij})x_{ij}$$
$$E(s) = - \sum_{i=1}^N P(x_i)\ln P(x_i)$$
QUBO Formulation
- Constraints
$$x_{ij} = 1 \iff m_{ik}=1 k \in S_{ij}$$
$$\sum_{i=1}^N m_{ik} = M \ \ \text{for} \ \ k=1,\dots,K $$
Set of measures in phrase \(j\) of \(i\)’th track
Number of measures
Number of tracks after reduction
- Similar to fixed interval scheduling
binary variable
Future work
- Compatibility between selected phrases
- Constraints about note ranges
- Dissonance criteria
- Track support for selected phrase
Guitar: 4 octaves
Piano: 8 octaves