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

x_{i,j} = \begin{cases}% 1 & \text{note at position $i$ is $j$,}\\ 0 & \text{otherwise.} \end{cases}

$$\sum_{j \in P} x_{i,j} = 1$$

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
y_{i,j} = \begin{cases}% 1 & \text{note at position $i$ has duration $j$,}\\ 0 & \text{otherwise.} \end{cases}

Rhythm Generation

$$ \sum_{j \in D} y_{i,j} = 1$$

    $$D = \{1,2,3,4\}$$

    $$j \in D$$

Harmonization

x_{i,j} = \begin{cases}% 1 & \text{chord at position $i$ contains note $j$,}\\ 0 & \text{otherwise.} \end{cases}

$$\sum_{j \in P} x_{i,j} = 3$$

$$(1-x_{i,p_{i_j}})$$

G

E

D

G

B

B

G

Boléro (Ravel, Maurice)

?

?

?

?

?

?

?

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

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Quantum Annealing and Music Reduction

By ludmilaasb

Quantum Annealing and Music Reduction

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