Information Dynamics and Aspects of Musical Perception

Abstract

Musical experience has been often suggested to be related to forming of expectations, their fulfillment or denial. In terms of information theory, expectancies and predictions serve to reduce uncertainty about the future and might be used to efficiently represent and “compress” data. In this chapter we present an information theoretic model of musical listening based on the idea that expectations that arise from past musical material are framing our appraisal of what comes next, and that this process eventually results in creation of emotions or feelings. Using a notion of “information rate” we can measure the amount of information between past and present in the musical signal on different time scales using statistics of sound spectral features. Several musical pieces are analyzed in terms of short and long term information rate dynamics and are compared to analysis of musical form and its structural functions. The findings suggest that a relation exists between information dynamics and musical structure that eventually leads to creation of human listening experience and feelings such as “wow” and “aha”.

Appendix

Probability of the observations (data) depends on a parameter that describes the distribution and probability of occurrence of the parameter itself

$$P(x_1,\ldots,x_n) = \int P(x_1,\ldots,x_n| \theta) P (\theta) d\theta.$$

((7.7))

Considering an approximation of probability around an empirical distribution θ, [17],

$$\begin{array}{lll} P(x_1^n) & =& P(x_1^n | \theta) \int \frac{P\left(x_1^n | \alpha \right)}{P\left(x_1^n | \theta\right)} P(\alpha) d\alpha \\ &=& P(x_1^n | \theta) \int \exp\left[- \log \frac{P\left(x_1^n | \theta \right)}{P\left(x_1^n | \alpha \right)} \right]P(\alpha) d \alpha \\ &\approx& P(x_1^n | \theta) \int e^{-n D(\theta ||\alpha)}P(\alpha) d\alpha,\end{array}$$

((7.8))

the entropy of a block of samples can be written in terms of conditional entropy given model θ and logarithm of partition function⁸ $Z_n(\theta) = \int P(\alpha) e^{-nD(\theta||\alpha)} d\alpha$,

$$\begin{array}{lll} H(x_1^n) & = & -\int P(\theta)\int P\left(x_1^n | \theta\right) \mathit{log} P\left(x_1^n | \theta \right) dx_1^n d \theta \\ & - & \int P(\theta) \int P(x_1^n | \theta) \cdot \mathit{log} \left[\int e^{-nD(\theta||\alpha)} P(\alpha) d \alpha\right] dx_1^n d \theta \\ & = & H(x_1^n | \theta) - E_\theta[\log Z_n(\theta)],\end{array}$$

((7.9))

where we used the fact that $\int P(x_1^n | \theta) dx_1^n = 1$ independent of θ. With entropy of a single observation expressed in terms of conditional entropy and mutual information

$$H(x_n) = H_\theta(x_n | \theta) + I(x_n,\theta).$$

((7.10))

we express IR in terms of data, model and configuration factors

$$\begin{array}{lll} \rho(x_1^n) & = & H(x_n) + H(x_1^{n-1}) - H(x_1^n) \\ & = & \rho_{\theta}(x_1^n) + I(x_n,\theta) + E_\theta\left[\log \frac{Z_{n}(\theta)}{Z_{n-1}(\theta)}\right]\end{array}$$

((7.11))

Assuming that the space of models comprises of several peaks centered around distinct parameter values, the partition function $Z_n(\theta)$ can be written through Laplace’s method of saddle point approximation in terms of a function proportional to its argument at an extremal value $\theta = \theta^*$. This allows writing the right hand of (7.11) as

$$\log \frac{Z_{n}(\theta)}{Z_{n-1}(\theta)} \approx -D(\theta||\theta^*).$$

((7.12))

resulting in equation of model-based IR

$$\rho(x_1^n) \approx \rho_{\theta}(x_1^n)+ I(x_n,\theta) - E_\theta[D(\theta||\theta^*)].$$

((7.13))