Music, point processes and 1/f noise

There is interesting observations in the music by the great classical composers - statistical properties of their time series appear to be as complex as social phenomena considered here on Physics of Risk. Their music may seem to be both - at certain times easily anticipated and predictable, while at the other times have large unexpected deviations. Their music behaves as a pink or 1/f noise [1, 2]! In [1] it was shown that the intensity time series of the music by the classical composers and human speech time series have 1/f region in their spectral densities. While in [2] these ideas are applied towards musical rhythm. To us [2] is especially interesting as this paper considers our own model, [3, 4], as a proper model for the 1/f noise in the spectral density of musical rhythm.

Musical rhythm is an excellent example of the point process. Namely we can see each individual key press as unit impulse at a certain moment. In such case the time series is defined by the sum of such impulses:

\begin{equation} I(t) = \sum_i \delta(t-T_i) . \end{equation}

Note that the time series is also fully described by the event times, \( T \). Alternatively by assuming that our time series starts at the origin we can describe the same time series using inter-event times:

\begin{equation} \tau_i = T_{i+1} - T_i . \end{equation}

The inter-event times prove to be useful in the empirical analysis of varying complex systems - be it financial markets or music. In [3] it was noted that the empirical financial market data has specific statistical features, which can be modeled using the following equation as the model for inter-trade times:

\begin{equation} \tau_{s,k+1} = \tau_{s,k} + \gamma \tau_{s,k}^{2 \mu-1} + \tau^\mu_{s,k} \zeta_{k} , \end{equation}

here \( \tau_{s,k} \) is the k-th scaled time between two consecutive trades, \( \gamma \) is related to the relaxation of the time series (note that here we use slightly different form than in [3, 4]), while \( \zeta_{k} \) is a standard normal random variable (zero mean and unit variance). It should be evident that the above equation is difference equation thus it can be rewritten as Langevin equation and that for further analysis we can use stochastic calculus.

In this text we won't consider more delicate topics related to the point process model. We just have to mention that the spectral density of the point process model in a certain range of frequencies can be approximated by the power law:

\begin{equation} S(f) \sim 1/f^\beta , \quad \beta = 1 + \frac{2 \gamma -2 \mu}{3 - 2 \mu} , \quad 0.5<\beta <2 . \end{equation}

From this expression it should be evident that we will obtain pink noise signal, if the inter-event times will change according the Brownian motion. Namely if \( \gamma = \mu = 0 \), the observed power law will be approximately equal to 1, while our difference equation will be the same as for the Brownian motion. While it is not the only case we can obtain 1/f noise with the point process model, but this case is almost surely the most interesting one.


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