$1/f$ in Music

Voss and Clarke conducted some studies on some selected musical compositions. In the first experiment the audio signal was run through a bandpass filter of 0.1-10KHz. The output of the filter was squared to obtain a power function, and that signal was run through a low pass filter with the cutoff at 20Hz. The data from this filter was plotted and it was reported that almost all kinds of music (ranging from a recording of Bach's First Brandenburg Concerto to arbitrary selections of signals from different types of radio stations) behaved like $1/f$ noise. With this experiment they concluded that the ``audio power fluctuations'' of music, which they called loudness, varies according to $1/f$ noise. We would like to point out that the structures observed were actually the rhythmical structures in the fast regions (about 0.25 to 8 seconds) and the formal structures in the slow regions (greater than 8 seconds). One way to interpret this data is that it describes the uniformity of the loudness between these two regions.

Voss and Clarke also studied the ``instantaneous pitch'' fluctuation of music. The ``instantaneous pitch'' was measured by counting the number of zero crossings of the audio signal in specific periods of time. Thus, a new signal $Z(t)$ was extracted from the audio signal $V(t)$, which they assumed, in this case, follows the melody of the music. $Z(t)$ was passed through a low pass filter at 20 Hz and then its power spectrum was measured. Again they found that $Z(t)$ for many different kinds of music and radio stations behaved as $1/f$ noise. In this study they also produced some sounds using white, $1/f$, and $1 / {f}^{2}$ noises. For every one of the samples the same process was used to control the pitch as well as the duration of every note. The pitches were rounded off to different musical scales such as pentatonic, major, or 12 tone chromatic. These examples were played to several hundreds of listeners, and it was reported that listeners classified the ``compositions'' according to: white noise, too random, $1 / {f}^{2}$ noise too correlated, and $1/f$ closest to what listeners expected of music.

They argued that even though low-level Markov models, or deterministic constraints imposed on white noise, can create some local correlations, they fail to provide a long-term correlation. They suggested that $1/f$ noise is the natural way of adding long-term correlations to stochastic compositions.