Effect of Changing the Average Duration

In this section we will examine random pitch signals. The values have been chosen from a logarithmic scale of frequencies with various quantization levels. Later we will use the same method for analyzing some pieces according to their MIDI encryptions. The pitch signals are stored as sound files. The frequency of the middle C, or the MIDI note number 60, is used as a reference point. We can have up to 273 quantization levels per semitone. The value of the pitches are restricted to 20 to 2100 Hz. Unless noted, in all the signals the pitches are quantized to frequency values of the well-tempered scale. Once the random signal is generated the average value of the signal in time is subtracted from all the samples.

Figure 4-3 shows the power spectrum and the first 30 seconds of a random signal with average note duration of 0.1 second. We can see that the power spectrum for this random signal is flat, which means that there are as many fast oscillations (structures) as there are slow oscillations. The power spectrum is shown on a log-log scale and for having a reference, the line which represents the $1/f$ spectrum is plotted on top of all the plots in this section. Figure 4-4 shows the power spectrum and the first 10 seconds of random signals with average note durations of 0.5, 2, and 200 seconds (for all the signals 1000 seconds of the random signal was generated.) Notice how these signals start to show a ``slope'' on the high frequency spectrum. This slope indicates some temporal correlation in that region. Obviously a constant value is more correlated than a random signal; therefore with a higher value of average note duration, the signal becomes more correlated. In fact we can characterize these functions as a $1/{f}^{\beta}$ spectrum, while in the case of the flat spectrum $\beta = 0$ for all the regions and for the other cases $\beta = 2$ in the region of correlation and $\beta = 0$ in other regions.

Figure 4-3: The time domain and power spectrum of white noise with average duration of 0.1 seconds. (a) illustrates the first 30 seconds of the signal and (b) is the power spectrum of such a signal in log-log scale. The line representing the $1/f$ is also drawn for reference. Notice that the power spectrum for this signal is flat for the area of our inspection which is between 0.001 and 5 Hz.

Figure 4-4: The time domain and (log-log scale) power spectrum of random signals with average duration of 0.5, 2, and 200 are illustrated. The line representing the $1/f$ line is also drawn for reference. Notice that the power spectrums show a slope steeper than the $1/f$ line in the area of correlation while the rest of the spectrum stays flat.