Recreated Results

In this section we will present the result of our analysis of the pieces we had access to. Rather than looking at the audio signal, we took a different route for our analysis. We used the data from 57 pieces which were coded in MIDI file format. We extracted a top voice from these pieces. The top voice is defined as the highest sounding pitch at any moment. Silences were eliminated by extending the last highest pitch. The data was stored as described in section 4.4.1. The tempo was set by the first tempo marking and all other tempo changes during the piece were ignored. The DC value of the pitch signal was subtracted from all samples and the power spectrum of the resulting signal was computed. We would like to emphasize the fact that we are not saying that such a signal is ``the'' melody of the piece; however, we are assuming that with the defined procedure we will obtain ``a'' melody which has some musical integrity. Audio example 3 is the resynthesis of the first 30 seconds of the pitch signal extracted from the J. S. Bach's 3rd Brandenburg concerto. As it can be clearly heard, there are still problems in the extraction method which, due to not having enough time, we did not solve. Figure 4-7 shows the first 30 seconds of the extracted pitch signal and the power spectrum computed for the duration of piece. The problems of the extraction method can be seen as the vertical spikes in the figure. As it can be seen, the power spectrum of this signal is best fitted by the $1/f$ line. Appendix A contains the result of all the pieces whose power spectrum were systematically computed.

Figure 4-7: (a) is the first 30 seconds of the ``top voice'' signal extracted from J. S. Bach's 3rd Brandenburg Concerto. (b) is the the power spectrum of the ``top voice'' signal for the duration of the piece. Notice that the line representing the $1/f$ line fits the slope of the power spectrum.

Almost all pieces behaved very closely to the $1/f$ noise. It is worth noting that we were able to find the fault of our extraction method by looking at the resulting power spectrums, and that shows that the power spectrum does carry useful analysis information. For example, the power spectrum of Prelude 11 from the first book of the Well-tempered Clavier (see figure 4-8-b) was the most odd looking spectrum. When we listened to the extracted signal we found that the many trills of the dotted quarters (which are scattered throughout the piece) mixed with the bottom voice created a ``noisy'' melody which accounts for the flat section of the spectrum between .1 to 5 Hz. The slope of the power spectrum is a good measure of how much material is coded in the melody. For example, the spectrum of Prelude 8 (see figure 4-8-a) showed a slope steeper than other pieces, which should mean that the melody of the extracted signal should be more correlated than the others. When we looked at the score for that piece, we noticed that much of the melody is coded in other voices rather than the top voice, and the highest pitch is kept for long periods of time; in one case (measures 32 to 34) the highest note is kept sounding for 3 full measures.

Figure 4-8: The spectrum of two of the odd cases of the analysis is shown. Figure (a) is the power spectrum of the ``top voice'' signal of the 8th prelude from J. S. Bach's Well-tempered Clavier Part I. Notice that the slope of the spectrum is sharper than the $1/f$ line and that can be explained by the static melody of the top voice in that piece. Figure (b) is the power spectrum of the 11th prelude from J. S. Bach's Well-tempered Clavier Part I. Notice that the spectrum is flat in the 0.1-5 Hz region. This effect was caused by the way we extracted the top voice. The interaction between the half note trills and our down-sampling of the MIDI data created a noisy melody which is characterized by a flat spectrum.