# Digital Signal Processing of the Inharmonic Complex Tone
## 4 Experimental Results
### Signal Processing Chain of the Recording Expected to Exhibit Inharmonicity  
The first experiment follows the processing of the recorder harp signal As4. The signal is recorded at the Faculty of Music, University of Arts in Belgrade, in a room typically used for practicing and teaching. The harp was played by the faculty professor. The recording is part of a set dedicated to evaluating the properties of harp tones. The signal and the segments' beginning and end points are plotted in Figure 8. The first segment corresponds to the entire recording. After the signal is preprocessed by the submodule MuPL_S, the spectrum content is plotted, and \(f_{0}\) and inharmonicity coefficient \(B\) are set using slider controls, visually matching the positions of the red help lines with the signal spectrum. The filter bank was designed as a 17-channel bank (the number of prominent partials was set to 15 by the user). The MuPL_B GUI window is shown in Figure 9. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline CPU & 13 th Gen Intel(R) Core(TM) i7-13650 HX \\  & 2.60 GHz \\ RAM & 16.0 GB \\ Operating system & 64-bit Windows 11 Home \\ Graphics & NVIDIA GeForce RTX 4050, 6 GB \\ \hline \end{tabular}
\end{table} 
Table 3: The specifications of the computer used for the presented experiments. Figure 8: The time-domain plot of the recorded signal used for further analysis. The dashed lines indicate starting and ending samples of signal segments, blue for the whole length signal and red for the second segment. The signal can be further processed, yielding additional results. In Figure 10, the results obtained by the module implementing the algorithm [38] are presented. The calculation is repeated for 50 frames for both selected segments of the signal and for the whole-length signal. The estimated values differ slightly from the value evaluated manually by the user. The variations from frame to frame in the first 50 frames of the signal (blue line) are probably due to the frequency glide, i.e., the spectrum content is slightly different in different frames. The second segment of the signal starts at 1 s, and at least some of the spectrum components have already vanished before this point. It can be assumed that the larger variation of the extracted inharmonicity coefficient values is due to the fact that spectral maximums are calculated automatically, regardless of whether the prominent sinusoidal component is even present. Figure 10: Values of the inharmonicity coefficient \(B\) calculated by the automatic algorithm [38] for 50 frames for two different segments (blue is whole signal and red is segment selected in MuPL_S). Solid lines corresponding to the mean values. Figure 9: The look of MuPI_B GUI window after the bank was designed. The bank is fitted to the signal content, i.e., each subchannel corresponds to a single signal partial. The red helplines correspond to partials of the signal for the set values of the fundamental frequency \(f_{0}\) and inharmonicity coefficient \(B\). Dashed black lines are gains of filters of the designed bank. The next step is to investigate the time envelopes of the signal partials. The GUI of the third submodule, representing one of the signal partials analysis, is presented in Figure 11. The 6 th partial is chosen because the spectrum in that range contains two very close components, as indicated by the green curve in the spectrum plot, with the lower one (in frequency) probably arising from the phantom partial. Those two components impact the shape of the time envelope. The black line in the time plot indicates the extracted envelope, and the green-black line indicates the fitted \"corrected\" envelope. Each component of the signal, each partial, can be included in the summing signal, excluded, or replaced with a synthesized partial in the form of (19). In this example, the partials 6, 7, 8, and 9 were replaced with synthesized ones. The shape of the \"corrected\" 6 th partial spectrum is shown in the spectrum plot, Figure 11, with a green-black line.