## 6 Practical Synthesis Usage  
In practical sound synthesis applications, a single folding stage is rarely used, as the timbral variety it can produce is quite limited. Most analog designs, for example the Intellijel \(\mu\)Fold and the aforementioned Yusynth Metalizer, employ several wavefolding stages arranged in series. The number of stages varies according to the design, but typically cascades of two to six stages are used. As mentioned in Section 3, the Serge middle VCM utilizes six identical folding stages. Figure 16 shows a simplified block diagram representation of the Serge middle VCM based on the original design [63]. Blocks labeled \"SWF\" correspond to the proposed Serge wavefolder model. An ad hoc gain factor of four, not present in the original circuit, has been added to compensate for the scaling of the signal introduced by the cascade of wavefolders. In cascaded wavefolder structures like the one shown in Figure 16, timbral control can be achieved in two manners. The first is by adjusting the gain of the input waveform (using \(G_{\mathrm{S}}\) in this case). This parameter controls the amount of folds introduced, allowing the overall brightness of the sound to be varied. It can be modulated in real-time to provide articulation to a sound similar to filtering in subtractive synthesis or modulation index in FM synthesis. The second way to control timbre is by adding a dc offset to the input of the wavefolder. This breaks the aforementioned symmetry of the folding function and introduces even harmonics. When modulated by using, for example, a low-frequency oscillator, this parameter provides an effect reminiscent of pulse-width modulation. Figure 17a shows the transfer function of the Serge middle VCM model for the case of zero dc offset. This plot was generated by defining \(V_{\mathrm{in}}\) to have a constant value of 1 V and sweeping through values of \(G_{\mathrm{S}}\) between \(-8\) and \(8\). Figure 17b shows the output of the Serge middle VCM when driven by a  
Figure 16: Block diagram representation of the Serge middle VCM. Blocks labeled “SWF” indicate the Serge wavefolder model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Amplitude (V)**} & **Audio Rate** & **Audio Rate** & **OSx2** & **OSx4** & **OSx8** & **OSx2** \\  & **(ms)** & **w/Antialiasing (ms)** & **(ms)** & **(ms)** & **(ms)** & **w/ Antialiasing (ms)** \\ \hline
1 & 11.5 & 12.5 & 23.4 & 46.6 & 92.9 & 25.4 \\
5 & 11.6 & 12.6 & 23.3 & 46.8 & 92.7 & 25.3 \\
10 & 11.5 & 12.6 & 23.7 & 46.7 & 92.9 & 25.5 \\
15 & 11.5 & 12.7 & 23.5 & 46.5 & 92.9 & 25.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Averaged computation times (in milliseconds) for the proposed Lockhart wavefolder model (\(R_{\mathrm{L}}=50\,\mathrm{k}\Omega\)) implemented in C for a 1-s 100-Hz sinusoidal input sampled at different oversampling (OS) rates and with different peak amplitude levels. 100-Hz sinusoid with \(G_{\rm S}=6\). For simplicity, in this section, we assume the range of \(V_{\rm in}\) to be fixed at \(\pm 1\,V\); therefore, all gain modulation is done using \(G_{\rm S}\) only. The spectrogram in Figure 18a shows the effect of increasing \(G_{\rm S}\) from 0 to 6 for a 150-Hz sinusoidal input. This plot effectively depicts the rich harmonic patterns introduced by the system, which are far more complex than those introduce by traditional waveshaping methods and offer a wide timbral palette for sound synthesis. The fluctuations in energy at the fundamental and first few harmonics indicate the gain values at which each new fold is introduced. Figure 17b shows the effect of introducing a dc offset at the input of the system for a constant 200-Hz sinusoidal input. This result shows how the use of a dc offset can extend the timbral possibilities of the system even further, by introducing complex patterns consisting of both even and odd harmonics. Now, although the Lockhart wavefolder was originally designed to operate as a standalone unit, it can be adapted into a series topology with relative ease.