## 3 Case Study: Recorder  
### The Physical Model  
In order to demonstrate the integration of the optimization framework with PM of musical instruments, we present the case of a recorder. The recorder constitutes a wind instrument with a flute-like (air-jet) excitation mechanism and a cylindrical resonator. The player produces various pitches by changing the fingering (i.e., arrangements of closed or open toneholes), or by overblowing. The instrument in our case study is a typical commercial recorder: Hohner's melody recorder with baroque fingering (type 1-095.143-1011) and eight toneholes (seven regular toneholes in the upper part of the acoustic pipe and one fingerhole in the bottom part). The method used for the physical modeling of the instrument is the Digital Waveguides (DWGs), a technique introduced by [PERSON] [11], which simulates traveling waves by digital delay lines [12]. In this work, we have chosen to demonstrate our framework on a computationally cheap PM that enables a fast calculation runtime of a significantly high number of iterations during the optimization process (here 10 k, see Section 3.3). However, every PM technique (e.g., FEM) is built upon parameters that can potentially be tuned with the use of correction factors, hence, it can be integrated with the optimization framework. The only prerequisite is the available computation power to enable the optimizer to perform several iterations. Figure 2 demonstrates the block diagram of our recorder's PM based on established approaches [13; 14; 15]. Recorder's excitation phenomenon is based on the effect of an air jet blown that strikes a sharp edge (labium) [16]. In Figure 2, we simulate the air jet traveling from the player's lips to the labium by a delay line (jet-delay). The mouth pressure (forming the air-jet) is simulated as a constant pressure enriched with vibrato and noise content. In the real world, this constant mouth pressure is not reached and released instantly thus,  
Figure 1: Block diagram of the Physical Modeling and Optimization framework integration, showing the input parameters and the looping procedure to output the optimal set of correction factors resulting in the maximum similarity between the synthesized and the recorded signal. in our model, a dynamic envelope is applied to provide the duration of the attack, the sustain, and the release. The air jet is modeled by a static non-linear element using a sigmoid function [17]. Here we use the \(y=x-x^{3}\) sigmoid function as proposed in [17]. When the air blown by the player enters the instrument's bore the air particles inside the resonator's cavity start to vibrate. The bore effect is simulated as a one-dimensional DWG by using delay lines (one delay line for the right and one for the left-going part of the wave, noted as \(z^{-M}\) in Figure 2) [12]. The length of the digital delay lines (in samples), which depends on the speed of the acoustic waves, corresponds to the bore's physical length (in meters). A more accurate modeling of wind instruments should also take into consideration the end corrections [18; 19] in order to tune the generated pitch. However, our initial model neglects end corrections (and therefore creates a lower--than the more accurate model--correlation due to the frequencies mismatch, see Section 3.3 and Figure 3) as it is the proposed optimization framework that chooses the optimal one itself. The pressure waves travel from the mouthpiece along the tube towards the other end (assumably, right-going). When reaching the end so-called the bell, a portion of the wave is reflected towards the mouthpiece (assumably, left-going) and the other portion is transmitted outside the instrument. The superposition of the right- and left-going pressure waves forms a standing wave inside the resonator's cavity. In particular, the effect of the bell is to radiate out of the instrument the high frequencies and to reflect back the low frequencies. This reflection is simulated as a lowpass filter, the RL(z), which is, in our case, a first-order averaging filter.