## 3 Case Study: Recorder
### Analysis by Synthesis Model  
We identified eleven internal parameters which are part of all the components of the block diagram in Figure 1 and affect the synthesized signal in both time and frequency domain. More specifically, three parameters affect the dynamic envelope of the mouth pressure (the duration of the attack, the sustain, and the release), three parameters affect the properties of the input (the frequency, the content, and the noise of the vibrato), one parameter affects the length of the delay lines, one parameter affects the interpolation used to achieve accurate tuning, and three parameters affect the filters' coefficients (the transmission of the tube into the mouth, and the reflections at both of its open ends). The number of correction factors is, thus, set to eleven. The chosen modification type is a multiplication (\(P_{modified}=P_{initial}\cdot CF\)), which, after several trials, proved to derive the best results and enabled the setting of initial generic logical boundaries (i.e., the range of values the optimizer is permitted to assign to the parameters in every iteration). The initial value of all the correction factors is set to one, which corresponds to the synthesized signal generated by the unmodified PM before the optimization framework integration. The determination of the logical boundaries of eleven parameters is not a straightforward task. Choosing a single modification type (in our case, this type is the multiplication) makes it easier to deal with this task by enabling the initial setting of generic logical boundaries as a starting point before their individual specification. These generic logical boundaries have been set to half and double the initial values for the lower and upper boundary, respectively. After several trials to ensure that all the extreme values are within logical limits and unwanted alterations of the PM algorithm's core elements are avoided, these boundaries were set for every individual correction factor. The core part of the optimization framework integration with the PM is the comparison between the synthesized and the real sound of the relative musical instrument. To make this comparison possible, we recorded samples of the commercial Hohner recorder mentioned above. The reference signals (nine signals for nine fingerings) are the recordings of the Hohner recorder. The recordings took place at the audio recording studio of the National and Kapodistrian University of Athens, Department of Informatics and Telecommunications, using an electroacoustic chain with a flat frequency response (microphone: SD Systems LCM 85 MK II with \"LP\" Preamp Power Supply, soundcard: apogee duet, computer: MacBook air 2019). The distance (recorder-microphone) was approximately 1m and the microphone was placed off-axis from the instrument's bell. We want to note at this point, that in order to cross-correlate our method's performance, we recorded each note of the recorder 15 times and calculated all the possible Pearson correlation coefficients between the 15 recordings of the same note. As a reference signal to evaluate our model we chose for each note this recording that had the highest average Pearson correlation coefficient between itself and the rest 14 recordings of the same note. The total average Pearson correlation coefficient was found to be 0.7, with a standard deviation of 0.16. The degree of similarity between the synthesized and recorded signals is defined by the Pearson correlation coefficient (Equation (1)), which measures the linear correlation between two variables [21] and takes values between \(-1\) and \(+1\) (\(+1\) corresponds to total positive linear correlation, 0 to no linear correlation, and \(-1\) to total negative linear correlation). As it is here non-relevant whether the correlation is positive or negative, we take the absolute value of this coefficient to define the objective (Equation (2)).