## 3 Case Study: Recorder
### Results and Discussion  
In this work, we studied the enhancement of the generated signal of the PM of a [PERSON] melody recorder with baroque fingering using the optimization framework. We studied the fingering system, which results from the sequential opening of the toneholes (i.e., the one that starts with having all toneholes closed and lifting the fingers one by one, beginning with the closest to the bell-end). The recorder's eight toneholes result in nine notes, which correspond to the sequence of all toneholes closed (note 1) to all open (note 9). The proposed model's inputs are (a) the building information, i.e., the geometrical details to synthesize nine audio files (9 notes), (b) the nine relative recordings of the real instrument, (c) the initial values for the correction factors along with their upper and lower boundaries and outputs nine sets of correction factors, one individual set per note. In order to calculate the optimal set of correction factors, we put in practice two optimization techniques. We compared their efficiency and embedded the winner to our model. The mathematical optimizers tested here are the Nelder-Mead (NM) [22; 23] and the Simulated Annealing (SA) [24; 25], which have been both used in acoustic-related studies [26; 27]. To benchmark their performance, we run the relevant algorithms ten times for 10 k iterations per time. After several trials, this number of maximum iterations per time proved to be adequately high to satisfy the need for accurate tuning (the deviation between the recorded and the synthesized signals' pitch to be less than 10 cents). Both techniques achieved the best costs (maximum correlation factors) of similar values (\(\pm\)5% maximum deviation); however, NM was found to be more efficient than SA since it came back with the best cost value much faster (NM: 100-400 iterations, SA 1 k-5 k iterations). Our framework significantly enhanced the similarity between the synthesized and the recorder signals (Figure 3). In the case of the PM synthesized signals (i.e., prior to the optimization integration--the initial value of all the correction factors equals one), the average Pearson correlation factor was 0.13 (the minimum and maximum are 0.03 and 0.48, respectively) and in the case of PM-OPT synthesized signals (PM integrated with the optimization framework), the correlation factor has reached the average value of 0.67 (the minimum and maximum are 0.59 and 0.76 respectively). The model resulted in a significant increase in the degree of similarity (Pearson correlation factor > 0.59) for all the notes, even for the ones with a low initial value (Pearson correlation factor < 0.1, notes 1, 3, 5, 7-9). The improvement in terms of the degree of similarity resulted in the synthesis of more accurately tuned signals as per the relevant reference recorded signals (Figure 4). The average absolute deviation of the fundamental frequency of the synthesized from the recorded signals reduced from 40 cents in the case of PM signals, which corresponds to an interval of almost half semitone (a half-semitone deviation is 50 cents) to only 2 cents in the case of PM-OPT signals (which is a non-noticeable difference [28]). In 5 out of 9 notes, the PM-OPT model led to the synthesis of perfectly tuned signals with the relevant recordings (0 cent deviation, Figure 4 notes 2-5, 7). Moreover, significant improvement in sound color resemblance was observed. The partials of the synthesized and the recorded signals initially deviated (e.g., PM vs. Recording case in Figure 5), and now they match (e.g., PM-OPT vs. Recording case in Figure 5). In order to measure the sound color resemblance, we studied the matching of the recorded and synthesized signals' frequency content by taking into consideration the first five partials (the fundamental and the first four overtones, Tables 1-3). The sound color resemblance per note between the recorded and the synthesized signals is determined by their first five partials average absolute deviation (the two columns on the right of Table 3).