## 3 Experiments and Discussion  
The main goal of the experiments we conducted was a multi-criteria comparison of the method to find keys based on the signature of fifths with the strictly correlational key-finding approaches that implement well-known key-profiles (i.e., Krumhansl-Kessler [36, 37], Temperley [38], and Albrecht-Shanahan [33]). Of course, many more key-profiles are known and could be used in the study. The selected ones, however, seemed to provide the best reference. The experiments were performed with the following three groups of pieces of music: the preludes by [PERSON] from the collection The Well-Tempered Clavier (Book I); the preludes of [PERSON] from Op. 28; and songs by The Beatles from the album A Hard Day's Night. It is worth mentioning that the analyzed collections of [PERSON] and [PERSON], individually, covered all 24 keys. It needs to be emphasized that the key detection method that implemented the signature of fifths required the computation of only two correlation coefficients for any given music sample--the correlation of the obtained weighted pitch-class representation with the Albrecht-Shanahan [28] key-profiles corresponding to the two relative keys associated with the main directed axis of the signature of fifths. In other approaches, the key was determined based on the values of the correlation coefficients, which were individually  
Figure 6: The results of the music key detection obtained for the method implementing the signature of fifths (SF) as well as for the strictly correlational approaches based on the music key-profiles of Krumhansl–Kessler (K-K), [PERSON] (T), and Albrecht–Shanahan (A-S), observed for the successively increased lengths of an analyzed music fragment—the weighting of individual pitch-classes was realized with the multiplicities of their occurrences (incorrectly detected keys are marked in red). Figure 7: The results of the music key-detection obtained for the method implementing the signature of fifths (SF) as well as for the strictly correlational approaches based on the music key-profiles of Krumhansl–Kessler (K-K), [PERSON] (T), and Albrecht–Shanahan (A-S), observed for the successively increased length of an analyzed music fragment—the weighting of individual pitch-classes was realized with their aggregate durations (incorrectly detected keys are marked in red). calculated for all 24 key-profiles of Krumhansl-Kessler, Temperley, or Albrecht-Shanahan. In all cases, the largest correlation coefficient indicated the key of a piece of music. The performed experiments involved the analysis of fragments of music extracted from the beginning (Beginning) and the end (End) of a given composition. Evaluation of the key was based on the first and last bars of a piece. The experiments were also conducted on whole (Whole) music compositions as well as on the concatenations of their first and last bars (Beginning_End). For each of the above-mentioned sample selection criteria (Beginning, End, Beginning_End, and Whole), the following weighted pitch-class representations of an analyzed fragment were used:  
* the multiplicities of occurrences of individual pitch-classes (\(R_{\text{m}}\));
* the aggregate durations of individual pitch-classes (\(R_{D}\)). In all the experiments, the quarter-note resolution was applied. Let us first examine the results of the conducted experiments in terms of the applied pitch-class representations. Tables 2-5 present the overall efficacies of all the considered key detection approaches, obtained for different weighted pitch-class representations in all three collections of compositions, regardless of the analyzed sample extraction method. The tables contain the results associated with the method implementing the signature of fifths as well as the approaches based on the strictly correlational analysis with popular key-profiles, i.e., Krumhansl-Kessler, Temperley, and Albrecht-Shanahan.