## 2 Materials and Methods
We will also define [Y \(\to\text{Z}\)] as the so-called characteristic value of the directed axis \(\text{Y}\to\text{Z}\), equal to \(\text{K}_{\text{R}}\)-\(\text{K}_{\text{L}}\), where \(\text{K}_{\text{R}}\) and \(\text{K}_{\text{L}}\) are the sums of lengths of the vectors found on the right and left sides of the directed axis \(\text{Y}\to\text{Z}\), respectively. The axis with the largest characteristic value will be called the main directed axis of the signature of fifths. **Definition 2** ([35]). : _The main directed axis of the signature of fifths._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Pitch-Classes** & \(R_{M}\) & \(R_{D}\) \\ \hline \(\text{x}_{\text{A}}\) & 1 & 1 \\ \(\text{x}_{\text{D}}\) & 2 & 8 \\ \(\text{x}_{\text{G}}\) & 0 & 0 \\ \(\text{x}_{\text{C}}\) & 0 & 0 \\ \(\text{x}_{\text{F}}\) & 0 & 0 \\ \(\text{x}_{\text{B}\text{}}\) & 0 & 0 \\ \(\text{x}_{\text{E}\text{}}\) & 0 & 0 \\ \(\text{x}_{\text{A}\text{D}}\) & 1 & 1 \\ \(\text{x}_{\text{B}}\) & 0 & 0 \\ \(\text{x}_{\text{B}}\) & 0 & 0 \\ \(\text{x}_{\text{E}}\) & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of the quarter-note multiplicities of occurrences (\(R_{m}\) ) and aggregate durations (\(R_{D}\) ) associated with individual pitch-classes of the chromatic scale, obtained for the fragment of music shown in Figure 1.  
Figure 2: The analyzed fragment of a sample music piece. The signatures of fifths representing the fragment of the music piece shown in Figure 1: (**a**) obtained for the multiplicities of occurrences of individual pitch-classes; (**b**) obtained for the aggregate durations of individual pitch-classes. A directed axis of the circle of fifths \(Y\to Z\), for which [\(Y\to Z\)] assumes the maximum value, is called the main directed axis of the signature of fifths. The main directed axes obtained for the signatures of fifths shown in Figure 2 are illustrated in Figure 3. The values on the outer edges of the drawn plots correspond to the characteristic values of individual directed axes. The dashed arrow present in each plot represents the main directed axis of a given signature of fifths. For the considered fragment of music, the directions of these axes happen to be the same. There are multiple ways to detect the tonality of a piece of music [18; 24; 36; 37]. One simple and effective method is based on key-profiles, which depict a particular key by assigning weights to individual pitch-classes. It is worth mentioning, though, that the key-profiles currently available tend to assign higher weights to diatonic tones than to non-diatonic ones. The first key-profiles were developed by [PERSON] and [PERSON] [36; 37] as a result of an experiment in which listeners were asked to rate the degree to which each tone of the chromatic scale matched a seven-tone major and a seven-tone minor scale. The degree to which the tones were related to each other was also examined and the obtained relationships were expressed as the stability of a given tone in a particular key. Findings indicated that the perceived stability of a tone was dependent on the tones that preceded it. This aspect is important because the same tones are characterized by different harmonic relationships in different keys. For example, the \"C\" tone in the key of C-major is a stable tonic, whereas in the D-major key it is an unstable tone (i.e., extraneous in the harmonic context). The obtained matching degrees comprise the normalized 12-element vectors featuring a given key and correspond to the weights associated with particular pitch-classes. Key-profiles representing any tonality can be obtained by relating the largest weight of such a vector with a tonic. An algorithm for key detection using key-profiles was presented by [PERSON] in [37]. It involved the calculation of the Pearson's linear correlation coefficients between the weights of the individual pitch-classes found in the analyzed fragment of music and the weights associated with 12 major and 12 minor key-profiles.