## 2 Materials and Methods  
Musical compositions can be represented with tones corresponding to the twelve pitch-classes of the chromatic scale: C; C\(\sharp\)/D\(\flat\); D; D\(\sharp\)/E\(\flat\); E; F; F\(\sharp\)/G\(\flat\); G; G\(\sharp\)/A\(\flat\); A; A\(\sharp\)/B\(\flat\); B. Of course, the tones comprising a particular piece may belong to different octaves. Knowing the number of occurrences of notes associated with individual pitch-classes in a given fragment of music, or their durations, it is possible to define a set of values X (1):  
\[\text{X}=\left\{\text{x}_{\text{C}},\text{ x}_{\text{C}\sharp\text{/D}\flat}, \text{ x}_{\text{D}},\text{ x}_{\text{D}\sharp\text{/E}\flat},\text{ x}_{\text{E}},\text{ x}_{\text{F}},\text{ x}_{\text{F}\sharp\text{/G}\flat},\text{ x}_{\text{G}},\text{ x}_{\text{G}\sharp\text{/A}\flat},\text{ x}_{\text{A}},\text{ x}_{\text{A}\sharp\text{/B}\flat},\text{ x}_{\text{B}}\right. \tag{1}\]  
where x\({}_{i}\) is the multiplicity of occurrences of tones associated with a particular (\(i\)-th) pitch-class or the aggregate duration of tones corresponding to that pitch-class. Using (1) we define the vector K (2):  
\[\text{K}=\left[k_{A},\,k_{D},\,k_{G},\,k_{C},\,k_{F},\,k_{B\flat},\,k_{E\flat},\,k_{A\flat},\,k_{D\flat},\,k_{F\sharp},\,k_{B},\,k_{E}\right] \tag{2}\]  
where:  
\[k_{i}\,=\,\frac{x_{i}}{max(X)} \tag{3}\]  
The elements of the vector K are sorted in accordance with the succession of the pitch-classes defining the circle of fifths, beginning from the A and proceeding counterclockwise. **Definition 1**. : _The signature of fifths._  
The signature of fifths (also referred to as the music signature [35]), corresponding to a given fragment of music, is a set of twelve polar vectors \(\left\{\overset{\rightarrow}{k_{i}}\,:\,i=A,\,D,\,G,\,\ldots,\,E\right\}\), whose coordinates \((r_{i},\,\phi_{i})\) are determined with the following assumptions:  
* the length of the \(i\)-th polar vector is equal to the \(i\)-th value defining the vector K, which can be obtained either using the multiplicities of occurrences of tones associated with individual pitch-classes or the aggregate duration of tones corresponding to those pitch-classes: \[r_{i}=\left|\overset{\rightarrow}{k_{i}}\right|=k_{i}\] (4)
* the angle of the \(i\)-th vector is determined with the following relationship: \[\phi_{i}=j\cdot 30^{\circ}\] (5) where \(j=0|_{i=A},\,j=1|_{i=D},\,\) and so on. **Example 1**. : _The two ways of creating the signature of fifths corresponding to the fragment of music shown in Figure 1. In the first case, we assume that the length of each constituent vector of the signature of fifths reflects the multiplicity of occurrences of notes associated with a particular pitch-class. In the second case, the lengths of the vectors will reflect the aggregate durations of individual pitch-classes. In either case, a certain time resolution of the analysis needs to be assumed, e.g., eight-note or quarter-note. In this example, we will create the quarter-note signatures of fifths corresponding to the fragment of music shown in Figure 1. The multiplicities of occurrences and aggregate durations of individual pitch-classes comprising the chromatic scale, obtained for this fragment, are presented in Table 1.  
The vectors K corresponding to the two ways of creating the signature of fifths are depicted with Equations (6) and (7), whereas the signatures themselves are illustrated in Figure 2:  
\[\text{K}_{R_{m}}=[0.33\,0.67\,0\,0\,0\,0\,0\,0\,0.33\,1\,0\,0.33] \tag{6}\]  
\[\text{K}=\left[k_{A},\,k_{D},\,k_{G},\,k_{C},\,k_{F},\,k_{B\flat},\,k_{E\flat},\,k_{A\flat},\,k_{D\flat},\,k_{F\sharp},\,k_{B},\,k_{E}\right] \tag{2}\]0  
Let us next define \(\text{Y}\to\text{Z}\) as the directed axis of the circle of fifths, connecting two opposite pitch-classes. The axis points from Y to Z, where: (\(\text{Y}\); Z) \(\in\) {(C, F\(\oplus\)); (F, B); (B\(\oplus\), E); (E\(\oplus\), A); (A\(\oplus\), D); (D\(\oplus\), G); (F\(\oplus\), C); (B, F); (E, B\(\oplus\)); (A, E\(\oplus\)); (D, Ab); (G, D\(\oplus\))}.