## 3 Materials and Methods
### Application of the Structural and Quantitative Metric of Network Similarity to Music Comparison  
The method described in the above subsection can be used to measure the similarity of graphs representing musical pieces. Nevertheless, given the specificity of this field, it should be additionally noted that:  
* The analyzed graphs always have 13 vertices. In European classical musical harmony, an octave is built of 12 semitones + there is a separate vertex for rests in the graph. Vertices representing notes that are not present in the piece are isolated vertices (i.e., they are not interlinked to other vertices--they have degree 0). As far as graph visualization and arrangement generation are concerned, they could be removed from the structure; however, in the problem of examining similarity, their presence is dictated by the desire to ensure the stability of the graph mapping into the adjacency matrix. This makes the matrix have a constant size of 13 \(\times\) 13, and using Formula (6), one can always specify that the first row is for note C, the tenth for note A, and the thirteenth for the rest. * In the problem under examination, only one vertex function (describing the frequency of the note represented by this vertex occurring in the piece) and exactly one edge function is used (describing the frequency of the sequence of the note represented by the directed edge's antecedent occurring in the piece before the note represented by the consequent). This slightly simplifies the method, because the matrix vectors \(V_{k},E_{k}\) are always single-element. * The vertex and edge functions described in the graph representation of a piece of music are logically interlinked. They both describe the multiplicity of selected \"phenomena in the piece\" (respectively, the occurrence of a specific note, the occurrence of a specific sequence of notes). Specifically, the value of the vertex label is equal to the sum of the weights of the labels entering it, i.e., \(\mu(v)=\sum\limits_{e_{i}\in E\in(v)}\xi(e_{i})\) (where \(E^{\prime}(v)\)--the set of directed edges entering vertex \(v\) or undirected edges incident with vertex \(v\)). It can therefore be assumed that their values are interrelated, and, consequently, the calculated similarity is similar for both measures of quantitative similarity. The only difference may occur in the vertex corresponding to the first note of the piece (the value of the vertex label will be 1 point greater than the sum of the label values of the directed edges entering that vertex). In regard to the large pieces of music analyzed in this article, this is a negligible difference, as shown in Section 4. It is worthwhile to note that depending on the parameters assumed, the method will reach values in the range of \([-1;1]\). For the parameters \((1,0,0)\) corresponding to the structural analysis of the graph, the set of achieved values is \([0;1]\), where 1 stands for pieces identical in terms of the structure and 0 denotes pieces dissimilar to each other. On the other hand, the construction of the objective function causes achieving negative values (range \([-1;0]\)) for the assumed parameters \((0,1,0)\) or \((0,0,1)\), where \(-1\)--pieces divergent in the quantitative sense, 0--pieces identical in terms of quantity. When using the meta-criterion in the form of a weighted sum for any parameter values, the objective function is a linear combination of the values of individual partial criteria and the weights assigned to them. Thus, for example, for parameters \(\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)\), the achieved function values originate from the range \(\left[-\frac{2}{3};\frac{1}{3}\right]\), where \(\frac{1}{3}\)--pieces that are, to the maximum extent, similar to each other in terms of structure and quantity, \(-\frac{2}{3}\)--pieces that are completely dissimilar to each other.