Harmonic Complex Networks
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We report the possibility of obtaining complex networks with diverse topology, henceforth called harmonic networks, by taking into account the consonances and dissonances between sound notes as defined by scale temperaments. Temperaments define the intervals between musical notes of scales. In real-world sounds, several additional frequencies (partials) accompany the respective fundamental, influencing the consonance between simultaneous notes. We use a method based on Helmholtz's consonance approach to quantify the consonances and dissonances, between each of the pairs of notes in a given scale temperament. We adopt two distinct partials structures: (i) harmonic; and (ii) shifted, obtained by taking the harmonic components to a given power β, which is henceforth called the anharmonicity index. When these estimated consonances/dissonances are taken along several octaves, respective harmonic complex networks can be obtained, in which nodes and weighted edge represent notes, and consonance/dissonance, respectively. We consider five scale temperaments (i.e., equal, meantone, Werckmeister, just, and Pythagorean). The obtained results can be organized into two major groups, those related to complex networks and musical implications. Regarding the former group, we have that the harmonic networks can provide, for varying values of β, a wide range of topologies spanning the space comprised between traditional models. The musical interpretations of the results include the confirmation of the more regular consonance pattern of the equal temperament, obtained at the expense of a wider range of consonances such as that obtained in the meantone temperament. We also have that scales derived for shifted partials tend to exhibit a wide range of consonance/dissonance behavior depending oh the adopted temperament and anharmonicity strength.
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