[1910.11047] Harmonic Complex Networks
These results suggest that the proposed harmonic network could correspond to a universal way to generate networks with diverse topology, especially when considering other temperaments and anharmonicity transformations

Abstract: 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 $\beta$, 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 $\beta$, 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.

‹Figura 1. Example network of equal temperament, where items (a), (b), and (c) represent variations of β parameter. Interestingly, a small difference of β can lead to a substantial change in the network topology. (Introduction)Figura 3. Example of PCA for the data organized into two dimensions, in which the computed PCA axes reflect the dispersion of the data. (Principal component analysis)

Figura 4. PCA projections respective to consonance and dissonance obtained for each of the considered temperaments, allowing a comparison of the non-shifted scale-based networks with some traditional network models. (Non-Shifted Partials)Figura 8. Example of consonance networks for some selected parameters of shifted partials and the respective positions in PCA. Here, we considered the equal temperament. In this projection, PC1 and PC2 represent 48.61% and 23.83% of the data variance, respectively. (Shifted Partials)Figura 2. Two notes (a) and (b) to have their consonance/dissonance quantified, with the latter being taken as reference. For each partial fi belonging to the note (b), the partials of (a) comprised between the intervals ∆Max and ∆Min are identified. In this case, the pink and green frequencies of (a) are considered consonant and dissonant, respectively. (Harmonic Connectivity)Figura 5. Visualization of consonance and dissonance networks we created from non-shifted partials, for all of the considered temperaments. (Non-Shifted Partials)Figura 9. Example of consonance networks for some selected parameters of shifted partials and the respective positions in PCA. Here, we considered the just temperament. In this projection, PC1 and PC2 represent 48.61% and 23.83% of the data variance, respectively. (Shifted Partials)Figura 6. PCA projection of the consonance networks (with shifted partials) for all of the considered temperaments (squares). The circles represent the network models. (Shifted Partials)Figura 7. PCA projection of the dissonance networks (with shifted partials) for all of the considered temperaments (squares). The circles represent the network models. (Shifted Partials)›