Abstract
This contribution introduces the concept of musical harmony as a geometric, physical mirror of human biologic proportionality. Although this idea is rather ubiquitous in many aspects and epochs of music theory, mathematical direct modelling is relatively a novelty within the field of dynamical systems. Furthermore, a hypothesis of atomic-molecular harmonicity is provided in order to explain how biologic proportionality is physically biased to perform harmonic patterns eventually codified by culture. This hypothesis is grounded on the topological properties of carbon, and its mapping and embedding within the characteristic geometry of music; from the graphene-Tonnetz analogy, to the map of musical harmony using the Arnold tongues analogy. The topological features of carbon are, then, conceived as crucially influential for the rising of human language and music, and for the development of an associated Euclidean intuition.
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Notes
- 1.
We should remark that, despite its two-dimensional nature, graphene has three phonon modes (LA, TA, in-plane modes with linear dispersion relation; and ZA with quadratic dispersion relation), a fact closely related to the electroacoustic properties of carbon migrating from two to three dimensions.
- 2.
A clear introductory explanation in [15]: “Coordination of the activity within and between the brain’s cellular networks achieved through synchronization has been invoked as a functional feature of normal and abnormal temporal dynamics, the integration and segregation of information, and of the emergence of neural rhythms.”
- 3.
See: [14] pp. 242–244, for a general introduction to the concept of fractional noise 1/f and its interpretation within music theory; and pp. 250–251 for its relationship with carbon.
- 4.
Carrillo suggests an infinite harmony embedded within a geometry of series of square roots of 2. During the last twenty years of his life, he attempted to represent this idea by a hexagonal lattice (original manuscripts and blueprints nowadays at the Carrillo Museum, San Luis Potosí, SLP, Mexico).
- 5.
In this relationship I obviously include the psychoacoustic shades between scales of (micro)tones and timbral roughness, always taking into account instrumental variables in terms of sound color.
- 6.
I owe special thanks to Kraig Grady for his personal communication [2015], emphasizing the nesting of the Novaro’s triangles within the Wilson reinterpretation of the Stern–Brocot tree.
- 7.
Here we suggest that the tree diagrams built by Charles S. Peirce, Heinrich Schenker and Fred Lerdahl, although quite distinct by their methods, belong to a same tradition of analogical hermeneutics, a concept formalized in recent times by philosopher Mauricio Beuchot (1950–).
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Pareyon, G. (2017). Tuning Systems Nested Within the Arnold Tongues: Musicological and Structural Interpretations. In: Pareyon, G., Pina-Romero, S., Agustín-Aquino, O., Lluis-Puebla, E. (eds) The Musical-Mathematical Mind. Computational Music Science. Springer, Cham. https://doi.org/10.1007/978-3-319-47337-6_23
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