Abstract
This paper has two aims. On the one hand, it aims to offer a brief overview of the history of graph theory applications in the fields of music theory and musicology. On the other hand, it presents an original modelisation by the Authors—developed within the distinctive framework of graph theory in order to represent music objects related to harmony—that could be of particular value to music theorists and composers and that aims to generalize many models so far introduced.
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Notes
- 1.
It comes at not surprise that also a basic dictionary of music relies on a mathematical terminology: durations are expressed by fractions, scale degrees by ordinals, intervals by cardinal numbers or by the count of semitones, etc.
- 2.
Euler’s Tentamen is characterized by a mathematical approach to music theory that, despite still largely being metaphisical in nature, presents a modern approach that considers the importance of mind and of experience. In this respect, this work can be distinguished from previous treatises aided by mathematics and it paves the way to a more scientific approach to the subject.
- 3.
Although edges have no arrow, in the 1739 drawing they are intended to connect vertices from top to bottom, as explained by Euler himself in his treatise.
- 4.
In fact, Marin Mersenne in his Harmonie universelle proved also that the fundamental frequency of a string of length L, of mass per unit length \(\mu \) and kept in tension by a constant force F, is given by the following formula: \(f_0=\frac{1}{2L}\sqrt{\frac{F}{\mu }}\). Thus, in Pythagoras’ monochord the ratio between the frequency of two pitches is inversely proportional to their length.
- 5.
A group action is said to be free if only the identity element fixes any element and transitive if it possesses only a single group orbit. It means that given two elements \(p,q\in P\) there is one and only one element \(i\in I\) such as \(\varphi (i,p)=q\). Moreover, this implies that \(|P|=|I|\).
- 6.
- 7.
As many as they are the subsets of 3 elements of a set of n ones.
- 8.
Triads are trichords that can be stacked in alternating major and minor thirds. Starting from the root of the chord, a major third followed by a minor one form a major triad, while a minor third followed by a major one form a minor triad. Augmented triads are obtained by two major thirds.
- 9.
For instance, in [2] the Authors introduced chord-networks as iterated line-graphs of tone-networks.
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Albini, G., Bernardi, M.P. (2019). Graph Theory and Music: A Mathematical Tool for Musicians. In: Cocchiarella, L. (eds) ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-319-95588-9_162
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