Abstract
This paper aims to refine the formalization of David Lewin’s Generalized Interval System (GIS) by the means of tropical semirings. Such a new framework allows to broaden the GIS model introducing a new operation and consequently new musical and conceptual insights and applications, formalizing consistent relations between musical elements in an original unified structure. Some distinctive examples of extensions of well-known infinite GIS for lattices are then offered and the impossibility to build tropical GIS in the finite case is finally proven and discussed.
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Notes
- 1.
According to Lewin’s exact definition, as given in [14], the group action on the set is required to be “simply transitive”. That is equivalent to the requirement of a free and transitive group action.
- 2.
The definition of a monoid requires only associativity and the existence of the identity element for the binary operation.
- 3.
However, outside the framework of lattices the ring structure can be successfully used to refine a GIS. Lewin gave in [15] an example of a GIS that calls out to be extended to a ring, although he did not carry out the extension himself. In fact, with respect to the GIS of Babbitt’s lists investigated in the aforesaid paper, the transformation group can be easily and meaningfully extended to a ring. See Example 3 in Sect. 5 of this paper.
- 4.
See also [8], Proposition 2.1.
- 5.
In fact, every element of a finite group generates a cyclical subgroup.
- 6.
For instance, it would be meaningful to order the elements in \(\mathbb {Z} /12\mathbb {Z}\) from 0, the minimum, to 11, the maximum, or, alternatively, in the following sequence: 0, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, in which it is taken the distance from 0 on both side, favoring the right one in case of the same value.
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Acknowledgments
We thank Claudio Bernardi (Università di Roma “La Sapienza”, Department of Mathematics) and the reviewers for their useful suggestions and remarks.
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Albini, G., Bernardi, M.P. (2019). Tropical Generalized Interval Systems. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_6
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