FOUR AXIOMS FOR A THEORY OF RHYTHMIC SETS AND THEIR IMPLICATIONS

Öz

In a recent article (Lugos Abarca, 2023) an equation was proposed that allows us to know the number of measures that a song has μ_mar from the musical variables of tempo Τ, song duration t and time signature β. Also, it was found that that by solving the equation μ_mar for the variable t yields a formula capable of expressing the duration in minutes of any rhythmic figure. Proceeding with this line of research, four axioms are presented whose purpose is to function as a basis for the construction of a set theory for rhythmic figures, during this process the consequences of the third axiom that establishes the non-commutativity in the sum of certain sets that have the same elements but with different order are studied, and whose most relevant consequence is to introduce the theorem that determines the existence of different types of empty sets.

Anahtar Kelimeler

• set theory
• axioms
• rhythmic figures
• commutativity
• empty set

Kaynakça

1. Burns, J. (2010). Rhythmic archetypes in instrumental music from Africa and the diaspora. Music Theory Online, 16(4).
2. Camiruaga, J. (2000). Manual de Aprendizaje: Rítmica y métrica. Udelar. CSE.
3. Cannas, S. & Andreatta, M. (2018). A generalized dual of the Tonnetz for seventh chords: mathematical, computational and compositional aspects. In Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture (pp. 301-308).
4. Chahine, I., & Montiel, M. (2015). Teaching modeling in algebra and geometry using musical rhythms: Teachers’ perceptions on effectiveness. Journal of Mathematics Education, 8 (2), 126-138.
5. Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G. T., Winograd, T., & Wood, D. R. (2009). The distance geometry of music. Computational geometry, 42(5), 429-454.
6. Ferreirós, J. (2000). ¿ Antinomia o trivialidad? La paradoja de Russell. Las matemáticas del siglo XX. Una mirada en 101 artículos, 59-64.
7. Forte, A. (1974). Structure of atonal music. Yale University Press.
8. Gómez-Martín, F. (2022). A review of Godfried Toussaint's the geometry of musical rhythm. Journal of Mathematics and Music, 16(2), 239-247.

9. Herrera, E. (2022). Teoria musical y armonia moderna(vol. 1). Antoni Bosch Editor.
10. Honing, H. (2013). Structure and interpretation of rhythm in music. The psychology of music, 3, 369-404.
11. Hsü, K. J., & Hsü, A. J. (1990). Fractal geometry of music. Proceedings of the National Academy of Sciences, 87 (3). 938-941.
12. Jones, S. M., & Pearson Jr, D. (2013). Music: Highly engaged students connect music to math. General Music Today, 27(1). 18-23.
13. Kania, A. (2010). Silent music. The Journal of Aesthetics and Art Criticism, 68 (4), 343-353.
14. Lopez Mateos, M. (2017). Conjuntos, Lógica Y Funciones: Edición Blanco Y Negro. Createspace Independent Publishing Platform.
15. Lovemore, T. S., Robertson, S. A., & Graven, M. (2021). Enriching the teaching of fractions through integrating mathematics and music. South African Journal of Childhood Education, 11(1), 899.
16. Lugos Abarca, J. A. (2023). Sobre la matemática de los compases musicales y su relación con la geometría. Ricercare, (16), 27–58. https://doi.org/10.17230/ricercare.2023.16.2
17. Mall, P., Spychiger, M., Vogel, R., & Zerlik, J. (2016). European Music Portfolio (EMP), Maths:'Sounding Ways Into Mathematics'.. Teacher's Handbook. Universitätsbibliothek Johann Christian Senckenberg.
18. Maor, E. (2020). Music by the numbers: From Pythagoras to Schoenberg. Princeton University Press.
19. Margulis, E. H. (2007). Silences in music are musical not silent: An exploratory study of context effects on the experience of musical pauses. Music Perception, 24(5), 485-506.
20. Mehta, R., Mishra, P., Henriksen, D., & Deep-Play Research Group. (2016). Creativity in mathematics and beyond–Learning from fields medal winners. TechTrends, 60, 14-18.
21. Mora F. W. (2012). Introducción a la teoría de números. Ejemplos y algoritmos. Instituto Tecnológico de Costa Rica.
22. Pearsall, E. (1997). Interpreting music durationally: a set-theory approach to rhythm. Perspectives of New Music, 205-230.
23. Perle, G. (1972). Serial composition and atonality: an introduction to the music of Schoenberg, Berg, and Webern. Univ of California Press.
24. Rahn, J. (1979, April). Logic, set theory, music theory. In College Music Symposium (Vol. 19, No. 1, pp. 114-127). College Music Society.
25. Schmeling, P. (2011). Berklee music theory. Berklee Press.
26. Schoenberg, A. (2014). Style and idea. Open Road Media.
27. Schoenberg, A. (2016). Theory of harmony (2nd ed.). Berkeley, CA: University of California Press.
28. Schönberg, A. (1994). Fundamentos de la composición musical. Real Musical
29. Schuijer, M. (2008). Analyzing atonal music: Pitch-class set theory and its contexts. University of Rochester Press.
30. Suppes, P. (1972). Axiomatic Set Theory. Dover Publications.
31. Toussaint, G. (2005). The geometry of musical rhythm. In Discrete and Computational Geometry: Japanese Conference, JCDCG 2004, Tokyo, Japan, October 8-11, 2004, Revised Selected Papers (pp. 198-212). Springer Berlin Heidelberg.
32. Toussaint, G. T. (2019). The geometry of musical rhythm the geometry of musical rhythm: What makes a “good” rhythm good? (second edition (2nd ed.). CRC Press.
33. Tymoczko, D. (2011). A geometry of music: Harmony and counterpoint in the extended common practice. Oxford University Press.
34. Zaldívar, F. (2014). Introducción a la teoría de números. Fondo de Cultura Economica.