Axiomatic of the set of musical frequencies in an octave
Axiomática del conjunto de las frecuencias musicales en una octava
Article Sidebar
The Journal offers Open Access under a Creative Commons Attibution License
This work is licensed under a Creative Commons Atribución 4.0 Internacional.
Main Article Content
Most of the Works that relate mathematics to musical frequencies are based on the physical phenomenon of harmonics, tuning systems and the theory of well-formed scales. This paper studies the link between musical notes (frequencies) in the frame of an octave and axiomatic set theory, algebra of groups and real analysis. It is demonstrated that the set of musical notes (frequencies) in an octave is well ordered and fulfills the supreme axiom. Furthermore, the set of intervals of musical notes (frequencies) in an octave, with the internal sum operation, has a commutative group structure. Other results from algebraic group theory are also shown.
Downloads
Article Details
Abbott, S. (2015). Understanding Analysis. New York: Springer. DOI: https://doi.org/10.1007/978-1-4939-2712-8
https://ndl.ethernet.edu.et/bitstream/123456789/88631/1/2015_Book_UnderstandingAnalysis.pdf
Artin, M. (2011). Algebra. Boston: Pearson.
https://indaga.ual.es/discovery/fulldisplay/alma991001482049704991/34CBUA_UAL:VU1
Assayag, G., Feichtinger, H. G. y Rodrigues, J. F., eds. (2020). Mathematics and Music:
A Diderot Mathematical Forum. Cham, Switzerland: Springer.
https://link.springer.com/book/10.1007/978-3-662-04927-3
Bartle, R. G. y Sherbet, D. R. (2011). Introduction to Real Analysis. Hoboken, New Jersey:
John Willey & Sons.
https://sowndarmath.files.wordpress.co./2017/10/real-analysis-by-bartle.pdf
Benson, D. J. (2006). Music: A Mathematical Offering. Cambridge: Univ. Press.
https://doi.org/10.1017/CBO9780511811722 DOI: https://doi.org/10.1017/CBO9780511811722
https://logosfoundation.org/kursus/music_math.pdf
Bukovský, L. (2011). The Structure of the Real Line. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-3-0348-0006-8
https://doi.org/10.1007/978-3-0348-006-8
https://link.springer.com/book/10.1007/978-3-0348-0006--8
Castrillón, M. y Domínguez, M. (2013). Un encuentro entre las matemáticas y la teoría
de escalas musicales: Escalas bien formadas. La Gaceta de la RSME, 16, 87-106.
https://gacetarsme.es/abris.php?id=1130
Damschroder, D. y Russell, D. (2013). Music Theory from Zarlino to Schenker: A
Bibliography and Guide. New York: Pendragon Press, Stuyvesant.
https://search.worldcat.org/es/formats-editions/21195293
Fauvel, J. Raymond, F. y Wilson, R. (2003). Music and Mathematics: From Pythagoras DOI: https://doi.org/10.1093/oso/9780198511878.001.0001
to Fractals. Oxford: Oxford University Press.
https://global.oup.com/academic/product/music-and-mathematics-9780199298938
Foreman, M. y Kanamori, A. (eds., 2010). Handbook of Set Theory. New York: Springer. DOI: https://doi.org/10.1007/978-1-4020-5764-9
https://link.springer.com/book/10.1007/978-1-4020-5764-9
Gallian, J. A. (2019). Contemporery Abstract Algebra. Boston: Cengage Learning.
Goldáraz Gainza, J. J. (1992). Afinación y temperamento en la música occidental. Madrid:
Alianza Música.
https://riunet.upv.es/bitstream/handle/10251/18491/Memoria.pdf?sequence=1
Harkleroad, L. (2006). The Math behind the Music. Cambridge: Cambridge University
Press.
https://archive.org/details/mathbehindmusic0000hark/mathbehindmusic0000hak
Kunen, K. (2011). Set Theory: An Introduction to Independence Proofs. Boca Raton:
Chapman and Hall.
Maor, E. (2018). Music by the Numbers: From Pythagoras to Schöenberg. Princeton, DOI: https://doi.org/10.23943/9781400889891
New Jersey: Princeton University Press.
https://ieeexplore.ieee.org/document/9453279
Mazzola, S. (2018). The Topos of Music I: Geometric Logic of Concepts, Theory and DOI: https://doi.org/10.1007/978-3-319-64364-9
Performance. Cham, Switzerland: Springer.
https://search.app.goo.gl/YG6G5yY
Mazzola, S. (2018). The Topos of Music II: Performance. Theory, Software and Case DOI: https://doi.org/10.1007/978-3-319-64444-8
Studies. Cham, Switzerland: Springer.
https://link.springer.com/book/10.1007/978-3-319-64444-8
Tymoczko, D. (2011). A Geometry of Music: Harmony and Counterpoint in the
Extended Common Practice. Oxford: Oxford University Press.
Wright, D. (2015). The Mathematics of Music: Theory and Compositions. Boca Raton,
Florida: CRC Press.
