Generalized Fibonacci Numbers and Music
DOI:
https://doi.org/10.24297/jam.v14i1.7323Keywords:
Fibonacci numbers, golden ratio, Simson's identity, Binet formulaAbstract
Mathematics and music have well documented historical connections. Just as the ordinary Fibonacci numbers have links with the golden ratio, this paper considers generalized Fibonacci numbers developed from generalizations of the golden ratio. It is well known that the Fibonacci sequence of numbers underlie certain musical intervals and compositions but to what extent are these connections accidental or structural, coincidental or natural and do generalized Fibonacci numbers share any of these connections?
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References
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23. Whitford, A.K. 1977. “Binet’s Formula Generalized.†The Fibonacci Quarterly. 15(1): 21, 24, 29.
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27. Kuijken, Barthold. 2013. The Notation is Not the Music. Bloomington, IN: Indiana University Press, Ch.1.
28. Shannon, Anthony G, and Jean V. Leyendekkers. 2018. The Fibonacci Numbers and Integer Structure. New York: Nova Science Publishers, Chs1,6.
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2. Markovsky, George. 2018. Misconceptions about the Golden Ratio. College Mathematics Journal. 23 (1): 2-19.
3. Tatlow, Ruth. 2006. The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today. Understanding Bach. 1: 69-85.
4. Benson, David I. 2006. Music: a Mathematical Offering. Cambridge: Cambridge University Press, Ch.6.
5. Putz, J.F. 1995. “The golden section and the piano sonatas of Mozart.†Mathematics Magazine. 68 (4): 275-282.
6. Barbour, J.M. 1948. “Music and ternary continued fractions.†American Mathematical Monthly. 55 (9): 545-555.
7. Huntley, H.E. 1970. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover, Ch.II.
8. Kramer, Jonathan. 1973. “The Fibonacci series in twentieth-century music.†Journal of Music Theory. 17 (1): 110-148.
9. Farey, John. 1807. “On a new mode of equally tempering the musical scale.†Philosophical Magazine, 27: 65–66.
10. Larcombe, Peter J. 2018. “A few thoughts on the aesthetics of mathematics in research and teaching.†Palestine Journal of Mathematics. 7 (1): 1-8.
11. Kepler, Johannes. 2014. Harmonies of the World. (Translated by Charles Glenn Wallis, 1939; original 1619). @GlobalGrey2014, pp.25-68.
12. Boetius. 2007. Consolation of Philosophy. (Translated by H.R. James). Adelaide: eBooks@Adelaide, Book II,Ch.4.
13. Anderson, Warren D. 1994. Music and musicians in ancient Greece. Ithaca, NY: Cornell University Press.
14. Stewart, Ian. 2004. Another fine math you’ve got me into… New York: Dover, Ch.15.
15. van Gend, Robert. 2014. “Fibonacci numbers and music.†Notes on Number Theory and Discrete Mathematics. 20 (1): 72-77.
16. Goldennumber.net. 2012. Acoustics. Online at http://www.goldennumber.net/acoustics/.
17. Yurick, S. 2012. Music and the Fibonacci Series and Phi. online: http://www.goldennumber.net/music.
18. Howat, R. 1983. Debussy in Proportion: A Musical Analysis. Cambridge: Cambridge University Press, p. 1.
19. Leyendekkers, J.V., and A.G. Shannon. 2014. The Decimal String of the Golden Ratio. Notes on Number Theory and Discrete Mathematics. 20 (1): 27-31.
20. Hoggatt, V. E. Jr. 1969. Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin.
21. Filipponi, P. 1991. “A Note on a Class of Lucas Sequences.†The Fibonacci Quarterly. 29 (3): 256–263.
22. Monzingo, M. G. 1980. “An Observation Concerning Whitford’s ‘Binet’s Formula Generalized’.†In A Collection of Manuscripts Related to the Fibonacci Sequence edited by V.E. Hoggatt Jr and Marjorie Bicknell-Johnson. Santa Clara, CA: The Fibonacci Association, pp.93-94.
23. Whitford, A.K. 1977. “Binet’s Formula Generalized.†The Fibonacci Quarterly. 15(1): 21, 24, 29.
24. Knott, R. 2015. “Fibonacci Numbers and the Golden Section.†Available online at: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html
25. Lucas, E. 1969. The Theory of Simply Periodic Numerical Functions. (Edited by D.A. Lind; translated by S. Kravitz.) San Jose: The Fibonacci Association.
26. Haylock, Derek. 1978. “Golden ratio and Beethoven’s 5th.†Mathematics Teaching. 84: 56-57.
27. Kuijken, Barthold. 2013. The Notation is Not the Music. Bloomington, IN: Indiana University Press, Ch.1.
28. Shannon, Anthony G, and Jean V. Leyendekkers. 2018. The Fibonacci Numbers and Integer Structure. New York: Nova Science Publishers, Chs1,6.
29. Falcon, Sergio. 2018. “Some new formulas on the k-Fibonacci numbers.†Journal of Advances in Mathematics. 14 (1): 7439-7445.
30. Montiel, Mariana, and Francisco Gómez (eds). 2018. Theoretical and Practical Pedagogy of Mathematical Music Theory. New Jersey, Singapore, Hong Kong, London: World Scientific, Section I. [https://www.worldscientific.com/ worldscibooks/10.1142/10665.
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Published
2018-04-30
How to Cite
Shannon, A. G., Klamka, I., & Gend, R. van. (2018). Generalized Fibonacci Numbers and Music. JOURNAL OF ADVANCES IN MATHEMATICS, 14(1), 7564–7579. https://doi.org/10.24297/jam.v14i1.7323
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