Fractals and music
Abstract
Many natural phenomena we find in our surroundings, are fractals. Studying and learning about fractals in classrooms is always a challenge for both teachers and students. We here show that the sound of musical instruments can be used as a good resource in the laboratory to study fractals. Measurement of fractal dimension which indicates how much fractal content is there, is always uncomfortable, because of the size of the objects like coastlines and mountains. A simple fractal source is always desirable in laboratories. Music serves to be a very simple and effective source for fractal dimension measurement. In this paper, we are suggesting that music which has an inherent fractal nature can be used as an object in classrooms to measure fractal dimensions. To find the fractal dimension we used the box-counting method. We studied the sound produced by different stringed instruments and some common noises. For good musical sound, the fractal dimension obtained is around 1.6882.
References
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Carmo, E. Do, & Hõnnicke, M. G. (2021). Fractal dimension analysis with popcorn grains and popped popcorn grains. Revista Brasileira de Ensino de Fisica, 43. https://doi.org/10.1590/1806-9126-RBEF-2021-0115
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Ching, W. K., Erickson, M., Garik, P., Hickman, P., Jordan, J., Schwarzer, S., & Shore, L. (1994). Overcoming resistance with fractalsâ€â€A new way to teach elementary circuits. The Physics Teacher, 32(9). https://doi.org/10.1119/1.2344109
Creffield, C. E. (2021). Fractals on a benchtop: Observing fractal dimension in a resistor network. https://doi.org/10.48550/arxiv.2107.02322
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Das, Atin, & Das, P. (2006). Fractal analysis of different eastern and western musical instruments. Fractals, 14(3), 165–170. https://doi.org/10.1142/S0218348X06003192
De Jong, M. L. (1992). Chaos and the simple pendulum. The Physics Teacher, 30(2). https://doi.org/10.1119/1.2343491
Duan, Q., An, J., Mao, H., Liang, D., Li, H., Wang, S., & Huang, C. (2021). Review about the application of fractal theory in the research of packaging materials. In Materials (Vol. 14, Issue 4). https://doi.org/10.3390/ma14040860
Feldman, D. P. (2019). Chaos and Dynamical Systems. In Chaos and Dynamical Systems. Princeton University Press, Princeton, NJ. https://doi.org/10.2307/j.ctvc5pczn
Feng, J., Wang, E., Huang, Q., Ding, H., & Dang, L. (2021). Time-Varying Multifractal Analysis of Crack Propagation and Internal Fracture Process of Coal Under Dynamic Loading. Fractals, 29(4). https://doi.org/10.1142/S0218348X21500894
GarcÃÂa, E., & Liu, C. H. (1995). A Classroom Demonstration of Electrodeposited Fractal Patterns. Journal of Chemical Education, 72(9). https://doi.org/10.1021/ed072p829
graphics - Creating a Sierpinski gasket with the missing triangles filled in - Mathematica Stack Exchange. (n.d.).
Harrison, J. (1989). An introduction to fractals. In R. L. Devaney & L. Keen (Eds.), Chaos and Fractals: The Mathematics behind the Computer Graphics. American Mathematical Society, Providence, RI. https://doi.org/10.1090/psapm/039/1010238
Hartvigsen, G. (2000). The analysis of leaf shape using fractal geometry. American Biology Teacher, 62(9). https://doi.org/10.2307/4451007
Hsü, K. J. (1993). Fractal Geometry of Music: From Bird Songs to Bach. In Applications of Fractals and Chaos (pp. 21–39). Springer. https://doi.org/10.1007/978-3-642-78097-4_3
Hsu, K. J., & Hsu, A. (1991). Self-similarity of the “1/f noise†called music. Proceedings of the National Academy of Sciences of the United States of America, 88(8), 3507–3509. https://doi.org/10.1073/pnas.88.8.3507
Hsu, K. J., & Hsu, A. J. (1990). Fractal geometry of music. Proceedings of the National Academy of Sciences of the United States of America, 87(3). https://doi.org/10.1073/pnas.87.3.938
Hughes, J. R. (2003). Fractals in a first year undergraduate seminar. Fractals, 11(1). https://doi.org/10.1142/S0218348X03001410
Hurd, A. J. (1988). Resource Letter FRâ€Â1: Fractals. American Journal of Physics, 56(11). https://doi.org/10.1119/1.15761
Ibrahim, O., Kamel, A., & Khamis, E. (2021). Fractal Geometry as a Source of Innovative Formations in Interior Design. Journal of Design Sciences and Applied Arts, 2(2). https://doi.org/10.21608/jdsaa.2021.42275.1075
Karakus, F. (2016). Pre-Service Teachers’ Concept Images on Fractal Dimension. International Journal for Mathematics Teaching and Learning, 17(2).
Karakuş, F. (2013). A cross-age study of students’ understanding of fractals. Bolema - Mathematics Education Bulletin, 27(47), 829–846. https://doi.org/10.1590/S0103-636X2013000400007
Karakuş, F. (2015). Investigation into how 8th grade students define fractals. Educational Sciences: Theory & Practice, 15(3), 825–836. https://doi.org/10.12738/estp.2015.3.2429
Karakuş, F., & Kösa, T. (2010). Exploring fractal dimension by experiment: Pre-service teachers’ gains. Procedia - Social and Behavioral Sciences, 2(2). https://doi.org/10.1016/j.sbspro.2010.03.145
Karakus, Fatih, & Karatas, I. (2014). Secondary school students’ misconceptions about fractals. Journal of Education and Human Development, 3(3), 241–250. https://doi.org/10.15640/jehd.v3n3a19
Knutson, P., & Dahlberg, E. D. (2003). Fractals in the Classroom. The Physics Teacher, 41(7). https://doi.org/10.1119/1.1616477
Kuzovlev, Y. E. (2015). Why nature needs 1/f noise. Physics-Uspekhi, 58(7). https://doi.org/10.3367/ufne.0185.201507d.0773
Lartillot, O., Toiviainen, P., & Eerola, T. (2008). A matlab toolbox for music information retrieval. Studies in Classification, Data Analysis, and Knowledge Organization. https://doi.org/10.1007/978-3-540-78246-9_31
Madhushani, K. N. R. A. K., & Sonnadara, D. U. J. (2012). Fractal Analysis of Cloud Shapes. Proceedings of the Technical Sessions, 28, 59–64.
Mandelbrot, B. B. (1982). The fractal geometry of nature. W.H. Freeman, San Francisco.
Meyer, P. S. (1993). Fractal Dimension of Music. Columbia University.
Mitić, V. V., Lazović, G. M., Manojlović, J. Z., Huang, W. C., Stojiljković, M. M., Facht, H., & Vlahović, B. (2020). Entropy and fractal nature. Thermal Science, 24. https://doi.org/10.2298/TSCI191007451M
Mitić, V. V., Lazović, G., Radosavljevic-Mihajlovic, A. S., Milosević, D., Marković, B., Simeunović, D., & Vlahović, B. (2021). Forensic science and fractal nature analysis. Modern Physics Letters B, 35(32). https://doi.org/10.1142/S0217984921504935
Niklasson, M. H., & Niklasson, G. A. (2020). The fractal dimension of music: Melodic contours and time series of pitch.
Nishanth, P., Prasanth, P., Reshma, P., & Udayanandan, K. M. (2020). Fractals in leaves-An interdisciplinary project for undergraduates. Physics Education (IAPT), 36(4).
Oestreicher, C. (2007). A history of chaos theory. In Dialogues in Clinical Neuroscience (Vol. 9, Issue 3). https://doi.org/10.31887/dcns.2007.9.3/coestreicher
Ornes, S. (2014). Hunting fractals in the music of J. S. Bach. Proceedings of the National Academy of Sciences of the United States of America, 111(29), 10393. https://doi.org/10.1073/pnas.1410330111
Peitgen, H.-O., Jürgens, H., & Saupe, D. (1992). Fractals for the Classroom. In Fractals for the Classroom. Springer, New York Heidelberg. https://doi.org/10.1007/978-1-4757-2172-0
Selvam, A. M. (2017). Universal Inverse Power-Law Distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter-Annual Variability of Indian and USA Region Rainfall. Pure and Applied Geophysics, 174(1). https://doi.org/10.1007/s00024-016-1394-9
Shore, L. S., Garik, P., Stanley, E., Trunfio, P. A., Hickman, P., & Erickson, M. J. (1992). Learning Fractals by “Doing Scienceâ€Â: Applying Cognitive Apprenticeship Strategies to Curriculum Design and Instruction. Interactive Learning Environments, 2(3). https://doi.org/10.1080/1049482920020305
Shu, Z. R., Chan, P. W., Li, Q. S., He, Y. C., Yan, B. W., Li, L., Lu, C., Zhang, L., & Yang, H. L. (2021). Characterization of vertical wind velocity variability based on fractal dimension analysis. Journal of Wind Engineering and Industrial Aerodynamics, 213. https://doi.org/10.1016/j.jweia.2021.104608
Souza, P. V. S., Alves, R. L., & Balthazar, W. F. (2019). A Tool to Study Fractals in an Interdisciplinary Perspective. The Physics Teacher, 57(7). https://doi.org/10.1119/1.5126825
Swapna, M. S., Sreejyothi, S., Raj, V., & Sankararaman, S. (2021). Is SARS CoV-2 a Multifractal?â€â€Unveiling the Fractality and Fractal Structure. Brazilian Journal of Physics, 51(3). https://doi.org/10.1007/s13538-020-00844-w
Uahabi, K. L., & Atounti, M. (2015). Applications of fractals in medicine. Annals of the University of Craiova, Mathematics and Computer Science Series, 42(1).
Voss, R. F., & Clarke, J. (1975). “1/fnoise†in music and speech. Nature, 258(5533). https://doi.org/10.1038/258317a0
Vuidel, G. (n.d.). Fractal.yse - Fractal analysis software.
Wiesenfeld, K. (2001). Resource Letter: ScL-1: Scaling laws. American Journal of Physics, 69(9). https://doi.org/10.1119/1.1383601
Wu, J., Jin, X., Mi, S., & Tang, J. (2020). An effective method to compute the box-counting dimension based on the mathematical definition and intervals. Results in Engineering, 6. https://doi.org/10.1016/j.rineng.2020.100106
Xu, J., Jian, Z., & Lian, X. (2017). An application of box counting method for measuring phase fraction. Measurement: Journal of the International Measurement Confederation, 100. https://doi.org/10.1016/j.measurement.2017.01.008
Yan, B., Chan, P. W., Li, Q., He, Y., & Shu, Z. (2021). Dynamic analysis of meteorological time series in Hong Kong: A nonlinear perspective. International Journal of Climatology, 41(10). https://doi.org/10.1002/joc.7106
Zanoni, M. (2002). Measurement of the fractal dimension of a cauliflower. The Physics Teacher, 40(1). https://doi.org/10.1119/1.1457822
Zembrowska, K., & Kuźma, M. (2002). Some Exercises on Fractals for High School Students. The Physics Teacher, 40(8). https://doi.org/10.1119/1.1526617
Zmeskal, O., Dzik, P., & Vesely, M. (2013). Entropy of fractal systems. Computers and Mathematics with Applications, 66(2). https://doi.org/10.1016/j.camwa.2013.01.017
Biswas, H. R., Hasan, M. M., & Kumar Bala, S. (2018). Chaos Theory And Its Applications In Our Real Life. Barishal University Journal Part 1, 5(1&2), 123–140.
Carmo, E. Do, & Hõnnicke, M. G. (2021). Fractal dimension analysis with popcorn grains and popped popcorn grains. Revista Brasileira de Ensino de Fisica, 43. https://doi.org/10.1590/1806-9126-RBEF-2021-0115
Carpinteri, A., Lacidogna, G., & Accornero, F. (2018). Fluctuations of 1/f noise in damaging structures analyzed by Acoustic Emission. Applied Sciences (Switzerland), 8(9). https://doi.org/10.3390/app8091685
Ching, W. K., Erickson, M., Garik, P., Hickman, P., Jordan, J., Schwarzer, S., & Shore, L. (1994). Overcoming resistance with fractalsâ€â€A new way to teach elementary circuits. The Physics Teacher, 32(9). https://doi.org/10.1119/1.2344109
Creffield, C. E. (2021). Fractals on a benchtop: Observing fractal dimension in a resistor network. https://doi.org/10.48550/arxiv.2107.02322
Crutchfield, J. P. (2012). Between order and chaos. Nature Physics, 8(1). https://doi.org/10.1038/nphys2190
Das, A., & Das, P. (2005). Classification of Different Indian Songs Based on Fractal Analysis. Complex Systems, 15(3), 253–259.
Das, Atin, & Das, P. (2006). Fractal analysis of different eastern and western musical instruments. Fractals, 14(3), 165–170. https://doi.org/10.1142/S0218348X06003192
De Jong, M. L. (1992). Chaos and the simple pendulum. The Physics Teacher, 30(2). https://doi.org/10.1119/1.2343491
Duan, Q., An, J., Mao, H., Liang, D., Li, H., Wang, S., & Huang, C. (2021). Review about the application of fractal theory in the research of packaging materials. In Materials (Vol. 14, Issue 4). https://doi.org/10.3390/ma14040860
Feldman, D. P. (2019). Chaos and Dynamical Systems. In Chaos and Dynamical Systems. Princeton University Press, Princeton, NJ. https://doi.org/10.2307/j.ctvc5pczn
Feng, J., Wang, E., Huang, Q., Ding, H., & Dang, L. (2021). Time-Varying Multifractal Analysis of Crack Propagation and Internal Fracture Process of Coal Under Dynamic Loading. Fractals, 29(4). https://doi.org/10.1142/S0218348X21500894
GarcÃÂa, E., & Liu, C. H. (1995). A Classroom Demonstration of Electrodeposited Fractal Patterns. Journal of Chemical Education, 72(9). https://doi.org/10.1021/ed072p829
graphics - Creating a Sierpinski gasket with the missing triangles filled in - Mathematica Stack Exchange. (n.d.).
Harrison, J. (1989). An introduction to fractals. In R. L. Devaney & L. Keen (Eds.), Chaos and Fractals: The Mathematics behind the Computer Graphics. American Mathematical Society, Providence, RI. https://doi.org/10.1090/psapm/039/1010238
Hartvigsen, G. (2000). The analysis of leaf shape using fractal geometry. American Biology Teacher, 62(9). https://doi.org/10.2307/4451007
Hsü, K. J. (1993). Fractal Geometry of Music: From Bird Songs to Bach. In Applications of Fractals and Chaos (pp. 21–39). Springer. https://doi.org/10.1007/978-3-642-78097-4_3
Hsu, K. J., & Hsu, A. (1991). Self-similarity of the “1/f noise†called music. Proceedings of the National Academy of Sciences of the United States of America, 88(8), 3507–3509. https://doi.org/10.1073/pnas.88.8.3507
Hsu, K. J., & Hsu, A. J. (1990). Fractal geometry of music. Proceedings of the National Academy of Sciences of the United States of America, 87(3). https://doi.org/10.1073/pnas.87.3.938
Hughes, J. R. (2003). Fractals in a first year undergraduate seminar. Fractals, 11(1). https://doi.org/10.1142/S0218348X03001410
Hurd, A. J. (1988). Resource Letter FRâ€Â1: Fractals. American Journal of Physics, 56(11). https://doi.org/10.1119/1.15761
Ibrahim, O., Kamel, A., & Khamis, E. (2021). Fractal Geometry as a Source of Innovative Formations in Interior Design. Journal of Design Sciences and Applied Arts, 2(2). https://doi.org/10.21608/jdsaa.2021.42275.1075
Karakus, F. (2016). Pre-Service Teachers’ Concept Images on Fractal Dimension. International Journal for Mathematics Teaching and Learning, 17(2).
Karakuş, F. (2013). A cross-age study of students’ understanding of fractals. Bolema - Mathematics Education Bulletin, 27(47), 829–846. https://doi.org/10.1590/S0103-636X2013000400007
Karakuş, F. (2015). Investigation into how 8th grade students define fractals. Educational Sciences: Theory & Practice, 15(3), 825–836. https://doi.org/10.12738/estp.2015.3.2429
Karakuş, F., & Kösa, T. (2010). Exploring fractal dimension by experiment: Pre-service teachers’ gains. Procedia - Social and Behavioral Sciences, 2(2). https://doi.org/10.1016/j.sbspro.2010.03.145
Karakus, Fatih, & Karatas, I. (2014). Secondary school students’ misconceptions about fractals. Journal of Education and Human Development, 3(3), 241–250. https://doi.org/10.15640/jehd.v3n3a19
Knutson, P., & Dahlberg, E. D. (2003). Fractals in the Classroom. The Physics Teacher, 41(7). https://doi.org/10.1119/1.1616477
Kuzovlev, Y. E. (2015). Why nature needs 1/f noise. Physics-Uspekhi, 58(7). https://doi.org/10.3367/ufne.0185.201507d.0773
Lartillot, O., Toiviainen, P., & Eerola, T. (2008). A matlab toolbox for music information retrieval. Studies in Classification, Data Analysis, and Knowledge Organization. https://doi.org/10.1007/978-3-540-78246-9_31
Madhushani, K. N. R. A. K., & Sonnadara, D. U. J. (2012). Fractal Analysis of Cloud Shapes. Proceedings of the Technical Sessions, 28, 59–64.
Mandelbrot, B. B. (1982). The fractal geometry of nature. W.H. Freeman, San Francisco.
Meyer, P. S. (1993). Fractal Dimension of Music. Columbia University.
Mitić, V. V., Lazović, G. M., Manojlović, J. Z., Huang, W. C., Stojiljković, M. M., Facht, H., & Vlahović, B. (2020). Entropy and fractal nature. Thermal Science, 24. https://doi.org/10.2298/TSCI191007451M
Mitić, V. V., Lazović, G., Radosavljevic-Mihajlovic, A. S., Milosević, D., Marković, B., Simeunović, D., & Vlahović, B. (2021). Forensic science and fractal nature analysis. Modern Physics Letters B, 35(32). https://doi.org/10.1142/S0217984921504935
Niklasson, M. H., & Niklasson, G. A. (2020). The fractal dimension of music: Melodic contours and time series of pitch.
Nishanth, P., Prasanth, P., Reshma, P., & Udayanandan, K. M. (2020). Fractals in leaves-An interdisciplinary project for undergraduates. Physics Education (IAPT), 36(4).
Oestreicher, C. (2007). A history of chaos theory. In Dialogues in Clinical Neuroscience (Vol. 9, Issue 3). https://doi.org/10.31887/dcns.2007.9.3/coestreicher
Ornes, S. (2014). Hunting fractals in the music of J. S. Bach. Proceedings of the National Academy of Sciences of the United States of America, 111(29), 10393. https://doi.org/10.1073/pnas.1410330111
Peitgen, H.-O., Jürgens, H., & Saupe, D. (1992). Fractals for the Classroom. In Fractals for the Classroom. Springer, New York Heidelberg. https://doi.org/10.1007/978-1-4757-2172-0
Selvam, A. M. (2017). Universal Inverse Power-Law Distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter-Annual Variability of Indian and USA Region Rainfall. Pure and Applied Geophysics, 174(1). https://doi.org/10.1007/s00024-016-1394-9
Shore, L. S., Garik, P., Stanley, E., Trunfio, P. A., Hickman, P., & Erickson, M. J. (1992). Learning Fractals by “Doing Scienceâ€Â: Applying Cognitive Apprenticeship Strategies to Curriculum Design and Instruction. Interactive Learning Environments, 2(3). https://doi.org/10.1080/1049482920020305
Shu, Z. R., Chan, P. W., Li, Q. S., He, Y. C., Yan, B. W., Li, L., Lu, C., Zhang, L., & Yang, H. L. (2021). Characterization of vertical wind velocity variability based on fractal dimension analysis. Journal of Wind Engineering and Industrial Aerodynamics, 213. https://doi.org/10.1016/j.jweia.2021.104608
Souza, P. V. S., Alves, R. L., & Balthazar, W. F. (2019). A Tool to Study Fractals in an Interdisciplinary Perspective. The Physics Teacher, 57(7). https://doi.org/10.1119/1.5126825
Swapna, M. S., Sreejyothi, S., Raj, V., & Sankararaman, S. (2021). Is SARS CoV-2 a Multifractal?â€â€Unveiling the Fractality and Fractal Structure. Brazilian Journal of Physics, 51(3). https://doi.org/10.1007/s13538-020-00844-w
Uahabi, K. L., & Atounti, M. (2015). Applications of fractals in medicine. Annals of the University of Craiova, Mathematics and Computer Science Series, 42(1).
Voss, R. F., & Clarke, J. (1975). “1/fnoise†in music and speech. Nature, 258(5533). https://doi.org/10.1038/258317a0
Vuidel, G. (n.d.). Fractal.yse - Fractal analysis software.
Wiesenfeld, K. (2001). Resource Letter: ScL-1: Scaling laws. American Journal of Physics, 69(9). https://doi.org/10.1119/1.1383601
Wu, J., Jin, X., Mi, S., & Tang, J. (2020). An effective method to compute the box-counting dimension based on the mathematical definition and intervals. Results in Engineering, 6. https://doi.org/10.1016/j.rineng.2020.100106
Xu, J., Jian, Z., & Lian, X. (2017). An application of box counting method for measuring phase fraction. Measurement: Journal of the International Measurement Confederation, 100. https://doi.org/10.1016/j.measurement.2017.01.008
Yan, B., Chan, P. W., Li, Q., He, Y., & Shu, Z. (2021). Dynamic analysis of meteorological time series in Hong Kong: A nonlinear perspective. International Journal of Climatology, 41(10). https://doi.org/10.1002/joc.7106
Zanoni, M. (2002). Measurement of the fractal dimension of a cauliflower. The Physics Teacher, 40(1). https://doi.org/10.1119/1.1457822
Zembrowska, K., & Kuźma, M. (2002). Some Exercises on Fractals for High School Students. The Physics Teacher, 40(8). https://doi.org/10.1119/1.1526617
Zmeskal, O., Dzik, P., & Vesely, M. (2013). Entropy of fractal systems. Computers and Mathematics with Applications, 66(2). https://doi.org/10.1016/j.camwa.2013.01.017
Authors
Pothiyodath, N., & Murkoth, U. K. (2022). Fractals and music. omentum: hysics ducation ournal, 6(2), 119–128. https://doi.org/10.21067/mpej.v6i2.6796
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