On the Geometric Realisation of Equal Tempered Music
Since the time of Pythagoras (c.550BC), mathematicians interested in music have asked, “What governs the whole number ratios that emerge from derivations of the harmonic series?” Simon Stevin (1548-1620) devised a mathematical underlay (where a semitone equals 21/12) that gave rise to the equal temperament tuning system we still use today. Beyond this, the structure of formalised musical orderings have eluded many of us. Music theorists use the tools and techniques of their trade to peer into the higher-order musical structures that underpin musical harmony. These methods of investigating music theory and harmony are difficult to learn (and teach), as complex abstract thought is required to imagine the components of a phenomenon that cannot be seen. This paper outlines a method to understanding the mathematical underpinnings of the equal tempered tuning system. Using this method, musical structure can be quantitatively modelled as a series of harmonic elements at each pulse of musical time.
Kelley, R. (2002). "Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method" http://robertkelleyphd.com/home/teaching/music-theory/intro-to-post-tonal-music-analysis/
Fauvel, J.; Flood, R. & Wilson, R. (Eds.) (2006). Music and Mathematics: From Pythagoras to fractals. London: Oxford University Press.
Herlinger, J. (2006). Medieval Canonics. In Thomas Christensen (Ed.),The Cambridge history of Western music theory, UK: Cambridge University Press.
Mathieson, T., J. (2006). Greek Music Theory. In Thomas Christensen (Ed.), The Cambridge history of Western music theory, UK: Cambridge University Press.
Rasch, R. (2006). Tuning and Temperament. In Thomas Christensen (Ed.), The Cambridge history of Western music theory, UK: Cambridge University Press.
Ashton-Bell, R., L., T. (2015). Sound in Sight: An autoethnographical case study of teaching and learning western music theory. PhD Thesis: Monash University repository. http://arrow.monash.edu.au/hdl/1959.1/1134415.
Suits, B. H. (2015). “Note Frequencies”, “Speed of Sound” & “Physics of Music”. Physics Department, Michigan Technological University. https://pages.mtu.edu/~suits/notefreqs.html, https://pages.mtu.edu/~suits/SpeedofSound.html, https://pages.mtu.edu/~suits/Physicsofmusic.html.