Cool Math for Hot Music

Abstract

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Quantum approaches to music cognition

“…As this study is devoted to symmetries in quantum models of static and dynamic tonal attraction, we first give a brief introduction into mathematical musicology (Balzano 1980, Mazzola 1990, 2002, Mazzola et al 2016).…”

“…This fundamental principle of octave equivalence induces an equivalence relation of acoustic stimuli into pitch classes, or chroma (Parncutt 2011) which circularly wind up throughout different octaves. Western and also Chinese music divides this continuous pitch class circle into twelve distinguished tones, or degrees (Mazzola et al 2016), comprising the chromatic scale shown in Tab. 1.…”

“…These are only a few examples for symmetries of the chroma circle. Others are the dichotomy between consonances and dissonances (v. Helmholtz 1877, Mazzola et al 2016, or the emergence of the third torus as the direct group product Z 12 = Z 3 × Z 4 (Balzano 1980, Mazzola et al 2016 which is crucial for the canonical construction of tertian chords in harmony theory (Schönberg 1978). This decomposition maps both, the chroma circle and the circle of fifths onto a torus, generated by a minor third (3 steps) in one direction and a major third (4 steps) in the orthogonal direction.…”

Gauge models of musical forces

“…Alexandroff 2012). For applying basic ideas of group theory to music (Balzano 1980;Mazzola 2002;Mazzola, Mannone, and Pang 2016), it is essential that there are certain operations that allow the transformation of tones into other tones. For instance, one can increase the tones by a certain number of steps 0, 1, 2, .…”

cARTegory Theory: Framing Aesthetics of Mathematics

“…We can make music out of a snail shell, a mathematical conchoid: the music reproduces the "getting closer" of the shell's spirals, through shortening of silence and approaching of the pitch. 11 In this sense, we are keeping the main idea of the conchoid from the domain of visuals to the domain of sounds, through an opportune mapping of its mathematical definition via a 3-component parametric equation [18,28]. We can explain the eventual analogies between mathematical steps and visual steps through crossmodal correspondences [38] and the identification of audiovisual objects [14], as well as gestural similarity [19].…”