Tag: Euler
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Music and Mathematics: Tonnetz, Tuning, and 12-Tone
Harmony as a Mathematical Map Music and mathematics share a long, intertwined history. A striking illustration of their bond is the Tonnetz, a geometric diagram that encodes the harmonic relationships among the notes of the musical scale. First glimpsed in Euler’s Tentamen Novae Theoriae Musicae (1739), the Tonnetz organizes closely related tones into visual clusters.…
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Music and Mathematics: How Tonnetz, Tuning, and Tonal Systems Reveal a Harmonic Universe
Introduction: When Numbers Meet Notes Music and mathematics share a rich, long-standing dialogue. The Tonnetz, a geometric diagram that maps harmonic relationships between notes, offers a vivid illustration of how numbers and ratios shape the sounds we hear. From Euler’s early iterations in 1739 to modern interpretations, this diagram helps illuminate why certain chords feel…