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Why the chromatic scale has 12 tones

There is a simple (ahem) reason why the standard (Western) music scale divides an octave into twelve (roughly) equal steps or semitones. This is that it allows a particularly close approximation of both the second and third harmonics of a fundamental tone. This is based on the properties of a vibrating string. Other sound elements (such as surfaces like a drum membrane) or resonating volumes of various shapes have more complicated sequences of harmonics. Read more…

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e^{i\pi}+1=0

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