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Why 12 notes to the Octave?

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Any moderately curious person will ask themselves at some point why, in western music, is the octave divided into 12 'semi-tones'. From a mathematical point of view, we can easily explain why 12 works nicely.


(One thing to remember while reading this text is that not all semi-tones have the same real physical value in frequency terms. Those above the fifth are larger in real physical terms.  In fact, I don't believe that any two semi-tones have the same real value, but for practical purposes they are all said to be the same.)

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