How to derive the tempered scale
The frequency is inverse proportional with the string length, so let
us derive it frequency based.
The problem is to find a number that multiplied with it self 12 times
yields a doubling of the frequency corresponding to an octave.
The equation looks like this
X*X*X*X*X*X*X*X*X*X*X*X=2
or
X= 1.059463094 about 6% higher pitch.
In string length
this yields
or 94,38% shorter string for each semitone.
The square root of a number is the number that multiplied with itself
yields the number. The twelfth root is the number that multiplied with
itself twelve times yields the number. You may get the twelfth root on
a scientific calculator by taking the number to the power of 1/12th.
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