What are the advantages to using the 12th root of 2 over a simple fraction of 1/12 between keys/semitones when setting pitch frequencies on a piano?
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It will be completely out of tune (in any tuning system) if the frequencies are 1/12.
Just calculate it for 2 octaves and you’ll see. An octave is double the frequency. Let’s pick the notes 120, 240, 480 Hz. That’s 2 octaves.
For the 120-240 Hz octave, each semitone would be +10 Hz. For the 240-480 octave, each semitone would be +20 Hz.
Clearly, that is wrong since they are different steps. If you use 1/12th root increments, it will be the same (geometric increment) for both octaves.
Read this for more info: https://en.m.wikipedia.org/wiki/Musical_tuning#Tuning_systems
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