For anyone who might be interested, pianos are tuned by "tempering" intervals and "stretching" octaves. Careful listening for beats is absolutely essential for a good tuning. Pianos cannot be tuned according to "pure" intervals because of the effect of "harmonics". I know this because in my misspent (?) youth I started to learn to tune pianos but never really got anywhere.
Piano tuners are a skilled bunch - in more ways than one!
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