Piano Tuning:
Principle:
listen to beats between harmonics
Piano tuner knows correct number of beats for tempered intervals.
example:
after tuning A4 to 440 Hz, tune D4:
Since tempered fifth is not exact, there will be some beats
between third harmonic of D and second harmonic of A.
Calculation gives 0. 98 beats/sec.
Octaves are tuned so there are no beats.
But first overtone of A4= 440 Hz is not exactly 880 Hz,
but is sharp because of stiffness of string - say it is 882 Hz
Result: a ěstretched octaveî - slightly larger than 2:1 ratio
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