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If a piano were to be tuned perfectly with the equal temperament system, what would be the relationship between standing waves from one note to the next? How would the frequency and wavelength of sound be affected?

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Every twelve notes (semitones) span an octave i.e. the frequency doubles. In equal temprement tuning the ratio of the frquencies of each pair of semitones, call this $r$, is constant so:

$$ r^{12} = 2 $$

and therefore $r \approx$ 1.0595.

If by one note to the next you mean a full tone, i.e. two semitones, then the ratio of the frequencies is $r^2$ or about 1.1225.

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