> Okay, not quite. You actually also need to know that the frequency ratios of all pairs of keys n keys apart should be the same, for all n. We could have made

> Okay, not quite. You actually also need to know that the frequency ratios of all pairs of keys n keys apart should be the same, for all n. We could have made a different second assumption instead, and remembered that not only is the note an octave higher twice the frequency, but also, the note a perfect fifth higher is one-and-a-half times the frequency. Propagating these axioms to tune all the notes would lead to Pythagorean tuning, which has some nicer-sounding intervals but makes playing in certain keys sound bad, since you can't make all n-key intervals exactly the same size.