I was looking at frequencies of pitches and saw something I hadn’t noticed before: F# and G have very nearly integer frequencies.
To back up a bit, we’re assuming the A above middle C has frequency 440 Hz. This is the most common convention now, but conventions have varied over time and place.
We’re assuming 12-tone equal temperament (12-TET), and so each semitone is a ratio of 21/12. So the nth note in the chromatic scale from A below middle C to A above middle C has frequency
220 × 2n/12.
I expected the pitch with frequency closest to integer would be an E because a perfect fifth above 220 Hz would be exactly 330 Hz. In equal temperament the frequency of the E above middle C is 329.6 Hz.
The frequency of F# is
220 × 29/12 Hz = 369.9944 Hz.
The difference between this frequency and 370 Hz is much less than the difference between equal temperament and other tuning systems.
The frequency of G is
220 × 25/6 Hz = 391.9954 Hz
which is even closer to being an integer.
In more mathematical terms, stripped of musical significance, we’ve discovered that
23/4 ≈ 37/22
25/6 ≈ 98/55.