F# and G

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 2^1/12. So the nth note in the chromatic scale from A below middle C to A above middle C has frequency

220 × 2^n/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 × 2^9/12 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 × 2^5/6 Hz = 391.9954 Hz

which is even closer to being an integer.

In more mathematical terms, stripped of musical significance, we’ve discovered that

2^3/4 ≈ 37/22

and

2^5/6 ≈ 98/55.