Let me take you down

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] I just learned that people are listening to music pitched slightly down because it makes them feel better. Instead of the A above middle C being set at 440 Hz, they have it tuned down to 432 Hz. This strikes me as odd, but how you feel is how you feel. Do whatever you want, as long as it doesn’t hurt anyone. I was interested, though, in the math behind this pitch change. In the equal-tempered scale, the frequency ratio of a semitone, which I’ll call rs is the twelfth root of two: rs=212=2112≈1.059463 This is the ratio of frequencies of adjacent piano keys. The ratio of 440 Hz to 432 Hz is 440432=5554≈1.0185185 so the pitch difference you get from moving down to 432 Hz is distinctly less than a semitone. How can we characterize that difference? Small differences in pitch are measured in cents. There are 100 cents in a semitone, so the frequency…