Frequently, I want to mix two songs in the same key but different bpms. To transition one song into another, I need to match their bpms. For example, the currently playing song might be at 124 bpm, but the next song on deck might be 120 bpm. In this case, I'd increase the speed of the next song to 124 bpm. (This is known as beatmatching.)
Of course, the side effect of a tempo change is a pitch shift. A 120 bpm song played at 124 bpm sounds somewhere shy of a half step (100 cents) higher, roughly estimated by ear.
I'd like to know the approximate number of cents that pitch shifts from a 1 bpm increase or decrease, centered around 123 bpm, but I don't know how to calculate it.
I know there isn't an exact answer, since it's a logarithmic scale. But most of the songs I DJ are between 120 and 126 bpm, so I only need a practical measure, i.e. I want to be able to think, "4 bpm difference; that's about 4 times X cents higher," and turn the pitch knob accordingly to bring the next song closer to its original pitch.
Also, I'm curious about the math.