The physics
Blow across the mouth of a pipe. The air column inside forms a standing wave.
At an open end the air moves freely. So an open end is a displacement antinode. At a closed end the air cannot move. So a closed end is a node.
A pipe open at both ends fits half a wave in its lowest note. It sounds every harmonic.
open: fn = n v / 2L, n = 1, 2, 3 …
A pipe with one end closed fits a quarter wave. It sounds only the odd harmonics.
closed: fn = (2n − 1) v / 4L
The first overtone is the next note above the lowest. For an open pipe it is the 2nd harmonic. For a closed pipe it is the 3rd.
Two close notes together throb as beats, |f₁ − f₂| times a second. A pipe's notes grow with the speed of sound. A denser gas carries sound slower, so the pipe sounds lower.
beats = |f₁ − f₂| v = √(B / ρ)