The physics
The tension T and the mass per metre μ set a string's wave speed.
v = √(T/μ)
Two equal waves running opposite ways add to a standing wave. Some points never move. These are nodes, half a wavelength apart. The points that swing most are antinodes.
y = 2A sin kx cos ωt node to node = λ/2
A string clamped at both ends has a node at each end. So it rings only when a whole number of loops fits. Each loop is half a wavelength.
fn = n v/2L n = 1, 2, 3, …
A light ring on a smooth pole is a free end. It is an antinode. Then the string fits odd numbers of quarter waves, and only odd notes ring.
f = (2n − 1) v/4L
Two strings joined end to end ring only at a note both pieces share. Two next-door notes of one string differ by its lowest note.