Harmonics

Essential knowledge 14.6.B.2 does the numbering: a standing wave with the longest possible wavelength is called the fundamental or first harmonic, the second-longest wavelength is typically called the second harmonic, the third-longest the third harmonic, and so on. However, for a standing wave with a node at one end and an antinode at the other end, only odd harmonics can be established.

Read the numbering rule off that sentence. Harmonics are ordered by wavelength, longest first, not by counting up from whatever note you happen to hear. In a pipe closed at one end the allowed modes are $n = 1, 3, 5$ and upward, and the second harmonic does not exist at all. Its third harmonic is three times the fundamental frequency, and it is the second sound the pipe can make.

Why the set is discrete. EK 14.6.B.1.ii: the possible wavelengths of a standing wave are determined by the size and boundary conditions of the region to which it is confined. A node is forced at a fixed or closed end and an antinode at an open or loose end, and only certain wavelengths satisfy both ends at once.

There is no harmonic formula to look up. Neither $f_n = nv/2L$ nor $\lambda_n = 2L/n$ is printed in the AP Physics 2 CED or on its equation sheet. EK 14.6.B.3 says what to do instead: sketch the pattern, read the wavelength off the sketch, then convert with $\lambda = v/f$. The fundamental frequency entry has the longest-wavelength cases.