In fact the different notes emitted by a sounding pipe will approximately follow the harmonic progression only if the maker has given the pipe a suitable shape, and makers spend their lives in search of this ideal.
If the pipe shape is poor, the succession of possible notes is also poor.
With a pipe of a given length, for example, one can, by changing its shape, make the fundamental rise progressively by a whole octave without displacing the other notes in the same proportions.
So where are our "natural" harmonics?
The different notes emitted by a sounding pipe are not harmonics but partials.
Each one is separately tunable and behaves like a new fundamental with its own timbre which is itself made up of harmonics.
These few reflections will help instrumentalists to understand their instruments' tuning faults better and should clear up a particularly grey area in musicians' knowledge.