OK, that’s what I was trying to figure out. It sounds like you’ll want some sort of waveshaping function, unfortuantely that’s not really my area of expertise.
This is because your original signal contains an entire spectral envelope – fundamental frequency, whatever harmonics are already present, and probably some noise content – and a pitch shifter shifts this entire signal. So pitch shifting up one octave will reproduce the original fundamental, and the original harmonics present, and the original noise content all up one octave.
It could be possible to isolate just the original fundamental frequency before pitch shifting, which would give you more of a “pure” shifted frequency.
The issue here is that a “harmonic” naturally produced by an acoustic instrument is not an exact copy of the fundamental frequency at a different pitch, it’s more of an acoustic resonance that produces additional frequencies, but also has an auditory effect on the fundamental frequency itself.
It’s possible waveshaping is the answer. You could also look into physical modeling synthesis, to model the ways that an acoustic signal would resonate with its harmonics… ¯_(ツ)_/¯