I'll echo the suggestions for a sound card + PC for FFT, that QuantAsylum 400 looks pretty good.
I have a spectrum analyser (Advantest TR4171) that starts from 10Hz, but the problem is the spectrum analyser's distortion is in the range of -70dBc or so, way too high for audio work, it is basically useless to have a look at individual harmonics.
You can use a notch filter to reject the fundamental, but now the problem becomes the veeeery long sweep times, expect anywhere between 3-15 minutes just for a 1kHz or 10kHz span at 3Hz or 10Hz RBW. Not very practical for most uses in audio.
Another half measure would be to get a dedicated audio analyser with a monitor output, they're pretty standard so you should find them on basically everything. You can then perform FFT on the monitor output without any risk of blowing up your PC or sound card (they can usually take up to 150V input too, good for power amps), and depending on the audio analyser you can get very decent performance.
Sometimes all I really need to look at are the first 5 harmonics, and any good audio analyser can display them individually, no need to set up math functions on a scope or fire up a PC, 1 button and you're golden.
Attached is an FFT on the monitor out of a ShibaSoku AM51B audio analyser loopback using the crappy built in sound card from a cheap laptop, that second harmonic is at -139dB from the fundamental.
ShibaSoku make great audio analysers, one to look out for is the old 725B or 725C models, with a few simple mods (upgrading 25 year old op amps
) they will bring any entry level audio analyser today to their knees in terms of residual noise+distortion anyway. They'll measure down to -120dB THD+N @ 1kHz, 30kHz BW no problem, and for reference the new flagship Audio Precision APx555 goes to like -124ish I think, but they start at 27 grand.