phase mod x freq mod equivalence do any of you know how to convert a FM index to a phase mod index equivalent considering they both have the same carrier and modulator frequency? Is it even possible?

This is in respect for sound synthesis and I'm talking about one sinusoid modulating another sinusoidal oscillator.

I'm being able to get very similar sounds with these parameters:

FM: carrier freq 200Hz, modulator freq 480Hz, index 400*


this means the carrier freq oscillates 400Hz up then back to original frequency and then down 400Hz and back 480 times a second


Phase Mod: carrier freq 200Hz, modulator freq 480Hz, index pi/4 **

** this index means the phase goes up 1/8 of a cycle and then back, down the same amount and back to the original point

FM and Phase Mod are clearly two independent processes, but I've seen people mention it can be equivalent, but I've never read anything that would tell me how to get the same results with both. So I don't know if it can really be done even if it sounds practically the same thing or if I just got lucky.

thanks
phase is the integral (w.r.t. time) of frequency.  and they are definitely not two independent processes.  back in, i dunno, the 30s, Armstrong implemented FM for the first time using AM and a synchronized carrier that was 90° out of phase with the AM.  so he was first accomplishing phase modulation (PM) out of AM (for which they had the existing technology - they didn't really have VCOs back in that day), then he implemented FM out of the PM by integrating the audio modulation signal and applying that to the PM modulation input.

this is all in a basic communications systems textbook like that of AB Carlson.