With pure sine-wave-based Frequency Modulation, one always hears the base pitch if the carrier-frequency is the same as the modulator-frequency. Raising the index will add harmonics and change their strength, but the base tonality will stay very clear. However, with this audio sample of a clarinet, it would seem that actual audio, when FM is applied to it, is much more complicated. Having the modulator-frequency approximately equal the base-frequency of the clarinet produces a sound with said frequency, but only at certain Indexes. I assume this is due to the obvious complications of audio: 1) No pitch-tracker is perfect and 2) unlike a sinus-tone, an actual acoustic sound contains a whole harmonic series of frequencies, all of which are modulated.
I am wondering if anyone here has experience with this phenomenon. I would like to know if there is some way of calculating which Index will contribute to the pureness of a same-frequency carrier-modulator relationship. Or is this impossible and dependent on the individuality of each instrument and each sound produced?

Many Thanks.

P.S. [transpositiontest2] is the main patch.