why the harmonic frequencies are integer multiples of the fundamental frequency?
Pretty much all of the physical processes that create vibrations at a base frequency, also create vibrations at integer multiples of that base frequency. This includes the way the human vocal chords work.
You can artificially synthesize a sound with any frequencies you like, but you'll find it sounds somewhat "unnatural" unless you add some extra frequencies at integer multiples of the fundamental.
Or what's the relationship between harmonic frequencies and formant frequencies?
Imagine for a moment that the human vocal tract didn't exist. In other words, you just had a voice box, and nothing above that to alter the sound.
Based on the physics of the vocal chords, this would produce a "sawtooth wave", very much like the sound of a bowed string. If you Fourier-transform such a wave, the
nth harmonic's amplitude is roughly proportional to
But with the vocal tract in place, that's not what we see. The geometry of the vocal tract imposes an "envelope" on the wave, damping some parts and amplifying others. The peaks of the envelope are what we call the formants.
In this diagram, the blue line is a Fourier-transformed speech sample, with each peak being a harmonic. The red line is a reconstruction of the envelope, a polynomial that transforms a sawtooth wave into this particular speech sound—where the red line is high, it's amplifying the harmonics, and where it's low, it's damping them. And the green bars on top/blue bars on the bottom mark the formants.
(The diagram is from my own work, released under CC-BY; the units on the horizontal axis are Hz, and the units on the vertical axis don't really have any physical meaning.)