That's because most of the time we are dealing with fast rising wave fronts which cause many harmonics but the 3rd is usually the most pronounced so it causes the most distortion. This is not always the case but for many things it is.
3rd harmonic distortion (and other odd-numbered harmonics) is usually caused by symmetrical clipping of an amplifier. It sounds like a BUZZ.
2nd harmonic distortion (and other even-numbered harmonics) sounds fairly musical because the 2nd harmonic is exactly double (one octave higher) than the fundamental frequency.
3rd harmonic distortion, no matter what is the source, is much more noticeable than 2nd harmonic distortion and even a small amount causes music to sound "bad", for lack of a better word.