There are no semitones in a C major chord -- and some semitones are in fact integer ratios.
Two tones sounding simultaneously beat according to the difference of their frequencies. If you have one tone at 440 Hz and another at 441 Hz, you'll hear one beat per second. Raise the frequency of the higher tone to 442 Hz and you'll hear two beats per second.
Now an acoustically tuned A major triad comprising A4, C♯5, and E5 has the frequencies 440 Hz, 550 Hz, and 660 Hz. If there is interference, with what frequency should we expect to hear beating?
660 - 550 = 110
660 - 440 = 220
550 - 440 = 110
The beating is so fast that it sounds like another tone, or rather two other tones, one an octave below the root and the other two octaves below the root. These are known as difference tones. For more information, see the Wikipedia article on combination tones.
In equal temperament, no intervals are integer ratios aside from the octave. The interference in an equal-tempered triad is much harder to hear because the difference tones do not reinforce the chord. Instead of 550 Hz, C♯5 is 554.365 Hz, and instead of 660 Hz, E5 is 659.255 Hz. The difference tones are therefore
659.255 - 554.365 = 104.890
659.255 - 440.000 = 219.255
554.365 - 440.000 = 114.365
In the justly tuned case, the difference tones are precisely one or two octaves below the root of the triad. Here, however, they are not. Rather than being precisely at A2, the difference between E5 and C♯5 is closer to A♭2, but about 18 cents sharper than the equal-tempered A♭2. The difference between E5 and A4, instead of being precisely A3, is about 6 cents flat. Finally, the difference between C♯5 and A4, in contrast to the other two, is rather sharper than in just intonation, being a little less than 33 cents flat from B♭2. This means that the two difference tones that are precisely two octaves below the root in just intonation are instead 149 cents apart in equal temperament.
These difference tones are hard to hear because they don't reinforce each other, and they don't reinforce the harmonics of the actual tones. What one normally hears in an equal-tempered major triad, rather, is the beating caused by interference between the harmonics of the major third and of the other pitches. This is why even a piano that is in good tune will have a little vibrato in its major thirds. This can be heard very vividly by comparing the fourth-fret harmonic of an open guitar string with the corresponding fretted pitch on another string.