To those who ever tried tuning a guitar or a piano and wound up with some chords in tune while others all crazy sketch ball beating, you most likely have dealt with this "in tune" nuisance. Basically when you tune twelve just perfect fifths up in a row (3:2 x 12), you wind up with a different slightly different pitch then when tuning seven octaves in row (2/1 x 7). The difference between these pitches is the Pythagorean Comma (diatonic comma). Yes there really is a difference between C and B♯ in real world application :-) The frequency ratio is 531441:524288, approximately 23.46 cents. Almost a quarter of a semi tone difference which is very recognizable to even the untrained ear, and stands out like a sore thumb to the trained musician.
Tempered instruments, such as fretted or keyed instruments, at first glance do away with this comma issue by dividing the octave into 12 parts, all of which are equal on a logarithmic scale. This means that the perceived "distance" from every note to its nearest neighbor is the same for every note in the system. HOWEVER in doing so all the characteristic intervals in a chromatic scale are grossly misrepresented from the most simplified Just Intonated version. Hence we get a 12TET Major 3rd of 24/12 : 1 instead of the more pleasing to the ear JI 5 : 4 ratio. A difference of nearly 14 cents. Some of the other intervals are more poorly represented by ET, so much so that they can said to be not represented at all (the Minor 7th ET for instance is a quarter tone off (49 cents) from the equvilant JI minor 7th interval)
In actual practice real musicians in the real world typically can use the Pythagorean Comma & other microtonal comma oddities for expressive purposes . A fine example can be found in Sinead O'Connor's cover of the song "Nothing Compares 2 U," in which she uses another comma, the syntonic comma. The syntonic comma is equal to the frequency ratio 81:80, or around 21.51 cents (or 11/600ths of an octave). On a piano keyboard (typically tuned with 12-tone equal temperament) a stack of four fifths (700 * 4 = 2800 cents) is exactly equal to two octaves (1200 * 2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80), She uses this interval to great affect between the notes "to" and "you".