Music that uses tuning systems other than 12-tone equal temperament (12-TET) is often referred to as "microtonal". 12-TET arose as an approximation of Just Intonation so that it was possible to modulate to any key. However, it's not a very close approximation. Other tuning systems, often called "microtonal" systems, provide better approximations.
A very good explanation of the mathematical difference between 12-TET and Just Intonation can be found in Difference between equal temperament and just intonation. A Just Major 3rd and a 12-Tet Major 3rd are close mathematically, but still audibly different. A "microtonal Major 3rd" might be equal to, or at least closer to, a Just Major 3rd.
Many instances of this type of microtonality can be found on this site by searching the microtonality tag. One in particular that might be of use: