Any tuning that includes intervals smaller than a semitone is considered microtonal. This terminology can also be applied to scales with larger intervals that contain notes not found in the twelve-tone scale.
One of the most common examples of a microtonal scale is the quarter tone scale, due to its similarities to the twelve-tone scale. All this scale does is add another note in-between each semitone, creating a total of twenty-four tones.
Just using sharps and flats won't cut it for quarter tones if we want to keep using the regular staff. This scale requires notation for half sharps and flats, as well as a sharp and flat and a half. There are a few ways to notate this, one of the most popular of which can be seen below.
The 24-TET / quarter tone scale, going up and down, starting on C.
There are two main methods for tuning - equal temperament and just intonation. Equal temperament is by and large the most common form of tuning.
In equal temperament tuning, the frequency ratio between each note interval is the same. Because of the logarithmic way in which our ears perceive pitch, this allows each interval between notes to seem identical in distance.
Both tuning examples so far - twelve tone and twenty-four tone - have been equal temperament. These are often shortened to 12-TET and 24-TET. Sometimes the acronym EDO is used, which stands for Equal Division of the Octave. This is used to specify that intervals are divided within an octave, instead of some other arbitrary interval.
Just intonation is much less common in the modern world. This form of tuning generates frequency ratios between notes with relatively small numbers. This creates notes with very little tension, or dissonance, between them.