The relative quality of an interval is defined in terms of the ambient scale where it occurs, while its absolute quality is defined based on a fixed scale independent of the context where it occurs, usually the diatonic scale.
Quality can also be expanded to chords. Chord qualities are related to the qualities of the component intervals that define the chord.
Relative interval quality
Given a scale, a relative interval quality is a specific interval size that occurs in a given interval class of the scale. An interval class is the set of all intervals in the scale that span a given number of steps. For example, all intervals that span two steps of a scale are thirds or 2-steps. (The latter is often used to avoid confusion with absolute interval quality and to make interval arithmetic in unfamiliar scales easier.) Scales with a higher density of notes typically have smaller 2-steps; as a result, in a scale with more or fewer notes per octave than the diatonic scale, 2-steps may fall outside of the usual range for diatonic thirds (i.e. between 240 ¢ and 480 ¢).
In an equal scale, each interval class contains exactly one interval; in other words, every interval is perfect. Both 5\8 and 5\13 are perfect 5-steps (or perfect sixths) within their respective edo taken as a scale, even though they have significantly different sizes.
In moment of symmetry (MOS) scales, each interval class contains two intervals except for the unison class, which only contains the unison, and the period class, which only contains the period. The two interval classes that correspond to the bright and dark generators contain only perfect intervals except for one imperfect interval, which is either augmented or diminished depending on its size relative to the perfect generator. The other interval classes contain major and minor intervals.
Scales with higher interval variety have interval classes with more qualities. Although there are no standard labels yet, large, medium and small can be used for variety-3 interval classes.
The harmonic series taken as a scale theoretically contains infinitely many interval qualities for each interval class. For that reason, relative quality is rarely used in that context and other tools are used to describe the variety of intervals found in just intonation taken as a whole, such as absolute interval quality.
Absolute interval quality
Another way to generalize interval qualities from traditional theory is to define interval regions corresponding to the 12 tones of the chromatic scale. That way, similar intervals, such as 13/11 (~289 ¢), 3\12 (300 ¢), 6/5 (~316 ¢), and 3\11 (~327 ¢), can all be considered different flavours of minor thirds. These regions can further be subdivided into finer categories; for instance, smaller minor thirds could be qualified as "narrow minor thirds", "subminor thirds", etc.
Interval regions can also be defined using a different framework than the diatonic scale. However, the diatonic scale is often preferred because it offers a familiar point of reference for many musicians.