Aristotle in the Poetics famously says that the effect of tragedy is catharsis, purification or purgation, particularly of pity and fear, but (also famously) does not give us much information about what this purification is. The word usually means a medical release of fluids. Some interpretations take this very seriously, and thus regard tragedy as being a healing venture, releasing poisonous passions; others think this is heavy-handed use of an obvious figure of speech. There are many other interpretations.
One of the things that I think hampers many interpretations of Aristotle on catharsis is a failure to take into account its social character. The Poetics is not the only place in which Aristotle talks about poetic catharsis. He also mentions it in the Politics, in which he actually refers to the Poetics. The passage (1341b-1342a) is worth quoting at length, but I will break it up a bit and add a few comments, always with the caveat that this is difficult and controversial material.
And since we accept the classification of melodies made by some philosophers, as ethical melodies, melodies of action, and passionate melodies, distributing the various harmonies among these classes as being in nature akin to one or the other, and as we say that music ought to be employed not for the purpose of one benefit that it confers but on account of several
Aristotle had previously noted three kinds of object for music, and raised the question of what it is suitable for:
(1) to relax and amuse, like sleeping and drinking and perhaps dancing, which are all things devoted to what is merely pleasant and removes care;
(2) to cultivate virtue, like athletics (gymnastics), since music affects character by accustoming us to feel rightly;
(3) to contribute to intellectual leisure and culture.
As he says here, he thinks that music can contribute effectively to all three of these, allowing people to occupy their leisure in noble ways. This is all part of a discussion of education for civic life, so Aristotle's concern is how music might aid the young in becoming good citizens and soldiers.
（for it serves the purpose both of education and of purgation—the term purgation we use for the present without explanation, but we will return to discuss the meaning that we give to it more explicitly in our treatise on poetry—and thirdly it serves for amusement, serving to relax our tension and to give rest from it）,
It is somewhat ironic that, despite the reference to the poetics, this is actually the closest thing we have in the Aristotelian corpus to an explanation of catharsis; the reference seems to be to the second book of the Poetics, which, of course, is lost. Keep an eye out for it if you visit any ancient monasteries; it would be one of the greatest finds of all time.
He had a little before this passage said that we should not teach the young the flute, because the flute does not teach young people to pay attention, and thus does not moderate but excite, and should only be used for purgation rather than education.
it is clear that we should employ all the harmonies, yet not employ them all in the same way, but use the most ethical ones for education, and the active and passionate kinds for listening to when others are performing （for any experience that occurs violently in some souls is found in all, though with different degrees of intensity—for example pity and fear, and also religious excitement; for some persons are very liable to this form of emotion, and under the influence of sacred music we see these people, when they use tunes that violently arouse the soul, being thrown into a state as if they had received medicinal treatment and taken a purge; the same experience then must come also to the compassionate and the timid and the other emotional people generally in such degree as befalls each individual of these classes, and all must undergo a purgation and a pleasant feeling of relief; and similarly also the purgative melodies afford harmless delight to people）.
So here we see that Aristotle himself seems to take the medical metaphor quite seriously. Passionate kinds of music violently arouse the soul, so that people who hear them are"thrown into a state as if they had received medicinal treatment and taken a purge" involving a "pleasant feeling of relief" and thus giving "harmless delight to people".
It turns out that he classifies the Dorian mode as an educational melody, because it is sedate and manly, and strongly disagrees with Plato's inclusion of the Phrygian mode as educational, because it is too exciting, as shown in its association with Dionysus. (I know nothing about music, but Wikipedia says that the modern Dorian mode is probably closest to the Greek Phrygian mode.) We seem to have no clear idea what was going on with ancient Greek modes, but here is an attempt to present something like what the ancient Greeks would have recognized as Dorian mode.
We don't get much clarity about musical education. But we do get some indication of the importance of catharsis as a social feature; it is so important that it needs to be considered as a matter of how to educate the young to be participants in their society. Hans-Georg Gadamer, I think, is probably on the right track when he says in Truth and Method that the key to tragic catharsis is that the experience is truly common (note Aristotle's explicit statement that "any experience that occurs violently in some souls is found in all, though with different degrees of intensity"), and thus we see ourselves in the tragic characters, and thus our own limitations and subjection to fate. The commonality of the experience seems quite crucial. We are all in some sense caught up together, and, releasing our passions in a civilized way, find relief, restoration, cure. This is very much the Neoplatonist view of the matter. In discussing the Mysteries, Iamblichus happens to comment that human passions, when exercised moderately, provide delight and are calmed, so that in tragedy and comedy we contemplate the passions of others and become more moderate in our own passions, thereby curing ourselves.
In any case, I think it's clear enough that any interpretation of catharsis has to recognize it as having a clear social character, and any interpretation that does not recognize this is probably wrong.