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Katharsis is the broader ritual process of which khernips is one common tool: where miasma names the state of impurity, katharsis names the act of resolving it. It could take many forms depending on the severity and source of the pollution - a simple hand-washing with khernips for everyday miasma, fumigation with sulphur or purifying herbs, ritual bathing in sea water or a river (sea water was considered especially potent for purification), or, in cases of serious pollution such as bloodguilt, formal rites of exile and ritual cleansing performed by a priest or specialist.
Katharsis wasn't only personal - it applied to spaces and objects too. A temple or altar could require ritual purification after contact with death or bloodshed before sacrifice could resume there; a sacrificial animal was itself purified with khernips before being offered, ensuring the offering itself was fit to approach the gods.
Beyond strict cult practice, the word had a wider life in Greek thought - most famously in Aristotle's use of katharsis in the Poetics to describe the emotional purging effect of tragic drama on an audience. That usage draws on the same underlying idea: a build-up of something (pity and fear, in Aristotle's case) that needs to be released and cleared for a state of balance to be restored.
For modern practitioners, katharsis is usually enacted as a simple pre-ritual habit - washing hands, briefly bathing, or even just a moment of conscious mental "clearing" - before prayer or offering, marking the shift from ordinary time into sacred practice.