Disharmony is a Greek word with a Latin prefix meaning “absence of harmony” or “bad harmony”. So why not spell it dysharmony, as one spells dysfunction or dyspepsia?
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Dis- is a Latin prefix, whereas dys- is a Greek one. Much of English stems from Latin, so it certainly makes sense for dis- to be the more common prefix, though you are correct that dys- as a prefix has a meaning similar to dis-.
Dys- is often used in a scientific context (where Greek gets a fair amount of use in addition to Latin) for terms like dysgraphia and dyslexia. Dys- tends to be used more to mean “bad, improper” rather than “the opposite of”, as you'll note from my examples; dysgraphia is not the opposite of writing, but rather means that one’s writing is impaired.