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What is the word that describes a society or family-system where power / responsibility is shared equally among males and females?
Specifically I was wondering if it might end with -riarchal.
For example, "As an egalitarian, I don't believe in a matriarchal OR patriarchal system, but rather (blank)."
The combining form parri- is the gender independent equivalent to matri- and patri-, as in the word parricide.¹ So you could use the term parriarchy if you want a strict parallel with patriarchy and matriarchy.
One difficulty is that parriarchy is not in wide circulation. Parriarch is found in the English corpus by Google search, but most hits seem to be the result of text recognition errors (“rr” for “tr”).
Also, parriarchy’s neutral position between patriarchy and matriarchy is a bit weak, etymologically speaking. According to Online Etymology Dictionary and Vocabulary.com, the word patriarch comes from the Greek patria (clan), not the Latin pater (father). By contrast, the eighteenth century coining matriarch does come from the Latin mater (mother), and the prefix parri- probably is a cognate of the Latin parus (relative). This is not to deny that patriarchy implies a male ruler; patria is related to pater (father).
Ultimately there is no simple answer, because the correct term actually depends on who does hold the power. For example, the term egalitarian is often used of a society that is not patriarchal or matriarchal but is characterized by decentralized power, the term gerontocracy is used when power is in the hands of elders but independent of their gender, and so on.
