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A quadrogy is a one more then a trilogy. It is usually meant to justify the 4th pillar or central importance to a particular work of art.
"If this record had a trilogy of songs, it would actually have a quadrogy. This song would be the first pillar in the quadrogy."

-The Juliana Theory on Deadbeat Sweetheartbeat
by Variations of a Dream September 14, 2005