What written system was cuneiform under?

Cuneiform wasn't "under" a written system in the way that, say, the Roman alphabet is under the Latin script. Cuneiform is the written system itself. It's a system of writing that uses wedge-shaped marks, hence the name "cuneiform" (from the Latin word for "wedge").

It was used for many languages, notably Sumerian, Akkadian, Babylonian, Assyrian, Elamite, and Hittite, and spanned over 3,000 years of history.