Why is a rooster sound spelled cock-a-doodle-doo?

As a kid I often wondered why the onomatopoeia for a rooster call seems to have zero relation to how it actually sounds. An actual rooster call sounds like "er, er-er, er-er", so why is it described as "cock-a-doodle-doo"? The answer is unlike many animal calls, the defining attribute of a rooster call is more in its rhythm than in its timbre. They needed a way to capture this rhythmic aspect, and using commas or dashes to denote rhythm would've rendered the onomatopoeia rather inelegant and hard to remember. What's interesting about "cock-a-doodle-doo" is that it is practically impossible to say it without the swing-like rhythm which was intended by its inventors. English speakers are then meant to infer that this combination of words represents only the rhythm of a rooster call rather than its actual sound.