Advice on Bulding a Pergola

With a heavy structure (like a house) the foundations are there to stop it sinking into the ground. With a light weight pergola you are more concerned with the structure getting out of shape and falling over, especially in high winds. So you do need a bit of depth to stop the uprights from moving about. The more weight you put down low the better it will be, so in your clay soil I'd go down 18" at least and half fill with hard core and a very dry concrete mix, then build up above ground level by 6" with the concrete mix (using your cardboard formers or similar). That should help keep the uprights dry.

Shiney’s structure is a bit different to yours, it’s just resting on the paving, the structure stays in place due to its weight and retains it shape because of those reinforcing braces at the top.