Tilde spec - befuzzled

The documentation for caret and tilde specs is making my head hurt, even though they always _do_ exactly what I want. And I thought I understood them until I read more carefully.

   A suffix '{caret}' to a revision parameter means the first parent of
   that commit object.  '{caret}<n>' means the <n>th parent (i.e.
   '<rev>{caret}'

So far, so good.

'<rev>{tilde}<n>', e.g. 'master{tilde}3'::
   A suffix '{tilde}<n>' to a revision parameter means the commit
   object that is the <n>th generation grand-parent of the named
   commit object, following only the first parents.

Hang on, *grand*-parents?

So HEAD~1 won't give me the *parent* commit of HEAD, but the *grandparent* commit of HEAD (following only the first parents) ?

How do I get to the *parent* commit of HEAD? Does that mean that HEAD~ != HEAD^ And why does HEAD~1 always look exactly what I would naively call the 'parent' of HEAD?

I'm pretty sure I'm missing something very obvious. I think it must confuse other people as well though, as it's quite easy to find webpages around that claim that tildes give the _parent_ commit.

Thanks Luke