The way I imagine it, the magical version would be a ring, kind of like the rings on a baby stacking toy. Can I cast that whole note (ring) around someone's wrist and it acts like a manacle?
Yes, you can make a solid note shaped like a whole note. And yes, you can put it around someone's wrist like a manacle, though it won't necessarily be a very good manacle.
Here's an interesting observation: You're assuming that a solid note takes the shape of the symbol for a note in modern staff notation. There is no reason to believe this is the case. Here's the relevant text from the spell description you linked:
The exact appearance of the solid note depends on your melody. You can make it spiked or smooth, simple or convoluted, and with any color pattern you wish, however, it always has a size approximately equal to that of your outstretched hand.
Nothing in there says it has a shape based on modern staff notation, or indeed, on any form of musical notation; All the spell specifies is that you give a musical note a solid form of your choosing. You could shape it like a leering face if you wanted. (I'm personally imagining it'd appear as a sort of neon line drawing that constantly wobbles and vibrates.)
There is, theoretically speaking, nothing preventing you from placing a solid note around someone's hand - but since the cited uses of the spell make it seem like the spell isn't usually used offensively, I suspect many GMs would allow a victim a saving throw to avoid being manacled as the note takes shape.
Even if you do manacle someone, note that it's relatively easy to break out of at low caster levels; It disappears if successfully shoved or struck, and the AC and strength score it uses to resist such attempts start at 11. This means that there's more ways to get out of solid note than standard manacles. The fact that it's anchored in space is a nice feature that normal manacles don't have, though.
The physical form is not of a part of Western-Earth musical notation. As the spell says, it can be of any form you can imagine, so long as it is no larger than your hand.
So yes, it can be a ring. It won't be very effective as manacles until your caster level is higher though, since an opposed strength check against Str 10 + CL can destroy it, but you can do that.