Ok, before getting into a technical discussion of POV, I think the main issue here is actually not POV.
The main issue is, how many major characters can you have without confusing the reader or dissipating the story's momentum? Regardless of point of view, the only way it could work to draw so much of the reader's attention to
seven different characters is:
(1) if there are seven different, closely connected storylines. There need to be seven; if you've only got six storylines, you simply don't need that seventh major character. He can either disappear or be relegated to a bit part or supporting role in one of the six storylines. Or...
(2) if there's one storyline in which all characters are involved
and some of those seven characters are antagonists, so your main characters can be grouped into protagonists and antagonists, and it makes sense in the story that there would be teams of protagonists and antagonists...
The Da Vinci Code is a good example of this. (Not saying it's a great book, but it is a good example of "teams" of protagonists and antagonists. Ditto
Lord of the Rings, about which more below.)
(3) if you're trying to write something like
The Seven Samurai in which the issue of point of view IS a big theme in your book. If your
point is "different people can experience and remember the same events in completely different and contradictory ways," then sure, illustrate that by having a bunch of characters go through the same experience and remember it differently.
But basically...
even The Lord of the Rings doesn't have seven main characters who each get chapters where they're the main focus. Even the
Lord of the Rings, which is a three-volume epic! If you count each of the three main hobbits as a major character you could hit seven, but really, Samwise and Merry have no stories of their own; they're just there as Frodo's friends, participating in his journey--i.e. in his story. They're not
main characters. The MC's are Frodo, Gollum, Gandalf, Lord Sauron, maybe Aragorn,
maybe Legolas (but that's a stretch). You have to throw in the dwarf Gimli to hit seven, and I can't see any rationale for calling Gimli a major or main character. And even counting this way, you still won't find one chapter per each of those characters; it would be confusing to go off into Gimli-land for a chapter, then off into Aragorn-land, etc.
So if even a sprawling three-volume fantasy epic with easily a hundred characters in it (if you count each elf, hobbit etc. that's mentioned by name), if even that kind of book sticks to a handful of main characters and would likely be confusing or unfocused if it didn't, then you have to wonder, how on earth can the OP's book work with seven main characters?
Actually, ideagirl, 7 POV's does not make omniscient. Omniscient is very distant and not as tight into a person's head as a third person limited.
Omniscient (a.k.a. "third person omniscient") CAN be distant and not in a person's head--and it's the only POV that can do that--but it's not necessarily distant. By definition, an omniscient narrator can "get into" any character's head, whereas a third-person limited narrator can only get into the head of one character. So what's going on here is either seven different third-person limited POV's, or one third-person omniscient POV that shifts its focus between seven different characters.
Basically, "omniscient," "first person," or "third-person limited" tells you the POSSIBLE scope of the narration; it tells you where the limits are, but it doesn't tell you exactly what the narrator does within those limits. An omniscient narrator can discuss absolutely anything, from the innermost feelings of a character to the formation of galaxies 10 billion years before the character's birth... anything. But just as you don't use all the words in the English language every time you write a book, an omniscient narrator doesn't do in one single book EVERYTHING that an omniscient character COULD do. An omniscient narrator is free to focus on just one character, and limit itself so much that you could almost call it third-person limited--but if that narrator, even just once, talks about something that is not or could not be in the consciousness of that character, then it's an omniscient narrator.
Third-person limited is limited to the awareness of the focus character; it's like the narrator is sitting on that character's shoulder or in their head--they CAN'T pull back to some other character's consciousness or to things outside the knowledge of the focus character. Third-person omniscient can be distant, sure, but it can also feel to the reader almost the same as third-person limited, with this difference: it CAN move around if it wants to, and it can talk about things outside the focus character's knowledge.
Here's an example--the Golden Compass trilogy, by Philip Pullman. The first book focuses on Lyra, while the second focuses on the boy who will eventually become her friend/comrade/etc., and as I recall in the second book there are also some chapters that switch back to focusing on Lyra. Both books are VERY focused on their main characters (Lyra, and then the boy and Lyra) in almost every scene... but not quite every scene. There are scenes in each book where the focus character does not appear. The narrator describes a few things that the focus character is not aware of. The books are therefore third-person omniscient, but with a very strong focus on those two characters.
If we want to get technical, here's the clincher: the OP said the narrator spends each chapter focused on one of the seven characters. Given that premise, if at any point in the book the narrator mentions ANYTHING that the chapter's "focus character" doesn't know and is never going to know, then what you have is an omniscient narrator. (When I say "is never going to know," I mean that, for example, if the MC is in the hospital, a 3d person limited narrator could write, "It was months before he discovered that while he was in the hospital, his best friend was emptying his bank account and going on a shopping spree in Rio." But a 3d-person limited narrator could not write about something that the MC never finds out about.)