Oliver,
The general problem of N players and X rounds would be a very hard problem to automate well. The only easy solutions are the Whist designs when N is divisible by 4 and there are N-1 rounds (to see some of these follow the link above to "Schedules").
Moving on to your specific problem above, I would choose 15 rounds as this seems to me to offer the best opportunity for balance.
Rnd Table 1 Table 2 Bye
1 (0 1) v (6 8) (2 7) v (3 9) (4 5)
2 (2 3) v (8 0) (4 9) v (5 1) (6 7)
3 (4 5) v (0 2) (6 1) v (7 3) (8 9)
4 (6 7) v (2 4) (8 3) v (9 5) (0 1)
5 (8 9) v (4 6) (0 5) v (1 7) (2 3)
6 (1 2) v (7 9) (3 8) v (4 0) (5 6)
7 (3 4) v (9 1) (5 0) v (6 2) (7 8)
8 (5 6) v (1 3) (7 2) v (8 4) (9 0)
9 (7 8) v (3 5) (9 4) v (0 6) (1 2)
10 (9 0) v (5 7) (1 6) v (2 8) (3 4)
11 (3 6) v (4 7) (8 1) v (9 2) (5 0)
12 (0 3) v (1 4) (5 8) v (6 9) (2 7)
13 (2 5) v (3 6) (7 0) v (8 1) (4 9)
14 (4 7) v (5 8) (9 2) v (0 3) (6 1)
15 (1 4) v (2 5) (6 9) v (7 0) (3 8)
Above, I have used 0..9 to represent the 10 players. The schedule has the following properties.
Each player has one bye in each block of 5 rounds.
For any one player there is at least 2 rounds of play between byes.
All pairs of players partner each other at least once and at most twice.
All pairs of players oppose each other at least twice and at most three times.
All players sit at the same table with all other players (either as a partner or an opponent) exactly four times. This "social" balance is the reason why I chose 15 rounds in the first place.
Hope that's useful for the tourney your are planning,
Ian.
PS note all the repeated partnerships occur in rounds 11-15.
