I think it likely that you will need 11 series to get a balanced schedule and that the optimal schedule you are looking for with 6 series may well be impossible (but I have not proof of that). With 11 series you can have all 11 partners exactly twice, and all 11 opponents exactly three times.
(11 0 1 v 2 4 7) ( 3 8 10 v 5 6 9)
(11 1 2 v 3 5 8) ( 4 9 0 v 6 7 10)
(11 2 3 v 4 6 9) ( 5 10 1 v 7 8 0)
(11 3 4 v 5 7 10) ( 6 0 2 v 8 9 1)
(11 4 5 v 6 8 0) ( 7 1 3 v 9 10 2)
(11 5 6 v 7 9 1) ( 8 2 4 v 10 0 3)
(11 6 7 v 8 10 2) ( 9 3 5 v 0 1 4)
(11 7 8 v 9 0 3) (10 4 6 v 1 2 5)
(11 8 9 v 10 1 4) ( 0 5 7 v 2 3 6)
(11 9 10 v 0 2 5) ( 1 6 8 v 3 4 7)
(11 10 0 v 1 3 6) ( 2 7 9 v 4 5 8)
The schedule above is Example 18.2 from page 9 of this paper by Abel et al on Generalized Whist Designs (http://www.sciencedirect.com/science/article/pii/S0012365X02007434) which is freely downloadable. I have used 0 to 11 for the players so you can see more easily how the first series given in the paper is translated into the full schedule.
Hope that helps.
Ian.
Here are some more balanced schedules for tournaments with games of 3 vs 3. They all have an odd number of players, with the same number of rounds as players. In each round there is one player with a bye (player 0 is the bye in the first round, player 1 is the bye in the 2nd round, etc..)
In general 3 vs 3 schedules will be possible for 6n players (like the 12 player schedule above), or 6n+1 players. Unfortunately no such schedule exists for 6 players and the existence of the 18 player schedule is an open problem.
I have sent you an e-mail regarding software/algorithms.
7 players
( 1 2 4 v 3 5 6)
( 2 3 5 v 4 6 0)
( 3 4 6 v 5 0 1)
( 4 5 0 v 6 1 2)
( 5 6 1 v 0 2 3)
( 6 0 2 v 1 3 4)
( 0 1 3 v 2 4 5)
13 players
(12 4 1 v 3 10 9) ( 2 7 11 v 6 8 5)
( 0 5 2 v 4 11 10) ( 3 8 12 v 7 9 6)
( 1 6 3 v 5 12 11) ( 4 9 0 v 8 10 7)
( 2 7 4 v 6 0 12) ( 5 10 1 v 9 11 8)
( 3 8 5 v 7 1 0) ( 6 11 2 v 10 12 9)
( 4 9 6 v 8 2 1) ( 7 12 3 v 11 0 10)
( 5 10 7 v 9 3 2) ( 8 0 4 v 12 1 11)
( 6 11 8 v 10 4 3) ( 9 1 5 v 0 2 12)
( 7 12 9 v 11 5 4) (10 2 6 v 1 3 0)
( 8 0 10 v 12 6 5) (11 3 7 v 2 4 1)
( 9 1 11 v 0 7 6) (12 4 8 v 3 5 2)
(10 2 12 v 1 8 7) ( 0 5 9 v 4 6 3)
(11 3 0 v 2 9 8) ( 1 6 10 v 5 7 4)
19 players
( 2 5 6 v 1 18 10) (11 15 8 v 3 9 17) (12 7 13 v 14 4 16)
( 3 6 7 v 2 0 11) (12 16 9 v 4 10 18) (13 8 14 v 15 5 17)
( 4 7 8 v 3 1 12) (13 17 10 v 5 11 0) (14 9 15 v 16 6 18)
( 5 8 9 v 4 2 13) (14 18 11 v 6 12 1) (15 10 16 v 17 7 0)
( 6 9 10 v 5 3 14) (15 0 12 v 7 13 2) (16 11 17 v 18 8 1)
( 7 10 11 v 6 4 15) (16 1 13 v 8 14 3) (17 12 18 v 0 9 2)
( 8 11 12 v 7 5 16) (17 2 14 v 9 15 4) (18 13 0 v 1 10 3)
( 9 12 13 v 8 6 17) (18 3 15 v 10 16 5) ( 0 14 1 v 2 11 4)
(10 13 14 v 9 7 18) ( 0 4 16 v 11 17 6) ( 1 15 2 v 3 12 5)
(11 14 15 v 10 8 0) ( 1 5 17 v 12 18 7) ( 2 16 3 v 4 13 6)
(12 15 16 v 11 9 1) ( 2 6 18 v 13 0 8) ( 3 17 4 v 5 14 7)
(13 16 17 v 12 10 2) ( 3 7 0 v 14 1 9) ( 4 18 5 v 6 15 8)
(14 17 18 v 13 11 3) ( 4 8 1 v 15 2 10) ( 5 0 6 v 7 16 9)
(15 18 0 v 14 12 4) ( 5 9 2 v 16 3 11) ( 6 1 7 v 8 17 10)
(16 0 1 v 15 13 5) ( 6 10 3 v 17 4 12) ( 7 2 8 v 9 18 11)
(17 1 2 v 16 14 6) ( 7 11 4 v 18 5 13) ( 8 3 9 v 10 0 12)
(18 2 3 v 17 15 7) ( 8 12 5 v 0 6 14) ( 9 4 10 v 11 1 13)
( 0 3 4 v 18 16 8) ( 9 13 6 v 1 7 15) (10 5 11 v 12 2 14)
( 1 4 5 v 0 17 9) (10 14 7 v 2 8 16) (11 6 12 v 13 3 15)