I think 19 rounds would be the most sensible as this will give everyone exactly 16 games (with 16 different partners) and three byes.
(17 19 v 5 4) ( 6 9 v 15 10) (18 11 v 16 8) (13 3 v 14 1) (12 2 7)
(18 1 v 6 5) ( 7 10 v 16 11) (19 12 v 17 9) (14 4 v 15 2) (13 3 8)
(19 2 v 7 6) ( 8 11 v 17 12) ( 1 13 v 18 10) (15 5 v 16 3) (14 4 9)
( 1 3 v 8 7) ( 9 12 v 18 13) ( 2 14 v 19 11) (16 6 v 17 4) (15 5 10)
( 2 4 v 9 8) (10 13 v 19 14) ( 3 15 v 1 12) (17 7 v 18 5) (16 6 11)
( 3 5 v 10 9) (11 14 v 1 15) ( 4 16 v 2 13) (18 8 v 19 6) (17 7 12)
( 4 6 v 11 10) (12 15 v 2 16) ( 5 17 v 3 14) (19 9 v 1 7) (18 8 13)
( 5 7 v 12 11) (13 16 v 3 17) ( 6 18 v 4 15) ( 1 10 v 2 8) (19 9 14)
( 6 8 v 13 12) (14 17 v 4 18) ( 7 19 v 5 16) ( 2 11 v 3 9) ( 1 10 15)
( 7 9 v 14 13) (15 18 v 5 19) ( 8 1 v 6 17) ( 3 12 v 4 10) ( 2 11 16)
( 8 10 v 15 14) (16 19 v 6 1) ( 9 2 v 7 18) ( 4 13 v 5 11) ( 3 12 17)
( 9 11 v 16 15) (17 1 v 7 2) (10 3 v 8 19) ( 5 14 v 6 12) ( 4 13 18)
(10 12 v 17 16) (18 2 v 8 3) (11 4 v 9 1) ( 6 15 v 7 13) ( 5 14 19)
(11 13 v 18 17) (19 3 v 9 4) (12 5 v 10 2) ( 7 16 v 8 14) ( 6 15 1)
(12 14 v 19 18) ( 1 4 v 10 5) (13 6 v 11 3) ( 8 17 v 9 15) ( 7 16 2)
(13 15 v 1 19) ( 2 5 v 11 6) (14 7 v 12 4) ( 9 18 v 10 16) ( 8 17 3)
(14 16 v 2 1) ( 3 6 v 12 7) (15 8 v 13 5) (10 19 v 11 17) ( 9 18 4)
(15 17 v 3 2) ( 4 7 v 13 8) (16 9 v 14 6) (11 1 v 12 18) (10 19 5)
(16 18 v 4 3) ( 5 8 v 14 9) (17 10 v 15 7) (12 2 v 13 19) (11 1 6)
