A light twin and a single engine have a NMAC at Isla Grande.

Narrative

I was flying the Tango Transition through San Juan International (TJSJ) airspace. After crossing overhead I was told to squawk VFR and contact Isla Grange (TJIG) tower; my destination. Isla Grande Tower advised me of Aircraft Y in the pattern and told me to report Candado Plaza. The tower frequency became very congested. I tersely reported 'abeam Candado' but never got a response from the Tower. I saw Aircraft Y on crosswind. I intended to report a farther reporting point; the Carib Hilton; then the Capital Building; but the tower frequency was going non-stop. I had even slowed down considerably in order to maximize my time opportunity to talk to the tower. There is a pretty well defined ground reference point for traffic coming from the east to turn base. Just as I was arriving at this point; called El Morro; the tower advised that they were going to ground frequency. I hated to transmit when I knew that he was talking/listening on ground but did so nonetheless. I told him I was turning left base. As I turned base there was Aircraft Y right below me; as close as I've ever been to another airplane in the air. I took immediate evasive action to the right. As I rolled out and tried to find Aircraft Y again he was still right there! Not as close as before but still close. This time he was taking evasive action to his left. I resumed my evasive action to the right; determined to do a 360 this time. I was already well into the turn when the tower advised me to do a 360. When I rolled out I was farther downwind than normal. The tower told me to follow Aircraft Y; #2. I advised that I didn't see Aircraft Y. I had one plane far out at one o'clock and another single-engine; close in at three o'clock. The tower advised that the first plane was Aircraft Y. Upon landing the tower said; 'Thanks for your help'. I think he had no knowledge of the near mid-air. It was a bit hard at first to explain the circumstances that would account for these facts. The only thing that makes sense to me is that we were on converging flight paths. I saw him before he saw me. He therefore continued on his original flight path while I was making the evasive turn. He saw me when I was in the process of recovering so then he took evasive action. Aircraft Y had an instructor with presumably a student aboard. I know how easy it is to get distracted from outside scanning when you're busy with a student. If we were on converging flight paths at similar speeds; I would never see him until I lowered a wing.