BE36 pilot reported a near mid-air collision in the traffic pattern when they mistook an aircraft on the runway for the aircraft they were supposed to follow and turned base in front of the aircraft they should have followed. Tower controller vectored both aircraft away from each other.

Narrative

Flying VFR from ZZZ1 to ZZZ at 1700 ft.; obtained clearance from ZZZ tower to enter right traffic for Runway XXL when I was northeast of airport by approximately 4 miles. Entered a standard right downwind leg for XXL and told tower. I was then slightly north of the north end of XXL. Given clearance to land following a Cessna on base leg for the same runway. Continued my prelanding sequence by lowering gear; flaps; and slowed to 110 mph. Looked for other aircraft but did not see him. Proceeded to a 45 degree offset and began my base leg after noticing an aircraft on XXL doing what looked like a touch and go and I assumed due to timing that it was the aircraft ahead of me. Normal timing for an aircraft on a base leg would give them plenty of time to land if the aircraft behind them is just entering a downwind leg unless that plane used an unusually extended downwind prior to entering the base leg. Since I was told only that the Cessna was on a base leg when I entered the downwind; I assumed it was a normal pattern and that by me slowing to my pattern speed that a normal pattern for me would put me in the correct position well behind the Cessna. I probably had a TA and RA on my TCAS; but the fact that it has no aural tone and that I was in a place that required my eyes outside of the cockpit; I had no idea the other plane was that close. The Cessna was a training flight with an instructor and equipped with a TCAS with visual and aural tones that would have been alerting the Cessna of my presence. When the tower noticed both of use; he had the Cessna turn left and do a 360 while telling me to proceed with my landing. Once landed I was told to call the tower and was told that they would pull the tapes and look at the circumstances for this near miss. If they felt they needed more information they would call me and the tower controller told me that she would call within the next hour if she felt I needed to file a report or if they felt I was at fault. They did not call.I think I should have made certain with the tower that the aircraft on the ground was indeed the one that was ahead of me prior to turning base; but I also felt that since the Cessna was on base when I entered downwind that they would be well clear of the pattern by the time I turned a standard base. The tower should have mentioned that they were on an extended downwind base when I was cleared so I was made aware of the non-standard pattern. And the training Cessna should have seen me and been notified by TCAS that I was there. So I think that we were both at fault and the controller was also part of the problem. I also think it should be mandatory that TCAS systems are aural. It is too easy to miss an RA when in a busy pattern.