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1F801100h+N*10h - Timer 0..2 Current Counter Value (R/W)

  0-15  Current Counter value (incrementing)
  16-31 Garbage

This register is automatically incrementing. It is write-able (allowing to set it to any value). It gets forcefully reset to 0000h on any write to the Counter Mode register and when reaching counter overflow condition (either when reaching FFFFh, or when reaching the selected target value).
Writing a Current value larger than the Target value will not trigger the condition of Mode Bit4, but make the counter run until FFFFh and wrap around to 0000h once, before using the target value.

1F801104h+N*10h - Timer 0..2 Counter Mode (R/W)

  0     Synchronization Enable (0=Free Run, 1=Synchronize via Bit1-2)
  1-2   Synchronization Mode   (0-3, see lists below)
         Synchronization Modes for Counter 0:
           0 = Pause counter during Hblank(s)
           1 = Reset counter to 0000h at Hblank(s)
           2 = Reset counter to 0000h at Hblank(s) and pause outside of Hblank
           3 = Pause until Hblank occurs once, then switch to Free Run
         Synchronization Modes for Counter 1:
           Same as above, but using Vblank instead of Hblank
         Synchronization Modes for Counter 2:
           0 or 3 = Stop counter at current value (forever, no h/v-blank start)
           1 or 2 = Free Run (same as when Synchronization Disabled)
  3     Reset counter to 0000h  (0=After Counter=FFFFh, 1=After Counter=Target)
  4     IRQ when Counter=Target (0=Disable, 1=Enable)
  5     IRQ when Counter=FFFFh  (0=Disable, 1=Enable)
  6     IRQ Once/Repeat Mode    (0=One-shot, 1=Repeatedly)
  7     IRQ Pulse/Toggle Mode   (0=Short Bit10=0 Pulse, 1=Toggle Bit10 on/off)
  8-9   Clock Source (0-3, see list below)
         Counter 0:  0 or 2 = System Clock,  1 or 3 = Dotclock
         Counter 1:  0 or 2 = System Clock,  1 or 3 = Hblank
         Counter 2:  0 or 1 = System Clock,  2 or 3 = System Clock/8
  10    Interrupt Request       (0=Yes, 1=No) (Set after Writing)    (W=1) (R)
  11    Reached Target Value    (0=No, 1=Yes) (Reset after Reading)        (R)
  12    Reached FFFFh Value     (0=No, 1=Yes) (Reset after Reading)        (R)
  13-15 Unknown (seems to be always zero)
  16-31 Garbage (next opcode)

In one-shot mode, the IRQ is pulsed/toggled only once (one-shot mode doesn't stop the counter, it just suppresses any further IRQs until a new write to the Mode register occurs; if both IRQ conditions are enabled in Bit4-5, then one-shot mode triggers only one of those conditions; whichever occurs first).
Normally, Pulse mode should be used (Bit10 is permanently set, except for a few clock cycles when an IRQ occurs). In Toggle mode, Bit10 is set after writing to the Mode register, and becomes inverted on each IRQ (in one-shot mode, it remains zero after the IRQ) (in repeat mode it inverts Bit10 on each IRQ, so IRQ4/5/6 are triggered only each 2nd time, ie. when Bit10 changes from 1 to 0).
The "free run" mode is simply saying that the counter will not reset at a given threshold value.

1F801108h+N*10h - Timer 0..2 Counter Target Value (R/W)

  0-15  Counter Target value
  16-31 Garbage

When the Target flag is set (Bit3 of the Control register), the counter increments up to (including) the selected target value, and does then restart at 0000h.


For more info on dotclock and hblank timings, see:
GPU Timings
Caution: Reading the Current Counter Value can be a little unstable (when using dotclk or hblank as clock source); the GPU clock isn't in sync with the CPU clock, so the timer may get changed during the CPU read cycle. As a workaround: repeat reading the timer until the received value is the same (or slightly bigger) than the previous value.

Reset and Wrap

When resetting the Counter by writing the Mode register, it will stay at 0000h for 2 clock cycles before counting up.
When writing the Current value, it will stay at the written value for 2 clock cycles before counting up or checking against Target overflows.
When wrapping around at FFFFh(Mode Bit3 not set), it will stay at 0000h for only 1 clock cycle.
When being reset to 0000h by reaching the Target value(Mode Bit3 set), it will stay at 0000h for 2 clock cycles.
Example behavior with Target Value of 0001h and Mode Bit3 set:

clock cycle 0 - Counter Value = 0000h
clock cycle 1 - Counter Value = 0000h
clock cycle 2 - Counter Value = 0001h
clock cycle 3 - Counter Value = 0000h
clock cycle 4 - Counter Value = 0000h
clock cycle 5 - Counter Value = 0001h