When is the 300 hit window?

In osu!, we gain different point values for hitting notes at different times, where 300 is closest to the actual beat within a small window as set by the difficulty, followed by 100 for being a bit out, and 50 a little further out.

Now, I'm wondering how does this match to the visuals. Is this hit window, before, around or after the approach circle touches the beat? So:

Assuming X is the 300 window, is it:

• X before the circle touches up to when the circle does touch,
• Between X/2 before and after the circle touches, or
• From when the circle touches to X after it does so.

I believe it's centred on precisely when the approach circle touches the hit circle or slider. There is a window on either side of that time where you can hit it and score 300. Without mods, this will range from 19.5 ms to 79.5 ms on either side. This gives a total window of 39 to 159 ms. A full chart is available on the osu! wiki.

It's actually possible to enable a view to show this more immediately in-game. In the options menu, set the Score meter type to Hit error, and a bar will appear at the bottom of your screen that reflects your most recent hits, relative to perfect timing.

Note that the precision required varies by the map settings (specifically, the OD or Overall Difficulty), and that sliders are more forgiving than hit circles in terms of timing precision.

After creating a test map and looking at the replays, the 300 hit window starts before the approach circle touches the beat.

Note that this is done on a map with OD0 and with EZ mod.