To be honest I don't do mine in the tmx. I'm making it using a 'continuous' system. I have a large glass drinks dispenser with a spigot (tap, in case you like me didn't know what a spigot was
). If you do use is system make sure your container isn't plastic as the GBP has a go at processing that along with the ginger, etc and releases chemical into your brew. Also, as I'm sure is obvious, the lid shouldn't be airtight.
I started off by doing a normal mix similar to the one you describe. After a couple of days I added more lumps of peeled ginger direct, put in some sugar, a squeeze of lemon and some filtered water. I just top it up this way intermittently. I don't measure (although I probably should). I put in a mug of sugar and a filter jug full of water, roughly, plus squeeze in half a lemon. I wondered about just dropping in the lemon, but thought it might eventually become bitter from the pith?
Every couple of top ups I add a tsp of cream of tartar. I'm thinking of adding the clean shell from a boiled egg to add calcium - good for us and the GBP
Every 3 months or so I drain it off, reserve the GBP (which I have contained in a muslin bag for ease), sterilise the container, and then start again.
I used to do a green tea version rather than a ginger version (same recipe just skip ginger and use green tea instead of water) which was lovely, but I make that with a kombucha scoby now to vary the health benefits.