Actually, a 12AWG, 6-circuit multicable rated at 90 degrees C with 12 current-carrying conductors is sized per NEC table 520.44. While each circuit ends up with a rating of 24.5A, there are two conditions required:Not very much!
16 gauge cable is usually used for audio, rather than lighting. Yes, the connectors fit, but....
12 gauge multicable is usually rated at 2k, 14 gauge at 1k; I think you'd be lucky to pull 300w off each circuit. The capacity of each conductor drops when it's bundled all together, as opposed to run in free air. At that distance, also, voltage drop along the cable will knock you down to 110v or so output.
You're better off checking with the manufacturer, though; most manufacturers carry a listing on their website.
Almost all modern gear in the US will run on 120-240v(if does not burn tungsten), . This can mean running 12 soca cables instead of 24, so there is a benefit to needing less power distribution, less copper,less deck space space, cleaner cable management, and a faster loadin/out.Just how do you avoid 120 volt gear in North America?
To actually answer the original question, I think 16ga is generally used up to 10 amps, so 30A total at any time across all 6 circuits would be the limit. Depending on the length, you may need to lower this a bit.
You don't just send off a 1000 foot roll of 16/37 to be recycled! You have to have a funeral for it firstIf of help, this year I finally parted 10 years later with a 1,000' spol of 16/37 SO cable. Was to use it on some Pantographs - and it will have been hard to do but the project demanded it. And the project got cancelled after started. Sat on the spool for ten years... to scrap in no longer taking up space.