Update the bios with a character remapping Razer Tartarus V2?

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You are correct in that it does not have any onboard memory. You do not, however, have to have Synapse running in the background. So it does have a “default” setting. This default setting is what I believe the OP is asking to change, since he is referring to “the bios”. Personally I do not think it is possible, but it would indeed have been good if it were. As Synapse 3 is a truly horrible piece of software, and the Tartarus actually works far better without Synapse than it does with Synapse. I have had soo many issue with the Tartarus and Synapse, all of which have been directly proven to be in Synapse itself, not that actual Device. So I stopped using it. If you use it for work, Rewasd is a good option, as it works far better than synapse, however quite a few games have banned it since it can be abused to gain advantages in gaming.