Disclaimer...I do not know the answer to your question.
Here are some observations: If it is a Mutron III, it was originally designed to run on 18V (two 9 volt batteries). Do you know how it was converted to 9 V? If there is some internal bump up to 18V, then I would not try to run it at 12V. If there is not...it may not be running at the right voltage. But see the first sentence of this post...
If it is modded with a 9v to 9v+/9v- converter it already has a lot of headroom, it was designed to have that. Those voltage converter-IC's are very sensitive and could easily break if you put 12v on it.
Yes it is a Mutron III. Modded by shad damron to accept boss type 9v plug adaptor. Thanks for responses. Why wouldn't it be like some other pedals (Analogman, Timmy) where you can try 12v to see if it sounds better or gives more headroom...is it because it's an old pedal? Thanks
Because they already took high-headroom into account when they designed it, so there is really no need trying a higher voltage. It runs at -9v/+9v which already makes 18v. I have just worked on a Mu-Tron III that also had the 9v adapter mod. This was done by using an LT1054, which can take up to 15V. However the often used MAX1044 voltage converter will fry when the voltage goes higher than 10V, so make sure which chip is in yours.
What I forgot to mention, even if the voltage converter could handle the higher voltage it would also increase the voltage going to the photocoupler, altering the sweep-response and perhaps even damage it, without increasing the current limiter resistor.