Unfortunately there is nothing connected to the TV other than the HDMI to the DTV box. The same thing happened when I had nothing connected but an antenna, or nothing connected but my xbox... I think it is just how this TV is.
The high-pitched whine it makes when it is off is VERY similar to the whine traditional picture tube TVs make when they are on, but a bit louder. It happens constantly when the TV is turned off. Not all people can hear that noise... When I was a teenager my parents would always turn off the cable box but not the TV, so the TV was left on with a black screen. I could be upstairs with doors shut and could tell the basement TV was on because of the high-pitched sound. They couldn't hear it and I think they both thought I was nuts, but I was always right!
The noise it makes when it is on is a humming noise, like an electrical transformer or something. If I adjust the contrast manually, the sound changes in relation to the contrast level, so it is definitely something related to that. This noise is not constant. It comes and goes.
