Nice! Singer 27 the bobbin winder is located low, Singer 127 bobbin winder is located high. Singer 127 has a shutter ejector while the 27 doesn't.
http://en.wikipedia.org/wiki/Singer_Model_27_and_127
You have a straight-leg treadle for added info.
And you should be able to treadle your Redeye without a spoke wheel. I treadle a Singer 201 that doesn't have a spoke wheel.