It needs the input signal to track from, but that original signal does not need to be sent to the audio output. In other words, yes, you can simply have the octave down sounding on its own. In fact, many players do this, the classic example being the OC-2 as its octave down sound is quite synthy-sounding.
Having said that, having some of the dry signal mixed can give more definition to the sound. If you dial in a subtle amount, you can get it to sound more like overtone colouration than an actual note. It sounds fat.