What is the state called in which muscles...?

Question

are partially contracted to allow the body to remain in an upright position?

Answer 1

Prolonged contraction by muscles is called tetanus. Where the action potential is prolonged due to the absence of refractory periods brough on by frequent stimulus. In other words high frequency of stimulation prevents the relaxation of muscle cells, thus they remain contracted.

The degree or strength of contraction depends on the number of motor units (groups of muscle cells) are being stimulated. In other words, it depends on how many nerons are being firing action potentials.  

Answer 2

I think you are after what the normal state of skeletal muscle undergoing constant partial contractions to keep your head up and back and legs straight even when otherwise relaxed. This is known as a person's "muscle tone" but I'm not sure of it being referred to by a specific state name otherwise.

http://en.wikipedia.org/wiki/Muscle_tone

Answer 3

Tonus