Correlations among responses to a forced-choice scale

I have data from participants who rate emotional content of pictures. Specifically, people see 20 pictures of abstract paintings (one at a time) and are asked, "what emotions is expressed in the picture?" There are 5 response options: anger, fear, sadness, happiness, and none. The key info here is that for each picture people choose only one of the emotion choices. I then add number of pictures rated as expressing anger, fear, sadness etc. and have 5 continuous variables. The 5 continuous variable are obviously not independent because if someone rated X pictures as expressing anger and Y as expressing happiness, then they could not choose more than 20-(X+Y) pictures as expressing fear or sadness; hence the variables have negative correlations. My question now is, how can I examine correlations among these responses (emotions) if they are not independent of one another?