ComposeList not building the expected list of compositions

Question

Starting with:

In[79]:= a[z_] := z + 1/2
r[z_] := 1/z

tuples = Tuples[{a, r}, 3]

Out[81]= {{a, a, a}, {a, a, r}, {a, r, a}, {a, r, r}, {r, a, a}, {r, 
  a, r}, {r, r, a}, {r, r, r}}

I want to get a ComposeList[] of each of the element ordered triples over x, ending with a list of list of the compositions, e.g. {{x,a[x],a[a[x]],a[a[a[x]]]},...

ComposeList[##, x] & @@@ tuples does not work. Why, and how should this be written?

That gives me the (erroneous) result:

{ComposeList[a, a, a, 3/2], ComposeList[a, a, r, 3/2], 
 ComposeList[a, r, a, 3/2], ComposeList[a, r, r, 3/2], 
 ComposeList[r, a, a, 3/2], ComposeList[r, a, r, 3/2], 
 ComposeList[r, r, a, 3/2], ComposeList[r, r, r, 3/2]}