Vowels 10 and 12, /ʌ/ versus /ə/ 4 pairs [humhm.html]
Since the schwa has to be unstressed while /ʌ/ is likely to be prominent, only one set of pairs was found in the dictionary, and h'm is of doubtful status as a word that can enter meaningful contrasts. It is in any case often spelled hem or hum in narrative, as in "He hemmed and hawed in embarrassment."
Two other pairs were suggested by Ben Holmberg, upend/append and discuss/discus, but in both cases there is a stress difference as well as the vowel.
The density figure is too small to be worth calculating.