Here's what rhymes with mediumhigh.
                        
                        
                    
                                high  thy  vi  hi  shy  sigh  fi  hy  thigh  vie  psi  hai  cy  fy  sy  thei  phi  by  my  why  die  lie  y  buy  fly  sky  guy  tie  bi  bye  chi  dye  nigh  pi  pie  spy  thai  dei  fry  ly  quae  rye  sly  ty  wir  awry  lye  mai  muy  nai  wry  kai  lai  luy  mei  tai  untie  wai  sty  bonsai  deify  amebae  stye  slier  apply  dry  try  cry  defy  ply  pry  tri  trie  pri  amoebae  aquae  spry  turfy  mummify  shanghai  banzais  supply  thereby  deny  imply  nearby  rely  comply  hereby  modify  notify  underlie  nullify  pacify  unify  belie  terrify  typify  dignify  edify  mollify  overlie  ramify  ossify  reunify  lanai  scurfy  versify  beatify  speechify  overbuy  yuppify  reply  justify  satisfy  whereby  nuclei  specify  signify  testify  verify  butterfly  certify  goodbye  purify  alibi  alveoli  amplify  magnify  ratify  samurai  beautify  decry  lullaby  pupae  termini  codify  descry  mortify  personify  resupply  vilify  acidify  dragonfly  firefly  mystify  retry  triply  ultrahigh  vivify  blowfly  horrify  misapply  petrify  stupefy  overfly  barfly  prettify  vitrify  drachmai  ripply  dandify  identify  occupy  clarify  multiply  qualify  alkali  classify  simplify  glorify  gratify  quantify  diversify  fortify  rectify  standby  falsify  indemnify  solidify  crucify  liquefy  demystify  stratify  stultify  calcify  emulsify  dissatisfy  liquify  scarify  gentrify  syllabify  solemnify  stimuli  intensify  objectify  oversupply  detoxify  electrify  preoccupy  reclassify  revivify  cumuli  declassify  hyperbolae  refortify  exemplify  disqualify  oversimplify  misidentify  transmogrify