Increment rhymes with:
excrement, sacrament, detriment, excrement, experiment, instrument, sacrament, temperament, adamant, argument, armament, complement, condiment, detriment, diamant, disarmament, element, excrement, experiment, filament, impediment, implement, instrument, integument, ligament, ornament, orpiment, parliament, predicament, rearmament, regiment, sacrament, sediment, sentiment, supplement, temperament, tenement, testament, tournament, vehement
Increment sounds like:
immigrant, incarnate, incoherent, incriminate, ingrained