Barthram rhymes with:
anagram, bartram, beltram, buttram, cablegram, centigram, centram, coram, diagram, diaphragm, electrocardiogram, engram, epigram, histogram, hologram, kilogram, logogram, mailgram, mammogram, microgram, milgram, milligram, monogram, pegram, polygram, program, programme, reprogram, sonogram, telegram, tungsram, wolfgram, wolfram
Barthram sounds like:
bartram, bertram, bertran, boardroom, brethren