A person who uses alliteration frequently or skillfully, or a scholar who studies alliteration.
From 'alliteration' plus the noun suffix '-ist,' meaning 'one who practices or specializes in.' This rare term creates a professional identity for those devoted to alliterative language.
While this is a rare term, poets like Gerard Manley Hopkins who obsessed over the musical qualities of repeated sounds could definitely be called alliterationists—he used alliteration so intensely that some of his verses feel almost like tongue twisters!
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