The article presents a critical edition, with a translation and an introduction, of the "refutation" of chapters 14-16 of Book III of Ptolemy's Harmonica, which were not handed down in the manuscript tradition. Chapters 14 and 15 were added in Byzantine times, the origin of chapter 16 is uncertain. The "refutation" was authored by the 14 th-century scholar and polemicist Barlaam of Seminara, one of the two leading characters in the Palamite and hesychast controversies. In this way, the last item of Barlaam's scientific writings can now be read in a critical edition. The present edition explains in detail the background of Barlaam's work, describes all of its manuscript witnesses, and reconstructs a stemma codicum.