Originally depicted a person carrying an axe while singing. Now a phonosemantic compound. 欠 (pictograph of a person opening their mouth) represents the meaning and 哥 represents the sound.
Shuowen Jiezi
《說文》:“歌,詠也。从欠,哥聲。謌,謌(歌)或从言。”
Components
哥(sound) — Originally depicted an axe being carried. Later reanalyzed into phonetic component 哥.