carry a tune in a bucket

Net Forceから取り上げる今日の表現も初めて見るものですが、意味も文脈から簡単に推測できます。
"Has he been drinking?" Maj heard someone whisper.
She shook her head one last time. It didn't seem matter, for drunk or sober, Alain plainly couldn't carry a tune in a bucket. He apparently couldn't manage to carry his plate either. He dropped it, busy waving his arms.
"couldn't carry a tune in a bucket" は音程が狂う意味（つまり音痴）だと推測できますが、何故 "bucket" が出てくるのが不思議です。辞書に説明があるでしょうか？
・Collins Dictionary: to be able to sing in tune: A little-known fact about Hamm: She might carry her sport, but she can't carry a tune.
・McGraw-Hill Dictionary of American Idioms and Phrasal Verbs: can't carry a tune and cannot carry a tune; can't carry a tune in a bushel basket; can't carry a tune in a bucket; can't carry a tune in a paper sack: Fig. [to be] unable to sing a simple melody; lacking musical ability. I wish that Tom wouldn't try to sing. He can't carry a tune. I don't know why Mary's in the choir. She can't carry a tune in a bushel basket. Joe likes to sing in the shower, though he can't carry a tune in a bucket. I'd try to hum the song for you, but I can't carry a tune in a paper sack.
"bucket" が出てくる訳を説明している辞書は見つかりませんでしたが、最後に引用した辞書に "can't carry a tune in a paper sack" とも言うと書かれている事から、動詞に "carry" が使われていることから来た表現の様です。