It is said.... 

The following paragraph seems to have a logic error in it:

"It is said that, incensed at having been outplayed, the Sirens flung themselves down into the water and perished. Varying traditions associate this event with their encounters with Jason or Odysseus, though the incident appears in neither Homer's Odyssey nor Apollonius Rhodius's Argonautica."

Since no other music was played in the Odyssey, the Sirens must have thrown themselves into the sea for some other reason, such as having failed in their goal.