…are called “Hennin”. Thanks, Wikipedia*.
I first came across this particular headgear when I was a child listening to a Story Teller tape of The Faery Flag by Beryl Maude-Jones. The accompanying book was full of pictures of pointy cone-hats with long veils. The funny thing was I was scared by the story. So, in true child-like spirit, I decided it would be better if I recorded over this story by singing Happy Birthday onto the tape instead. I earnestly informed my mother that I would re-record the story when I was older and it no longer scared me (as if I could possibly recapture the talent of reader Annette Crosby (she of One Foot in the Grave fame)). Luckily my rendition of Happy Birthday was only long enough to record over the title of the story. And the last time I was able to listen to the cassette tape, as an adult, I laughed at my little five-year-old self singing to unscarify the audio. My husband now informs me that we no longer have any equipment capable of playing those old tapes.