From Japan, Masaki Taniguchi writes
I have been thinking about the teasing melody for a week. I have traced my memory of childhood. Now I clearly remember a teasing melody that my friends and I used to use in the early 1960's, which went |so so mi mi | so mi mi |. I also remember two versions of lyrics to go with this melody…
I have just contacted my childhood friend who used to live in the same town in Nagasaki Prefecture. He says that he vaguely remembers such a melody. He also says that today bullying is considered very inhumane and probably such a song has disappeared.
Nobody else that I have asked knows or remembers such a melody. I have asked a professor of music, and he recommended another who, he said, would be knowledgeable in such matters. I asked her, but she said she knew nothing of the sort, except for /ja:i ja:i/ (pitch: HL HL). This, I am sure, is known to all Japanese. It is certainly used for teasing.
I have also conducted a small survey. In a class with 12 students, I sang the melody you introduced in your blog, using the sounds used there, and told them that it was used among English children. I asked them what kind of impression they perceived and what kind of feeling they thought children would mean to express. Out of the 12, seven said it sounded "lively", four said, "teasing", and one said it sounded like a "lullaby". After that, I showed them your blog and we discussed it.
I think this negative evidence is important, because it tends to show that the claim of universality is false.
Meanwhile, from England, Jill House confirms that the phenomenon is extensively discussed in Mark Liberman’s 1978 dissertation ‘The intonational system of English’ (Indiana University Linguistics Club). She also points out that the tune underlies not only ‘Bye baby bunting’ but also the song for the children’s game ‘Ring-a-ring-a-roses’. That’s true for the version I sang as a child, but — as Wikipedia reveals — there are or were other variants of the tune which differ.
Anyhow, thanks to all correspondents.