This simple question seems to have a complicated answer. The most common answer -- at least among experts on alt.music.jewish -- is Moshe Nathanson.
But Barry Cohon has weighed in with a strong counterargument for Abraham Zevi Idelsohn.
Since Idelsohn and Yudelson could well have been the same name in the Old Country, my vote is for Idelsohn, at least until I'm able to research the matter for myself.
Meanwhile, here is Cohon's history of Havah Nagilah:
The man largely responsible for the song's existence in its present form is Abraham Zevi Idelsohn, and he was the father of Jewish Musicology.
As a young cantor, he left his native Latvia, worked in Germany and South Africa, then went to Jerusalem early in this century to pursue his dream of collecting the oral traditions of his people and making them available to the world of music.
In the course of his research he visited a group of Sadigura Hasidim there, in 1915, and wrote down some of their Nigunim. This was one of them. It was a wordless "bim-bom" melody, a mystic chant.
Then came World War I. Idelsohn became a bandmaster in the Turkish Army.
Three years later he was back in Jerusalem again, leading a chorus in a victory concert. The Turks were out, the British were in, there was a Balfour Declaration, and the yishuv (Jewish community) was celebrating. He needed a good crowd-pleasing number to end his concert, and he didn't have one. But he had a file. So he browsed, and as luck would have it his hand fell on this Sadigura Nigun.
He arranged it in four parts, put some simple Hebrew lyrics to it, and performed it. The rest, as you know, is history, as this became the best-known Jewish song in the world.
Idelsohn documented this part of the transmigration of this melody in Volume 9 of his "Thesaurus of Hebrew Oriental Melodies" page XXIV. I know a little more about it, because he was my first teacher of music. In recent years, long after his death, the Government of Israel finally awarded his family some royalties. Also after his death, Moshe Nathanson claimed authorship, since he was a boy in one of Idelsohn's Hebrew classes at the time I think. But to my knowledge, Israel never accepted his claim.
Interestingly enough, recordings of Havah Nagilah made in Europe in the 20's go at a relatively slow pace. The Hora rhythm was added later, came from a Rumanian folkdance brought to the yishuv by the Halutzim.
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