Considering the Gemara:
“בבלי ביצה ל:א אין מטפחין ואין מספקין ואין מרקדין”
Why are we allowed to dance on Simchas Torah? When and why was this relaxed?
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Tosfot (Beitzah 30a s.v. Tenan) writes that the gezerah of Chazal only applied then when they were experts in fixing musical instruments but it wouldn’t apply to us since we’re not experts in that area.
The Bet Yosef 339:3 writes that the implication of all the poskim who simply copy the prohibition of the Mishna is that they do not hold of the logic of Tosfot. The Shulchan Aruch 339:3 rules clearly that it’s forbidden to clap or dance on Shabbat.
The Rama writes that the minhag is to be lenient based on the opinion of Tosfot that there’s no prohibition anymore of coming to fix a musical instrument. The implication of the Rama is that this minhag isn’t proper but it’s better not to inform people of the prohibition so that they only violate it unintentionally and not deliberately.
Aruch Hashulchan 339:5-9 writes that the logic to be lenient would be that our singing and dancing nowadays is different and would never bring one to fixing an instrument and thus the gezera of Chazal not to dance and clap wouldn't apply to our dancing or clapping.
The Mishna Brurah 339:10 writes that one shouldn’t rely on this minhag except in cases of mitzvah. This is also the opinion of the Shemirat Shabbat KeHilchata 16:43-4. Sh"t Igrot Moshe 2:100 writes that even though most rishonim forbid, since the rama quotes tosfot and that is the minhag there is what to rely on but a baal nefesh should be strict. 39 Melachos (Rabbi Ribiat, vol 4, pg 1160) writes that based on the Rama many people permit themselves to clap and drum their fingers while singing (and quotes Igrot Moshe in the footnote). Sh"t Minchat Elazar 1:29 justifies the practice of dancing on Yom Tov for someone who becomes very excited from dancing because it is like the leniency for the simchat mitzva of dancing with the torah on Simchat Torah
They also cite "The 39 Melochos" of Rabbi Dovid Ribiat (vol 4, pg 1160) and Beit Yosef 339 quoting the Mahari Kolon, who are lenient with regard to Simchat Torah because of the honor of the Torah.
Yalqut Yosef states (in Se'if 5) that Simhhat Torah was not part of the Rabbinic decree (of not clapping/dancing) to safeguard against fixing tools. He states that, out of honor for the Torah, HaZa"L did not include Simhhat Torah celebrations in the decree.