Is one allowed to clap one's hands on Shabbos? Does it matter whether one is clapping to a beat or trying to get someone's attention?
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Generally, one is not allowed to clap to a beat on Shabbos (S.A. O.C. 339:3) Some have said that nowadays the gezeirah doesn't apply since we're not adept at instrument repair and the incidence is not common. The Rema brings this for even dancing. Others only permit for hand-clapping. I don't even know if this heter would apply now since instrument repair has become pretty easy.
Some other areas which are debated are:
Clapping not to a musical beat (e.g. applauding, attention getting). Magen Avraham there permits. The Mishna Berurah there brings down those who argue, but even the M.B. sounds like it's a chumrah (compare 339:9 to 338:1)
For a mitzvah. The M.A. above and Taz in Y.D.282 disapprove of bells on a sefer torah. The Mishna Berurah brings the Shach that argues with the Taz, but I didn't see where the Shach says that.
Generally, I have seen the bnei yeshiva clapping to zemiros. I don't know if it's based on mitzva, or a combination of heterim.
In practice sefaradim prohibit and some ashkenazim permit for sake of a mitsva, like singing during meals or the bet hacneset.
Even for sefaradim would be permitted to clap with the back of the hand; since is an isur miderabanan you can do it with shinui.
I have heard the sefer Minchas Elazar writes about this subject (and other Chasidic minhagim), in support of a heter.
Yalkut Yosef 338:1
In a circumstance of happiness and singing, it is prohibited.
To wake people up it is permissible because it is not in a way of music.