The Sages enacted a prohibition against playing music on Shabbat, due to the concern that the instrument might break and the person will then repair it, in violation of Shabbat. Playing music in itself – without the use of electric instruments, amplification and the like – does not constitute an act of Shabbat desecration, but repairing a broken instrument – such as replacing a violin string – does entail a violation of Shabbat ("Makeh Be’patish"). The Sages thus forbade playing music as a safeguard against Shabbat desecration. This applies to all forms of making music, including clapping, snapping or banging to a beat, even while singing for a Misva, such as while singing Bakashot or Pizmonim. However, it is permissible to clap, snap or bang to call people to attention, or to awaken somebody. Since the intention is only to make noise, and not to play music, this is permissible. Thus, for example, it is permissible to strike one’s silverware against a glass to call people to the table or to ask for silence, since there is no intent to play a melody.
In discussing this Halacha, the Shulhan Aruch (Orah Haim 338:2) presents a ruling that will sound startling to most of us. The ruling is based on the general principle of "Shebut De’shebut Be’makom Misva" – that when necessary for the sake of a Misva, one may ask a non-Jew to perform an action that for Jews is forbidden Mi’de’rabbanan – by a decree of the Sages. As long as the act in not forbidden by Torah law, and is forbidden only by the Rabbis, one may ask a non-Jew to perform the act if this is necessary for the purpose of a Misva. As such, the Shulhan Aruch rules, one may bring non-Jews to play music on Shabbat for the celebration of a Sheba Berachot. Since playing music at a Sheba Berachot fulfills the Misva of Simhat Hatan Ve’kalla – rejoicing with the bride and groom – and playing music (without the use of electricity) is forbidden by the Sages, and not by Torah law, one may have non-Jews play music at a Sheba Berachot on Shabbat. The Rama (Rav Moshe Isserles of Cracow, 1530-1572) goes even further, and permits asking a non-Jew on Shabbat to repair an instrument that is needed for music at a Sheba Berachot.
However, while this is the explicit ruling of the Shulhan Aruch, the widespread custom throughout the generations is not to put this Halacha into practice. The Rabbis were, understandably, concerned about the potential effects of having music played on Shabbat, as people who do not understand the mechanics of this Halacha – specifically, the concept of "Shebut De’shebut Be’makom Misva" – will reach wrong conclusions and permit music in other situations when Halacha forbids it. They might bring non-Jews to play music in contexts that do not involve a Misva, and they might bring non-Jews to play music with electric instruments and amplification systems, which is clearly forbidden. Indeed, the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes (Parashat Shofetim, Shana Rishona, 18) that the practice in Baghdad was to have music played for a newlywed couple on Shabbat, but he stopped this practice out of concern of what it might it lead to.
It is worth noting in this context the remark made by Rav Moshe Feinstein (1895-1986) that Shabbat in our generation is like a "Gosses" – a person about to die. Halacha forbids touching a "Gosses," even gently, as even the gentlest touch could hasten the person’s death. Likewise, Rav Moshe warned, we may not "touch" Shabbat in any way, by accepting leniencies that could potentially lead people to be even less careful with regard to Shabbat observance than they already are.
Therefore, although technically, it is permissible to hire non-Jews to play music at a Sheba Berachot on Shabbat, this should not be done as a matter of practice.
Summary: It is forbidden to play music on Shabbat, even if no electricity is used, and one may not hire non-Jews to play music on Shabbat, even for the purpose of a Misva, such as a Sheba Berachot celebration.