Daf Yomi - Sukkah 55 - Shir shel Yom by Mincha

It is evident from the Gemora Rosh Hashana (30b) that the Leviim would sing a shira by the korban tamid in the afternoon as well. The Maharam Alshich is bothered as to why we do not recite a shir shel yom nowadays by mincha? He answers that since it is ruled that if the Leviim did not sing the shirah in the afternoon, the korban will become passul, they could not institute that we who do not have the korban should recite the shir (in the morning, it is not meakev).

The Magen Avrohom (132:14) offers two answers. Firstly, he cites Tosfos who rules that if the libations of the afternoon were not brought until the evening, they could be offered the entire night, however the shirah cannot be sung then since they didn't sing at night. Since there would be times in the Beis Hamikdosh that they did not sing the shirah in the afternoon (when the nesachim were delayed until the evening), the Chachamim didn't institute that we should recite it by mincha.

Secondly, he answers, that the halacha was that after the korban tamid was offered on the mizbeach, they didn't sing the shirah. Shirah was only sung before the tamid was completed. The Chachamim could not institute that we should recite the shir shel yom after mincha since mincha is corresponding to the korban tamid and after the tamid, they could not sing shirah anymore.

The Chasam Sofer in Beitza (4b) answers that there are two reasons as to why we say the parsha of korbanos and the avodos that were performed in the Beis Hamikdosh nowadays. Firstly, we recite these parshiyos based on the verse which says that our lips are regarded as the offering of the korbanos. There is another reason as well. The reciting of these tefilos is indicating our desire and anxiousness for the rebuilding of the Beis Hamikdosh in our time. In the morning, we do not have the second explanation in mind. The halacha is that the mizbeach is only inaugurated with the offering of the aftrenoon tamid. Even if the Beis Hamikdosh would be built today, we would not be able to offer the morning tamid. Our primary kavanah in the morning is that our tefilos should be accepted as if we were offering the korbanos. In the afternoon, the primary kavanah we should have is to signify our desire to see the Beis Hamikdosh built speedily and if the Beis Hamikdosh would be built at that moment (which is what we should be thinking), there would not be a shirah sung, since the Beis Yosef (51) rules that shirah will not be sung in the times of the third Beis Hamikdosh (except mizmor l'sodah). It is for this reason that we do not recite the shir shel yom by mincha for it will prevent us from having our correct kavanos.