The Gemara in Berachos 26b brings a dispute regarding the origin of our obligation to daven.
Rabi Yosi ben Rabi Chanina says that the avos where mEsaken the tefillos. The Gemara quotes a braisa that supports this view and cites a pasuk from which we derive that Avraham was mesaken Shacharis, and a pasuk in this week’s parsha that Yitzchak was mesaken Minchah, and another that Yaakov was mesaken Ma’ariv. Rabi Yehoshua ben Levi disagrees, saying that tefillah was instituted to correspond to the korbanos.
There were two korbanos that were brought daily, the tamid shel Shacharis and the tamid shel bein ha’arbayim. The tefillos of Shacharis and Minchah correspond to these two korbanos respectively, while Ma’ariv corresponds to the burning of the eimurim from the korbanos – which was performed at night. The tefillah of Musaf was instituted to correspond to the korban musaf that was brought on specific days. The braisa adds that the times to daven these tefillos correspond to the times that the korbanos were brought.
Regarding the time to daven Minchah the Rambam writes (Hilchos Tefillah 3:2) that the tamid shel bein ha’arbayim was brought at nine and one half hours into the day from sunrise. (All of the hours in this article refer to sha’os zimaniyos – where the amount of daylight hours are divided into twelve equal parts. Only on the equinox are these hours 60 minutes each.)Therefore, the proper time to daven Minchah is at nine and one half hours into the day. The Rambam adds that since whenever erev Pesach would fall on erev Shabbos they would bring the tamid shel bein ha’arbayim at six and one half hours into the day, one may daven Minchah all year round at that time, although it is not the optimal time.
It is evident from the Rambam that the tefillos were instituted to correspond to the time that the korbanos were actually brought, as opposed to the time that one was permitted to bring a korban. It was not sufficient that the tamid shel bein ha’arbayim was permitted to be brought from six and one half hours until shkiyah, it was required that the korban was actually brought at that time as well. If the korban was never brought at that time even if it was permitted to be brought at that time, we would not be permitted to daven then.
On the Tukachinsky Calendar, it says that when erev Pesach falls out on erev Shabbos, it is optimal to daven Minchah from six and one half hours into the day. This is because on that day the tamid shel bein ha’arbayim was brought at that time.
The implication of the Shulchan Aruch (Orach Chaim 233:2) is that one may not daven Minchah before six and one half hours into the day. However, the Mishnah Berurah there cites several Acharonim that opine that if one davens from chatzos (six hours into the day) he has fulfilled his obligation. This is because the tamid shel bein ha’arbayim was permitted to be brought from chatzos. Perhaps the Shulchn Aruch is following the opinion of the Rambam that it is not sufficient that a korban could have been brought at that time, rather the korban had to actually have been brought at that time. Since the tamid shel bein ha’arbayim was never brought before six and one hours we may not daven Minchah then.
The Rema (Orach Chaim 89:1) rules that one may not daven Shacharis after chatzos. The Taz there disagrees, saying that since one cannot yet daven Minchah he should be allowed to daven Shacharis. The Magen Avraham there answers the Rema by saying that in fact chatzos begins the time that one can daven Minchah, and therefore one may no longer daven Shacharis. The reason why we may not daven Minchah until a half hour after chatzos is because we are not exactly sure when chatzos is.