This question already has an answer here:
- Rosh-Hashanah is clearly one day from the Torah.
- As it falls on the first day of the month, in the days of keeping the Real New Moon, there's no way of informing the other places outside Jerusalem, so practically everywhere it was kept two days as in doubt (Rambam).
- Later some rulings were issued as to make Elul always lacking and sending messengers to inform about the RH (Mishnah) 29 days in advance.
After the fixed calendar was accepted, why keeping the second day does not count as the transgression of adding a day to a one-day holiday (בל תוסיף), just as adding a day for Succot? Does someone discuss this option?
Note that Rosh Hashannah can be compared to a two day Rosh Chodesh (which we know from Shmuel I existed as far back as the reign of Shaul Hamelech). However, the two days of Rosh Chodesh are the last day of the previous month (30th) and the first day of the following month. Rosh Hashannah on the other hand is the first two days of Tishrei. Elul is always 29 days. The answer should include the reason.