> As for the lengths of the months, imagine if you were trying to make the case for the different lengths of months and you were trying to get buy in from your friends in accounts. Surely they would want equal amounts of days in each quarter? Imagine trying to persuade them that different lengths were better, what plausible possible reason could there be?
The year is 365 + epsilon days long. The epsilon means we need to periodically add or remove days to synchronize solar time with the seasonal calendar; in the Gregorian calendar, we sprinkle 97 such days every 400 years. Armed with this complexity, you have several options:
1. Accept that a month is going to need an extra day about once every 4 years.
2. Have the extra day be not part of the regular cyclical rotation (i.e., be intercalated). This is probably even more complex for bookkeeping than option #1.
3. Insert an extra month's worth of days on a longer period of time (this is essentially what lunar calendars do, but the effect of leap days themselves are less noticeable since the lunar month is quite out of whack with the solar calendar).
As for months themselves, you run into the problem that 365 and 366 are not amenable for evenly dividing the year into months: their factors are 5×73 and 2×3×61, respectively. 360 is much nicer (2³×3²×5, i.e., is divisible by every number between 2 and 10 but 7); unsurprisingly, a few cultures opted to use 360 as their basis for month divisions. The Mesoamericans used a calendar that was 18 months of 20 days, with 5 extra days that were intercalated.
As it turns out, the lunar month is about 29.5 days long, and the moon is a very obvious cyclical timekeeper (much more obvious than keeping track of days than the sun's seasonal positioning changes). Using a lunar month is going to give you a natural 29/30 alternating month period. However, you end up short by around 11 days when you do this. If you sprinkle them around the months, you find yourself naturally having 7 months of 30 days and 5 months of 31 days.
In terms of calendrical complexity, no one beats the Mayans. They opted for a calendar that has three different lengths of the year: a ceremonial calendar that counts 13 months of 20 days, a solar calendar that is 18 months of 20 days with a 19th month of 5 days, another calendar for long-term record keeping that counts from an epoch in base-20, except that one digit is base-18 instead (for a "year" of 360 days). On top of this, they have a "week" of 9 days, and they counted the current day of the lunar cycle, the current lunar cycle number, and whether the current lunar cycle was 29 or 30 days long. And there's another 819-day cycle that we still don't know what it corresponds to. Oh, and you count months and days like "January 1, February 2, March 3, ..., December 12, January 13, February 14, July 31, August 1, ..."