Babylonian calendar was a kind of mix between solar and lunar calendars. It was based on the accidental fact that 235 lunar months (235·29.5306 days = 6939.691 days) is approximately equal to 19 solar years (19·365.2422 = 6939.602 days). This calendar lost about 0.089·100/19 = 0.47 days/century to the solar century, which is less than the Julian calendar (0.78 days/century) but more than the Gregorian calendar (0.03 days/century). Moreover, the length of year was jumping around in a fixed pattern during this 19-year cycle, because 1 lunar month had to be added to 7 out of the 19 years. The calendar was periodicaly gaining on and falling behind the Sun and it was never more than about 15 days (half a month) out of step with the Sun.

Moslem calendar, on the other hand, is strictly a lunar calendar. It is based on the accidental fact that 30 lunar years (12 lunar month each) almost exactly equal to an integer number of days: 30·12·29.5306 = 10631.016 days. Compare this with 30 lunar calendar years, where the length of a calendar month alternates between 29 and 30 days, for 29.5 days on the average. 30 lunar calendar years equal to 30·12·29.5 = 10620 days, which is just 11 days short of the 30 true lunar years. Again, these 11 days are scattered through the 30 lunar calendar years in a fixed pattern. The calendar gains 0.016·100/30 = 0.053 days to a lunar century (approximately 97 solar years). The Gregorian calendar with a loss of 0.03 days/century is still better than that.

You might like to google for it, perhaps including Omar Khayyam in the search terms, since he is popularly supposed to have invented it.

I'm currently holidaying in the US, so don't have my reference books with me, so can't add much more now.

Persian calendar
The modern Persian calendar uses quite complicated leap year rules, defining a 2820-year cycle with 683 leap years, which results a in mean length of a year of 365 683/2820 = 365,2422 days. Considering the length of the tropical year as being constant, the remaining error would amount to a day in more than 2 million years.

The 2820-year cycle is divided into 21 subcycles of 128 years each, and a 132-year subcycle at the end of each 2820-year cycle. A 128-year subcycle consists of a 29-year sub-subcycle, followed by 3 sub-subcycles of 33 years each. Finally, the 132-year subcycle consists of one sub-subcycle of 29 years, followed by two 33-year sub-subcycles and a final sub-subcycle of 37 years.
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The 1st paragraph stands, is relevant to any calendar, including Persian, and I repeat it:

We know the solar year more accurately than anybody else in history. The calendar year in a particular time zone must start and end at midnight, i.e., the calendar year must have an integer number of days. We can always skip a (century) leap year if the calendar falls out of the step with the solar year by more than, say, 0.5 of a day. This flexibility allows the calendar to be in step with the Sun ± 0.5 day and you cannot do better than that.

Compare this to the complicated and rigid pattern of leap years in the Persian calendar. Moreover, making calendar for more than a few millenia is quite unjustified, certainly with the present experimental data (which is better than any other experimental data in history). We even know how the Earth rotation is slowing down - about 5.33 sec/millenium at present:

Scientific article:
One of the latest (1990) experimetal formula for the length of tropical year is

t = 365.242189669781 - 6.161870·10^-6·T - 6.44·10^-10·T².

where T is the time reckoned from the year 2000 and measured in Julian centuries of 365.25 ephemeris days. The formula valid over about 8000 years centered at the present. A comparison of the Gregorian calendar with a perfect solar calendar suggests that the former will be adequate at least during the nearest one to two thousand years. Due to high uncertainty in the Earth rotation it is premature at present to suggest any reform that would reach further than a few thousand years into the future.

Hold on, I think I'm on to something here.

L/P » t - 365 » 0.2421897 days

as close as possible. Then P is the period of the calendar in years and L is the number of leap years in the calendar period P. There are some reasonable requirements on the calendar period:

1. The period should be comparable to or greater than the average human lifespan. Then the exception from the simple rule of the Julian calendar "one leap year in 4 years" occurs at most once in anyone's lifetime.

2. The simple rule of the Julian calendar "years divisible by 4 are leap years" should be valid in any particular period. Then even if the exception from the rule "one leap year in 4 years" occurs someone's lifetime, things are the same after the exception.

3. The period should preferably be a simple round number. Then the calculations of remote dates and time periods between 2 remote dates are simple.

The Gregorian calendar uses L = 97 and P = 400, L/P = 0.2425. All 3 requirements are satisfied, although one could do better on the requirement #1 - the accuracy is a loss of 0.031 days/millenium = 7 hours 27 minutes/millenium.

The modern Persian calendar was proposed in 1925. At that time, the best experimental value of the solar year length was 365.242199 days. So the 2 integers L and P of the Persian calendar are selected as L = 683, P = 2820 years, L/P = 0.2421986. The accuracy (requirement #1) is excellent - loss of 12 min 49 sec/millenium. However, neither of the remaining 2 requirements are satisfied (128 and 132 years cycles, 29, 33, and 37 years subcycles).

It turns out that the best present day experimental value for the solar year length is a little shorter - about 365.2421897 days. Let's propose a new CTK Exchange calendar. On the first attempt, L = 15, P = 62 years, L/P = 0.2421935 - after 60 years of "one leap year in 4 years", there is a gap of 1 leap year in 6 years. The accuracy is excellent (loss of 5 min 28 sec/millenium, better than the Persian calendar), the period is simple and comparable to the average human lifespan. However, the rule "years divisible by 4 are the leap years" or "double even years are leap years" has to alternate with the rule "single even years are leap years" every 62 years, which would be inconvenient. The correct this defect, let's double the calendar period to P = 124 years and the number of leap years to L = 30. You simply skip a leap year every 124 years. What could be simpler than that?

(365.2425 * 3200 - 1)/3200 = 365.2425 - 1/3200 = 365.2421875 days

and it would gain only

(365.24220 - 365.2421875)·1000 = 0.0125 days/millenium = 18 minutes/millenium

to the solar millenium and a one full day to 80 solar millenia:

(365.24220 - 365.2421875)·80000 = 1 day/80 millenia