Ah, but my Computations, People say
Reduced the Year to better reckoning – Nay,
'Twas only striking from the Calendar
Unborn To-morrow and dead Yesterday
Rubaiyat, verse #57, by Omar Khayyam
A calendar which tracks the solar year even better than the Gregorian one is the Persian (Iranian) calendar, the first version of which was devised by Omar Khayyam (1044-1123), author of the famous “Rubaiyat" poems, masterfully translated in 1839 into English by Edward Fitzgerald. It is also called the Jalali calendar, after the king Malik Shah Jalaludin who in 1074 assigned Omar and 7 other scholars to devise a new calendar.
Though the count of Persian years starts, like the Islamic one, from the flight of Mohammed to Medina in 622, establishing there the first strong base of Islam, the new year starts at the spring equinox, March 21, with the holiday of Nowruz.
The Persian year itself has 12 months --- the first 6 have 31 days, the next 5 have 30 days, and the last has 28 or 29, depending on whether the year is or isn't a leap year. Each month corresponds to a sign of the zodiac. The number of days in each month (if not the order of months) is therefore the same as in the Western civil calendar. The difference is in the rule for determining leap year, which is more complex. Even the original Jalali calendar was more accurate than the Gregorian one; the current version assigns 683 leap years in a cycle of 2820 years and would take two million years before it shows a one-day inaccuracy!
An interesting related calendar is used by the Coptic Christian church in Ethiopia, with 12 months of 30 days each, plus a 13th short month of 5 days. A tourist brochure once lured visitors with a promise “Come to Ethiopia and enjoy 13 months of sunshine a year."