Here at Mah Rabu, we spend a fair amount of time on Hebrew calendar minutiae because, well, we can. But I'm sure the other calendar geeks out there can identify with the feeling of the excitement level of this stuff reaching a plateau. Once you think you know all the fun facts -- that July 4, 1776 fell on 17 Tammuz, or that the Hebrew and Gregorian year numbers will be equal starting in the year 317,760,208 (on both calendars!) -- it seems like there aren't going to be any more surprises.
On Sunday, I found out I was wrong. It's still possible to blow my mind about this stuff, like finding out that there was a solar calendar (though not a very good one) that was fully compatible with the biblical holidays.
I heard a lecture on Sunday by Dr. Michael Segal, author of a new book about the Book of Jubilees. The Book of Jubilees is a Jewish sectarian text from the 2nd century BCE. It was written in Hebrew, but only fragments of the Hebrew original have survived (at Qumran). Since the book is part of the canon of the Ethiopian Church, the only language in which the full text has survived is Ge'ez, and it has since been translated back into Hebrew and other languages.
The calendar in the Book of Jubilees is a solar calendar, with a 364-day year. This means the year divides exactly into 52 weeks, so a given date falls on the same day of the week every year. Furthermore, the year is divided into 4 quarters of 91 days, or exactly 13 weeks. Each quarter is divided into 3 months, of 30, 30, and 31 days. Thus the calendar of each quarter looks exactly the same!
You're probably wondering what happens to the extra 1 1/4 days every year. And it's a good question; we're all worried about the Hebrew lunisolar calendar because it's off by 1 day every ~200 years, but an error of more than one day every year has to be much worse! But the answer is that no one really knows. It's an important feature of this calendar that dates fall on the same day every year, but they can't simply add a day or two at the end of the year without a day of the week (as in the proposed World Calendar), because we would have heard about it if there had been Jewish sects observing Shabbat on a different day! That would make Shavuot (a major concern for the rabbis) look like peanuts.
Like the rabbinic solar calendar which we use to determine the date of birkat hachamah, the Book of Jubilees solar calendar relies on the fact that the sun was created on Wednesday (the fourth day of creation) in the biblical narrative. Unlike the rabbinic solar calendar, you don't have to wait 28 years for the anniversary of the sun's creation to cycle back to Wednesday; since the year is a whole number of weeks, it happens every year! Nay, every quarter! Thus, the first day of each quarter falls on Wednesday, corresponding theoretically to the solstices and equinoxes (but, I imagine, having little resemblance to astronomical reality after a very short time, thanks to the aforementioned 1 1/4-day error). So the 1st, 4th, 7th, and 10th months always begin on Wednesday; the 2nd, 5th, 8th, and 11th months begin on Friday; and the 3rd, 6th, 9th, and 12th months begin on Sunday; and that's all you need to know.
Now let's look at the holidays. Rosh Hashanah (1st day of the 7th month) is always on Wednesday, and Yom Kippur (10th of the month) is on Friday. That has to be annoying. (Our calendar is rigged so that Yom Kippur can never fall on Friday).
The 1st month (what we would call Nisan, but the Book of Jubilees, like the Torah, refers to the months only by number) begins on Wednesday, so the first day of Pesach (15 Nisan) is also on Wednesday. So when does the omer begin, to count 7 weeks toward Shavuot? The Torah says "the day after the sabbath", but what day is "the sabbath"? The Book of Jubilees agrees neither with the rabbis/Pharisees ("the sabbath" = the 1st day of Pesach, 15 Nisan) nor with the Sadducees/Boethusians (the Shabbat during Pesach, whatever date that may be); it understands "the sabbath" as the Shabbat after the end of Pesach, which is always 25 Nisan. So the omer begins on Sunday, 26 Nisan. Add 7 weeks (recalling that the 1st and 2nd month both have 30 days), and Shavuot always falls on Sunday, 15 Sivan! So all three pilgrimage festivals are on the 15th of the month! Ironic that this only happens in the solar calendar, in which the 15th of the month has no astronomical significance. BTW, the Book of Jubilees also has Shavuot as both the agricultural festival of the first fruits and the time of receiving the Torah, and is the earliest extant text that explicitly links Shavuot to the revelation at Sinai.