Dear Amos and Calendar People
I then thought of adding 7 more stations to the Metonic cycle to make a total of 28 stations and have a leap month on every 4th station. Then all the even-numbered years of the Metonic cycle can have two stations and the odd-numbered years have one station. Then we get leap months thus:
Station 4 at end of 3rd year
Station 8 at middle of 6th year (1st station in 6th year)
Station 12 at end of 8th year (2nd station in 8th year)
Station 16 at end of 11th year
Station 20 at middle of 14th year (1st station in 14th year)
Station 24 at end of 16th year (2nd station in 16th year) not 17th year
Station 28 at end of 19th year.
This reminds me of the Quarter-Moon Calendar and I find that if each leap month were split into four quarter moons and one of these quarter moons were added to each station, one would get a quarter-moon calendar.
A similar idea could be applied to the “Babylonian Tournament Year Calendar” with moon-thirds, so each year has 37 moon-thirds except the 8th and 17th years of a Metonic cycle, which has 38 moon-thirds. Most moon-thirds would have 10 days, but one in six or seven would have 9 days.
Dear Calendar People
Helios introduced the idea that a lunisolar calendar has a few tournament years, so that no tournament year has a leap month and leap months occur once every 2 years not counting any tournament year. A 19-year cycle would have 5 tournament years, so that 14 years are counted to provide the 7 leap months.
I’ve thought of reversing this idea and instead of not counting the tournament year, one counts it twice by giving it two places, where a leap month can occur. Other years have just one place, where a leap month can occur. I call these places stations. We can have a leap month occur once every 3 stations. The 19-year cycle would have 2 tournament years (instead of 5) to provide 21 stations for its 7 leap months.
Suppose every year had a station at the end and the tournament years also have a station in the middle. Then if the 8th and 17th years of the Metonic cycle were its two tournament years, the 7 leap months would occur as follows:
Station 3 at end of 3rd year
Station 6 at end of 6th year
Station 9 at end of 8th year (2nd station in 8th year) 3rd leap month
Station 12 at end of 11th year
Station 15 at end of 14th year
Station 18 at middle of 17th year (1st station in 17th year) 6th leap month
Station 21 at end of 19th year.
And so I seem to have described the Babylonian Calendar.
Also the leap month years are exactly the same as in the Hebrew Calendar.
Also both the tournament years and leap month years are symmetrically placed in a cycle starting at the 13th year.
Then after renumbering the so the 13th year becomes year 1, we get leap month years
(2, 5, 7, 10, 13, 15, 18) and tournament years (5, 15).
The idea can be applied to more accurate lunisolar calendars, in which case, the octaeteris and the 11-year cycle each have one tournament year and so for example the 353-year cycle has 37 tournament years. The 315-year and 372-year cycles can have tournament year cycles on third of their length of 11 tournament years in 105 years and 13 tournament years in 124 years respectively. The 353-year cycle can then be made up of 105+124+124 years of 11+13+13 tournament years. The sidereal 160-year cycle would have 17 tournament years.
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