A Planting Calendar Counted in Moons: The Drift Bound and the Intercalation Rule That Holds It

A calendar of twelve lunations contains 354.36708 days, which is 10.87512 days short of the tropical year of 365.2422 days, so a planting date fixed by counting moons falls earlier against the seasons by that amount every year. This paper computes how quickly such a calendar fails, and designs and tests a correction rule that a community can apply with no instrument beyond watching one seasonal sign. Without correction, a fixed moon-count planting date is half a lunation (14.765 days) wrong after 1.358 years and a full lunation (29.53059 days) wrong after 2.715 years. The rule proposed here is to keep a running count of the accumulated drift and to insert a thirteenth month ahead of the planting date in any year in which the drift, as estimated from the seasonal sign, reaches half a lunation, 14.765 days. In a 200-year simulation with the sign judged to plus or minus 3 days, the rule inserted 74 intercalary months and held the planting-date error to a maximum of 17.24 days with a mean absolute error of 7.47 days, inside the stated success window of plus or minus 20 days and inside the analytic worst-case bound of half a lunation plus the observation error, 17.77 days. With a perfectly judged sign the maximum error was 14.75 days and the mean 7.38 days. The guarantee behind the rule reaches the edge of the 20-day window when the sign-judging error reaches 5.235 days, and in the error sweep the first simulated failures of the window appear at a judging error of 7.5 days, where the worst maximum error over the trials run was 21.92 days. The fixed 19-year pattern of 7 intercalations, computed for comparison, accumulates almost no long-run drift (0.0868 days per 19-year cycle, 0.91 days over 200 years) but lets the planting date swing further within each cycle under the definitions used here, reaching a maximum error of 26.35 days with a mean of 12.19 days, because it waits for the count rather than responding to the observed drift. This study is simulated, not measured.