Oh, is variation about when the year starts, up to half the lunar cycle fine? 'Cause the only way I know to go about this is to vary the number of months in the year back and forth.
In this case, most years having 16 months, which moves the beginning of the year forward ~2 days each year, up until it's 10 (or 12) days sooner than it should be, then you have a year with 17 months, which sets it back to 12 (or 10) days later than it should, until it gradually catches back up again, and then starts over.
Re: *math-related despair*
In this case, most years having 16 months, which moves the beginning of the year forward ~2 days each year, up until it's 10 (or 12) days sooner than it should be, then you have a year with 17 months, which sets it back to 12 (or 10) days later than it should, until it gradually catches back up again, and then starts over.