The Gregorian system in a new conception and layout, as a new style which combines solar and lunar cycles. The gregorian year is structured in 13 "holy nights" (the time between the 24 of December and 6 of January) and 13 real sidereal or tropical lunar month. This new calendar recombine loyalty to seasons with genuine lunar months, soothes the blurring of all other systems by the month "hug" of the 13 nights rounding. See
The year starts on 6 of January. 13 sidereal or tropical lunar month follow with alternately 27/27/28 days, ending on 24 of December - more 13 nights (12 days) complete the year.
(Well, I know that the difference between 13 sidereal month (13*27,322=355,186) and a solar year is around about 10 days, not 12. So the days beside have to belong both to the sidereal lunar months and to the 13 holy nights, too.)