30-31-day month calendars

Months have either 30 or 31 days. Patterns can be simple, i.e. 30:31 and 31:30, quarter-based, i.e. 31:30:30, 30:31:30 and 30:30:31, more complicated ,or totally irregular as in the Gregorian calendar.

On the one hand, six times 61 makes 366 days, thus either a 31st or a 30th day must be left out in leap years. On the other hand, four time 91 makes 364 days, thus either a 31st or a 32nd day must be added in common years, and another 31st, 32nd or 33rd in leap years, or a leap week has to be introduced which may make one month 37 or 38 days long in leap years.

End-of-year leap day[]

Month lengths in days, values for leap years shown in parentheses
Month	J/G	31:30	30:31	31:30…30:31	30:31…31:30	31:30:30	30:31:30	30:30:31
January	31	±0	−1	±0	−1	±0	−1	−1
February	28 (29)	+2 (+1)	+3 (+2)	+2 (+1)	+3 (+2)	+2 (+1)	+3 (+2)	+2 (+1)
March	31	±0	−1	±0	−1	−1	−1	±0
April	30	±0	+1	±0	+1	+1	±0	±0
May	31	±0	−1	±0	−1	−1	±0	−1
June	30	±0	+1	±0	+1	±0	±0	+1
July	31	±0	−1	−1	±0	±0	−1	−1
August	31	−1	±0	±0	−1	−1	±0	−1
September	30	+1	±0	±0	+1	±0	±0	±0
October	31	−1	±0	±0	−1	±0	−1	−1
November	30	+1	±0	±0	+1	±0	+1	±0
December	31	−2 (−1)	−1 (±0)	−1 (±0)	−2 (−1)	±0 (+1)	±0 (+1)	+1 (+2)

Month starts in ordinal days, values for leap years shown in parentheses
Month	J/G	31:30	30:31	31:30…30:31	30:31…31:30	31:30:30	30:31:30	30:30:31
January	001	±0	±0	±0	±0	±0	±0	±0
February	032	±0	−1	±0	−1	±0	−1	−1
March	060 (061)	+2 (+1)	+2 (+1)	+2 (+1)	+2 (+1)	+2 (+1)	+2 (+1)	+1 (±0)
April	091 (092)	+2 (+1)	+1 (±0)	+2 (+1)	+1 (±0)	+1 (±0)	+1 (±0)	+1 (±0)
May	121 (122)	+2 (+1)	+2 (+1)	+2 (+1)	+2 (+1)	+2 (+1)	+1 (±0)	+1 (±0)
June	152 (153)	+2 (+1)	+1 (±0)	+2 (+1)	+1 (±0)	+1 (±0)	+1 (±0)	±0 (−1)
July	182 (183)	+2 (+1)	+2 (+1)	+2 (+1)	+2 (+1)	+1 (±0)	+1 (±0)	+1 (±0)
August	213 (214)	+3 (+2)	+1 (±0)	+1 (±0)	+2 (+1)	+1 (±0)	±0 (−1)	±0 (−1)
September	244 (245)	+2 (+1)	+1 (±0)	+1 (±0)	+1 (±0)	±0 (−1)	±0 (−1)	−1 (−2)
October	274 (275)	+1 (±0)	+1 (±0)	+1 (±0)	+2 (+1)	±0 (−1)	±0 (−1)	±0 (−1)
November	305 (306)	+1 (±0)	+1 (±0)	+1 (±0)	+1 (±0)	±0 (−1)	−1 (−2)	−1 (−2)
December	335 (336)	+1 (±0)	+1 (±0)	+1 (±0)	+2 (+1)	±0 (−1)	±0 (−1)	−1 (−2)

31:30...30:31 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	31	001	±0	±0
February	30	032	+2 (+1)	±0
March	31	062	±0	+2 (+1)
April	30	093	±0	+2 (+1)
May	31	123	±0	+2 (+1)
June	30	154	±0	+2 (+1)
July	30	184	−1	+2 (+1)
August	31	214	±0	+1 (±0)
September	30	245	±0	+1 (±0)
October	31	275	±0	+1 (±0)
November	30	306	±0	+1 (±0)
December	30 (31)	336	−1 (±0)	+1 (±0)

30:31...31:30 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	30	001	−1	±0
February	31	031	+3 (+2)	−1
March	30	062	−1	+2 (+1)
April	31	092	+1	+1 (0)
May	30	123	−1	+2 (+1)
June	31	153	+1	+1 (±0)
July	31	184	±0	+2 (+1)
August	30	215	−1	+2 (+1)
September	31	245	+1	+1 (±0)
October	30	276	−1	+2 (+1)
November	31	306	+1	+1 (±0)
December	29 (30)	337	−2 (−1)	+2 (+1)

31:30 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	31	001	±0	±0
February	30	032	+2 (+1)	±0
March	31	062	±0	+2 (+1)
April	30	093	±0	+2 (+1)
May	31	123	±0	+2 (+1)
June	30	154	±0	+2 (+1)
July	31	184	±0	+2 (+1)
August	30	215	−1	+2 (+1)
September	31	245	+1	+1 (±0)
October	30	276	−1	+2 (+1)
November	31	306	+1	+1 (±0)
December	29 (30)	337	−2 (−1)	+2 (+1)

30:31 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	30	001	−1	±0
February	31	031	+3 (+2)	−1
March	30	062	−1	+2 (+1)
April	31	092	+1	+1 (0)
May	30	123	−1	+2 (+1)
June	31	153	+1	+1 (±0)
July	30	184	−1	+2 (+1)
August	31	214	±0	+1 (±0)
September	30	245	±0	+1 (±0)
October	31	275	±0	+1 (±0)
November	30	306	±0	+1 (±0)
December	30 (31)	336	−1 (±0)	+1 (±0)

31:30:30 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	31	001	±0	±0
February	30	032	+2 (+1)	±0
March	30	062	−1	+2 (+1)
April	31	092	+1	+1 (±0)
May	30	123	−1	+2 (+1)
June	30	153	±0	+1 (±0)
July	31	183	±0	+1 (±0)
August	30	214	−1	+1 (±0)
September	30	244	±0	±0 (−1)
October	31	274	±0	±0 (−1)
November	30	305	±0	±0 (−1)
December	31 (32)	335	±0 (+1)	±0 (−1)

30:31:30 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	30	001	−1	±0
February	31	031	+3 (+2)	−1
March	30	062	−1	+2 (+1)
April	30	092	±0	+1 (±0)
May	31	122	±0	+1 (±0)
June	30	153	±0	+1 (±0)
July	30	183	−1	+1 (±0)
August	31	213	±0	±0 (−1)
September	30	244	±0	±0 (−1)
October	30	274	−1	±0 (−1)
November	31	304	±0	−1 (−2)
December	31 (32)	335	±0 (+1)	±0 (−1)

30:30:31 Month lengths with changes from Julian/Gregorian in days, values for leap years shown in parentheses
Month	Length	Start	Length	Position
January	30	001	−1	±0
February	30	031	+2 (+1)	−1
March	31	061	±0	+1 (±0)
April	30	092	±0	+1 (±0)
May	30	122	−1	+1 (±0)
June	31	152	+1	±0 (−1)
July	30	183	−1	+1 (±0)
August	30	213	−1	±0 (−1)
September	31	243	±0	−1 (−2)
October	30	274	−1	±0 (−1)
November	30	304	±0	−1 (−2)
December	32 (33)	334	+1 (+2)	−1 (−2)