Lunar Month of 19 Cells and Solar year normally of 235 Cells

Dear Helios and Calendar People

I recall that Helios introduced the idea of a year divided into 235 cells and a lunar month divided into 19 of these cells. Such a calendar would form a 19-year Metonic cycle. I suggested correcting the Metonic cycle by having just 234 cells in some years. Only one such short year would be needed every three or four centuries.

Several ways of defining the number of days in the cells were discussed. The one I liked most was a fortnightly 9-cell cycle of 14 days. The 9 cells would have lengths of 2,1,2,1,2,1,2,1,2 days. Each year would have 26 of these 9-cell cycles totalling 234 cells and then have one more cell, which would have one day in a common year or two days in a leap year. This gives rise to 235 cells lasting 365 or 366 days.

Each lunar month of 19-cells would begin on the 1^st, 3^rd, 4^th, 6^th, 7^th, 9^th, 10^th, 12^th or 13^th day of a of a fortnight counting from new year’s day. Furthermore next lunar month not counting the month that includes the 235^th cell would begin on the next day of the fortnight cyclically. This gives rise to a 9-month yermette of  30,29,30,29,30,29,30,29,30 days not counting the month with the 235^th cell. The month with the 235^th cell simply has 29 days in a common year or 30 days in a leap year and so corrects the rather short yermette. 

Suppose each of the 26 fortnights were named A to Z, the 235^th cell is named ZZ and also the first lunar month began on new years day of year 1, then you’d get a 19-year cycle of lunar month start dates thus:

year 01:  A01 C03 E04 G06 I07 K09  M10 O12 Q13;T01 V03 X04 Z06

year 02:  B06 D07 F09 H10 J12 L13; O01 Q03 S04 U06 W07 Y09

year 03:  A09 C10 E12 G13;J01 L03  N04 P06 R07 T09 V10 X12 Z13

year 04:  B13;E01 G03 I04 K06 M07  O09 Q10 S12 U13;X01 Z03

year 05:  B03 D04 F06 H07 J09 L10  N12 P13;S01 U03 W04 Y06

year 06:  A06 C07 E09 G10 I12 K13; N01 P03 R04 T06 V07 X09 Z10

year 07:  B10 D12 F13;I01 K03 M04  O06 Q07 S09 U10 W12 Y13

year 08:  A13;D01 F03 H04 J06 L07  N09 P10 R12 T13;W01 Y03

year 09:  A03 C04 E06 G07 I09 K10  M12 O13;R01 T03 V04 X06 Z07

year 10:  B07 D09 F10 H12 J13;M01  O03 Q04 S06 U07 W09 Y10

year 11:  A10 C12 E13;H01 J03 L04  N06 P07 R09 T10 V12 X13 ZZ1;

year 12:  C01 E03 G04 I06 K07 M09  O10 Q12 S13;V01 X03 Z04

year 13:  B04 D06 F07 H09 J10 L12  N13;Q01 S03 U04 W06 Y07

year 14:  A07 C09 E10 G12 I13;L01  N03 P04 R06 T07 V09 X10 Z12

year 15:  B12 D13;G01 I03 K04 M06  O07 Q09 S10 U12 W13;Z01

year 16:  B01 D03 F04 H06 J07 L09  N10 P12 R13;U01 W03 Y04

year 17:  A04 C06 E07 G09 I10 K12  M13;P01 R03 T04 V06 X07 Z09

year 18:  B09 D10 F12 H13;K01 M03  O04 Q06 S07 U09 W10 Y12

year 19:  A12 C13;F01 H03 J04 L06  N07 P09 R10 T12 V13 Y01;

The last month of the 11^th year begins on ZZ1 the first day of the 235^th cell . If that cell were omitted as is occasionally done to correct the 19-year cycle, that month would begin on the first day of next year, just like year 01 and so the 19-year cycle is effectively truncated to 11 years.  Hence, if the 235^th cell is omitted once every 334 years, you get a 334-year cycle. If it is omitted once every 353 years you get a 353-year cycle and likewise for 19N+11 years for any non-negative integer N.

The leap years would have to occur a little more often to compensate for occasional 364-day year arising from omission of the 235^th  cell (ZZ) to correct the 19-year cycle.

Finally to get the 13-month years to follow a Helios cycle, one counts the month containing the 235^th cell (or where it would be if omitted), as belonging to the year the majority of its 19 cells belong to (rather than the year of its first day). This gives rise to

year 01:  A01 C03 E04 G06 I07 K09  M10 O12 Q13;T01 V03 X04

year 02:  Z06 B06 D07 F09 H10 J12  L13;O01 Q03 S04 U06 W07 Y09

year 03:  A09 C10 E12 G13;J01 L03  N04 P06 R07 T09 V10 X12

year 04:  Z13 B13;E01 G03 I04 K06  M07 O09 Q10 S12 U13;X01

year 05:  Z03 B03 D04 F06 H07 J09  L10 N12 P13;S01 U03 W04 Y06

year 06:  A06 C07 E09 G10 I12 K13; N01 P03 R04 T06 V07 X09

year 07:  Z10 B10 D12 F13;I01 K03  M04 O06 Q07 S09 U10 W12 Y13

year 08:  A13;D01 F03 H04 J06 L07  N09 P10 R12 T13;W01 Y03

year 09:  A03 C04 E06 G07 I09 K10  M12 O13;R01 T03 V04 X06

year 10:  Z07 B07 D09 F10 H12 J13; M01 O03 Q04 S06 U07 W09 Y10

year 11:  A10 C12 E13;H01 J03 L04  N06 P07 R09 T10 V12 X13

year 12:  ZZ1;C01 E03 G04 I06 K07  M09 O10 Q12 S13;V01 X03

year 13:  Z04 B04 D06 F07 H09 J10  L12 N13;Q01 S03 U04 W06 Y07

year 14:  A07 C09 E10 G12 I13;L01  N03 P04 R06 T07 V09 X10

year 15:  Z12 B12 D13;G01 I03 K04  M06 O07 Q09 S10 U12 W13;Z01

year 16:  B01 D03 F04 H06 J07 L09  N10 P12 R13;U01 W03 Y04

year 17:  A04 C06 E07 G09 I10 K12  M13;P01 R03 T04 V06 X07

year 18:  Z09 B09 D10 F12 H13;K01  M03 O04 Q06 S07 U09 W10 Y12

year 19:  A12 C13;F01 H03 J04 L06  N07 P09 R10 T12 V13 Y01;

Karl

10(08(13

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