Lunar Calendar Cycles
Less Than 1885 Months

by Karl Palmen

One day Peter Meyer wrote a computer program to find all lunar calendar structures of the following type:

(i) The calendar period (after which the structure of the calendar repeats itself) is composed of n1 29-day months and n2 30-day months (with both n1 and n2 less than 1000), for a total of n1 + n2 months.

(ii) The length of the average calendar month, namely ( n1*29 + n2*30 ) / ( n1 + n2 ) days, is within 0.0001% of either 29.530589 days (the synodic month in 2000 CE) or 29.530583 days (the estimated synodic month in 3000 CE).

These cycles do not necessarily use a year of 12 or 13 months.

Peter Meyer sent me the results and I saw that all the cycles listed could be constructed by variations of my Yerm Calendar, where each yerm has 17 or 15 months alternating between 30 and 29 days, so that the first and last month have 30 days.

Each of these cycles comes in one of three categories:

1. Basic Yerm Cycle of 3 yerms, 49 months and 1447 days
2. Pure Yerm Cycles: Various adjustments of the Yerm Calendar Cycle of 3.n+1 yerms, 49.n+17 months and 1447.n+502 days.
3. Mixed Yerm Cycles: Mixtures of pure yerm cycles.
I marked the entries that were mixed yerm cycles with that mixture at the end. Then I ordered the entries according to the length of the mean month. The result is shown below. It forms a comprehensive list of relatively short lunar calendar cycles.

The number of short months (29 days) is n1 and long months (30 days) is n2. Since every yerm has exactly one long month more than short months, n2-n1 is the number of yerms in the cycle.

```    n1   n2  n1+n2  n2-n1    av. mon.   diff.2000   diff.3000
653  738  1391    85   29.530554  -0.000120%  -0.000100% (19:22:22:22)
822  929  1751   107   29.530554  -0.000119%  -0.000098% (19:22:22:22:22)
169  191   360    22   29.530556  -0.000113%  -0.000093%
868  981  1849   113   29.530557  -0.000108%  -0.000088% (22:22:22:22:25)
699  790  1489    91   29.530557  -0.000107%  -0.000087% (22:22:22:25)
530  599  1129    69   29.530558  -0.000105%  -0.000085% (22:22:25)
361  408   769    47   29.530559  -0.000101%  -0.000081% (22:25)
553  625  1178    72   29.530560  -0.000097%  -0.000077% (22:25:25)
745  842  1587    97   29.530561  -0.000095%  -0.000075% (22:25:25:25)
192  217   409    25   29.530562  -0.000090%  -0.000070%
791  894  1685   103   29.530564  -0.000085%  -0.000065% (25:25:25:28)
599  677  1276    78   29.530564  -0.000084%  -0.000063% (25:25:28)
407  460   867    53   29.530565  -0.000081%  -0.000060% (25:28)
622  703  1325    81   29.530566  -0.000078%  -0.000057% (25:28:28)
837  946  1783   109   29.530566  -0.000076%  -0.000056% (25:28:28:28)
215  243   458    28   29.530568  -0.000072%  -0.000052%
883  998  1881   115   29.530569  -0.000068%  -0.000048% (28:28:28:31)
668  755  1423    87   29.530569  -0.000067%  -0.000047% (28:28:31)
453  512   965    59   29.530570  -0.000065%  -0.000044% (28:31)
691  781  1472    90   29.530571  -0.000062%  -0.000042% (28:31:31)
238  269   507    31   29.530572  -0.000058%  -0.000037%
737  833  1570    96   29.530573  -0.000053%  -0.000033% (31:31:34)
499  564  1063    65   29.530574  -0.000051%  -0.000031% (31:34)
760  859  1619    99   29.530574  -0.000049%  -0.000029% (31:34:34)
261  295   556    34   29.530576  -0.000046%  -0.000025%
806  911  1717   105   29.530577  -0.000042%  -0.000022% (34:34:37)
545  616  1161    71   29.530577  -0.000040%  -0.000020% (34:37)
829  937  1766   108   29.530578  -0.000039%  -0.000018% (34:37:37)
284  321   605    37   29.530579  -0.000036%  -0.000015%
875  989  1864   114   29.530579  -0.000033%  -0.000012% (37:37:40)
591  668  1259    77   29.530580  -0.000031%  -0.000011% (37:40)
307  347   654    40   29.530581  -0.000027%  -0.000007%
637  720  1357    83   29.530582  -0.000023%  -0.000003% (40:43)
330  373   703    43   29.530583  -0.000020%   0.000001%
683  772  1455    89   29.530584  -0.000016%   0.000004% (43:46)
353  399   752    46   29.530585  -0.000013%   0.000007%
729  824  1553    95   29.530586  -0.000010%   0.000010% (46:49)
376  425   801    49   29.530587  -0.000008%   0.000013%
775  876  1651   101   29.530588  -0.000005%   0.000015% (49:52)
399  451   850    52   29.530588  -0.000003%   0.000018%
821  928  1749   107   29.530589  -0.000000%   0.000020% (52:55)
422  477   899    55   29.530590   0.000002%   0.000022%
867  980  1847   113   29.530590   0.000004%   0.000024% (55:58)
445  503   948    58   29.530591   0.000006%   0.000026%
468  529   997    61   29.530592   0.000009%   0.000030%
491  555  1046    64   29.530593   0.000013%   0.000033%
514  581  1095    67   29.530594   0.000016%   0.000036%
537  607  1144    70   29.530594   0.000018%   0.000039%
560  633  1193    73   29.530595   0.000021%   0.000041%
583  659  1242    76   29.530596   0.000023%   0.000043%
606  685  1291    79   29.530596   0.000025%   0.000046%
629  711  1340    82   29.530597   0.000027%   0.000047%
652  737  1389    85   29.530598   0.000029%   0.000049%
675  763  1438    88   29.530598   0.000031%   0.000051%
698  789  1487    91   29.530599   0.000032%   0.000053%
721  815  1536    94   29.530599   0.000034%   0.000054%
744  841  1585    97   29.530599   0.000035%   0.000055%
767  867  1634   100   29.530600   0.000036%   0.000057%
790  893  1683   103   29.530600   0.000038%   0.000058%
813  919  1732   106   29.530600   0.000039%   0.000059%
836  945  1781   109   29.530601   0.000040%   0.000060%
859  971  1830   112   29.530601   0.000041%   0.000061%
882  997  1879   115   29.530601   0.000042%   0.000062%
23   26    49     3   29.530612   0.000079%   0.000099% (basic)
```
Of these the following have a whole number of weeks
```    n1   n2  n1+n2  n2-n1    av. mon.   diff.2000   diff.3000
453  512   965    59   29.530570  -0.000065%  -0.000044% (28:31)=(19:40)
307  347   654    40   29.530581  -0.000027%  -0.000007%
775  876  1651   101   29.530588  -0.000005%   0.000015% (49:52)=(40:61)
468  529   997    61   29.530592   0.000009%   0.000030%
629  711  1340    82   29.530597   0.000027%   0.000047%
790  893  1683   103   29.530600   0.000038%   0.000058%
```
and the following have a whole number of 12-month "years".
```    n1   n2  n1+n2  n2-n1    av. mon.   diff.2000   diff.3000
169  191   360    22   29.530556  -0.000113%  -0.000093%
445  503   948    58   29.530591   0.000006%   0.000026%
721  815  1536    94   29.530599   0.000034%   0.000054%
```
The first of these, which has 22 yerms of 360 months, produces the 30-year cycle of 19 common (354-day) years and 11 leap (355-day) years. This forms the tabular Islamic calendar in which months are worked out by rules rather than by the first visible crescent. The second more accurate cycle, produces 79 years with 29 leap years. Combining these two produces two 40-yerm cycles of 109 years with 40 leap years. This cycle has a whole number of weeks.

The 40-yerm cycle would be a good adjustment of my Yerm calendar for the year 3000 CE.