Mixing the Primary Lunisolar Cycles

Dear Calendar People

Some time ago I stated to the list that all accurate lunisolar calendar cycles can be made up three of three primary lunisolar cycles:

(1)     391 years, 4836 months, 142,810 days

(2)     334 years, 4131 months, 121,991 days

(3)     315 years, 3896 months, 115,051 days

For examples of lunisolar cycles see http://www.the-light.com/cal/kp_Lunisolar_xls.html  some of which are shown as a mixture of primary cycles in http://www.the-light.com/cal/Lunisolar_333.html .

I’ve found a way of deriving a equation that the numbers of each type of primary cycle must satisfy for a given mean year or for a given mean month.

Suppose a given lunisolar cycle has P 391-year cycles (1), Q 334-year cycles (2) and R 315-year cycles (3),  then for a mean year of 365 + E/A days

P*(E*391 – A*95) + Q*(E*334 – A*81) + R*(E*315 – A*76) = 0

This gives rise to examples such as

Mean year 365 8/33 days:   12*R = 7*P + Q

Mean year 365 31/128 days: 37*R = 39*P + 14*Q

Mean year 365 39/161 days: 49*R = 46*P + 15*Q

Note that the third equation is the sum of the first two. This applies in general whenever the numerator and denominator of the fractional part of the mean year both add up (as in this case). Other examples are

Mean year 365 71/293 days:   97*R = 74*P + 19*Q

Mean year 365 97/400 days:  155*R = 73*P + 2*Q

Mean year 365 25/103 days:   47*R = 10*P – 7*Q

Similarly, for a mean lunar month of (59 + F/J)/2 days:

P*(F*4836 – J*296)/2 + Q*(F*4131 – J*253)/2 + R*(F*3896 – J*238)/2 = 0

This gives rise to examples such as

Mean month 29 451/850 days (F=52, J=850): 146*R =  64*P + 119*Q

Mean month 29 512/965 days (F=59, J=965):  97*R = 158*P + 208*Q

The latter equation arises from the Vij tithi of 2/59 lunation equated to 966/965 days. Every lunisolar cycle based strictly on this tithi is made up of primary lunisolar cycles whose numbers (P, Q and R) satisfy this latter equation.

The lunar equations add up in a similar manner to the solar equations. The following equation is useful in this adding.

Mean month 29 26/49 days (F=3, J=49):      13*R =   2*P -   2*Q   

One can find a lunisolar cycle with a desired mean year and mean month by solving the corresponding solar and lunar equations as simultaneous equations. The resulting cycle is unique when common divisors have been cancelled out, but probably  very long (perhaps millions of years).

Karl

10(07(23

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