Lunisolar multiples of the 850-month cycle RE: Mean Year of 353-year lunisolar cycle

Dear Helios and Calendar People

Now that Helios has found a lunisolar multiple of the 899-month cycle of
55 yerms, could we find a similar lunisolar multiple of the 850-month
cycle of 52 yerms as in my Yerm Lunar Calendar
http://www.hermetic.ch/cal_stud/palmen/yerm1.htm ?

I'm ware that the 334-year cycle has 4131=17*243 months and the
850-month cycle has 17*50 months and so we can construct 50 334-year
cycles with 243 850-month cycles. The resulting mean year is about
365.2420958 days. The resulting cycle of 16,700 years is rather long, so
how could I find a shorter cycle?

Using http://the-light.com/cal/converter/ , I found that 29 850-month
cycles is within a day or two of 1993 years.
 21-01(01(01 = 11 Nov 1996
 50-01(01(01 = 12 Nov 3989
 79-01(01(01 = 13 Nov 5982
108-01(01(01 = 14 Nov 7975.
Can we get a shorter cycle by adding this rough 1993-year cycle to the
16,700-year cycle and divide out a common divisor?
The answer is in the determinant of the two cycles, which is 29*16,700 -
243*1993 = 1.
The answer is no, because the determinant of 1 indicates that no common
divisor can occur and that the 16,700-year and 1993-year cycles are
mixer cycles for all cycles whose mean year lies in between.

Can we get a much shorter cycle by subtracting this rough 1993-year
cycle to the 16,700-year cycle and divide out a common divisor?
The resulting mean year would be less than the 365.2420958 days of the
16,700-year cycle.
I think the determinant of 1 indicates an answer no, but I'm not yet so
sure.

So it seems that the 16,700-year cycle is the best multiple of the
850-month cycle you can get. I investigate this further:

I think the determinant of 1 indicates that the 16,700-year cycle is
made up of a number of 1993-year cycles and just one other shorter
cycle, such that both cycles are mixer cycles. I also think the shorter
cycle is 756 years of 11 859-month cycles.
Indeed, 16,700 = 8*1993 + 756 and 243 = 8*29 + 11, so confirming what I
also think. Also 29*756 - 11*1993 = 1 and so the 1993-year and 756-year
cycles are indeed mixers. The mix needs to be close to 8 to 1 for an
accurate cycles and therefore an accurate cycle cannot be much shorter
than 16,700 years.

Similar work could be done to see if the 6707-year, 6251-year or
5515-year cycles can be shortened much while keeping them as multiples
of the same lunar calendar cycle without much loss of accuracy. I expect
the answer is no in all three cases.

Karl

10(08(18

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 11 May 2009 14:18
To: CALNDR-L@...
Subject: Re: Mean Year of 353-year lunisolar cycle

There is one more lunisolar cycle made up of the same repetition of
one lunar calendar cycle listed in Karl P,'s "Lunar Calendar Cycles less
than 1885 Months"

This is the case of the 55 yerm era with 899 months with
mean month = 29 & 477 / 899 days
mean year = 365 & 1513 / 6251 days

6251 years = [ 2*43 | 2*2*6637 ] = 77314 months

The 6251-year cycle is also a multiple of 19-years but not listed in
http://www.the-light.com/cal/Lunisolar19.html of
http://www.the-light.com/cal/kp_Lunisolar_xls.html .


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