Re: 5515-Year Luni-Solar Cycle

Dear Helios and Calendar People

Helios's 5515-year cycle is a multiple of the 1749-month cycle that he
has previously mentioned, which can be made up from a 52-yerm cycle and
a 55-year cycle. There are 39 1749-month cycles in a 5515-year cycle.

Month M=1 is the first month of yerm 27 of a 52-yerm cycle in the
alternating 52-yerm 55-yerm cycles.

The 5515-year cycle can be generated in my lunisolar spread sheets
http://www.the-light.com/cal/kp_Lunisolar_xls.html 
with A=5515, B=2031, C=1071.
The spreadsheet then tells me (in column F) that the lunar calendar of M
has 4173 yerms in each 5515-year cycle as counted by Helios.

Also it tells us that a solar calendar with the same mean year would
have 1336 leap days every 5515-year cycle.
This shows that it is NOT made up of 166 33-year cycles and a 37-year
cycle, which would have 166*8+9=1337 leap days. Instead it is made up of
32 29-year cycles and 139 33-year cycles.

If a solar calendar were run alongside the lunisolar calendar with year
1 starting on the same day and a year Y is a leap year if

(1136*Y + 2757) MOD (5515) < 1136

Then the lunisolar new year day would be the nearest new month day to
solar new year day, with possible exceptions when solar new year day is
very close to a full moon.


While Helios's suggestion has minimum lunar jitter, it suffers from the
difficulty of determining the month number M for a given month of year
Y. I expect that the third formula can be converted into a formula give
M for month N of year Y. This can then be substituted into the second
formula. To find the number of days in the months of year Y, if is
sufficient to when the next yerm after the new year begins.

What is a trecena?

Karl

10(02(26

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 22 November 2008 02:29
To: CALNDR-L@...
Subject: 5515-Year Luni-Solar Cycle

5515-Year Luni-Solar Cycle

Here's the decomposition is smaller, more well-known cycles;
5515 years = 166*( 33 years ) + 37 years
5515 years = 7*( 334 years ) + 9*( 353 years )

The cycle of entirety
= 4173 yerms
= 5515 years
= 68211 months
= 154947 trecena
= 2014311 days

Mean year = 365 & 1336 / 5515 days = 365.2422484 days
Mean month = 29 & 928  / 1749 days = 29.5305889 days
13-month years = 2031

For a month count M and year count Y...

A luni-solar year has 13 months if
( 2031*Y + 2757 )MOD( 5515 ) < 2031

An ecclesiastical month has 30 days if
( 928*M + 874 )MOD( 1749 ) < 928

A month is a new year's month if
( 5515*M + 65453 )MOD( 68211 ) < 5515

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