Eclimpiad in Years RE: Year/Pentalunex Ratio

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Eclimpiad in Years RE: Year/Pentalunex Ratio

Karl Palmen
Dear Helios and Calendar People

I've named the mean period of time each pentalunex occurs later than the previous pentalunex, an eclimpiad, because it is nearly 4 years.

I have previously calculated this as follows.

Mean eclimpiad = m/( 6*m/e - 1), where e is the mean eclipse season and m the mean synodic month.

For a mean synodic month of 29.530588 days and a mean eclipse season interval of 173.31004 days, we get:

mean eclimpiad = 44.7426.... months = 1321.276... days = 3.61753... Gregorian years = 7.62377... eclipse seasons.

Helios came up with the approximation of 123/34 years from a continued fraction method applied to an unknown value.
This works out as 3.617647... years. Given it has 1321.276 days this implies a year of 365.23077.. days, which is somewhat short for this value of the eclimpiad.

Helios gives the 4551-year cycle as an example that uses 123/34, it has 56288/1258 = 44.7440...  months per eclimpiad. For a mean month of 29.530588 days this works out as 1321.31775... days and so the mean year is then 365.24231... days, which is accurate.

I now do calculations for some of the lunisolar eclipse cycles I recently listed:

372 years, 4601 months, 784 seasons, 103 eclimpiads; Eclimpiad = 3.61165... years
391 years, 4836 months, 824 seasons, 108 eclimpiads; Eclimpiad = 3.62037... years
1154 years, 14273 months, 2432 seasons,  319 eclimpiads; Eclimpiad = 3.61755... years
4160 years, 51452 months, 8767 seasons, 1150 eclimpiads; Eclimpiad = 3.61739... years
4551 years, 56288 months, 9591 seasons, 1258 eclimpiads; Eclimpiad = 3.61765... years

I see that while 4551 has 123/34, 4160 has 416/115. The 416/115 is a mix of three 123/34 and one 47/13.

In an earlier note, I worked out that by using the approximation
Years = 18.03*S + 28.945*I, where S is the number of Saros cycles and I the number of Inex cycles.
That a Saros has a mean eclimpiad of 3.606 years and an Inex had a mean eclimpiad of 3.618125 years.

This led me to find simpler approximations, 65/18 and 47/13, both have a mean eclimpiad less than 123/34. There is no larger one with a smaller denominator less than 76/21 = 3.619..., which exceeds the Saros-Inex range reckoned above.



-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 03 December 2015 10:33
To: [hidden email]
Subject: Year/Pentalunex Ratio

Dear Calendar People,

A number  I've meant to calculate is the Year/ Pentalunex Ratio. For
example, 19 / 5 = 3.8 years. After inspecting the recent "accurate eclipse"
list, I approximated this ratio and looked at the continued fraction
make-up. It turns out that there's really no significant improvement better
than 123/34. There are 34 Pentalunex every 123 years.

This is within one of the listed;

4551 years = 1662218 days = 56288 months = 9591 E-seasons

total Pentalunexs = 1258 = ( 34/123 )*4551

P = 6*E - M

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