Eclipse Seasons Epacts RE: Nodetides

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Eclipse Seasons Epacts RE: Nodetides

Karl Palmen
Dear Helios and Calendar People

I thought about how the eclipse cycle tithi scheme mentioned below could be applied to the Gregorian epacts.

Firstly I decided that it would be inconvenient to count the eclipse seasons. Instead the leap eclipse seasons could be determined by the year number and the leap tithi can be inserted at the end of the year. About one in 7.3 years would have a leap tithi.

Then each year would have an eclipse season epact between 0 and 175 to determine the age of the eclipse season at the start of the year.
The epact of days table can be used to schedule the tithis throughout each year with one tithi on each day with one epact and two tithis on each day with two epacts.

Secondly, I thought it would be convenient to have 371 eclipse cycle tithis in every year without exception, so that the eclipse epact increments don't depend on the lunar epact increments. This would change the leap eclipse season rule, reducing their number, so enabling a simpler rule for them. We'd then have an eclipse epact increment of 19 mod 176 after each year without a leap tithi. A leap tithi reduces this increment to 18 mod 176.

I then look at the 372-year, 391-year and 4160-year cycles to see how often the leap eclipse seasons should occur.

372 years, 784 seasons, 138,012 tithis,  28 leap eclipse seasons  
391 years, 824 seasons,  145,061 tithis,  37 leap eclipse seasons
4160 years, 8767 seasons,  1,543,360 tithis, 368 leap eclipse seasons

This gives years per leap eclipse season of 13.28... , 10.56... and 11.30... respectively.
This suggests 1 in 11 years to have a leap tithi inserted to form a leap eclipse season.

Then the total epact increment over 11 years is 208, so giving 23 32/176 = 23 2/11 eclipse seasons to 11 years.
121 years then have 255 eclipse seasons and so we then have the 121-year cycle, which Helios has mentioned.

I show some years whose epact [] relative to year 0 is near 0 for the 121-year cycle symmetrical about 0.
000[0] 009[-6] 019[+7] 028[+1] 037[-4] 047[+9] 056[+3] 065[-3] 074[-9] 84[+4] 93[-1] 102[-7] 112[+6] 121[0].


It would be convenient to have a formula for the number L of leap eclipse seasons in a cycle of Y years and N nodetides.

The number of tithis is 371*Y
The number of eclipse seasons is 2*Y + N
So the number of tithis is also 176*(2*Y + N) + L

352*Y + 176*N + L = 371*Y
L = 371*Y - 352*Y - 176*N

L = 19*Y - 176*N

I check this with Y=121 and N=13 and get the expected 11.

Now I apply it to Y=698 and N=75 and get 62 leap eclipse seasons.
This would be achieved with a 349-year cycle in which
26 leap eclipse cycles occur 11 years after previous and
5 leap eclipse cycles occur 12 years after previous.


Helios may ask, why not have 373 instead of 371 tithis in a year and 177 tithis instead of 176 tithis in a common eclipse season, then the corrections such as leap eclipse seasons would occur only once every 53 or 54 years. The answer is that one cannot then use the Gregorian epact of days table nor even take the tithi to be 1/30 lunar month within each year.


Karl

16(03(25

-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 04 November 2016 13:10
To: 'East Carolina University Calendar discussion List'
Subject: RE: Nodetides

Dear Helios and Calendar People

Helios said in this note
" We could also fashion something like a tithi and epact for a 1/177th
eclipse season reckoning."

I did this on 19 September 2013, but with the tithi normally 1/176 eclipse season. I used the conventional tithi of 1/30 synodic month. A year normally has 371 tithis, but one in every 20 or 21 years on average has an additional tithi, which is called a saltus lunae (jump of the moon).

If every month has 30 tithis and every eclipse season has 176 tithis, the Tzolkinex of 88 months = 15 eclipse seasons would result.
For a more accurate eclipse cycle, we can occasionally have a leap eclipse season of 177 tithis.

If 1 in every 16 eclipse seasons is a leap eclipse season, the calendar would follow the Short Callippic Cycle of 939 months = 160 eclipse seasons.

If 2 in every 31 eclipse seasons is a leap eclipse season, the calendar would follow the Half-Babylonian period of 2729 months = 465 eclipse seasons.

Karl

16(03(05

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 04 November 2016 07:36
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

From the equation;

( 75*Y + 34 )MOD( 698 ) < 75

Karl said he expected 37 instead of 34 and I don't know why.

I've also looked at the approximate relation,

373 eclipse season  = 177 years = 19 nodetides

in which we can see, amid a 19-day period, where and what year a node
alignment will occur. These modular numbers look like,

047 019 168 140 112 084 056 028 000 149 121 093 065 037 009 158 130 102 074

with year 000 at the center. A correction would be needed every 53 - 54
years. We could also fashion something like a tithi and epact for a 1/177th
eclipse season reckoning.




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View this message in context: http://calndr-l.10958.n7.nabble.com/Nodetides-tp17237p17255.html
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Re: Eclipse Seasons Epacts RE: Nodetides

Karl Palmen
Dear Helios and Calendar People

I said
" Now I apply it to Y=698 and N=75 and get 62 leap eclipse seasons.
This would be achieved with a 349-year cycle in which
26 leap eclipse cycles occur 11 years after previous and
5 leap eclipse cycles occur 12 years after previous."

The last two lines were wrong and should be:

23 leap eclipse seasons occur 11 years after previous and
8 leap eclipse seasons occur 12 years after previous.

To create half a 698-year cycle.

11*23 + 8*12 = 253 + 96 = 349 = 698/2.


I then think it is not very important to have the leap eclipse seasons spaced as smoothly as possible. An eclipse season starting a day or two early or late won't matter, because there are about 35 days of an eclipse season in which an eclipse can occur.

Then we can have
30 leap eclipse seasons occur 11 years after previous and
1 leap eclipse seasons occurs 19 years after previous.
This has the advantage of preserving some of the 121-year sub-cycles of the 698-year cycle.

Also the 19-year interval normally has no change in lunar epact and the 11-year interval has a small change of normally 1 or 2 in the lunar epact. This leads to a slow drift in the lunar epact of the leap eclipse season years.


Karl

16(03(26

-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 24 November 2016 13:22
To: 'East Carolina University Calendar discussion List'
Subject: Eclipse Seasons Epacts RE: Nodetides

Dear Helios and Calendar People

I thought about how the eclipse cycle tithi scheme mentioned below could be applied to the Gregorian epacts.

Firstly I decided that it would be inconvenient to count the eclipse seasons. Instead the leap eclipse seasons could be determined by the year number and the leap tithi can be inserted at the end of the year. About one in 7.3 years would have a leap tithi.

Then each year would have an eclipse season epact between 0 and 175 to determine the age of the eclipse season at the start of the year.
The epact of days table can be used to schedule the tithis throughout each year with one tithi on each day with one epact and two tithis on each day with two epacts.

Secondly, I thought it would be convenient to have 371 eclipse cycle tithis in every year without exception, so that the eclipse epact increments don't depend on the lunar epact increments. This would change the leap eclipse season rule, reducing their number, so enabling a simpler rule for them. We'd then have an eclipse epact increment of 19 mod 176 after each year without a leap tithi. A leap tithi reduces this increment to 18 mod 176.

I then look at the 372-year, 391-year and 4160-year cycles to see how often the leap eclipse seasons should occur.

372 years, 784 seasons, 138,012 tithis,  28 leap eclipse seasons  
391 years, 824 seasons,  145,061 tithis,  37 leap eclipse seasons
4160 years, 8767 seasons,  1,543,360 tithis, 368 leap eclipse seasons

This gives years per leap eclipse season of 13.28... , 10.56... and 11.30... respectively.
This suggests 1 in 11 years to have a leap tithi inserted to form a leap eclipse season.

Then the total epact increment over 11 years is 208, so giving 23 32/176 = 23 2/11 eclipse seasons to 11 years.
121 years then have 255 eclipse seasons and so we then have the 121-year cycle, which Helios has mentioned.

I show some years whose epact [] relative to year 0 is near 0 for the 121-year cycle symmetrical about 0.
000[0] 009[-6] 019[+7] 028[+1] 037[-4] 047[+9] 056[+3] 065[-3] 074[-9] 84[+4] 93[-1] 102[-7] 112[+6] 121[0].


It would be convenient to have a formula for the number L of leap eclipse seasons in a cycle of Y years and N nodetides.

The number of tithis is 371*Y
The number of eclipse seasons is 2*Y + N
So the number of tithis is also 176*(2*Y + N) + L

352*Y + 176*N + L = 371*Y
L = 371*Y - 352*Y - 176*N

L = 19*Y - 176*N

I check this with Y=121 and N=13 and get the expected 11.

Now I apply it to Y=698 and N=75 and get 62 leap eclipse seasons.
This would be achieved with a 349-year cycle in which
26 leap eclipse cycles occur 11 years after previous and
5 leap eclipse cycles occur 12 years after previous.


Helios may ask, why not have 373 instead of 371 tithis in a year and 177 tithis instead of 176 tithis in a common eclipse season, then the corrections such as leap eclipse seasons would occur only once every 53 or 54 years. The answer is that one cannot then use the Gregorian epact of days table nor even take the tithi to be 1/30 lunar month within each year.


Karl

16(03(25