The 4016 Luni-solar cycle

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The 4016 Luni-solar cycle

Helios
Dear Calendar People,

The 4016 year luni-solar cycle is a convergent continued fraction value that is comparable to the mean year of the Gregorian calendar. It's the longest cycle that today's lunar tables can contain. The cycle has 49671 months.
We can fashion an Octaeteris lunisolar calendar that adds 3 leap months every 8 years.
This requires that a leap month be omitted every 18 & 16/27 octaeterides. That's 27 times every 4016 years.

Similarly, a year can be divided into 99 parts of eighth-months. One of these eighth-months drop off occasionally.

1/[ 99 - 8*( 49671/4016  )] = 18 & 16/27 years

That's 27 times every 502 years.

Then the formulation goes as follows,

[ 27*O + 40 ]MOD( 502 ) < 27
0, 19, 37, 56, 74, 93, 112, 130, 149, 167, 186, 205, 223, 242, ...

for the correction of both cases, years or octaeterides.
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Re: The 4016 Luni-solar cycle

Irv Bromberg
From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Helios [[hidden email]]
Sent: Monday, December 26, 2016 22:00

The 4016 year luni-solar cycle is a convergent continued fraction value that is comparable to the mean year of the Gregorian calendar.
...
The cycle has 49671 months.
...
We can fashion an Octaeteris lunisolar calendar that adds 3 leap months every 8 years. This requires that a leap month be omitted every 18 & 16/27 octaeterides. That's 27 times every 4016 years.



Irv replies:

Helios didn't specify the mean month that he used to reckon that this cycle has a mean year comparable to the Gregorian calendar.

The exact value = 4016 years * (365+97/400 days per year) / 49671 months = 29+73208/137975 days per month = 29d 12h 44m 2+4850/5519 seconds (about 2.879 seconds). This would require an inconveniently long lunar cycle of 137975 months having 73208 full months per lunar cycle = 8441 yerms. That mean month is a tad too long for the present era, and therefore is inappropriate for use into the future.

For comparison, the following two currently accurate short lunar cycles yield mean years that are slightly shorter than the Gregorian mean year:

52 yerms = 29+451/850 days per month = 29d 12h 44m 2+14/17 seconds or about 2.824 seconds for mean year 365d 5h 49m 11+1351/4267 seconds or about 11.317 seconds.

49 yerms = 29+425/801 days per month = 29d 12h 44m 2+62/89 seconds or about 2.697 seconds for mean year 365d 5h 49m 9+16689/22339 seconds or about 9.747 seconds.

With the currently intentionally slightly short mean month (for better long-term lunar cycle accuracy) of the 25-Saros cycle the mean year is more accurate (relative to the mean northward equinoctial year) but still slightly longer than ideal:

341 yerms = 29+2958/5575 days per month = 29d 12h 44m 2+82/223 seconds or about 2.368 seconds for mean year 365d 5h 49m 5+38003/55973 seconds or about 5.679 seconds, which is almost the same as the mean year of the Dee calendar.

A smoothly spread lunisolar leap cycle would have mainly metonic subcycles with the occasional subcycle truncated to remove an octaeteris, so I suspect that Helios' proposed octaeteris-based cycle would appreciably increase the jitter (on top of the already large ~30-day minimum jitter of a smoothly spread lunisolar cycle).

-- Irv Bromberg, Toronto, Canada

http://www.sym454.org/lunar/
http://www.sym454.org/leap/
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Re: The 4016 Luni-solar cycle

Karl Palmen

Dear Irv, Helios and Calendar People

 

Here I reply to another note sent while I was away.

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Irv Bromberg
Sent: 27 December 2016 19:12
To: [hidden email]
Subject: Re: The 4016 Luni-solar cycle

 

From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Helios [[hidden email]]
Sent: Monday, December 26, 2016 22:00

The 4016 year luni-solar cycle is a convergent continued fraction value that is comparable to the mean year of the Gregorian calendar.
...
The cycle has 49671 months.
...
We can fashion an Octaeteris lunisolar calendar that adds 3 leap months every 8 years. This requires that a leap month be omitted every 18 & 16/27 octaeterides. That's 27 times every 4016 years.



Irv replies:

Helios didn't specify the mean month that he used to reckon that this cycle has a mean year comparable to the Gregorian calendar.

 

KARL REPLIES:  At first I thought Helios was referring to the 4160-year cycle, but this is not a typo, because the number of months 49671 fits in with it. The cycle is equal the 216 Metonic cycles of which 11 are truncated by removing an octaeteris. This works out to be 378.1818… years per truncation and so suits a long mean year such as 365.2425 days and with that, it has a mean month of 29.53058887… days, which I reckon is 2.879 seconds more than 29 days 12 hours and 44 minutes as Irv calculated next.


IRV CONTUNED:
The exact value = 4016 years * (365+97/400 days per year) / 49671 months = 29+73208/137975 days per month = 29d 12h 44m 2+4850/5519 seconds (about 2.879 seconds). This would require an inconveniently long lunar cycle of 137975 months having 73208 full months per lunar cycle = 8441 yerms. That mean month is a tad too long for the present era, and therefore is inappropriate for use into the future.

 

KARL REPLIES:  The complete cycle is 100,400 years equal to twenty-five 4016-year cycles and also nine 137,975-month lunar cycles as found by Irv.

 

Helios’s error was to go too far with the continued fraction approximation to today’s value of the mean synodic month, so getting the excessively long lunisolar cycle 100,400 years. If he stopped at an earlier convergent, he might have found one of the cycles listed in

http://the-light.com/cal/Lunisolar400.html

 


IRV CONTINUED:
For comparison, the following two currently accurate short lunar cycles yield mean years that are slightly shorter than the Gregorian mean year:

52 yerms = 29+451/850 days per month = 29d 12h 44m 2+14/17 seconds or about 2.824 seconds for mean year 365d 5h 49m 11+1351/4267 seconds or about 11.317 seconds.

49 yerms = 29+425/801 days per month = 29d 12h 44m 2+62/89 seconds or about 2.697 seconds for mean year 365d 5h 49m 9+16689/22339 seconds or about 9.747 seconds.

 

KARL REPLIES: These two cycles would form longer lunisolar cycles with a mean year of exactly 365.2425 days. For 52 yerms, it would be 1,115,600 years = 16,233 52-yerm cycles. The 16,233 is equal to the number of days in the Gregorian 400-year cycle divided by 9, which is the highest common divisor with the number of days in the 52-yerm cycle. The cycle for 49 yerms would be longer, because its number of days has no common divisor with the number of days in a Gregorian 400-year cycle.

 


IRV CONTINUED:
With the currently intentionally slightly short mean month (for better long-term lunar cycle accuracy) of the 25-Saros cycle the mean year is more accurate (relative to the mean northward equinoctial year) but still slightly longer than ideal:

341 yerms = 29+2958/5575 days per month = 29d 12h 44m 2+82/223 seconds or about 2.368 seconds for mean year 365d 5h 49m 5+38003/55973 seconds or about 5.679 seconds, which is almost the same as the mean year of the Dee calendar.

 

KARL REPLIES: For a lunisolar calendar, one would like a small multiple of the lunar cycle to be a whole number of years.

 

Irv omitted to state that this cycle is one quarter of a 1803-year cycle as listed in

http://the-light.com/cal/LunisolarA.htm  The number of yerms and months tally.

This 341-yerm cycle has 8 eras; 7 of 43-yerms and one of 40-yerms.

Yerms: 341 = 7*43 + 40.

Months: 5575 = 7*703 + 654.

 

Helios has found numerous lunisolar cycles that are each a multiple of a shorter lunisolar cycle and I put them with some other such cycles into

http://the-light.com/cal/LunisolarML.htm

I’ve also listed some cycles that are an exact multiple of the Dee 33-year cycle at

http://the-light.com/cal/Lunisolar33.html

 

The columns in these  links are explained at

http://the-light.com/cal/kp_Lunisolar_xls.html which also links other lists of lunisolar cycles.

 


IRV CONTINUED:
A smoothly spread lunisolar leap cycle would have mainly metonic subcycles with the occasional subcycle truncated to remove an octaeteris, so I suspect that Helios' proposed octaeteris-based cycle would appreciably increase the jitter (on top of the already large ~30-day minimum jitter of a smoothly spread lunisolar cycle).

 

KARL REPLIES:

I addressed this issue in the thread about the Olympic calendar, which started soon after the note I’m replying to.

 

If the Octaeteris was corrected 27 times per 4016-year cycle as in the Olympic Calendar, the number of days in the 4016-year cycle would be

2923.5*502 – 27*29 = 1466814, giving mean year of 365.24252988… days and a mean month of 29.53059129… days.

This is 0.12 more days than the 1466813.88 days for the Gregorian mean year.

 

I think there is a unique cycle that has a mean year of exactly 365.2425 days and can be generated by the Olympic Calendar technique.

It would satisfy (3/8)*A – B = C – (3/16)*A = number of corrections of octaeteris, where A B & C are the first three columns of my spreadsheets.

This simplifies to (9/16)*A – B = C.

I find it’s the 11,600-year cycle, which I use for my Annuary Calendar (A=11600, B=4272, C=2253). It has 78 corrections (one per 18 & 23/39 octaeterides).

http://www.hermetic.ch/cal_stud/palmen/anry.htm

 

Karl

 

16(07(02

-- Irv Bromberg, Toronto, Canada

http://www.sym454.org/lunar/
http://www.sym454.org/leap/