13 x 353 = 4589-year lunisolar cycle

[Irv Bromberg]

Dear Calendar People:

I was unable to find a reasonably accurate intentionally slightly  
short mean month and mean year that yielded an exact integer number of  
days in the 353-year, 4366 lunar month cycle.

Thirteen repeats of that cycle = 4589 years or 56758 lunar months,  
however, can contain exactly 1676097 days (not divisible by 7),  
yielding a mean month of 1676097/56758 = 29+30115/56758 days or 29d  
12h 44m 2+17882/28379s (the fraction is just under 2/3 second) and a  
calendar mean year of 1676097/4589 = 365+1112/4589 days = 365d 5h 48m  
56+1496/4589s (the fraction is just under 1/3 second), or about  
365.242318588 days.

The stable calendar season in the present era is at an ecliptic  
longitude of about 3°, very close to the target northward equinox.

With that mean month, 4366 lunar months (353 years) contains  
128930+7/13 days, so 13 repeats are necessary to make the fraction go  
away (adding up to one week).  In other words, in the full 4589-year  
cycle, 353-year subcycles having 128931 days alternate with 353-year  
subcycles having 128930 days, for a total of 7 + 6 = 13 of the 353-
year subcycles, respectively.

The mean month fraction numerator 30115 indicates it has that many  
full lunar months per cycle, and 56758-30115=26643 deficient lunar  
months.  The difference 30115-26643=3472 is the number of yerms per  
4589-year cycle.  The full 4589-year cycle contains a mix of 61 eras  
of 40 yerms with 24 eras of 43 yerms.

The time unit 1/56758 of a day is just a tad more than 1+1/2 seconds.
Compare with the 1/25920 of day "part" used for the traditional Hebrew  
calendar molad, which is exactly 3+1/3 seconds.

If this 4589-year cycle were used for reform of the traditional Hebrew  
calendar, with a fixed revised molad interval of 29+30115/56758 days,  
then the cycle would nicely approximate the mean northward equinox for  
at least a millennium longer than the 353-year cycle with traditional  
molad interval, but not as long as the 353-year cycle with  
progressively shorter molad interval.  Correspondence with the mean  
lunar cycle would be improved compared to the traditional molad while  
retaining similar arithmetic simplicity, but the progressively shorter  
molad would have much less drift.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Irv Bromberg]

Dear Calendar People:

The calendar mean year of the 13 x 353 = 4589-year cycle as described  
in my previous message is 259200/1344577 of a second (almost 1/5 of a  
second) shorter than a 293-year cycle having 71 leap days or 52 leap  
weeks.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Karl Palmen]

Dear Irv and Calendar People

Two 353-year cycles is not too bad. I didn't find out about it for some time and have it listed in
http://www.the-light.com/cal/Lunisolar_333.html
It has 257861 days and is formed by combining the primary cycles of 315 and 391 years.
Its mean year is 365.24221 days and mean month is 29.53057719 days.

The cycle of 13 353-year cycles with 1676097 days can be constructed on my lunisolar spreadsheets by setting the number of abundant years (column C) to 891 and is formed from the three primary cycles, by combining six 391-year cycles, two 334-year cycles and five 315-year cycles.
Its mean year is 365.24232 days and its mean month is 29.53058600 days.
As stated in another note by Irv, its mean year is very close to that of the 293-year cycle of 71 leap days or 52 leap weeks.

Karl

10(08(06 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 01 May 2009 02:51
To: CALNDR-L@...
Subject: 13 x 353 = 4589-year lunisolar cycle

Dear Calendar People:

I was unable to find a reasonably accurate intentionally slightly  
short mean month and mean year that yielded an exact integer number of  
days in the 353-year, 4366 lunar month cycle.

Thirteen repeats of that cycle = 4589 years or 56758 lunar months,  
however, can contain exactly 1676097 days (not divisible by 7),  
yielding a mean month of 1676097/56758 = 29+30115/56758 days or 29d  
12h 44m 2+17882/28379s (the fraction is just under 2/3 second) and a  
calendar mean year of 1676097/4589 = 365+1112/4589 days = 365d 5h 48m  
56+1496/4589s (the fraction is just under 1/3 second), or about  
365.242318588 days.

The stable calendar season in the present era is at an ecliptic  
longitude of about 3°, very close to the target northward equinox.

With that mean month, 4366 lunar months (353 years) contains  
128930+7/13 days, so 13 repeats are necessary to make the fraction go  
away (adding up to one week).  In other words, in the full 4589-year  
cycle, 353-year subcycles having 128931 days alternate with 353-year  
subcycles having 128930 days, for a total of 7 + 6 = 13 of the 353-
year subcycles, respectively.

The mean month fraction numerator 30115 indicates it has that many  
full lunar months per cycle, and 56758-30115=26643 deficient lunar  
months.  The difference 30115-26643=3472 is the number of yerms per  
4589-year cycle.  The full 4589-year cycle contains a mix of 61 eras  
of 40 yerms with 24 eras of 43 yerms.

The time unit 1/56758 of a day is just a tad more than 1+1/2 seconds.
Compare with the 1/25920 of day "part" used for the traditional Hebrew  
calendar molad, which is exactly 3+1/3 seconds.

If this 4589-year cycle were used for reform of the traditional Hebrew  
calendar, with a fixed revised molad interval of 29+30115/56758 days,  
then the cycle would nicely approximate the mean northward equinox for  
at least a millennium longer than the 353-year cycle with traditional  
molad interval, but not as long as the 353-year cycle with  
progressively shorter molad interval.  Correspondence with the mean  
lunar cycle would be improved compared to the traditional molad while  
retaining similar arithmetic simplicity, but the progressively shorter  
molad would have much less drift.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
--
Scanned by iCritical.

[Irv Bromberg]

On 2009 May 1, at 04:22 , Palmen, KEV (Karl) wrote:

Two 353-year cycles is not too bad. I didn't find out about it for some time and have it listed in
http://www.the-light.com/cal/Lunisolar_333.html
It has 257861 days and is formed by combining the primary cycles of 315 and 391 years.
Its mean year is 365.24221 days and mean month is 29.53057719 days.

Irv replies:  I discounted the 706-year cycle because its mean year = 365d 5h 48m 47s (close to the so-called mean tropical year) and mean month = 29d 12h 44m 1.87s are both too short.

My targets were intentionally slightly shorter than the mean northward equinoctial year and slightly shorter than the mean synodic month.

On the other hand, 9 x 353 = 3177 years has has a mean year = 365d 5h 49m 1s and mean month = 29d 12h 44m 2.97, which are both too long.

Odd-numbered higher multiples (x11, x13, x15, x17...) make the mean year and mean month progressively shorter.

The x11 multiple is closer to the present-era targets (365d 5h 48m 58s and 29d 12h 44m 2.77s) but I prefer intentionally slightly short mean year and month because both the mean northward equinoctial year and the mean synodic month are getting progressively shorter.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear Irv and Calendar People

 

Thank you Irv for making the reasons behind your choice explicit.

This is necessary to avoid misleading other calendar people into believing that other multiples do not exist.

 

I note that Irv stated that this 1x353-year cycle has a mean year very close to that of the 293-year cycle. There exists a 293x353-year cycle with exactly the same mean year, I expect it would be made up of twenty 13x353-year cycles  and just three 11x353-year cycles.

 

Karl

 

10(08(07

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 01 May 2009 15:32
To: CALNDR-L@...
Subject: Re: 2 x 353 = 706-year and 13 x 353 = 4589-year lunisolar cycles

 

On 2009 May 1, at 04:22 , Palmen, KEV (Karl) wrote:

Two 353-year cycles is not too bad. I didn't find out about it for some time and have it listed in
http://www.the-light.com/cal/Lunisolar_333.html
It has 257861 days and is formed by combining the primary cycles of 315 and 391 years.
Its mean year is 365.24221 days and mean month is 29.53057719 days.

 

Irv replies:  I discounted the 706-year cycle because its mean year = 365d 5h 48m 47s (close to the so-called mean tropical year) and mean month = 29d 12h 44m 1.87s are both too short.

 

My targets were intentionally slightly shorter than the mean northward equinoctial year and slightly shorter than the mean synodic month.

 

On the other hand, 9 x 353 = 3177 years has has a mean year = 365d 5h 49m 1s and mean month = 29d 12h 44m 2.97, which are both too long.

Odd-numbered higher multiples (x11, x13, x15, x17...) make the mean year and mean month progressively shorter.

 

The x11 multiple is closer to the present-era targets (365d 5h 48m 58s and 29d 12h 44m 2.77s) but I prefer intentionally slightly short mean year and month because both the mean northward equinoctial year and the mean synodic month are getting progressively shorter.

-- Irv Bromberg, Toronto, Canada

 

<http://www.sym454.org/>

 

---

Scanned by iCritical.

[Irv Bromberg]

On 2009 May 1, at 12:14 , Palmen, KEV (Karl) wrote:

I note that Irv stated that this 1x353-year cycle has a mean year very close to that of the 293-year cycle. There exists a 293x353-year cycle with exactly the same mean year, I expect it would be made up of twenty 13x353-year cycles  and just three 11x353-year cycles.

Irv replies:  In order for the 293*353=103,429 year cycle to have a mean year of 365+71/293 days there would have to be 37,776,648 days = 5,396,664 weeks in the cycle.

With 293*4366=1,279,238 months the mean month would be 37776648/1279238 = 29+339373/639619 days = 29d 12h 44m 2+413002/639619s (the fraction is just shy of 2/3 of a second), which is very accurate for the present era.

If one takes the ratio of the mean year to the mean month:

(365 + 71/293) / (29 + 339373/639619) = 4366 / 353

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear Irv and Calendar People

 

The 13x353-year cycle has 1,676,097 days and the 11x353-year cycle has 257,861 days fewer.

 

So twenty 13x353 plus three 11x353 has 23x1,676,097 – 3x257, 861 = 37,776,648 days, which is the number Irv calculates for the 293x353.

Also 37,776,648=294*364*353 as expected.

 

This verifies my guess based on the number of years.

 

Karl

 

10(08(10 till noon

 

PS: I meant 13x353 when I typed 1x353.

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 01 May 2009 22:35
To: CALNDR-L@...
Subject: Re: 2 x 353 = 706-year and 13 x 353 = 4589-year lunisolar cycles

 

On 2009 May 1, at 12:14 , Palmen, KEV (Karl) wrote:

I note that Irv stated that this 1x353-year cycle has a mean year very close to that of the 293-year cycle. There exists a 293x353-year cycle with exactly the same mean year, I expect it would be made up of twenty 13x353-year cycles  and just three 11x353-year cycles.

 

Irv replies:  In order for the 293*353=103,429 year cycle to have a mean year of 365+71/293 days there would have to be 37,776,648 days = 5,396,664 weeks in the cycle.

 

With 293*4366=1,279,238 months the mean month would be 37776648/1279238 = 29+339373/639619 days = 29d 12h 44m 2+413002/639619s (the fraction is just shy of 2/3 of a second), which is very accurate for the present era.

 

If one takes the ratio of the mean year to the mean month:

 

(365 + 71/293) / (29 + 339373/639619) = 4366 / 353

 

-- Irv Bromberg, Toronto, Canada

 

<http://www.sym454.org/>

---

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