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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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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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Re: 5515-Year Luni-Solar Cycle
Here's the correction to the 5k515 ( or V_DXV ) lunisolar and solar cycle
5515 years = 7*( 334 years ) + 9*( 353 years )
5515 years = 32*( 29 years ) + 139*( 33 years )
The cycle of entirety
= 4173 yerms
= 5515 years
= 68211 months
= 154947 trecena*
= 2014311 days
*A trecena is a 13-day period, so there is,
A trecena based year that has 29 trecena if
( 527*Y + 2757 )MOD( 5515 ) < 527
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Lunisolar Cycles made up of Short Lunar Cycles RE: 5515-Year Luni-Solar Cycle
Dear Helios, Victor and Calendar People
I wonder what other lunisolar cycles (besides Helios's 5515-year cycle)
has
(1) months made up of a repetition of one lunar calendar cycle listed in
"Lunar Calendar Cycles less than 1885 Months"
http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm ,
(2) fewer than 10,000 years and
(3) a mean year between 365.2416 and 365.2428 days.
The only other example I know of is
(a) 725-year cycle made up from 183 49-month-cycles of 1447 days (3
yerms),
Mean year 365.24276 days, mean month 29.5306122 days (A=725, B=267,
C=141).
I see that this example can also be made up of Helios Cycles. The
49-month cycle would then have yerms of 17,15,17 months in that order as
suggested in Sir Isaac Newton's unpublished lunar calendar.
Can any calendar person find another example?
A,B and C in my example are as defined in
http://www.the-light.com/cal/kp_Lunisolar_xls.html
In any example, the number of Yerms (column F) must have a large common
divisor with the number of months (column J), so that when both these
numbers are divided by this common divisor, you get the number of yerms
and months in one of the lunar calendar cycles listed. The same also
applies to the number of days (column I) and the number of months
(column J).
Karl
10(02(27
-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto: CALNDR-L@...] On Behalf Of Helios
Sent: 25 November 2008 04:40
To: CALNDR-L@...
Subject: Re: 5515-Year Luni-Solar Cycle
Here's the correction to the 5k515 ( or V_DXV ) lunisolar and solar
cycle
5515 years = 7*( 334 years ) + 9*( 353 years )
5515 years = 32*( 29 years ) + 139*( 33 years )
The cycle of entirety
= 4173 yerms
= 5515 years
= 68211 months
= 154947 trecena*
= 2014311 days
*A trecena is a 13-day period, so there is,
A trecena based year that has 29 trecena if
( 527*Y + 2757 )MOD( 5515 ) < 527
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Re: Lunisolar Cycles made up of Short Lunar Cycles RE: 5515-Year Luni-Solar Cycle
5515-Year Luni-Solar Cycle
Here's one more observation of the 5k515 ( or V_DXV ) lunisolar and solar cycle
The cycle can now include the
octaeteris year = 365 & 187 / 424 days
The cycle of entirety
= 4173 yerms
= 5512 octaeteris years
= 5515 years
= 68211 months
= 154947 trecena
= 2014311 days
This proportionality constant K = 5512 / 5515 could just serve to define this lunisolar year,
Y = K*( octaeteris year )
I don't know yet if the octaeteris leap-moon function
( 3*Y + 3 + ?? )MOD( 8 ) < 3
2, 5, 7, 10, 13, 15, 18, 21, ??, ??
can be tweaked and functionally related to K to yield the same output as
( 2031*Y + 2757 )MOD( 5515 ) < 2031
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Re: 5515-Year Luni-Solar Cycle
Dear Helios, Victor and Calendar People
So the Octaeteris new year would drift through all the seasons
(spring->summer->fall->winter) three times in the course of one
5515-year cycle.
The Octaeteris has eight years (an even-number) so cannot be made into a
Helios cycle. This may be why Helios cannot find a leap-moon function
for it. However the three leap months can be spread perfectly evenly
through the 99 months of Octeateris exactly one every 33 months. To form
a Helios cycle of months, select the 17th, 50th and 83rd month as leap
months in the 99-month cycle.
The 5515-year cycle has 2014311 = 3 * (13^2) * 29 * 137 days and 5512 =
(2^3) * 13 * 53. So there is a common divisor of 13 for the 5512
Octeateris years. So the octaeteris cycle would repeat once every 424
octaeteris years = 53 octaeterides = three 1749-month cycles. This
arises from the fact that 1749 is divisible by 33=99/3 so makes up
1749/99 = 17 2/3 octaeterides = 141 1/3 years, which is one third of 424
Octeateris years.
I realise now that a lunar calendar using the 1749-month cycle could
have a 13-month solar calendar run alongside it so that three 1749-month
cycles of 5247 lunar months are exactly equal to 5515 solar months. The
5155 solar months (424 years and three months) would have 527 days in
addition to 28 days per month. In this period, there would be
5515-5247=268 solar months that do not contain the first day of a lunar
month. A 527/5515 calendar like Victor's 43/450 or 28/293 calendar would
do this. However there may be a way of using the lunar calendar to
determine which solar month gets an extra day.
One possibility is to have two long (29-day) solar months begin in each
17-month yerm and one long solar month begin in each short 15-month
yerm. There'd be ten exceptional 17-month yerms every three 1749-month
(107-yerm) cycles with just one long solar month beginning in it. This
leads to 2*321 - 10 + 105 = 527 long solar months as required. Also
between any two consecutive exceptions, there'd be exactly 103 solar
months every 6 yerms of 98 lunar months.
Another possibility is to have a long solar month after each solar month
that misses either the 1st or 15th day of a lunar month, with three
exceptions every 1749-month cycle. This leads to 268*2 - 3*3 = 527 long
months as required.
Karl
10(03(04
-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto: CALNDR-L@...] On Behalf Of Helios
Sent: 01 December 2008 00:17
To: CALNDR-L@...
Subject: Re: Lunisolar Cycles made up of Short Lunar Cycles RE:
5515-Year Luni-Solar Cycle
5515-Year Luni-Solar Cycle
Here's one more observation of the 5k515 ( or V_DXV ) lunisolar and
solar
cycle
The cycle can now include the
octaeteris year = 365 & 187 / 424 days
The cycle of entirety
= 4173 yerms
= 5512 octaeteris years
= 5515 years
= 68211 months
= 154947 trecena
= 2014311 days
This proportionality constant K = 5512 / 5515 could just serve to define
this lunisolar year,
Y = K*( octaeteris year )
I don't know yet if the octaeteris leap-moon function
( 3*Y + 3 + ?? )MOD( 8 ) < 3
2, 5, 7, 10, 13, 15, 18, 21, ??, ??
can be tweaked and functionally related to K to yield the same output as
( 2031*Y + 2757 )MOD( 5515 ) < 2031
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 Re: 5515-Year Luni-Solar Cycle
Speaking of my 28/293 calendar, this month is the last in a series of
rectangular months because next month contains a special day. There won't be
another rectangular month until 7 more special days.
I define a rectangular month as a month where day 0 is on Sunday and the
last day of the month is on Saturday. Weeks are arranged Sunday through
Saturday as is the custom here in America.
The next special day is on -4-0-19, 09 Jan 2009 Gregorian. This is three
days after the troll in my yard solves a puzzle.
This isn't relevant to anything, but I find it interesting nonetheless. The
day number of the special day contains the same digits as the month and day
in the Gregorian calendar. Furthermore, the special day occurs on day 9 of
the Gregorian calendar and 9 days from the end of the month in the 28/293
calendar (the accumulator of the month is 9).
Victor
> -----Original Message-----
> From: East Carolina University Calendar discussion List [mailto:CALNDR-
> L@...] On Behalf Of Palmen, KEV (Karl)
> Sent: Monday, December 01, 2008 6:49 AM
> To: CALNDR-L@...
> Subject: Re: 5515-Year Luni-Solar Cycle
>
> Dear Helios, Victor and Calendar People
>
> So the Octaeteris new year would drift through all the seasons
> (spring->summer->fall->winter) three times in the course of one
> 5515-year cycle.
>
> The Octaeteris has eight years (an even-number) so cannot be made into a
> Helios cycle. This may be why Helios cannot find a leap-moon function
> for it. However the three leap months can be spread perfectly evenly
> through the 99 months of Octeateris exactly one every 33 months. To form
> a Helios cycle of months, select the 17th, 50th and 83rd month as leap
> months in the 99-month cycle.
>
> The 5515-year cycle has 2014311 = 3 * (13^2) * 29 * 137 days and 5512 =
> (2^3) * 13 * 53. So there is a common divisor of 13 for the 5512
> Octeateris years. So the octaeteris cycle would repeat once every 424
> octaeteris years = 53 octaeterides = three 1749-month cycles. This
> arises from the fact that 1749 is divisible by 33=99/3 so makes up
> 1749/99 = 17 2/3 octaeterides = 141 1/3 years, which is one third of 424
> Octeateris years.
>
>
> I realise now that a lunar calendar using the 1749-month cycle could
> have a 13-month solar calendar run alongside it so that three 1749-month
> cycles of 5247 lunar months are exactly equal to 5515 solar months. The
> 5155 solar months (424 years and three months) would have 527 days in
> addition to 28 days per month. In this period, there would be
> 5515-5247=268 solar months that do not contain the first day of a lunar
> month. A 527/5515 calendar like Victor's 43/450 or 28/293 calendar would
> do this. However there may be a way of using the lunar calendar to
> determine which solar month gets an extra day.
>
> One possibility is to have two long (29-day) solar months begin in each
> 17-month yerm and one long solar month begin in each short 15-month
> yerm. There'd be ten exceptional 17-month yerms every three 1749-month
> (107-yerm) cycles with just one long solar month beginning in it. This
> leads to 2*321 - 10 + 105 = 527 long solar months as required. Also
> between any two consecutive exceptions, there'd be exactly 103 solar
> months every 6 yerms of 98 lunar months.
> Another possibility is to have a long solar month after each solar month
> that misses either the 1st or 15th day of a lunar month, with three
> exceptions every 1749-month cycle. This leads to 268*2 - 3*3 = 527 long
> months as required.
>
>
>
> Karl
>
> 10(03(04
>
>
> -----Original Message-----
> From: East Carolina University Calendar discussion List
> [mailto: CALNDR-L@...] On Behalf Of Helios
> Sent: 01 December 2008 00:17
> To: CALNDR-L@...
> Subject: Re: Lunisolar Cycles made up of Short Lunar Cycles RE:
> 5515-Year Luni-Solar Cycle
>
> 5515-Year Luni-Solar Cycle
>
> Here's one more observation of the 5k515 ( or V_DXV ) lunisolar and
> solar
> cycle
>
> The cycle can now include the
> octaeteris year = 365 & 187 / 424 days
>
> The cycle of entirety
> = 4173 yerms
> = 5512 octaeteris years
> = 5515 years
> = 68211 months
> = 154947 trecena
> = 2014311 days
>
> This proportionality constant K = 5512 / 5515 could just serve to define
> this lunisolar year,
> Y = K*( octaeteris year )
>
> I don't know yet if the octaeteris leap-moon function
> ( 3*Y + 3 + ?? )MOD( 8 ) < 3
> 2, 5, 7, 10, 13, 15, 18, 21, ??, ??
> can be tweaked and functionally related to K to yield the same output as
> ( 2031*Y + 2757 )MOD( 5515 ) < 2031
>
> --
> View this message in context:
> http://www.nabble.com/5515-Year-Luni-Solar-Cycle-tp20632776p20764152.htm> l
> Sent from the Calndr-L mailing list archive at Nabble.com.
>
> --
> Scanned by iCritical.
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13-month Solar Calendar for 5515-Year Luni-Solar Cycle
Dear Helios, Victor and Calendar People
The 5512 octaeteris years in 5515 years lead me to realise that a
13-month solar calendar ran alongside a lunar calendar with the
1749-month cycle could produce a 5515-year cycle.
Both the lunar and solar calendars can run in a Helios cycle.
The lunar calendar can run as alternate 52-yerm and 55-yerm cycles ( see
http://www.hermetic.ch/cal_stud/palmen/yerm1.htm and
http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm )
The Helios cycle would begin with yerm 29 of each 55-yerm cycle and have
yerm 27 of each 52-yerm cycle in the middle. It is a Helios cycle for
both months and yerms.
The 13-month solar calendar can also be made into a Helios cycle.
Solar month M of a 5515-month cycle has an extra day (to 28 days) if and
only if
(527*M + 2757) mod (5515) < 527
I believe that the resulting years also form a Helios cycle, so year Y
has 366 days (rather than 365 days) if and only if
(1336*M + 2757) mod (5515) < 1336.
Each 5515-month cycle has exactly the same number of days as three
1749-month cycles and forms 1/13 of a 5515-year cycle.
Karl
10(03(05
-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto: CALNDR-L@...] On Behalf Of Helios
Sent: 01 December 2008 00:17
To: CALNDR-L@...
Subject: Re: Lunisolar Cycles made up of Short Lunar Cycles RE:
5515-Year Luni-Solar Cycle
5515-Year Luni-Solar Cycle
Here's one more observation of the 5k515 ( or V_DXV ) lunisolar and
solar
cycle
The cycle can now include the
octaeteris year = 365 & 187 / 424 days
The cycle of entirety
= 4173 yerms
= 5512 octaeteris years
= 5515 years
= 68211 months
= 154947 trecena
= 2014311 days
This proportionality constant K = 5512 / 5515 could just serve to define
this lunisolar year,
Y = K*( octaeteris year )
I don't know yet if the octaeteris leap-moon function
( 3*Y + 3 + ?? )MOD( 8 ) < 3
2, 5, 7, 10, 13, 15, 18, 21, ??, ??
can be tweaked and functionally related to K to yield the same output as
( 2031*Y + 2757 )MOD( 5515 ) < 2031
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Re: 5515-Year Luni-Solar Cycle
Dear Helios and Calendar People
Helios said about the 5515-year cycle:
The cycle of entirety
= 4173 yerms
= 5515 years
= 68211 months
= 154947 trecena (13-day periods)
= 2014311 days
But also
= 78 yerm eras
= 39 1749-month cycles
= 1248 drifts over 29-day cycle
The total number of days is 2,014,311 = 3 * (13^2) * 29 * 137.
So not only is the number (39) of 1749-month cycles is divisible by 13,
but the number of days (51649) in each 1749-month cycle.
Each 1749-month cycle
= 107 yerms
= 1749 months
= 3979 trecena
= 32 drifts over 29-day cycle
= 51649 days
= 2 yerm eras
Twenty 1749-month cycles contain a whole number of tzolkins:
= 2140 yerms
= 34980 months
= 3979 tzolkins (260-day periods)
= 640 drifts over 29-day cycle
= 40 yerm eras
= 15 1981-month cycles + 13 405-month cycles.
One curious thing is that a yerm era is approximately 100 tzolkins
(26,000 days) and this suggests the idea of starting a new yerm era at
the next opportunity after every 100 tzolkins. The resulting yerm eras
would all have either 52 yerms (25,101 days) or 55 yerms (26,548 days).
Karl
10(03(12
-----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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Re: 5515-Year Luni-Solar Cycle
A 39-week solar calendar can be devised such that
5515 weeks = 1749 months = 141 & 16 / 39 years
Each three weeks would group into 13th-of-year months ( A, B, C, ... )
In its symmetrical form...
---------
A 9, 10, 9
B 9, 10, 9
---------
C 10, 9, 9
D 10, 9, 9
E 10, 9, 9
F 10, 9, 9
---------
G 10, 9, 10
---------
H 9, 9, 10
I 9, 9, 10
J 9, 9, 10
K 9, 9, 10
---------
L 9, 10, 9
M 9, 10, 9
---------
Almost every solar month contains a new moon. Except that in 5515 years, there are 3484 orphan months, or drifts over the solar month.
There are 2031 years in which every solar month contains a new moon. This coincides with the 2031 13-month years on the luni-solar calendar.
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Re: 5515-Year Luni-Solar Cycle
Dear Helios, Victor and Calendar People
I too thought of dividing the year into 39 long weeks rather than 13
months, but didn't mention it, because the 13 months would be much
simpler than 39 long weeks.
I have some comments below.
-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto: CALNDR-L@...] On Behalf Of Helios
Sent: 12 December 2008 00:00
To: CALNDR-L@...
Subject: Re: 5515-Year Luni-Solar Cycle
A 39-week solar calendar can be devised such that
5515 weeks = 1749 months = 141 & 16 / 39 years
Each three weeks would group into 13th-of-year months ( A, B, C, ... )
In its symmetrical form...
---------
A 9, 10, 9
B 9, 10, 9
---------
C 10, 9, 9
D 10, 9, 9
E 10, 9, 9
F 10, 9, 9
---------
G 10, 9, 10
---------
H 9, 9, 10
I 9, 9, 10
J 9, 9, 10
K 9, 9, 10
---------
L 9, 10, 9
M 9, 10, 9
---------
Almost every solar month contains a new moon. Except that in 5515 years,
there are 3484 orphan months, or drifts over the solar month.
There are 2031 years in which every solar month contains a new moon.
This
coincides with the 2031 13-month years on the lunisolar calendar.
KARL SAYS: This nice pattern would be broken each leap year. Symmetry
would be preserved if the leap year rule were a Helios cycle and the
leap day added to the middle long week of month G making it into a
30-day month (10 10 10). The months themselves of would not follow a
Helios cycle. Their long weeks are not spread as evenly as possible.
Also the 30-day month could contain two new moons and so more than 2031
months may have no new moon.
If you want these long weeks to spread as evenly as possible, then every
10th or 11th month would be 10, 9, 10 as shown for G rather than every
13th, not counting leap days. The other months would follow a pattern
like that shown, but with usually three months of each other type
instead of always four.
If you had a 527/5515 calendar like Victor's 28/293 or 43/450 calendar
http://www.the-light.com/cal/ve28293.htmhttp://www.the-light.com/calendar.html ,
the type of month would be determined by its accumulator:
0000-0526: 10, 9, 10 (527 months)
0527-2189: 9, 9, 10 (1663 months)
2190-3851: 9, 10, 9 (1662 months)
3852-5514: 10, 9, 9 (1663 months)
The accumulator of month A of year 1 would be 2757+527=3284 for a Helios
cycle and the accumulator would increment 527 mod 5515 each month
thereafter.
Year 1 would go:
A (3284) 9, 10, 9
B (3811) 9, 10, 9
-----------------
C (4338) 10, 9, 9
D (4865) 10, 9, 9
E (5392) 10, 9, 9
-----------------
F (0404) 10, 9, 10
-----------------
G (0931) 9, 9, 10
H (1458) 9, 9, 10
I (1985) 9, 9, 10
-----------------
J (2512) 9, 10, 9
K (3039) 9, 10, 9
L (3566) 9, 10, 9
-----------------
M (4093) 10, 9, 9
Year 2 would be different etc. .
The number of months missing a new moon every 5515 months is
5515-3*1749=268. Over a whole 5515-year cycle this is 13*268=3484 as
calculated by Helios.
Karl
10(03(15
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Re: Lunisolar Cycles made up of Short Lunar Cycles RE: 5515-Year Luni-Solar Cycle
Here is a luni-solar cycle that misses Karl Palmen's criteria he mentioned earlier, but is near the nebulous boundry he set at 10,000 years.
(1) months made up of a 22-fold repetition of one lunar calendar cycle of 6097 months ( > 1885 )
(2) 10,845 year cycle of 134,134 months
(3) a mean year between 365.2416 and 365.2428 days.
= 154 yerm eras
= 8206 yerms
= 10845 years
= 134134 months
= 3961056 days = ( 2 ^ 5 )* 3 *( 11 ^ 3 )* 31
mean year = 365 & 877 / 3615
This mean year is close to the Gregorian year.
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Re: Lunisolar Cycles made up of Short Lunar Cycles RE: 5515-Year Luni-Solar Cycle
Dear Helios and Calendar People
Pity the lunar cycle is so long.
The number of months in the lunar cycle is odd and so the months can be
arranged into 22 Helios cycles.
The number of years is odd, so years can be arranged into a Helios
cycles according to the number of months each year has.
I can easily see that the number of months is divisible by 7, 11 and 13,
so the number of months in the lunar cycle is divisible by 7 and 13.
For my lunisolar spreadsheets
The number of years (A) is 10845.
The number of leap months (B) is 3994.
The abundance (C) is 2106
Upon putting that into my spreadsheet, I verified the number of yerms,
months and days is as stated by Helios. I see that the mean year is
365.2426 days and the mean month is 29.5305888 days. The number of
truncations of the 19-year cycle (to 11 years) is 29.
Each lunar 6097-month cycle is made up of seven yerm eras of which four
have 52 yerms and three have 55 yerms totalling 208+165=373=8206/22
yerms.
Now let's add a 725-year cycle (of 183 3-yerm cycles) to two 6097 month
cycles. We then get 2*373+549=7*185 yerms but still have 14 yerm eras,
which makes 92.5 yerms per era. This can be realised by alternating
yerm eras of 91 and 94 yerms. So we have a less accurate but shorter
lunar calendar cycle of 3023 months. 77 of these would then make up a
10845+11*725=18820 years.
= 154 yerm eras
= 77 lunar cycles
= 14245 yerms
= 18820 years
= 232771 months
= 6873867 days = (3^2) * (7^2) * 11 * 13 * 109
The mean year is 365.242667 days and the mean month is 29.5305987 days.
This cycle cannot be made into a Helios cycle, because the number of
years is even.
This shows a way that one could get a third cycle from two and give the
third a shorter lunar cycle than the first.
I also note that the number of months in seven yerm eras is always
divisible by seven.
Karl
10(03(19
-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto: CALNDR-L@...] On Behalf Of Helios
Sent: 16 December 2008 12:44
To: CALNDR-L@...
Subject: Re: Lunisolar Cycles made up of Short Lunar Cycles RE:
5515-Year Luni-Solar Cycle
Here is a luni-solar cycle that misses Karl Palmen's criteria he
mentioned
earlier, but is near the nebulous boundry he set at 10,000 years.
(1) months made up of a 22-fold repetition of one lunar calendar cycle
of
6097 months ( > 1885 )
(2) 10,845 year cycle of 134,134 months
(3) a mean year between 365.2416 and 365.2428 days.
= 154 yerm eras
= 8206 yerms
= 10845 years
= 134134 months
= 3961056 days = ( 2 ^ 5 )* 3 *( 11 ^ 3 )* 31
mean year = 365 & 877 / 3615
This mean year is close to the Gregorian year.
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Re: 5515-Year Luni-Solar Cycle
The symmetrical distribution of 13-month years in the 5515 year cycle we use
( 2031*Y + 2757 )MOD( 5515 ) < 2031
and a look at the very beginning and end years ( 13-month years )
2, 5, 7, 10, 13, 15, 18, 21
5495, 5498, 5501, 5503, 5506, 5509, 5511, 5514
evidently also obey the octaeteris accumulator
( 3*Y + 3 )MOD( 8 ) < 3
If I am counting correctly, I get 2068 years that satisfy
( 3*Y + 3 )MOD( 8 ) < 3
in the entire 5515 year cycle. We should like 2031 13-month years in the 5515 year cycle so 37 of these years we could remove the leap month to make an octaeteris calendar that is occasionally adjusted about every 149 years. These 37 months equal the 3 years that is the difference between 5515 years and 5512 octaeteris years.
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Re: 5515-Year Luni-Solar Cycle
On 2009 Apr 28, at 01:15 , Helios wrote: The symmetrical distribution of 13-month years in the 5515 year cycle we use...
Helios, why are you persisting with working with such a long cycle?
Due to tidal transfer of angular momentum Earth's mean solar day is getting progressively longer, which causes the mean solar year to get progressively shorter, and the lunar cycle is getting progressively shorter in terms of those progressively longer mean solar days. In addition there are periodic variations in the lengths of the solar year, depending on where one measures the year length from, due to precession of the equinoxes, the advance of perihelion, changing Earth orbital eccentricity, and changing Earth axial tilt (obliquity of the ecliptic). All of these long-term trends make it futile to employ any very long duration fixed arithmetic cycle for calendrical purposes.
In the present era the 353-year lunisolar cycle is the best fit for the mean northward equinoctial year and mean synodic month, with 4366 months per cycle. It is the equivalent of 18 repeats of the 19-year metonic cycle plus an 11-cycle subcycle (octaeteris omitted). For the northward equinox it works well to omit the octaeteris in the middle of the cycle = (9 x 19) + 11 + (9 x 19) = 353 years, which is easy to do with a single-step MOD expression.
What is the "calendar season" that you are targeting? (northward equinox?) What is your target calendar mean year? (mean northward equinoctial year presently = 365d 5h 49m) Where is your reference meridian? (UT, Jerusalem, Washington, somewhere else?) Where does your numbering of years begin?
-- Irv Bromberg, Toronto, Canada
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Re: 5515-Year Luni-Solar Cycle
Some parts of this message have been removed.
Learn more about Nabble's security policy.
Dear Irv, Helios and Calendar People
The only merit to the 5515-year cycle is that it is an exact
multiple of a lunar 1749-month cycle as explained in earlier notes about this
cycle.
Karl
10(08(04
From: East Carolina University Calendar
discussion List [mailto:CALNDR-L@...] On Behalf Of Irv
Bromberg
Sent: 28 April 2009 16:12
To: CALNDR-L@...
Subject: Re: 5515-Year Luni-Solar Cycle
On
2009 Apr 28, at 01:15 , Helios wrote:
The
symmetrical distribution of 13-month years in the 5515 year cycle we
use...
Helios,
why are you persisting with working with such a long cycle?
Due
to tidal transfer of angular momentum Earth's mean solar day is getting
progressively longer, which causes the mean solar year to get progressively
shorter, and the lunar cycle is getting progressively shorter in terms of
those progressively longer mean solar days. In addition there are
periodic variations in the lengths of the solar year, depending on where one
measures the year length from, due to precession of the equinoxes, the advance
of perihelion, changing Earth orbital eccentricity, and changing Earth axial
tilt (obliquity of the ecliptic). All of these long-term trends make it
futile to employ any very long duration fixed arithmetic cycle for calendrical
purposes.
In
the present era the 353-year lunisolar cycle is the best fit for the mean
northward equinoctial year and mean synodic month, with 4366 months per cycle.
It
is the equivalent of 18 repeats of the 19-year metonic cycle plus an 11-cycle
subcycle (octaeteris omitted).
For
the northward equinox it works well to omit the octaeteris in the middle of the
cycle = (9 x 19) + 11 + (9 x 19) = 353 years, which is easy to do with a
single-step MOD expression.
What
is the "calendar season" that you are targeting? (northward
equinox?)
What
is your target calendar mean year? (mean northward equinoctial year
presently = 365d 5h 49m)
Where
is your reference meridian? (UT, Jerusalem, Washington, somewhere else?)
Where
does your numbering of years begin?
-- Irv Bromberg, Toronto,
Canada
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Re: 5515-Year Luni-Solar Cycle
On 2009 Apr 28, at 11:28 , Palmen, KEV (Karl) wrote: The only merit to the 5515-year cycle is that it is an exact multiple of a lunar 1749-month cycle as explained in earlier notes about this cycle.
It's 39 repeats = 68211 lunar months, but by the end of one cycle the mean lunar cycle will be appreciably shorter, so why is this "exact multiple" considered to have any real merit?
By contrast, the 353-year cycle contains only 4366 lunar months, no multiples here.
-- Irv Bromberg, Toronto, Canada
From: East Carolina University Calendar discussion List [CALNDR-L@...] On Behalf Of Irv Bromberg Sent: 28 April 2009 16:12
On 2009 Apr 28, at 01:15 , Helios wrote: The symmetrical distribution of 13-month years in the 5515 year cycle we use...
Helios, why are you persisting with working with such a long cycle? Due to tidal transfer of angular momentum Earth's mean solar day is getting progressively longer, which causes the mean solar year to get progressively shorter, and the lunar cycle is getting progressively shorter in terms of those progressively longer mean solar days. In addition there are periodic variations in the lengths of the solar year, depending on where one measures the year length from, due to precession of the equinoxes, the advance of perihelion, changing Earth orbital eccentricity, and changing Earth axial tilt (obliquity of the ecliptic). All of these long-term trends make it futile to employ any very long duration fixed arithmetic cycle for calendrical purposes. In the present era the 353-year lunisolar cycle is the best fit for the mean northward equinoctial year and mean synodic month, with 4366 months per cycle. It is the equivalent of 18 repeats of the 19-year metonic cycle plus an 11-cycle subcycle (octaeteris omitted). For the northward equinox it works well to omit the octaeteris in the middle of the cycle = (9 x 19) + 11 + (9 x 19) = 353 years, which is easy to do with a single-step MOD expression. What is the "calendar season" that you are targeting? (northward equinox?) What is your target calendar mean year? (mean northward equinoctial year presently = 365d 5h 49m) Where is your reference meridian? (UT, Jerusalem, Washington, somewhere else?) Where does your numbering of years begin? -- Irv Bromberg, Toronto, Canada
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Re: 5515-Year Luni-Solar Cycle
Some parts of this message have been removed.
Learn more about Nabble's security policy.
Dear Irv, Helios and Calendar people
Try finding another such multiple of a such a short lunar
calendar cycle that is also a whole number of years.
The only other one I’ve found is the less accurate 725-year
cycle of 183 49-month cycles.
Karl
10(08(04
From: East Carolina University Calendar
discussion List [mailto:CALNDR-L@...] On Behalf Of Irv
Bromberg
Sent: 28 April 2009 17:46
To: CALNDR-L@...
Subject: Re: 5515-Year Luni-Solar Cycle
On
2009 Apr 28, at 11:28 , Palmen, KEV (Karl) wrote:
The only merit to the
5515-year cycle is that it is an exact multiple of a lunar 1749-month cycle as
explained in earlier notes about this cycle.
It's
39 repeats = 68211 lunar months, but by the end of one cycle the mean lunar
cycle will be appreciably shorter, so why is this "exact multiple"
considered to have any real merit?
By
contrast, the 353-year cycle contains only 4366 lunar months, no multiples
here.
-- Irv Bromberg, Toronto,
Canada
From: East Carolina University Calendar
discussion List [CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 28 April 2009 16:12
On 2009 Apr 28, at 01:15 , Helios wrote:
The symmetrical distribution of 13-month years in the 5515
year cycle we use...
Helios, why are you persisting with working with such a long cycle?
Due to tidal transfer of angular momentum Earth's mean solar day
is getting progressively longer, which causes the mean solar year to get
progressively shorter, and the lunar cycle is getting progressively
shorter in terms of those progressively longer mean solar days. In
addition there are periodic variations in the lengths of the solar year,
depending on where one measures the year length from, due to precession of the
equinoxes, the advance of perihelion, changing Earth orbital eccentricity, and
changing Earth axial tilt (obliquity of the ecliptic). All of these long-term
trends make it futile to employ any very long duration fixed arithmetic cycle
for calendrical purposes.
In the present era the 353-year lunisolar cycle is the best fit
for the mean northward equinoctial year and mean synodic month, with 4366 months
per cycle.
It is the equivalent of 18 repeats of the 19-year metonic cycle
plus an 11-cycle subcycle (octaeteris omitted).
For the northward equinox it works well to omit the octaeteris in
the middle of the cycle = (9 x 19) + 11 + (9 x 19) = 353 years, which is easy
to do with a single-step MOD expression.
What is the "calendar season" that you are targeting?
(northward equinox?)
What is your target calendar mean year? (mean northward
equinoctial year presently = 365d 5h 49m)
Where is your reference meridian? (UT, Jerusalem,
Washington, somewhere else?)
Where does your numbering of years begin?
-- Irv Bromberg, Toronto,
Canada
Scanned by iCritical.
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Re: 5515-Year Luni-Solar Cycle
All the 5515 year cycle has to do is survive to within one day all of the long-term physical effects mentioned by Irv B.. I think it's about right but it should expire when it's finished and not be used multiple times.
Have a look at these dates
----------------------------------------
Jan 30 3999 JD = 3181693.5
Mar 1 3999 JD = 3181723.5
Mar 30 3999 JD = 3181752.5
Apr 29 3999 JD = 3181782.5
May 28 3999 JD = 3181811.5
----------------------------------------
Now subtract 2014311 days
----------------------------------------
Feb 14 -1516 JD = 1167382.5 ( julian calendar, year 0 inclusive )
Mar 15 -1516 JD = 1167412.5
Apr 13 -1516 JD = 1167441.5
May 13 -1516 JD = 1167471.5
Jun 11 -1516 JD = 1167500.5
----------------------------------------
Now a look at NASA Moon phases data ( new moons )
----------------------------------------
Jan 30 15:42 3999
Mar 1 06:54 3999
Mar 30 19:43 3999
Apr 29 06:05 3999
May 28 14:31 3999
----------------------------------------
Feb 14 05:53 -1516 ( julian calendar, year 0 inclusive )
Mar 14 14:32 -1516
Apr 12 23:11 -1516
May 12 08:57 -1516
Jun 10 20:43 -1516
----------------------------------------
It looks to me that about 3/4 day tardiness has accrued over 5515 years. I know I have chosen just 5 datums but I think this shows a good ball-park estimate to show my point. Yes, the mean lunar cycle will be shorter but the term "appreciably" I dispute.
Karl P. identifies the cycle's "only merit" and Irv B. even disputes "any real merit" at all even though changes in physical world do not even accrue to one day difference. There are several merits to the cycle and each are where you find them. I began this with a look at the octaeteris and how it could be corrected. I have the opinion that the year which is 5512 / 5515 of the octaeteris has merit. Anyone is welcome to find a better method.
Look at the troubles when ones choose a "target year" first ( like the spring equinox ) and then try to find a luni-solar cycle to fit. Karl just showed a few post ago that sol'ns can be dreadful. Look what happens when you choose a longitude first ( like Jerusalem ) and then try to find a luni-solar cycle epoch. Heed the advice "Don't put the apple cart in front of the horse". Start with the cycle, then the epoch and then the longitude. This approach and using E. Brown's lunation numbers puts a latitude near Zurich, but this is a whole other story.
I will resume with the octaeteris study soon. I will name the 5515 year cycle the "Meritorious Cycle" ( just kidding ). I enjoy these forums and the good company here.
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Re: 5515-Year Lunisolar Cycle
Dear Helios, Irv and Calendar People
I think the most important property of the 5515-year cycle is that it is
a whole number (39) of an accurate lunar calendar cycles (of 1749
months). Helios does not seem to exploit this property enough, but
instead tries to exploit the property that the number of lunar months is
divisible by 99 and so it a whole number (689) of Octaeterides.
The whole number of lunar calendar cycles is divisible by 13. This
suggests to me using a solar calendar of 13 months in every year
alongside the lunar calendar with a 1749-month cycle.
I'm aware that if each solar year of the 5515-year cycle were divided
into 13 solar months, then 5515 solar months would have exactly three
cycles of 1749 lunar months. This is equal to 53 octaeterides and so 424
Octaeteris years. It is also equal to 424 3/13 solar years and so the
Octaeteris year falls one solar month behind the solar year exactly once
every lunar cycle of 1749 lunar months.
-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto: CALNDR-L@...] On Behalf Of Helios
Sent: 28 April 2009 06:15
To: CALNDR-L@...
Subject: Re: 5515-Year Luni-Solar Cycle
The symmetrical distribution of 13-month years in the 5515 year cycle we
use
( 2031*Y + 2757 )MOD( 5515 ) < 2031
and a look at the very beginning and end years ( 13-month years )
2, 5, 7, 10, 13, 15, 18, 21
5495, 5498, 5501, 5503, 5506, 5509, 5511, 5514
evidently also obey the octaeteris accumulator
( 3*Y + 3 )MOD( 8 ) < 3
KARL SAYS: The first year that does not satisfy the Octaeteris
accumulator is year 24, whose 5515-year cycle accumulator is 1866 which
is less than 2031 so is a 5515-year cycle leap month year, but its
Octaeteris accumulator is 3 which is not less than 3. Exceptions will
occur more frequently later on as the octaeteris drifts out of place.
Towards the end of the 5515-year cycle the Octaeteris would have drifted
a three whole years late so will align itself again with the 5515-year
cycle years.
There'll be similar agreement about 1/3 and 2/3 through the 5515-year
cycle.
HELIOS CONTINUES:
If I am counting correctly, I get 2068 years that satisfy
( 3*Y + 3 )MOD( 8 ) < 3
in the entire 5515 year cycle.
KARL SAYS:
There are (3/8)*13*424 = 3*689 = 2067 not 2068 of them and
only a minority of these also satisfy ( 2031*Y + 2757 )MOD( 5515 ) <
2031 .
HELIOS CONTINUES:
We should like 2031 13-month years in the
5515 year cycle so 37 of these years we could remove the leap month to
make
an octaeteris calendar that is occasionally adjusted about every 149
years.
These 37 months equal the 3 years that is the difference between 5515
years
and 5512 octaeteris years.
KARL SAYS:
It's 36 years (2067=2031+36) rather than 37 to remove 13th month from.
Then you'd also need to insert three years of 12 months to bring the
5512 years (from the 689 octaeterides) up to 5515 years. Seems
pointless.
The property of the 5515-year cycle having a number of lunar months
divisible by 99 is no longer exploited.
I think it would be better to have a solar calendar with 13 solar months
in every year.
The solar months (normally of 28 days) can be arranged in a Helios cycle
of 5515 solar months, so a solar month M has an extra (29th) day if and
only if
( 527*M + 2757 ) MOD (5515) < 527
Each solar 5515-month cycle is exactly equal to three lunar 1749-month
cycles.
The resulting solar years would also form a Helios cycle of 5515 years,
such that year Y is has 366 days if and only if
( 1336*Y + 2757 ) MOD (5515) < 1336.
Now if you run an Octeateris calendar against this using the lunar
1749-month cycle, you find that the Octeateris year runs one solar month
late (on average) every lunar 1749-month cycle elapsed.
Karl
10(08(05
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