Epacts for Leap Week Calendars

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Epacts for Leap Week Calendars

Karl Palmen - UKRI STFC
Epacts for Leap Week Calendars

Dear Calendar People

I have been thinking about how a leap week calendar could be made lunisolar.

One possibility is to use epacts like in the Julian and Gregorian lunar calendars. The epact effectively specifies how many days the lunar new year (or some other specified date in the lunar calendar year) begins before the solar new year (or some other specified day of the solar calendar year).

These leap day calendars have a epact rule that ignores leap years. This ignoring of leap years can be done because the jitter arising from the leap days is quite small.  This does not apply to leap week calendars. Hence an epact rule for a leap week calendar cannot ignore leap years.

I’ve realised that the following epact rule can normally be applied to a leap week calendar

(1)     After a common year (of 364 days) increment the epact by 10 mod 30

(2)     After a leap year (of 371 days) increment the epact by 16 mod 30

This would need correcting by decrementing by 1 unit about once every 60 years. Whenever such a correction is made, the otherwise constant  parity (even/odd) of the epacts changes. Also, the final digit of the epact does not change after any common year, except when there is a correction.

Firstly, let’s seem what happens if we follow those rules without any correction:

For a 28-year cycle with 5 leap weeks, as would arise from the Julian Calendar, 28 years have a total epact increment of 280+5*6=310 and so three 28-year cycles, which is 84 years have exactly 31 leap months. This 84-year cycle has been suggested for an Easter rule in Roman times.

For the  more accurate cycle of 62 years with 11 leap weeks, 62 years have a total epact increment of 620+11*6=686. Six 62-year cycles come close  a Gregoriana cycle of 372  years and 137 leap months. The total epact increment would have to be 137*30=4110, but is in fact 6*686=4116. This suggests one correction every 62-year cycle (so bringing the 686 down to 685).

In http://www.the-light.com/cal/Lunisolar7.html of http://www.the-light.com/cal/kp_Lunisolar_xls.html , we have several accurate lunisolar calendar cycles that are also a whole number of weeks and we could apply this to any of them.

Let A, B and C be the values of the first three columns (years, leap months, abundant years). Then the cycle has the equivalent of C + 30B - 11A leap days and so (C + 30B – 10A)/7 leap weeks. The total epact increment without correction would therefore be 10*A + (6*C + 180B – 60A)/7, which is  equal to (6*C + 180B + 10A)/7. If corrected this would be 30*B, hence the number of corrections must be

(6*C – 30B + 10A)/7

For the 6840-year, Meyer-Palmen cycle (A=6940, B=2519, C=1328), this works out to be 114, so giving one correction every 60 years exactly.

For the 210 128-year cycles cycle (A=26880, B=9899, C=5220), this works out to be 450, which is 15 corrections every 896-year cycle (one per 59.73333… years).

For 98 33-year cycles (A=3234, B=1191, C=628) (see http://www.the-light.com/cal/Lunisolar33.html ), this works out to be 54 corrections, so giving one correction every 59 8/9 years (539-year cycle with intervals 60,60,60,60,60,60, 60,60,59 ) .

Finally, here is what the epacts for 60 consecutive years could be:

00, 10, 20, 06, 16,  26, 06, 16, 26, 12,

22, 02, 12, 22, 02,  18, 28, 08, 18, 28,

14, 24, 04, 14, 24,  04, 20, 00, 10, 20,

29, 15, 25, 05, 15,  25, 05, 21, 01, 11,

21, 01, 11, 27, 07,  17, 27, 07, 23, 03,

13, 23, 03, 13, 29,  09, 19, 29, 09, 19,

The epacts of the leap years are shown in bold and the epact of the year after which a correction is made is shown in italic.

Karl

10(10(24


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Re: Epacts for Leap Week Calendars

Ryan Provost-2
Lunisolar leap week calendars? In my preseptive for my style of lunisolar calendar... The new year starts on the first new moon of the solar year (in that case, in the Gregorian calendar), and the new year on the lunisolar leap week calendar and the new month should start on a Monday. (ie The Monday on or following the new moon) The lunar part should be based astromically and the solar part should be based on the Gregorian calendar. That is NOT like the Chinese calendar. The lunisolar NON leap week and the Gregorian calendar will be used alongside with the lunisolar leap week calendar. The Gregorian leap week calendar would be used as well.

ELITE 3000

On 2009-07-16, at 8:28 AM, "Palmen, KEV (Karl)" <[hidden email]> wrote:

Dear Calendar People

I have been thinking about how a leap week calendar could be made lunisolar.

One possibility is to use epacts like in the Julian and Gregorian lunar calendars. The epact effectively specifies how many days the lunar new year (or some other specified date in the lunar calendar year) begins before the solar new year (or some other specified day of the solar calendar year).

These leap day calendars have a epact rule that ignores leap years. This ignoring of leap years can be done because the jitter arising from the leap days is quite small.  This does not apply to leap week calendars. Hence an epact rule for a leap week calendar cannot ignore leap years.

I’ve realised that the following epact rule can normally be applied to a leap week calendar

(1)     After a common year (of 364 days) increment the epact by 10 mod 30

(2)     After a leap year (of 371 days) increment the epact by 16 mod 30

This would need correcting by decrementing by 1 unit about once every 60 years. Whenever such a correction is made, the otherwise constant  parity (even/odd) of the epacts changes. Also, the final digit of the epact does not change after any common year, except when there is a correction.

Firstly, let’s seem what happens if we follow those rules without any correction:

For a 28-year cycle with 5 leap weeks, as would arise from the Julian Calendar, 28 years have a total epact increment of 280+5*6=310 and so three 28-year cycles, which is 84 years have exactly 31 leap months. This 84-year cycle has been suggested for an Easter rule in Roman times.

For the  more accurate cycle of 62 years with 11 leap weeks, 62 years have a total epact increment of 620+11*6=686. Six 62-year cycles come close  a Gregoriana cycle of 372  years and 137 leap months. The total epact increment would have to be 137*30=4110, but is in fact 6*686=4116. This suggests one correction every 62-year cycle (so bringing the 686 down to 685).

In http://www.the-light.com/cal/Lunisolar7.html of http://www.the-light.com/cal/kp_Lunisolar_xls.html , we have several accurate lunisolar calendar cycles that are also a whole number of weeks and we could apply this to any of them.

Let A, B and C be the values of the first three columns (years, leap months, abundant years). Then the cycle has the equivalent of C + 30B - 11A leap days and so (C + 30B – 10A)/7 leap weeks. The total epact increment without correction would therefore be 10*A + (6*C + 180B – 60A)/7, which is  equal to (6*C + 180B + 10A)/7. If corrected this would be 30*B, hence the number of corrections must be

(6*C – 30B + 10A)/7

For the 6840-year, Meyer-Palmen cycle (A=6940, B=2519, C=1328), this works out to be 114, so giving one correction every 60 years exactly.

For the 210 128-year cycles cycle (A=26880, B=9899, C=5220), this works out to be 450, which is 15 corrections every 896-year cycle (one per 59.73333… years).

For 98 33-year cycles (A=3234, B=1191, C=628) (see http://www.the-light.com/cal/Lunisolar33.html ), this works out to be 54 corrections, so giving one correction every 59 8/9 years (539-year cycle with intervals 60,60,60,60,60,60, 60,60,59 ) .

Finally, here is what the epacts for 60 consecutive years could be:

00, 10, 20, 06, 16,  26, 06, 16, 26, 12,

22, 02, 12, 22, 02,  18, 28, 08, 18, 28,

14, 24, 04, 14, 24,  04, 20, 00, 10, 20,

29, 15, 25, 05, 15,  25, 05, 21, 01, 11,

21, 01, 11, 27, 07,  17, 27, 07, 23, 03,

13, 23, 03, 13, 29,  09, 19, 29, 09, 19,

The epacts of the leap years are shown in bold and the epact of the year after which a correction is made is shown in italic.

Karl

10(10(24


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Re: Epacts for Leap Week Calendars

Karl Palmen - UKRI STFC

Dear Elite 3000 and Calendar People

 

This idea is intended to work with just ONE solar calendar, which is a leap week calendar. No reference to any other calendar or astronomy is needed.

It creates a lunar calendar that can be used alongside the solar leap week calendar and whose date can be calculated relatively easily for the leap week calendar date.

Using multiple calendars is very complicated in practice (although simple to specify) and would probably need a computer to make it workable.

 

Making all months of the lunar calendar begin on the same day of week would make it into a poor lunar calendar and could be achieved by rounding the epacts to the nearest multiple of seven, taking care to increment unrounded epacts.

 

Karl

 

10(10(27 till noon

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of ELITE 3000
Sent: 18 July 2009 17:06
To: [hidden email]
Subject: Re: Epacts for Leap Week Calendars

 

Lunisolar leap week calendars? In my preseptive for my style of lunisolar calendar... The new year starts on the first new moon of the solar year (in that case, in the Gregorian calendar), and the new year on the lunisolar leap week calendar and the new month should start on a Monday. (ie The Monday on or following the new moon) The lunar part should be based astromically and the solar part should be based on the Gregorian calendar. That is NOT like the Chinese calendar. The lunisolar NON leap week and the Gregorian calendar will be used alongside with the lunisolar leap week calendar. The Gregorian leap week calendar would be used as well.

ELITE 3000


On 2009-07-16, at 8:28 AM, "Palmen, KEV (Karl)" <[hidden email]> wrote:

Dear Calendar People

I have been thinking about how a leap week calendar could be made lunisolar.

One possibility is to use epacts like in the Julian and Gregorian lunar calendars. The epact effectively specifies how many days the lunar new year (or some other specified date in the lunar calendar year) begins before the solar new year (or some other specified day of the solar calendar year).

These leap day calendars have a epact rule that ignores leap years. This ignoring of leap years can be done because the jitter arising from the leap days is quite small.  This does not apply to leap week calendars. Hence an epact rule for a leap week calendar cannot ignore leap years.

I’ve realised that the following epact rule can normally be applied to a leap week calendar

(1)     After a common year (of 364 days) increment the epact by 10 mod 30

(2)     After a leap year (of 371 days) increment the epact by 16 mod 30

This would need correcting by decrementing by 1 unit about once every 60 years. Whenever such a correction is made, the otherwise constant  parity (even/odd) of the epacts changes. Also, the final digit of the epact does not change after any common year, except when there is a correction.

Firstly, let’s seem what happens if we follow those rules without any correction:

For a 28-year cycle with 5 leap weeks, as would arise from the Julian Calendar, 28 years have a total epact increment of 280+5*6=310 and so three 28-year cycles, which is 84 years have exactly 31 leap months. This 84-year cycle has been suggested for an Easter rule in Roman times.

For the  more accurate cycle of 62 years with 11 leap weeks, 62 years have a total epact increment of 620+11*6=686. Six 62-year cycles come close  a Gregoriana cycle of 372  years and 137 leap months. The total epact increment would have to be 137*30=4110, but is in fact 6*686=4116. This suggests one correction every 62-year cycle (so bringing the 686 down to 685).

In http://www.the-light.com/cal/Lunisolar7.html of http://www.the-light.com/cal/kp_Lunisolar_xls.html , we have several accurate lunisolar calendar cycles that are also a whole number of weeks and we could apply this to any of them.

Let A, B and C be the values of the first three columns (years, leap months, abundant years). Then the cycle has the equivalent of C + 30B - 11A leap days and so (C + 30B – 10A)/7 leap weeks. The total epact increment without correction would therefore be 10*A + (6*C + 180B – 60A)/7, which is  equal to (6*C + 180B + 10A)/7. If corrected this would be 30*B, hence the number of corrections must be

(6*C – 30B + 10A)/7

For the 6840-year, Meyer-Palmen cycle (A=6940, B=2519, C=1328), this works out to be 114, so giving one correction every 60 years exactly.

For the 210 128-year cycles cycle (A=26880, B=9899, C=5220), this works out to be 450, which is 15 corrections every 896-year cycle (one per 59.73333… years).

For 98 33-year cycles (A=3234, B=1191, C=628) (see http://www.the-light.com/cal/Lunisolar33.html ), this works out to be 54 corrections, so giving one correction every 59 8/9 years (539-year cycle with intervals 60,60,60,60,60,60, 60,60,59 ) .

Finally, here is what the epacts for 60 consecutive years could be:

00, 10, 20, 06, 16,  26, 06, 16, 26, 12,

22, 02, 12, 22, 02,  18, 28, 08, 18, 28,

14, 24, 04, 14, 24,  04, 20, 00, 10, 20,

29, 15, 25, 05, 15,  25, 05, 21, 01, 11,

21, 01, 11, 27, 07,  17, 27, 07, 23, 03,

13, 23, 03, 13, 29,  09, 19, 29, 09, 19,

The epacts of the leap years are shown in bold and the epact of the year after which a correction is made is shown in italic.

Karl

10(10(24

 

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Re: Epacts for Leap Week Calendars

Karl Palmen - UKRI STFC
In reply to this post by Ryan Provost-2

Dear Elite 3000 and Calendar People

 

If the leap week calendar is based on the Gregorian Calendar, then the Gregorian Epact system could be used. This would require one to refer to the Gregorian Calendar.

 

If instead you wanted to use a leap week calendar that follows the Gregorian 400-year cycle without referring to the Gregorian epact, one could create any lunisolar cycle based on the 400-year cycle including those listed in http://www.the-light.com/cal/Lunisolar400.html  of http://www.the-light.com/cal/kp_Lunisolar_xls.html .

 

I have found out that the number of corrections needed for a cycle is also equal to the number of abundant years( C ) less the number of leap weeks. This helps me calculate the number of corrections needed in the following examples (noting that there are 71 leap weeks per 400-year cycle):

 

29 Gregorian cycles (mean lunar month 29.5305913 days)  194 corrections (two per 29 Gregorian leap days).

45 Gregorian Cycles (mean lunar month 29.5305868 days) 300 corrections (once every 60 years exactly).

 

I like the idea of correcting once every 60 years exactly whatever the leap week calendar is.

 

I’ve also found out that if leap week calendar A is running on average x days behind leap week calendar B and both calendar use exactly the same epact rule of the type I have defined, then leap week calendar A will reckon the moon phase approximately x/7 days later (rather than x days later as for a leap day calendar) than leap week calendar B.

 

Karl

 

10(10(28

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of ELITE 3000
Sent: 18 July 2009 17:06
To: [hidden email]
Subject: Re: Epacts for Leap Week Calendars

 

Lunisolar leap week calendars? In my preseptive for my style of lunisolar calendar... The new year starts on the first new moon of the solar year (in that case, in the Gregorian calendar), and the new year on the lunisolar leap week calendar and the new month should start on a Monday. (ie The Monday on or following the new moon) The lunar part should be based astromically and the solar part should be based on the Gregorian calendar. That is NOT like the Chinese calendar. The lunisolar NON leap week and the Gregorian calendar will be used alongside with the lunisolar leap week calendar. The Gregorian leap week calendar would be used as well.

ELITE 3000


On 2009-07-16, at 8:28 AM, "Palmen, KEV (Karl)" <[hidden email]> wrote:

Dear Calendar People

I have been thinking about how a leap week calendar could be made lunisolar.

One possibility is to use epacts like in the Julian and Gregorian lunar calendars. The epact effectively specifies how many days the lunar new year (or some other specified date in the lunar calendar year) begins before the solar new year (or some other specified day of the solar calendar year).

These leap day calendars have a epact rule that ignores leap years. This ignoring of leap years can be done because the jitter arising from the leap days is quite small.  This does not apply to leap week calendars. Hence an epact rule for a leap week calendar cannot ignore leap years.

I’ve realised that the following epact rule can normally be applied to a leap week calendar

(1)     After a common year (of 364 days) increment the epact by 10 mod 30

(2)     After a leap year (of 371 days) increment the epact by 16 mod 30

This would need correcting by decrementing by 1 unit about once every 60 years. Whenever such a correction is made, the otherwise constant  parity (even/odd) of the epacts changes. Also, the final digit of the epact does not change after any common year, except when there is a correction.

Firstly, let’s seem what happens if we follow those rules without any correction:

For a 28-year cycle with 5 leap weeks, as would arise from the Julian Calendar, 28 years have a total epact increment of 280+5*6=310 and so three 28-year cycles, which is 84 years have exactly 31 leap months. This 84-year cycle has been suggested for an Easter rule in Roman times.

For the  more accurate cycle of 62 years with 11 leap weeks, 62 years have a total epact increment of 620+11*6=686. Six 62-year cycles come close  a Gregoriana cycle of 372  years and 137 leap months. The total epact increment would have to be 137*30=4110, but is in fact 6*686=4116. This suggests one correction every 62-year cycle (so bringing the 686 down to 685).

In http://www.the-light.com/cal/Lunisolar7.html of http://www.the-light.com/cal/kp_Lunisolar_xls.html , we have several accurate lunisolar calendar cycles that are also a whole number of weeks and we could apply this to any of them.

Let A, B and C be the values of the first three columns (years, leap months, abundant years). Then the cycle has the equivalent of C + 30B - 11A leap days and so (C + 30B – 10A)/7 leap weeks. The total epact increment without correction would therefore be 10*A + (6*C + 180B – 60A)/7, which is  equal to (6*C + 180B + 10A)/7. If corrected this would be 30*B, hence the number of corrections must be

(6*C – 30B + 10A)/7

For the 6840-year, Meyer-Palmen cycle (A=6940, B=2519, C=1328), this works out to be 114, so giving one correction every 60 years exactly.

For the 210 128-year cycles cycle (A=26880, B=9899, C=5220), this works out to be 450, which is 15 corrections every 896-year cycle (one per 59.73333… years).

For 98 33-year cycles (A=3234, B=1191, C=628) (see http://www.the-light.com/cal/Lunisolar33.html ), this works out to be 54 corrections, so giving one correction every 59 8/9 years (539-year cycle with intervals 60,60,60,60,60,60, 60,60,59 ) .

Finally, here is what the epacts for 60 consecutive years could be:

00, 10, 20, 06, 16,  26, 06, 16, 26, 12,

22, 02, 12, 22, 02,  18, 28, 08, 18, 28,

14, 24, 04, 14, 24,  04, 20, 00, 10, 20,

29, 15, 25, 05, 15,  25, 05, 21, 01, 11,

21, 01, 11, 27, 07,  17, 27, 07, 23, 03,

13, 23, 03, 13, 29,  09, 19, 29, 09, 19,

The epacts of the leap years are shown in bold and the epact of the year after which a correction is made is shown in italic.

Karl

10(10(24

 

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