34-33-34-33-34-33-34 RE: Cutting for late leap years

classic Classic list List threaded Threaded
4 messages Options
Reply | Threaded
Open this post in threaded view
|  
Report Content as Inappropriate

34-33-34-33-34-33-34 RE: Cutting for late leap years

Karl Palmen

Dear Walter and Calendar People

 

Reply below

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Walter J Ziobro
Sent: 20 September 2016 03:44
To: [hidden email]
Subject: Re: Cutting for late leap years

 

Dear Karl and Calendar List:


I have given some thought about the best distribution of the 7 leap months over the metonic cycle, and have concluded that the smoothest distribution can be obtained by having a leap month every 34th or 33rd month over the cycle, thereby creating a series of months in which there are 33 ordinary months, plus a leap month, alternating with 32 ordinary months, plus a leap month, like this:

34-33-34-33-34-33-34, for a total of 235 months.

This cycle can be maintained over a period of 10 or 11 metonic cycles, but then the occurrence of each leap month needs to be gradually shifted by a month, so that by the completion of 360 metonic cycles (6840 years) one leap month slips into the next metonic cycle. This can be done by having the first month of each 34 or 33 month leap  cycle be counted as the leap month, and then, after either 10 or 11  metonic cycles, having the second month of each leap cycle counted as the leap month, and so on throughout the entire 360 metonic cycles of 6840 years.

 

KARL REPLIES:  Walter’s idea runs into problems after the 33rd month of each 34 or 33 is the leap month. What happens after then?

One solution is to jump over the 34th to the 1st to smoothly drop a leap month. Then 33 of these postponements would be needed in a 6840-year cycle, 30 after 11 Metonic cycles and 3 after 3 Metonic cycles so following a sub-cycle of 120 Metonic cycles of 2280 years.

 

Another idea is to have exactly 30 permitted positions for the leap month in each 34 and 33, then postponement to the next permitted position would occur exactly once every 12 Metonic cycles of 228 years.

 

Whenever the 30th, 31st or 32nd month in each 33 & 34 were a leap month, the leap month years would follow the Hebrew 19-year cycle.

 

 

No such cycle would create the smoothest possible distribution of leap months, because some leap months will occur at least 35 months after the previous, whenever a postponement occurs. I expect it would add a day or two to the solar jitter of about one month.  If the leap months were spread as smoothly as possible, we’d have 168-month periods of 34-33-34-33-34 in addition to the 235-month periods of 34-33-34-33-34-33-34 all with fixed leap month.

 

 

Karl

 

16(01(19

 

 



-Walter Ziobro

 

 

 

-----Original Message-----
From: Karl Palmen <[hidden email]>
To: CALNDR-L <[hidden email]>
Sent: Wed, Sep 14, 2016 8:06 am
Subject: Cutting for late leap years

Dear Irv & Calendar People

 

I’ve already mentioned an algorithm for cutting calendar cycles where the leap years are spread as smoothly as possible into symmetrical parts, grouped in larger symmetrical parts.

 

Now I show a similar algorithm where the parts have the leap years as late as possible.

 

In the first iteration cut after any leap year followed by a common year. For example:

 

Hebrew 19-year cycle:  ccL|ccL|cL|ccL|ccL|ccL|cL| ,

Symmetrical 17-year leap week cycle: ccL|cccccL|cccccL|cc ,

33-year cycle: cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL|c,

Julian 4-year cycle: cccL|

                                                                                                                      

Then there should either be two different types of part (joining parts over end of cycle) or just one part in which case the algorithm has finished. The part with the greatest proportion of leap years and so shortest if leap years are in minority (shown in red) is  designated as leap year for the next iteration, even if such parts are in a majority. The examples then become:

 

ccL|ccL|cL||ccL|ccL|ccL|cL|| ,

ccL||cccccL|cccccL|cc ,

cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL||c

 

The second example has only one double cut and so the algorithm has finished. If the cycle is started after the double cut (at 4th year) the leap years will be as late as possible and if the year before the double cut (3rd year) is year Z as defined in earlier notes and if designated year 0, makes K=0.

The third example also has only one double cut and so year Z is the 32nd year and if started on the 33rd year, the leap years would occurs as late as possible.

 

The first example, which corresponds to the Hebrew 19-year cycle,  has a third iteration:

 

ccL|ccL|cL|||ccL|ccL|ccL|cL||

 

It has only one triple cut, which is between the 8th and 9th year. So the 8th year is year Z and the 9th year is the latest starting year and has subsequent leap years as late as possible.

 

 

I may look into Irv’s 353-year cycle with 130 leap years and K=269 in a later note. I’ll say at present that the second iteration cuts it into 11s and 8s. I expect the final cut to occur in the 4th iteration after the 169th year, which I reckoned to be year Z (year with 0 remainder).

 

For leap week calendars of sufficient accuracy, the first iteration cuts into 6s & 5s, the second iteration into 17s and 11s and the third iteration cuts into 62s and either 45s or 79s, except of course the 62-year cycle.

 

The number of iterations that this algorithm runs may be considered to be a measure of the structural complexity of the cycle.

 

The is also a similar mirror image algorithm, which cuts into parts that have the leap years as early as possible.

 

Karl

 

16(01(13

Reply | Threaded
Open this post in threaded view
|  
Report Content as Inappropriate

Re: 34-33-34-33-34-33-34 RE: Cutting for late leap years

Walter J Ziobro

Dear Karl

I thank you for that info

I like your second suggestion, that there be 30 leap month shifts, one every 228 years  This can be accomplished by skipping a designated leap month every 10 cycles, such that the designated leap month s in the 33-34 month groups would be 1-10, 12-21, and 23-32

Walter Ziobro

Sent from AOL Mobile Mail




On Tuesday, September 20, 2016 Karl Palmen <[hidden email]> wrote:

Dear Walter and Calendar People

 

Reply below

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Walter J Ziobro
Sent: 20 September 2016 03:44
To: CALNDR-L@...
Subject: Re: Cutting for late leap years

 

Dear Karl and Calendar List:


I have given some thought about the best distribution of the 7 leap months over the metonic cycle, and have concluded that the smoothest distribution can be obtained by having a leap month every 34th or 33rd month over the cycle, thereby creating a series of months in which there are 33 ordinary months, plus a leap month, alternating with 32 ordinary months, plus a leap month, like this:

34-33-34-33-34-33-34, for a total of 235 months.

This cycle can be maintained over a period of 10 or 11 metonic cycles, but then the occurrence of each leap month needs to be gradually shifted by a month, so that by the completion of 360 metonic cycles (6840 years) one leap month slips into the next metonic cycle. This can be done by having the first month of each 34 or 33 month leap  cycle be counted as the leap month, and then, after either 10 or 11  metonic cycles, having the second month of each leap cycle counted as the leap month, and so on throughout the entire 360 metonic cycles of 6840 years.

 

KARL REPLIES:  Walter’s idea runs into problems after the 33rd month of each 34 or 33 is the leap month. What happens after then?

One solution is to jump over the 34th to the 1st to smoothly drop a leap month. Then 33 of these postponements would be needed in a 6840-year cycle, 30 after 11 Metonic cycles and 3 after 3 Metonic cycles so following a sub-cycle of 120 Metonic cycles of 2280 years.

 

Another idea is to have exactly 30 permitted positions for the leap month in each 34 and 33, then postponement to the next permitted position would occur exactly once every 12 Metonic cycles of 228 years.

 

Whenever the 30th, 31st or 32nd month in each 33 & 34 were a leap month, the leap month years would follow the Hebrew 19-year cycle.

 

 

No such cycle would create the smoothest possible distribution of leap months, because some leap months will occur at least 35 months after the previous, whenever a postponement occurs. I expect it would add a day or two to the solar jitter of about one month.  If the leap months were spread as smoothly as possible, we’d have 168-month periods of 34-33-34-33-34 in addition to the 235-month periods of 34-33-34-33-34-33-34 all with fixed leap month.

 

 

Karl

 

16(01(19

 

 



-Walter Ziobro

 

 

 

-----Original Message-----
From: Karl Palmen <[hidden email]
>
To: CALNDR-L <[hidden email]>
Sent: Wed, Sep 14, 2016 8:06 am
Subject: Cutting for late leap years

Dear Irv & Calendar People

 

I’ve already mentioned an algorithm for cutting calendar cycles where the leap years are spread as smoothly as possible into symmetrical parts, grouped in larger symmetrical parts.

 

Now I show a similar algorithm where the parts have the leap years as late as possible.

 

In the first iteration cut after any leap year followed by a common year. For example:

 

Hebrew 19-year cycle:  ccL|ccL|cL|ccL|ccL|ccL|cL| ,

Symmetrical 17-year leap week cycle: ccL|cccccL|cccccL|cc ,

33-year cycle: cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL|c,

Julian 4-year cycle: cccL|

                                                                                                                      

Then there should either be two different types of part (joining parts over end of cycle) or just one part in which case the algorithm has finished. The part with the greatest proportion of leap years and so shortest if leap years are in minority (shown in red) is  designated as leap year for the next iteration, even if such parts are in a majority. The examples then become:

 

ccL|ccL|cL||ccL|ccL|ccL|cL|| ,

ccL||cccccL|cccccL|cc ,

cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL||c

 

The second example has only one double cut and so the algorithm has finished. If the cycle is started after the double cut (at 4th year) the leap years will be as late as possible and if the year before the double cut (3rd year) is year Z as defined in earlier notes and if designated year 0, makes K=0.

The third example also has only one double cut and so year Z is the 32nd year and if started on the 33rd year, the leap years would occurs as late as possible.

 

The first example, which corresponds to the Hebrew 19-year cycle,  has a third iteration:

 

ccL|ccL|cL|||ccL|ccL|ccL|cL||

 

It has only one triple cut, which is between the 8th and 9th year. So the 8th year is year Z and the 9th year is the latest starting year and has subsequent leap years as late as possible.

 

 

I may look into Irv’s 353-year cycle with 130 leap years and K=269 in a later note. I’ll say at present that the second iteration cuts it into 11s and 8s. I expect the final cut to occur in the 4th iteration after the 169th year, which I reckoned to be year Z (year with 0 remainder).

 

For leap week calendars of sufficient accuracy, the first iteration cuts into 6s & 5s, the second iteration into 17s and 11s and the third iteration cuts into 62s and either 45s or 79s, except of course the 62-year cycle.

 

The number of iterations that this algorithm runs may be considered to be a measure of the structural complexity of the cycle.

 

The is also a similar mirror image algorithm, which cuts into parts that have the leap years as early as possible.

 

Karl

 

16(01(13

Reply | Threaded
Open this post in threaded view
|  
Report Content as Inappropriate

Re: 34-33-34-33-34-33-34 RE: Cutting for late leap years

Karl Palmen

Dear Walter and Calendar People

 

Thank you Walter for your reply. I don’t understand what Walter means after “This can be accomplished by skipping”. It needs explaining in more detail.

 

I had suggested the 30 months of each 34 or 33 are permitted leap months as so 4 or 3 months of each 34 or 33 respectively are forbidden months.

These could be (5 15 24 34) in a 34 and (10 19 29) in a 33 forming a Helios cycle over each 34+33, which is interrupted, between 19-year cycles when there are two consecutive 34s.

 

This could be simplified to (4 14 24 34) in a 34 and (9 19 29) in a 33.

 

It could be simplified further to (4 14 24 34) in a 34 and (4 14 24) in a 33.

 

This further simplification ensures that for any 228-year cycle,  the leap months occur in the same position (counting from start) in both the 34s and the 33s, whereas my earlier suggestions would for some 228-year cycles, have the leap months occurring 1 month earlier in a 33 than in a 34, assuming they initially occurred at the first month of both.

 

In this simplest suggestion a month would be skipped every nine 228-year cycles within a 6840-year cycle starting with the 4th (from 685th year), then the 13th (from 2737th year), the 22nd (from 4789th year). Between 6840-year cycles, month 34 is skipped in 34s only and the interval between skips, not counting this skip of 34, is thirteen 228-year cycles.

 

Karl

 

16(01(20

 

 

From: Walter J Ziobro [mailto:[hidden email]]
Sent: 21 September 2016 04:37
To: [hidden email]; Palmen, Karl (STFC,RAL,ISIS)
Subject: RE: 34-33-34-33-34-33-34 RE: Cutting for late leap years

 

Dear Karl

I thank you for that info

I like your second suggestion, that there be 30 leap month shifts, one every 228 years  This can be accomplished by skipping a designated leap month every 10 cycles, such that the designated leap month s in the 33-34 month groups would be 1-10, 12-21, and 23-32

Walter Ziobro

Sent from AOL Mobile Mail

 


On Tuesday, September 20, 2016 Karl Palmen <[hidden email]> wrote:

Dear Walter and Calendar People

 

Reply below

 

From: East Carolina University Calendar discussion List [[hidden email]] On Behalf Of Walter J Ziobro
Sent: 20 September 2016 03:44
To: CALNDR-[hidden email]
Subject: Re: Cutting for late leap years

 

Dear Karl and Calendar List:


I have given some thought about the best distribution of the 7 leap months over the metonic cycle, and have concluded that the smoothest distribution can be obtained by having a leap month every 34th or 33rd month over the cycle, thereby creating a series of months in which there are 33 ordinary months, plus a leap month, alternating with 32 ordinary months, plus a leap month, like this:

34-33-34-33-34-33-34, for a total of 235 months.

This cycle can be maintained over a period of 10 or 11 metonic cycles, but then the occurrence of each leap month needs to be gradually shifted by a month, so that by the completion of 360 metonic cycles (6840 years) one leap month slips into the next metonic cycle. This can be done by having the first month of each 34 or 33 month leap  cycle be counted as the leap month, and then, after either 10 or 11  metonic cycles, having the second month of each leap cycle counted as the leap month, and so on throughout the entire 360 metonic cycles of 6840 years.

 

KARL REPLIES:  Walter’s idea runs into problems after the 33rd month of each 34 or 33 is the leap month. What happens after then?

One solution is to jump over the 34th to the 1st to smoothly drop a leap month. Then 33 of these postponements would be needed in a 6840-year cycle, 30 after 11 Metonic cycles and 3 after 3 Metonic cycles so following a sub-cycle of 120 Metonic cycles of 2280 years.

 

Another idea is to have exactly 30 permitted positions for the leap month in each 34 and 33, then postponement to the next permitted position would occur exactly once every 12 Metonic cycles of 228 years.

 

Whenever the 30th, 31st or 32nd month in each 33 & 34 were a leap month, the leap month years would follow the Hebrew 19-year cycle.

 

 

No such cycle would create the smoothest possible distribution of leap months, because some leap months will occur at least 35 months after the previous, whenever a postponement occurs. I expect it would add a day or two to the solar jitter of about one month.  If the leap months were spread as smoothly as possible, we’d have 168-month periods of 34-33-34-33-34 in addition to the 235-month periods of 34-33-34-33-34-33-34 all with fixed leap month.

 

 

Karl

 

16(01(19

 

 



-Walter Ziobro

 

 

 

-----Original Message-----
From: Karl Palmen <
[hidden email]>
To: CALNDR-L <[hidden email]
[hidden email]>
Sent: Wed, Sep 14, 2016 8:06 am
Subject: Cutting for late leap years

Dear Irv & Calendar People

 

I’ve already mentioned an algorithm for cutting calendar cycles where the leap years are spread as smoothly as possible into symmetrical parts, grouped in larger symmetrical parts.

 

Now I show a similar algorithm where the parts have the leap years as late as possible.

 

In the first iteration cut after any leap year followed by a common year. For example:

 

Hebrew 19-year cycle:  ccL|ccL|cL|ccL|ccL|ccL|cL| ,

Symmetrical 17-year leap week cycle: ccL|cccccL|cccccL|cc ,

33-year cycle: cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL|c,

Julian 4-year cycle: cccL|

                                                                                                                      

Then there should either be two different types of part (joining parts over end of cycle) or just one part in which case the algorithm has finished. The part with the greatest proportion of leap years and so shortest if leap years are in minority (shown in red) is  designated as leap year for the next iteration, even if such parts are in a majority. The examples then become:

 

ccL|ccL|cL||ccL|ccL|ccL|cL|| ,

ccL||cccccL|cccccL|cc ,

cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL||c

 

The second example has only one double cut and so the algorithm has finished. If the cycle is started after the double cut (at 4th year) the leap years will be as late as possible and if the year before the double cut (3rd year) is year Z as defined in earlier notes and if designated year 0, makes K=0.

The third example also has only one double cut and so year Z is the 32nd year and if started on the 33rd year, the leap years would occurs as late as possible.

 

The first example, which corresponds to the Hebrew 19-year cycle,  has a third iteration:

 

ccL|ccL|cL|||ccL|ccL|ccL|cL||

 

It has only one triple cut, which is between the 8th and 9th year. So the 8th year is year Z and the 9th year is the latest starting year and has subsequent leap years as late as possible.

 

 

I may look into Irv’s 353-year cycle with 130 leap years and K=269 in a later note. I’ll say at present that the second iteration cuts it into 11s and 8s. I expect the final cut to occur in the 4th iteration after the 169th year, which I reckoned to be year Z (year with 0 remainder).

 

For leap week calendars of sufficient accuracy, the first iteration cuts into 6s & 5s, the second iteration into 17s and 11s and the third iteration cuts into 62s and either 45s or 79s, except of course the 62-year cycle.

 

The number of iterations that this algorithm runs may be considered to be a measure of the structural complexity of the cycle.

 

The is also a similar mirror image algorithm, which cuts into parts that have the leap years as early as possible.

 

Karl

 

16(01(13

Reply | Threaded
Open this post in threaded view
|  
Report Content as Inappropriate

Re: 34-33-34-33-34-33-34 RE: Cutting for late leap years

Karl Palmen

Dear Walter and Calendar People

 

The original topic concerned cycles whose leap years are spread as smoothly as possible and here I’ll show what happens, if the 2519 leap months of the 6840-year cycle are spread as smoothly as possible. I reckon the 6840-year cycle would split into a number of 235-month periods of (34-33-34-33-34-33-34) equal to 19 years and 168-month periods of (34-33-34-33-34) equal to 13 & 7/12 years and a fixed month of each 34 or 33 is a leap month. I also mention similar cycles where the leap months are spread as smoothly as possible.

 

The 6840-year cycle has 360*19 years,  360*235 - 1 = 84599 months and 360*7 – 1 = 25219 leap months.

If we count the months from the start of a 6840-year cycle, then month M will be a leap month if and only if

 

Remainder of (2519*M + K) divided by 84599 is less than 2519

 

Over 34 months this remainder goes up by 1047 or down by 84599-1047.

So if month M has its remainder at least 1047, but less than 2519-1047=1472, then it is 34 months after the previous leap month and 34 months before the next leap month and is vice versa.  Each period of 235 or 168 months has exactly one such month, which occurs in its first 34.

 

There are 1472-1047=425 such months and hence 425 periods of 235 or 168 months. If all 425 has 235 months, and so 7 leaps months, we’d have 7*425=2975 leap months, which is 456 leap months too many and so we have 456/2=228 periods of 168 months and the remaining 425-228=197 periods have 235 months. So we have

 

197 periods (34-33-34-33-34-33-34) of 235 months = 19 years with 7 leap months &

228 periods (34-33-34-33-34) of 168 months = 13 & 7/12 years with 5 leap months.

 

Checking:

Months: 197*235 + 228*168 = 83599

Leap months: 197*7 + 228*5 = 2519.

 

These would be spread as smoothly as possible over the 6840-year cycle.

 

The number of 235s is roughly equal to the number of 168s, so it may be worth looking at the case, where they are equal. Then we’d have a 403-month cycle of 32 & 7/12 years = 33 & 7/12 lunar years, which I recently mentioned in reply to Aristeo.

 

(34-33-34-33-34) (34-33-34-33-34-33-34) of 403 months = 32 & 7/12 years with 12 leap months.

 

Twelve of these make up a 391-year cycle equal to 403 lunar months, which is the Grattan-Guinness cycle mentioned numerous times before.

 

Other more accurate cycles can be obtained by removing a small number of 235-month periods from these twelve 403-month cycles, so if leap months are spread as smoothly as possible producing a few yerm-like periods of (168-235-168-235….168).

 

391 years of 144 leap months:  168-235 alternating

372 years of 137 leap months: (168-235-168-235…168) of 23

353 years of 130 leap months: 2*(168-235-168-235…168) of 11

334 years of 123 leap months: 3*(168-235-168-235-168-235-168) of 7

315 years of 116 leap months: 4*(168-235-158-235-168) of 5

 

The 6840-year cycle would have 228-197=31 of these (168-235-168-235…168) of 13 or 15 spread as smoothly as possible. However the 6840-year does not have particular merit, when the leap months are spread as smoothly as possible and its better suited to earlier suggestions.

 

 

The following approximations of solar years to lunar years have been alluded to in this note

67 lunar years to 65 solar years in 34-33 of 67 months

168 lunar years to 163 solar years in (34-33-34-33-34) of 168 months

235 lunar years to 228 solar years in (34-33-34-33-34-33-34) of 235 months

403 lunar years to 391 solar years in (34-33-34-33-34) (34-33-34-33-34-33-34) = (168+235) of 403 months

Happy Equinox

 

Karl

 

16(01(21

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 21 September 2016 13:10
To: 'Walter J Ziobro'; [hidden email]
Subject: RE: 34-33-34-33-34-33-34 RE: Cutting for late leap years

 

Dear Walter and Calendar People

 

Thank you Walter for your reply. I don’t understand what Walter means after “This can be accomplished by skipping”. It needs explaining in more detail.

 

I had suggested the 30 months of each 34 or 33 are permitted leap months as so 4 or 3 months of each 34 or 33 respectively are forbidden months.

These could be (5 15 24 34) in a 34 and (10 19 29) in a 33 forming a Helios cycle over each 34+33, which is interrupted, between 19-year cycles when there are two consecutive 34s.

 

This could be simplified to (4 14 24 34) in a 34 and (9 19 29) in a 33.

 

It could be simplified further to (4 14 24 34) in a 34 and (4 14 24) in a 33.

 

This further simplification ensures that for any 228-year cycle,  the leap months occur in the same position (counting from start) in both the 34s and the 33s, whereas my earlier suggestions would for some 228-year cycles, have the leap months occurring 1 month earlier in a 33 than in a 34, assuming they initially occurred at the first month of both.

 

In this simplest suggestion a month would be skipped every nine 228-year cycles within a 6840-year cycle starting with the 4th (from 685th year), then the 13th (from 2737th year), the 22nd (from 4789th year). Between 6840-year cycles, month 34 is skipped in 34s only and the interval between skips, not counting this skip of 34, is thirteen 228-year cycles.

 

Karl

 

16(01(20

 

 

From: Walter J Ziobro [[hidden email]]
Sent: 21 September 2016 04:37
To: [hidden email]; Palmen, Karl (STFC,RAL,ISIS)
Subject: RE: 34-33-34-33-34-33-34 RE: Cutting for late leap years

 

Dear Karl

I thank you for that info

I like your second suggestion, that there be 30 leap month shifts, one every 228 years  This can be accomplished by skipping a designated leap month every 10 cycles, such that the designated leap month s in the 33-34 month groups would be 1-10, 12-21, and 23-32

Walter Ziobro

Sent from AOL Mobile Mail

 


On Tuesday, September 20, 2016 Karl Palmen <[hidden email]> wrote:

Dear Walter and Calendar People

 

Reply below

 

From: East Carolina University Calendar discussion List [[hidden email]] On Behalf Of Walter J Ziobro
Sent: 20 September 2016 03:44
To: CALNDR-[hidden email]
Subject: Re: Cutting for late leap years

 

Dear Karl and Calendar List:


I have given some thought about the best distribution of the 7 leap months over the metonic cycle, and have concluded that the smoothest distribution can be obtained by having a leap month every 34th or 33rd month over the cycle, thereby creating a series of months in which there are 33 ordinary months, plus a leap month, alternating with 32 ordinary months, plus a leap month, like this:

34-33-34-33-34-33-34, for a total of 235 months.

This cycle can be maintained over a period of 10 or 11 metonic cycles, but then the occurrence of each leap month needs to be gradually shifted by a month, so that by the completion of 360 metonic cycles (6840 years) one leap month slips into the next metonic cycle. This can be done by having the first month of each 34 or 33 month leap  cycle be counted as the leap month, and then, after either 10 or 11  metonic cycles, having the second month of each leap cycle counted as the leap month, and so on throughout the entire 360 metonic cycles of 6840 years.

 

KARL REPLIES:  Walter’s idea runs into problems after the 33rd month of each 34 or 33 is the leap month. What happens after then?

One solution is to jump over the 34th to the 1st to smoothly drop a leap month. Then 33 of these postponements would be needed in a 6840-year cycle, 30 after 11 Metonic cycles and 3 after 3 Metonic cycles so following a sub-cycle of 120 Metonic cycles of 2280 years.

 

Another idea is to have exactly 30 permitted positions for the leap month in each 34 and 33, then postponement to the next permitted position would occur exactly once every 12 Metonic cycles of 228 years.

 

Whenever the 30th, 31st or 32nd month in each 33 & 34 were a leap month, the leap month years would follow the Hebrew 19-year cycle.

 

 

No such cycle would create the smoothest possible distribution of leap months, because some leap months will occur at least 35 months after the previous, whenever a postponement occurs. I expect it would add a day or two to the solar jitter of about one month.  If the leap months were spread as smoothly as possible, we’d have 168-month periods of 34-33-34-33-34 in addition to the 235-month periods of 34-33-34-33-34-33-34 all with fixed leap month.

 

 

Karl

 

16(01(19

 

 



-Walter Ziobro

 

 

 

-----Original Message-----
From: Karl Palmen <
[hidden email]>
To: CALNDR-L <[hidden email]
[hidden email]>
Sent: Wed, Sep 14, 2016 8:06 am
Subject: Cutting for late leap years

Dear Irv & Calendar People

 

I’ve already mentioned an algorithm for cutting calendar cycles where the leap years are spread as smoothly as possible into symmetrical parts, grouped in larger symmetrical parts.

 

Now I show a similar algorithm where the parts have the leap years as late as possible.

 

In the first iteration cut after any leap year followed by a common year. For example:

 

Hebrew 19-year cycle:  ccL|ccL|cL|ccL|ccL|ccL|cL| ,

Symmetrical 17-year leap week cycle: ccL|cccccL|cccccL|cc ,

33-year cycle: cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL|c,

Julian 4-year cycle: cccL|

                                                                                                                      

Then there should either be two different types of part (joining parts over end of cycle) or just one part in which case the algorithm has finished. The part with the greatest proportion of leap years and so shortest if leap years are in minority (shown in red) is  designated as leap year for the next iteration, even if such parts are in a majority. The examples then become:

 

ccL|ccL|cL||ccL|ccL|ccL|cL|| ,

ccL||cccccL|cccccL|cc ,

cccL|cccL|cccL|cccL|cccL|cccL|cccL|cccL||c

 

The second example has only one double cut and so the algorithm has finished. If the cycle is started after the double cut (at 4th year) the leap years will be as late as possible and if the year before the double cut (3rd year) is year Z as defined in earlier notes and if designated year 0, makes K=0.

The third example also has only one double cut and so year Z is the 32nd year and if started on the 33rd year, the leap years would occurs as late as possible.

 

The first example, which corresponds to the Hebrew 19-year cycle,  has a third iteration:

 

ccL|ccL|cL|||ccL|ccL|ccL|cL||

 

It has only one triple cut, which is between the 8th and 9th year. So the 8th year is year Z and the 9th year is the latest starting year and has subsequent leap years as late as possible.

 

 

I may look into Irv’s 353-year cycle with 130 leap years and K=269 in a later note. I’ll say at present that the second iteration cuts it into 11s and 8s. I expect the final cut to occur in the 4th iteration after the 169th year, which I reckoned to be year Z (year with 0 remainder).

 

For leap week calendars of sufficient accuracy, the first iteration cuts into 6s & 5s, the second iteration into 17s and 11s and the third iteration cuts into 62s and either 45s or 79s, except of course the 62-year cycle.

 

The number of iterations that this algorithm runs may be considered to be a measure of the structural complexity of the cycle.

 

The is also a similar mirror image algorithm, which cuts into parts that have the leap years as early as possible.

 

Karl

 

16(01(13

Loading...