Cutting for late leap years

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Cutting for late leap years

Karl Palmen

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

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Re: Cutting for late leap years

Walter J Ziobro
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.

-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
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