Re: 400-yrs,834-yrs & 896-yrs RE: Symmetry Statement and Divide-by-Six RE:

Dear Brij and Calendar People

 

The Kepler’s leap week years of the 834-year cycle can be arranged symmetrically thus:

45^th, 141^st, 231^st, 327^th 417^th, 507^th, 603^rd, 693^rd, 789^th years. The intervals between the successive Kepler's leap weeks are either 90 or 96 years, including the interval over the end of the cycle.

This is much better than Brij’s suggestion of

81^st, 165^th, 249^th, 333^rd, 417^th, 501^st, 855^th, 669^th 753^rd  years, which suffers from a huge gap of 162 years over the end of the cycle. Even the simple 57^th, 147^th, 237^th, 327^th, 417^th, 507^th, 597^th, 687^th, 777^th is better with a gap of 114 years.

Brij seems to be unaware of the importance of having the all the intervals between Kepler's leap weeks with either 90 or 96 years to avoid excessive jitter. Also the intervals of 90 and 96 years need to be roughly equal in proportion with slightly more 90s than 96s.

 

The cycles listed although symmetrical do not have the same symmetry in the symmetry statement, which requires the first year to be of the same type as the last year. In this case year 1 has no leap week, but year 834 does has a leap week. So the statement that Irv and I have made does not apply to them.

What the symmetry of the 834-year cycle does tell you is that the middle of year 417 and  834 is time at the average of middles of all years.

 

 

About the 895-year cycle, Brij stated

 

During use of 896-Years Cycle (Div.6 Plan), In addition to Div.six(6) Leap Weeks, following Kepler's Leap Weeks are ADDED every896-year cycle. First KLWk is added during 87th-year, while rest nine are added ONCE every 90-years and then repeating every 896-year cycle ; to give Mean Year of 7*(52+159/896) =365.2421875 days. 1920 thro 2816 AD: 2007, 2097, 2187, 2277, 2367, 2457, 2547, 2637, 2727 & 2817th years

I hope this clarify my approach, to place Kepler's Leap Weeks in 400-year, 834-year & 896-years cycle. Multiples of these cycles could be used, in necessary to achieve better placing/results.

 

The 896-year cycle as stated is here erroneous. If he were to add 10 Kepler’s leap week years like this every 896-year cycle, then in three 896-year cycles, he have 3*896/6 = 448 leap weeks in years divisible by six and 30 Kepler’s leap weeks adding to 478 in total. But three 896-year cycle require just 3*159=477 leap weeks and so such a 896-year cycle would be one week out every 3 cycles of 2688 years.  Perhaps, Brij intends to omit one of these 10 Kepler’s leap weeks once every third 896-year cycle. This would correct the number of leap weeks, but produce extra jitter by leaving a gap of 180 or 186 years between two of the Kepler’s leap weeks.

 

Instead one can use a 2688-year cycle (equal to three 896-year cycles) made up of two 834-year cycles and one 1020-year cycle. The Kepler’s leap years can be as stated above on the 45^th, 141^st, 231^st, 327^th 417^th, 507^th, 603^rd, 693^rd and 789^th year of each 834-year cycle and also the 879^th and 975^th years of each 1020-year cycle. This gives rise to 9+9+11=29 Kepler’s leap weeks in a whole 2688-year cycle, which added to the 448 divide-by-six leap weeks gives the required 477=3*159 leap weeks. The 1020-year cycle cannot be made symmetrical in the same way as the 834-year cycle,  because the number of years is not twice an odd number. The same applies to the entire 2688-year cycle.

 

However, the 462-year cycle of 5 Kepler’s leap weeks and same mean year as 33-year cycle can have this symmetry with Kepler’s leap weeks on the 45^th, 141^st, 231^st, 321^st and 417^th year of each 462-year cycle.

 

Karl

 

10(08(03

 

 

 

Karl, Irv & CC:
>I now realise that if the 148 leap years were arranged in two identical cycles of 417 years and 79 leap years, then an odd number of
>moves would be needed to change this into the 834-year cycle described, where one move is a shift of one leap week by one year.
I had shown such distribution in my: http://www.brijvij.com/bbv_417-year-div.6.pdf

Thus 5th KLWk is placed at midway between 2*417-years i.e. 834 years. 0081, 0165, 0249, 0333, 0417, 0501, 0585, 0669, 0753.....

Distribution shown now in http://www.brijvij.com/bb_harappaTithi-Cycles.pdf is aslo (Y1920+417 =Y2337) as the mid year for symmetry.
During my discussion with list earlier (some 2 years ago), I had shown that Mean Year for Gregorian calendar could be attained using (3*400)-year cycle, with 13 Keplers' Leap Weeks. There, however, was difference of opinion for 896-year cycle (896/6 not being excatly divisible, and 3*896=2688 years cycle was considered). I believe, since 896-years are a little over 159 weeks (as pointed earlier) I attempted to charp cut close to 448th year and shown distribution as:

During use of 896-Years Cycle (Div.6 Plan), In addition to Div.six(6) Leap Weeks, following Keplers Leap Weeks are ADDED every896-year cycle. First KLWk is added during 87th-year, while rest nine are added ONCE every 90-years and then repeating every 896-year cycle ; to give Mean Year of 7*(52+159/896) =365.2421875 days. 1920 thro 2816 AD: 2007, 2097, 2187, 2277, 2367, 2457, 2547, 2637, 2727 & 2817th years

I hope this clarify my approach, to place Keplers' Leap Weeks in 400-year, 834-year & 896-years cycle. Multiples of these cycles could be used, in necessary to achieve better placing/results.
Regards,
Brij Bhushan Vij
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Date: Fri, 24 Apr 2009 14:38:15 +0100
From: karl.palmen@...
Subject: Re: Symmetry Statement and Divide-by-Six RE: using Primary cycles of 11,15,19&33-years RE:
To: CALNDR-L@...

 

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