Interval Cycles of Interval Cycles

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Interval Cycles of Interval Cycles

Karl Palmen

Dear Calendar People

 

Suppose one is presented with a cycle of leap years, such as a Tabular Islamic 30-year cycle of leap years (2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29) and one wants to know whether the leap years are spaced as smoothly as possible. One can do so by recursively examining the interval cycles.

 

In the example given the 10 intervals between the 11 leap years are (3, 2, 3, 3, 3, 2, 3, 3, 2, 3). To these add the interval between the last leap year of the cycle and the first leap year of the next cycle and we get:

 

(3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3)

 

There are two interval lengths and they differ by one. If there difference were more than 1 or there were 3 or more interval lengths, the leap years are not spread as smoothly as possible. In this case this is not so and so we go onto the next step, which is to regard each interval as a year and the interval of minority length as a leap year and then work out the intervals between these leap years.

 

In this example the three intervals of 2 count as leap years. The occur in the 2nd, 6th & 9th positions in 11. The resulting intervals are thus:

 

( 4, 3, 4 )

 

There are two interval lengths differing by 1 and so we treat the 3 as a leap year and so have one leap year every 3 years, so ending up with one interval of 3.

 

(3)

 

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps.

 

 

Here is a 30-year cycle that appears to have its leap years spread as smoothly as possible, but does not

(2, 4, 7, 10, 13, 16, 18, 21, 24, 26, 29)

It is the same as the previous example, but year 4 is a leap year instead of year 5. The intervals are thus

 

(2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3)

 

So we can go onto the next step (noting that the 2s are in the 1st, 6th & 9th positions in 11)  and we get:

 

(5, 3, 3)

 

Here the interval lengths differ by two and this shows that the 30-year cycle does not have its 11 leap years spread as smoothly as possible. The consecutive intervals of two [2, 4) and [16,18) are too far apart.

 

Karl

 

16(09(24

 

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 21 October 2016 13:03
To: '[hidden email]'
Subject: RE: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

 

Dear Calendar People

 

A summary of examples of the complexity of cycles whose leap years are spread as smoothly as possible.

 

Definition: The complexity of a cycle is how far its Ford circle is from either Ford circle of 0/1 or 1/1 in steps of touching Ford Circles.

See https://en.wikipedia.org/wiki/Ford_circle for Ford Circle.

The Ford circle of a cycle is the Ford circle of the fractional part of its mean year, which will serve as notation of the example cycles below.

 

For example, the 293-year leap week cycle 52/293 has its Ford circle touch 11/62, which in turn touches 3/17, which in turn touches 1/6, which in turn touches 0/1. There are 4 touchings, therefore 4 steps and also this is the minimum and so 52/293 has complexity 4. Note that the Ford circles of a/b & c/d touch if and only if ad & bc differ by 1.

 

A cycle has complexity 1 if it has only one leap year or one common year.

A cycle has complexity 2 if its number of years is one different from a proper multiple of the number of leap years or the number of common years.

 

Solar leap day calendars

Complexity 1: ¼ Julian Calendar

Complexity 2: 8/33 33-year cycle

Complexity 3: 31/128 & 71/400

 

Leap Week Calendars

Complexity 1: 1/6 & 1/5

Complexity 2: 3/17 & 2/11

Complexity 3: 14/79, 11/62, 8/45 & 5/28

Complexity 4: 69/389, 74/417, 52/293 & 41/231

Complexity 5: 159/896 & 71/400

 

Lunar Calendars by month

Complexity 1: ½

Complexity 2: 9/17 & 8/15 yerm

Complexity 3: 26/49 3-yerm cycle

Complexity 4: 399/752, 425/801, 451/850, 477/899, … and all other one-era cycles

Complexity 5: 876/1651 & 928/1749 some multi-era cycles

Complexity 6: 13752/25920 Hebrew Yerm Calendar

 

Lunar Calendars by 12-month year

Complexity 1: 1/3, ½

Complexity 2: 4/11, 3/8

Complexity 3: 11/30 Tabular Islamic

Complexity 4: 29/79 Suggested improvement of Tabular Islamic

 

Lunisolar Calendar leap month cycles

Complexity 1: 1/3, ½

Complexity 2: 4/11, 3/8

Complexity 3: 7/19 Metonic cycle

Complexity 4: 123/334 & 130/353 and other cycles made from 7/19s one of which is truncated to 4/11.

Complexity 5: 239/649, 383/1040 & 267/725

Complexity 6: 622/1689 was used in an unsmooth form in 2nd Goddess Lunar calendar

Complexity 7: 2519/6840 Meyer-Palmen cycle

 

28-day month Calendar leap month cycles

Complexity 3: 13/293

 

30-day month Calendar leap month cycles

Complexity 3: 7/40 Julian calendar mean year

Complexity 4: 18/103

 

Karl

 

16(02(20

 

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19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Brij Bhushan metric VIJ
Karl, list sirs:
Islamic lunar calendar 'lunar moons spread' is possibly the best so far, using a 30-year cycle 'symmetrically spread'.
>Ending with one interval proves that the >leap years in the cycle are spaced as >smoothly as possible and the number of >steps to get there is equal to the structural >complexity of the cycle as I defined in >October and show in the note below. In this >case the complexity is 3 from three steps.
While I may not be as qualified as Karl or some other members on the list; this has on a way triggered me to consider a 100-years cycle in (4*25-years) each having 9 'leap moons' spread as smoothly as possible:

01. 04. 07. 10. 13. 16. 19. 22. 25. (27) 29. 32. 35. 38. 41. 44. 47. 50. (52) 54. 57. 60. 63. 66  69 72. 75. (77) 79.  82. 85. 88. 91. 94. 97. (99).....(102).    

19/7=2.7143-years/leap moon

The above spread, I show is in line with my proposed [2 cycles of 448-years/5541 lunar moons] suggesting the distribution 'close to 3-years' between adjacent 'leap moons'.

The spread generally is: 

3 3,3,3,3,3,3,3,2 i.e. 36 leap moons per 100-years; at an average of 2.777777778 between two adjacent leap moons. 

Possibilities of 19-years spread can be:

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

There must be other permutations for calendar experts to consider!


image1.JPG
Regards, 
Ex-Flt.Lt. Brij Bhushan VIJ, Author
Brij-Gregorian Modified Calendar
Friday, 2017  May 19H09:12(decimal)

Sent from my iPhone

On May 19, 2017, at 5:03 AM, Karl Palmen <[hidden email]> wrote:

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps
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Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Karl Palmen

Dear Brij and Calendar People

 

While I don’t fully understand Brij’s ideas here. I can demonstrate my idea of intervals cycles of interval cycles on some of his examples:

 

First look at Brij’s 3 suggested 19-year cycles

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

and apply the intervals of intervals method to them.

3 3 3 3 3 3 1

3 3 3 3 3 2 2

3 3 3 2 2 3 3

Then the first fails because the intervals have length 3 & 1, which differ by 2. The interval of 1 is between the 19th year of one cycle and the 1st year of the next cycle.

Others can be carried on to the next step and then both have one interval of 6 and one interval of 1, so are not spread as smoothly as possible.

 

The Hebrew 19-year cycle is spread as smoothly as possible:

3. 6. 8. 11. 14. 17. 19.

It has intervals

3 2 3 3 3 2 3

And the intervals of intervals are

4 3

This has one interval of intervals of 2.

So this cycle is spread as smoothly as possible of complexity 3, because there are 3 steps to the single interval of 2.

There are 18 other possible such cycles formed by changing the start year. All of them have complexity 3.

One of these is symmetrical:

2. 5. 7. 10. 13. 15. 18.

 

 

I also see Brij presented a cycle of 36 leap years in a 100 years. The example he gave was not clear and the leap years were not spread as smoothly as possible, because like his 2nd  & 3rd  19-year cycle suggestions it had two consecutive intervals of 2 while the majority of intervals were 3. If the 36 leap years were spread as smoothly as possible, we’d have four cycles of 25 years with 9 leap years each.

 

The 25-year cycle of 9 leap years as 9 intervals that add up to 25. So we have seven intervals of 3 and two intervals of 2 (25 = 7*3 + 2*2). Therefore the interval of intervals cycle has two intervals that add up to 9.

Working my intervals of intervals method backwards, we can construct such a cycle

 

4, 5

 

3, 3, 3, 2, 3, 3, 3, 3, 2

 

1. 4. 7. 9. 12. 15. 18. 21. 23.

 

This is not the only 25-year cycle of 9 leap years spread as smoothly as possible. There are 24 others formed by changing the start year. They all have complexity 3.

One of these is symmetrical:

 

2. 5. 7. 10. 13. 16. 18. 21. 24.

 

If used as a lunisolar cycle, it is not as accurate as the 19-year cycle, but if all the years were to have 365 days, then its months would be more accurate than the those of the 19-year cycle and the Egyptians did use such a lunar calendar.

 

 

 

Karl

 

16(09(27

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Brij Bhushan metric VIJ
Sent: 19 May 2017 17:08
To: [hidden email]
Subject: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

 

Karl, list sirs:

Islamic lunar calendar 'lunar moons spread' is possibly the best so far, using a 30-year cycle 'symmetrically spread'.

>Ending with one interval proves that the >leap years in the cycle are spaced as >smoothly as possible and the number of >steps to get there is equal to the structural >complexity of the cycle as I defined in >October and show in the note below. In this >case the complexity is 3 from three steps.
While I may not be as qualified as Karl or some other members on the list; this has on a way triggered me to consider a 100-years cycle in (4*25-years) each having 9 'leap moons' spread as smoothly as possible:

 

01. 04. 07. 10. 13. 16. 19. 22. 25. (27) 29. 32. 35. 38. 41. 44. 47. 50. (52) 54. 57. 60. 63. 66  69 72. 75. (77) 79.  82. 85. 88. 91. 94. 97. (99).....(102).    

19/7=2.7143-years/leap moon

The above spread, I show is in line with my proposed [2 cycles of 448-years/5541 lunar moons] suggesting the distribution 'close to 3-years' between adjacent 'leap moons'.

The spread generally is: 

3 3,3,3,3,3,3,3,2 i.e. 36 leap moons per 100-years; at an average of 2.777777778 between two adjacent leap moons. 

Possibilities of 19-years spread can be:

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

There must be other permutations for calendar experts to consider!


image1.JPG
Regards, 

Ex-Flt.Lt. Brij Bhushan VIJ, Author

Brij-Gregorian Modified Calendar

Friday, 2017  May 19H09:12(decimal)

Sent from my iPhone


On May 19, 2017, at 5:03 AM, Karl Palmen <[hidden email]> wrote:

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps

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The Best option Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Brij Bhushan metric VIJ
Karl, list sirs:
I 😊 for examining the option, and have NO reservations.
Honestly, I have only heard about Ford circles, I only tried to put in my option as 'one among several options' using 100-year 'span' using 37,36,37,36 & 17 making the required 165 *leap moons* in 448-years/5541 Lunation cycle. Naturally, the best need be chosen. My input is 'just' a thought.
I thank you, Karl. 
Regards,
Ex-Flt Lt Brij Bhushan VIJ, Author
Brij-Gregorian Modified Caldndar
Monday, 2017 May 22H07:02 (decimal)

Sent from my iPhone

On May 22, 2017, at 5:28 AM, Karl Palmen <[hidden email]> wrote:

Dear Brij and Calendar People

 

While I don’t fully understand Brij’s ideas here. I can demonstrate my idea of intervals cycles of interval cycles on some of his examples:

 

First look at Brij’s 3 suggested 19-year cycles

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

and apply the intervals of intervals method to them.

3 3 3 3 3 3 1

3 3 3 3 3 2 2

3 3 3 2 2 3 3

Then the first fails because the intervals have length 3 & 1, which differ by 2. The interval of 1 is between the 19th year of one cycle and the 1st year of the next cycle.

Others can be carried on to the next step and then both have one interval of 6 and one interval of 1, so are not spread as smoothly as possible.

 

The Hebrew 19-year cycle is spread as smoothly as possible:

3. 6. 8. 11. 14. 17. 19.

It has intervals

3 2 3 3 3 2 3

And the intervals of intervals are

4 3

This has one interval of intervals of 2.

So this cycle is spread as smoothly as possible of complexity 3, because there are 3 steps to the single interval of 2.

There are 18 other possible such cycles formed by changing the start year. All of them have complexity 3.

One of these is symmetrical:

2. 5. 7. 10. 13. 15. 18.

 

 

I also see Brij presented a cycle of 36 leap years in a 100 years. The example he gave was not clear and the leap years were not spread as smoothly as possible, because like his 2nd  & 3rd  19-year cycle suggestions it had two consecutive intervals of 2 while the majority of intervals were 3. If the 36 leap years were spread as smoothly as possible, we’d have four cycles of 25 years with 9 leap years each.

 

The 25-year cycle of 9 leap years as 9 intervals that add up to 25. So we have seven intervals of 3 and two intervals of 2 (25 = 7*3 + 2*2). Therefore the interval of intervals cycle has two intervals that add up to 9.

Working my intervals of intervals method backwards, we can construct such a cycle

 

4, 5

 

3, 3, 3, 2, 3, 3, 3, 3, 2

 

1. 4. 7. 9. 12. 15. 18. 21. 23.

 

This is not the only 25-year cycle of 9 leap years spread as smoothly as possible. There are 24 others formed by changing the start year. They all have complexity 3.

One of these is symmetrical:

 

2. 5. 7. 10. 13. 16. 18. 21. 24.

 

If used as a lunisolar cycle, it is not as accurate as the 19-year cycle, but if all the years were to have 365 days, then its months would be more accurate than the those of the 19-year cycle and the Egyptians did use such a lunar calendar.

 

 

 

Karl

 

16(09(27

 

From: East Carolina University Calendar discussion List [[hidden email]] On Behalf Of Brij Bhushan metric VIJ
Sent: 19 May 2017 17:08
To: [hidden email]
Subject: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

 

Karl, list sirs:

Islamic lunar calendar 'lunar moons spread' is possibly the best so far, using a 30-year cycle 'symmetrically spread'.

>Ending with one interval proves that the >leap years in the cycle are spaced as >smoothly as possible and the number of >steps to get there is equal to the structural >complexity of the cycle as I defined in >October and show in the note below. In this >case the complexity is 3 from three steps.
While I may not be as qualified as Karl or some other members on the list; this has on a way triggered me to consider a 100-years cycle in (4*25-years) each having 9 'leap moons' spread as smoothly as possible:

 

01. 04. 07. 10. 13. 16. 19. 22. 25. (27) 29. 32. 35. 38. 41. 44. 47. 50. (52) 54. 57. 60. 63. 66  69 72. 75. (77) 79.  82. 85. 88. 91. 94. 97. (99).....(102).    

19/7=2.7143-years/leap moon

The above spread, I show is in line with my proposed [2 cycles of 448-years/5541 lunar moons] suggesting the distribution 'close to 3-years' between adjacent 'leap moons'.

The spread generally is: 

3 3,3,3,3,3,3,3,2 i.e. 36 leap moons per 100-years; at an average of 2.777777778 between two adjacent leap moons. 

Possibilities of 19-years spread can be:

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

There must be other permutations for calendar experts to consider!


<image001.jpg>
Regards, 

Ex-Flt.Lt. Brij Bhushan VIJ, Author

Brij-Gregorian Modified Calendar

Friday, 2017  May 19H09:12(decimal)

Sent from my iPhone


On May 19, 2017, at 5:03 AM, Karl Palmen <[hidden email]> wrote:

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps

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Re: The Best option Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Brij Bhushan metric VIJ
Sirs:
My apology for not reading before hitting...SEND command.
>, I only tried to put in my option as 'one among several options' using 100-year 'span' using 37,36,37,36 & 17 making the required 165 *leap moons*.......
....should read:
, "I only tried to put in my option as 'one among several options' using 100-year 'span' using 37,36,37,36 & 19 making the required 165 *leap moons*..."
Thanks & Regards,
Flt Lt Brij Bhushan Vij (Retd.)
Monday, 2017 May 22H09:07(decimal)

Sent from my iPhone

> On May 22, 2017, at 7:02 AM, Brij Bhushan metric VIJ <[hidden email]> wrote:
>
> , I only tried to put in my option as 'one among several options' using 100-year 'span' using 37,36,37,36 & 17 making the required 165 *leap moons*
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Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Walter J Ziobro
In reply to this post by Karl Palmen

Dear Karl Brij and Calendar List

Another approach to this issue would be to distribute month lengths in the Islamic calendar according to the Balinese ngunalatri method This will rresjult in 3 32 month cycles in 8 Islamic years One day only needs to be dropped in 120 Islamic years for it to have exactly as many days as in the tabular format

Walter Ziobro

Sent from AOL Mobile Mail




On Monday, May 22, 2017 Karl Palmen <[hidden email]> wrote:

Dear Brij and Calendar People

 

While I don’t fully understand Brij’s ideas here. I can demonstrate my idea of intervals cycles of interval cycles on some of his examples:

 

First look at Brij’s 3 suggested 19-year cycles

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

and apply the intervals of intervals method to them.

3 3 3 3 3 3 1

3 3 3 3 3 2 2

3 3 3 2 2 3 3

Then the first fails because the intervals have length 3 & 1, which differ by 2. The interval of 1 is between the 19th year of one cycle and the 1st year of the next cycle.

Others can be carried on to the next step and then both have one interval of 6 and one interval of 1, so are not spread as smoothly as possible.

 

The Hebrew 19-year cycle is spread as smoothly as possible:

3. 6. 8. 11. 14. 17. 19.

It has intervals

3 2 3 3 3 2 3

And the intervals of intervals are

4 3

This has one interval of intervals of 2.

So this cycle is spread as smoothly as possible of complexity 3, because there are 3 steps to the single interval of 2.

There are 18 other possible such cycles formed by changing the start year. All of them have complexity 3.

One of these is symmetrical:

2. 5. 7. 10. 13. 15. 18.

 

 

I also see Brij presented a cycle of 36 leap years in a 100 years. The example he gave was not clear and the leap years were not spread as smoothly as possible, because like his 2nd  & 3rd  19-year cycle suggestions it had two consecutive intervals of 2 while the majority of intervals were 3. If the 36 leap years were spread as smoothly as possible, we’d have four cycles of 25 years with 9 leap years each.

 

The 25-year cycle of 9 leap years as 9 intervals that add up to 25. So we have seven intervals of 3 and two intervals of 2 (25 = 7*3 + 2*2). Therefore the interval of intervals cycle has two intervals that add up to 9.

Working my intervals of intervals method backwards, we can construct such a cycle

 

4, 5

 

3, 3, 3, 2, 3, 3, 3, 3, 2

 

1. 4. 7. 9. 12. 15. 18. 21. 23.

 

This is not the only 25-year cycle of 9 leap years spread as smoothly as possible. There are 24 others formed by changing the start year. They all have complexity 3.

One of these is symmetrical:

 

2. 5. 7. 10. 13. 16. 18. 21. 24.

 

If used as a lunisolar cycle, it is not as accurate as the 19-year cycle, but if all the years were to have 365 days, then its months would be more accurate than the those of the 19-year cycle and the Egyptians did use such a lunar calendar.

 

 

 

Karl

 

16(09(27

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Brij Bhushan metric VIJ
Sent: 19 May 2017 17:08
To: CALNDR-L@...
Subject: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

 

Karl, list sirs:

Islamic lunar calendar 'lunar moons spread' is possibly the best so far, using a 30-year cycle 'symmetrically spread'.

>Ending with one interval proves that the >leap years in the cycle are spaced as >smoothly as possible and the number of >steps to get there is equal to the structural >complexity of the cycle as I defined in >October and show in the note below. In this >case the complexity is 3 from three steps.
While I may not be as qualified as Karl or some other members on the list; this has on a way triggered me to consider a 100-years cycle in (4*25-years) each having 9 'leap moons' spread as smoothly as possible:

 

01. 04. 07. 10. 13. 16. 19. 22. 25. (27) 29. 32. 35. 38. 41. 44. 47. 50. (52) 54. 57. 60. 63. 66  69 72. 75. (77) 79.  82. 85. 88. 91. 94. 97. (99).....(102).    

19/7=2.7143-years/leap moon

The above spread, I show is in line with my proposed [2 cycles of 448-years/5541 lunar moons] suggesting the distribution 'close to 3-years' between adjacent 'leap moons'.

The spread generally is: 

3 3,3,3,3,3,3,3,2 i.e. 36 leap moons per 100-years; at an average of 2.777777778 between two adjacent leap moons. 

Possibilities of 19-years spread can be:

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

There must be other permutations for calendar experts to consider!



Regards, 

Ex-Flt.Lt. Brij Bhushan VIJ, Author

Brij-Gregorian Modified Calendar

Friday, 2017  May 19H09:12(decimal)

Sent from my iPhone


On May 19, 2017, at 5:03 AM, Karl Palmen <[hidden email]> wrote:

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps

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Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Walter J Ziobro
In reply to this post by Karl Palmen

Dear Karl Brij and Calendar List

Thinking about the Subject Title of this message, It occurs to me that the best distribution of leap months in the 235 months of the 19 year cycle is one each 34-33-34-33-34-33-34 months

Walter Ziobro

Sent from AOL Mobile Mail




On Monday, May 22, 2017 Karl Palmen <[hidden email]> wrote:

Dear Brij and Calendar People

 

While I don’t fully understand Brij’s ideas here. I can demonstrate my idea of intervals cycles of interval cycles on some of his examples:

 

First look at Brij’s 3 suggested 19-year cycles

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

and apply the intervals of intervals method to them.

3 3 3 3 3 3 1

3 3 3 3 3 2 2

3 3 3 2 2 3 3

Then the first fails because the intervals have length 3 & 1, which differ by 2. The interval of 1 is between the 19th year of one cycle and the 1st year of the next cycle.

Others can be carried on to the next step and then both have one interval of 6 and one interval of 1, so are not spread as smoothly as possible.

 

The Hebrew 19-year cycle is spread as smoothly as possible:

3. 6. 8. 11. 14. 17. 19.

It has intervals

3 2 3 3 3 2 3

And the intervals of intervals are

4 3

This has one interval of intervals of 2.

So this cycle is spread as smoothly as possible of complexity 3, because there are 3 steps to the single interval of 2.

There are 18 other possible such cycles formed by changing the start year. All of them have complexity 3.

One of these is symmetrical:

2. 5. 7. 10. 13. 15. 18.

 

 

I also see Brij presented a cycle of 36 leap years in a 100 years. The example he gave was not clear and the leap years were not spread as smoothly as possible, because like his 2nd  & 3rd  19-year cycle suggestions it had two consecutive intervals of 2 while the majority of intervals were 3. If the 36 leap years were spread as smoothly as possible, we’d have four cycles of 25 years with 9 leap years each.

 

The 25-year cycle of 9 leap years as 9 intervals that add up to 25. So we have seven intervals of 3 and two intervals of 2 (25 = 7*3 + 2*2). Therefore the interval of intervals cycle has two intervals that add up to 9.

Working my intervals of intervals method backwards, we can construct such a cycle

 

4, 5

 

3, 3, 3, 2, 3, 3, 3, 3, 2

 

1. 4. 7. 9. 12. 15. 18. 21. 23.

 

This is not the only 25-year cycle of 9 leap years spread as smoothly as possible. There are 24 others formed by changing the start year. They all have complexity 3.

One of these is symmetrical:

 

2. 5. 7. 10. 13. 16. 18. 21. 24.

 

If used as a lunisolar cycle, it is not as accurate as the 19-year cycle, but if all the years were to have 365 days, then its months would be more accurate than the those of the 19-year cycle and the Egyptians did use such a lunar calendar.

 

 

 

Karl

 

16(09(27

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Brij Bhushan metric VIJ
Sent: 19 May 2017 17:08
To: CALNDR-L@...
Subject: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

 

Karl, list sirs:

Islamic lunar calendar 'lunar moons spread' is possibly the best so far, using a 30-year cycle 'symmetrically spread'.

>Ending with one interval proves that the >leap years in the cycle are spaced as >smoothly as possible and the number of >steps to get there is equal to the structural >complexity of the cycle as I defined in >October and show in the note below. In this >case the complexity is 3 from three steps.
While I may not be as qualified as Karl or some other members on the list; this has on a way triggered me to consider a 100-years cycle in (4*25-years) each having 9 'leap moons' spread as smoothly as possible:

 

01. 04. 07. 10. 13. 16. 19. 22. 25. (27) 29. 32. 35. 38. 41. 44. 47. 50. (52) 54. 57. 60. 63. 66  69 72. 75. (77) 79.  82. 85. 88. 91. 94. 97. (99).....(102).    

19/7=2.7143-years/leap moon

The above spread, I show is in line with my proposed [2 cycles of 448-years/5541 lunar moons] suggesting the distribution 'close to 3-years' between adjacent 'leap moons'.

The spread generally is: 

3 3,3,3,3,3,3,3,2 i.e. 36 leap moons per 100-years; at an average of 2.777777778 between two adjacent leap moons. 

Possibilities of 19-years spread can be:

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

There must be other permutations for calendar experts to consider!



Regards, 

Ex-Flt.Lt. Brij Bhushan VIJ, Author

Brij-Gregorian Modified Calendar

Friday, 2017  May 19H09:12(decimal)

Sent from my iPhone


On May 19, 2017, at 5:03 AM, Karl Palmen <[hidden email]> wrote:

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps

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Re: The Best option Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Karl Palmen
In reply to this post by Brij Bhushan metric VIJ

Dear Brij, Walter and Calendar People

 

A better way to construct a 448-year cycle of 165 leap months is from 23 nineteen-year cycles of seven leap months each and 1 nineteen-year cycle truncated to eleven years of four leap months

Years: 23*19 + 11 = 448

Leap months: 23*7 + 4 = 165.

This can have the leap month years spread as smoothly as possible so ensure that the new year moves no more than one month back and forth through the seasons.

 

The structural complexity of such a cycle is 4.

The intervals of intervals cycle (2nd step) is

4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3,  4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3,  4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4

The 3rd step gives the intervals between the 23 3s.

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3

And the final 4th step gives

23

 

 

Brij’s suggestion seems to be:

Put 37 leap months in the first 100 years, 36 in the second 100 years 37 in the third 100 years 36 in the fourth hundred years and 19 in the remaining 48 years.

If the 100 years were to have 36 fixed leap months sometimes with one additional leap month, then the new year would move at least two months back and forth through the seasons rather than one month in my suggestion above.

 

 

Walter has suggested placing the leap months at different times of the year at intervals of 34-33-34-33-34-33-34   months. The 448-year cycle can be constructed from 23 of these and a 34-34-34-34, which lasts 11 years.

An alternative to adding the 34-34-34-34 is to have three cycles of five 34-33-34-33-34-33-34   and four 34-33-34-33-34  .

 

34-33-34-33-34-33-34  

34-33-34-33-34   

34-33-34-33-34-33-34  

34-33-34-33-34   

34-33-34-33-34-33-34  

34-33-34-33-34   

34-33-34-33-34-33-34  

34-33-34-33-34   

34-33-34-33-34-33-34  

 

Each of these cycles has 5*7 + 4*5 = 55 leap months over 5*235 + 4*168 = 1847 months.

So three of these have 165 leap months and 5541 months and so (5541-165)/12 = 448 years.

 

 

The 448-year cycle is not very accurate and is about half a day out. The more accurate 315-year, 334-year, 353-year cycles can be constructed in a similar way but with 16, 17 & 18 nineteen-year cycles respectively instead of 23. Also the 334-year cycle can be constructed from three

 

34-33-34-33-34   

34-33-34-33-34-33-34  

34-33-34-33-34   

34-33-34-33-34-33-34  

34-33-34-33-34   

34-33-34-33-34-33-34  

34-33-34-33-34   

 

3*7 + 4*5 = 41 leap months 3*235 + 4*168 = 1377 months. Three have 123 leap months and 4131 months and so (4131-123)/12 = 334 years.

 

This raises the question, which cycle would be constructed from

 

34-33-34-33-34-33-34  

34-33-34-33-34   ?

 

We have 12 leap months and 235+168 = 403 months. So 12 of these make a Grattan-Guinness cycle of 391 years and 144 leap months = 403 lunar years. I think I have mentioned this one before.

 

Karl

 

16(09(28

 

 

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Brij Bhushan metric VIJ
Sent: 22 May 2017 15:02
To: [hidden email]
Subject: The Best option Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

 

Karl, list sirs:

I 😊 for examining the option, and have NO reservations.

Honestly, I have only heard about Ford circles, I only tried to put in my option as 'one among several options' using 100-year 'span' using 37,36,37,36 & 17 making the required 165 *leap moons* in 448-years/5541 Lunation cycle. Naturally, the best need be chosen. My input is 'just' a thought.

I thank you, Karl. 

Regards,

Ex-Flt Lt Brij Bhushan VIJ, Author

Brij-Gregorian Modified Caldndar

Monday, 2017 May 22H07:02 (decimal)


Sent from my iPhone


On May 22, 2017, at 5:28 AM, Karl Palmen <[hidden email]> wrote:

Dear Brij and Calendar People

 

While I don’t fully understand Brij’s ideas here. I can demonstrate my idea of intervals cycles of interval cycles on some of his examples:

 

First look at Brij’s 3 suggested 19-year cycles

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

and apply the intervals of intervals method to them.

3 3 3 3 3 3 1

3 3 3 3 3 2 2

3 3 3 2 2 3 3

Then the first fails because the intervals have length 3 & 1, which differ by 2. The interval of 1 is between the 19th year of one cycle and the 1st year of the next cycle.

Others can be carried on to the next step and then both have one interval of 6 and one interval of 1, so are not spread as smoothly as possible.

 

The Hebrew 19-year cycle is spread as smoothly as possible:

3. 6. 8. 11. 14. 17. 19.

It has intervals

3 2 3 3 3 2 3

And the intervals of intervals are

4 3

This has one interval of intervals of 2.

So this cycle is spread as smoothly as possible of complexity 3, because there are 3 steps to the single interval of 2.

There are 18 other possible such cycles formed by changing the start year. All of them have complexity 3.

One of these is symmetrical:

2. 5. 7. 10. 13. 15. 18.

 

 

I also see Brij presented a cycle of 36 leap years in a 100 years. The example he gave was not clear and the leap years were not spread as smoothly as possible, because like his 2nd  & 3rd  19-year cycle suggestions it had two consecutive intervals of 2 while the majority of intervals were 3. If the 36 leap years were spread as smoothly as possible, we’d have four cycles of 25 years with 9 leap years each.

 

The 25-year cycle of 9 leap years as 9 intervals that add up to 25. So we have seven intervals of 3 and two intervals of 2 (25 = 7*3 + 2*2). Therefore the interval of intervals cycle has two intervals that add up to 9.

Working my intervals of intervals method backwards, we can construct such a cycle

 

4, 5

 

3, 3, 3, 2, 3, 3, 3, 3, 2

 

1. 4. 7. 9. 12. 15. 18. 21. 23.

 

This is not the only 25-year cycle of 9 leap years spread as smoothly as possible. There are 24 others formed by changing the start year. They all have complexity 3.

One of these is symmetrical:

 

2. 5. 7. 10. 13. 16. 18. 21. 24.

 

If used as a lunisolar cycle, it is not as accurate as the 19-year cycle, but if all the years were to have 365 days, then its months would be more accurate than the those of the 19-year cycle and the Egyptians did use such a lunar calendar.

 

 

 

Karl

 

16(09(27

 

From: East Carolina University Calendar discussion List [[hidden email]] On Behalf Of Brij Bhushan metric VIJ
Sent: 19 May 2017 17:08
To: [hidden email]
Subject: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

 

Karl, list sirs:

Islamic lunar calendar 'lunar moons spread' is possibly the best so far, using a 30-year cycle 'symmetrically spread'.

>Ending with one interval proves that the >leap years in the cycle are spaced as >smoothly as possible and the number of >steps to get there is equal to the structural >complexity of the cycle as I defined in >October and show in the note below. In this >case the complexity is 3 from three steps.
While I may not be as qualified as Karl or some other members on the list; this has on a way triggered me to consider a 100-years cycle in (4*25-years) each having 9 'leap moons' spread as smoothly as possible:

 

01. 04. 07. 10. 13. 16. 19. 22. 25. (27) 29. 32. 35. 38. 41. 44. 47. 50. (52) 54. 57. 60. 63. 66  69 72. 75. (77) 79.  82. 85. 88. 91. 94. 97. (99).....(102).    

19/7=2.7143-years/leap moon

The above spread, I show is in line with my proposed [2 cycles of 448-years/5541 lunar moons] suggesting the distribution 'close to 3-years' between adjacent 'leap moons'.

The spread generally is: 

3 3,3,3,3,3,3,3,2 i.e. 36 leap moons per 100-years; at an average of 2.777777778 between two adjacent leap moons. 

Possibilities of 19-years spread can be:

 1. 4. 7. 10. 13. 16. 19.                   

 2. 5. 8. 11. 14. 17. 19

 3. 6. 9. 11. 13. 16. 19.

There must be other permutations for calendar experts to consider!


<image001.jpg>
Regards, 

Ex-Flt.Lt. Brij Bhushan VIJ, Author

Brij-Gregorian Modified Calendar

Friday, 2017  May 19H09:12(decimal)

Sent from my iPhone


On May 19, 2017, at 5:03 AM, Karl Palmen <[hidden email]> wrote:

Ending with one interval proves that the leap years in the cycle are spaced as smoothly as possible and the number of steps to get there is equal to the structural complexity of the cycle as I defined in October and show in the note below. In this case the complexity is 3 from three steps

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Re: The Best option Re: 19-years/7 Leap months Re: Interval Cycles of Interval Cycles

Brij Bhushan metric VIJ
Karl, Walter sirs:

>Years: 23*19 + 11 = 448

>Leap months: 23*7 + 4 = 165.

>This can have the leap month years spread >as smoothly as possible so ensure that the >new year moves no more than one month >back and forththrough the seasons.

image1.JPG

Excellent, this is in line with my 'original post' for aligning 448-years/5541 Lunation, short of 1/2 Tithi, in half my 896-years/11082 moons needing an 'extra' day getting Mean Lunation: 29d12h44m2s.98863. 

My 37,36,37,36 moons per 100-years & remaing 17 moons in 48-years to leap 165-moons- in (448x12+165=5541 moons) make the sense. But, leaving other 'parameters'; as I said, I have no reservations with Astro-experts!

I thank you Karl for considering my input,as yet another option - bridging cultures.

Regards,

Ex-Flt Lt Brij Bhushan VIj, Aughor

Brij-Gregorian Modified Calenfar

Tuesday, 2017 May 23H14:05 (decimal)


Sent from my iPhone

On May 23, 2017, at 5:04 AM, Karl Palmen <[hidden email]> wrote:

Years: 23*19 + 11 = 448

Leap months: 23*7 + 4 = 165.

This can have the leap month years spread as smoothly as possible so ensure that the new year moves no more than one month back and forth through the seasons.

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