Structural Complexity of all Cycles up to 20 years

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Structural Complexity of all Cycles up to 20 years

Karl Palmen

Dear Calendar People

 

Here I list every cycle up to 20 years and its structural complexity as defined in earlier notes. I show each cycle as a fraction L/C, where L is the number of leap years and C the number of years in the cycle. The fractional part of the mean year is therefore equal to L/C. Cycles more than ½ are not shown because the structural complexity of 1 – L/C is equal to the of L/C. The list forms half a Farey sequence and so the structural complexity of any two consecutive cycles differs by at most 1.

 

1/20: 1

1/19: 1

1/18: 1

1/17: 1

1/16: 1

1/15: 1

1/14: 1

1/13: 1

1/12: 1

1/11: 1

1/10: 1

2/19: 2

1/9:  1

2/17: 2

1/8:  1

2/15: 2

1/7:  1

3/20: 2

2/13: 2

3/19: 2

1/6:  1

3/17: 2

<tropical & sidereal year in weeks>

2/11: 2

3/16: 2

1/5:  1

4/19: 2

3/14: 2

2/9:  2

3/13: 2

4/17: 2

<tropical year in days>

1/4:  1 Julian Calendar

<sidereal year in days>

5/19: 2

4/15: 2

3/11: 2

5/18: 3

2/7:  2

5/17: 3

3/10: 2

4/13: 2

5/16: 2

6/19: 2

1/3:  1

7/20: 2

6/17: 2

5/14: 2

4/11: 2

<12-month lunar year in days>

<tropical year in synodic months>

7/19: 3 Metonic Cycle

<sidereal year in synodic months>

3/8:  2

5/13: 3

7/18: 3

2/5:  2

7/17: 3

<worst possible: 1/(sqrt(2)+1)>

5/12: 3 Piano keyboard

8/19: 3

3/7:  2

7/16: 3

4/9:  2

9/20: 3

5/11: 2

6/13: 2

7/15: 2

8/17: 2

9/19: 2

1/2:  1

 

Cycles less than 1/2,  have complexity 1, if the numerator is 1 and complexity 2, if the denominator is 1 different from the multiple of the numerator. The shortest cycle of complexity 4 has 29 years. I constructed the list in the order of the denominators and found that each fraction I inserted into the list was the mediant of the two fractions between which I inserted (numerators and denominator add up to it).

 

For leap week calendars, I show cycles between 3/17 & 2/11 up to 100 years:

 

3/17:  2

14/79: 3

11/62: 3  Most Accurate leap week cycle less than 100 years 

8/45:  3

13/73: 4

5/28:  3  Julian Calendar solar cycle

17/95: 4

12/67: 4

7/39:  3  A sidereal year cycle

16/89: 4

9/50:  3

11/61: 3

13/72: 3

2/11:  2

 

 

Karl

 

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