Dear Calendar People On the May 16 this year in reply to the issue of continued fractions I said: “I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.” I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles. If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar. Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion. I found one measure of structural complexity, which involves Ford circles: The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture. Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are … 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, … These cycles have complexity 1. The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions: 8/33 for solar leap day calendars 2/11 & 3/17 for leap week calendars (but not 5/28) 8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar 3/8 & 4/11 for the leap month years in a lunisolar calendar. You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions: Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128) 11/62 for leap week calendars (also 5/28, 8/45 & 14/79) 26/49 for months in for pure lunar calendars (3yerm cycle) 11/30 for 12month years in pure lunar calendars (but not 29/79) 7/19 for leap month years in a lunisolar calendar Complexity 4 includes the 334year cycle, 353year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions: 52/293 & 69/389 for leap week calendars (also 41/231) 451/850 for months of my yerm calendar or any other oneera cycle 29/79 for 12month years in a pure lunar calendars 123/334 & 130/353 for the leap month years of a lunisolar calendar. Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions: 71/400 & 159/896 for leap week calendars 928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multiera cycle 239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.
Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion. I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Both steps of mixing can be shown as cLccLcLccLccLccLcLccLccLccLcLc Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target. ½ [1]+ 1/3 [1] 3/8 [2]+ 4/11 [2] 11/30 [3] ½ [1] 8/15 [2]+ 9/15 [2] 26/49 [3] So not every iteration increases complexity. Karl 16(01(13 
Dear Calendar People I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year. I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a Cyear cycle with L leap years is I leap year if and only if
Remainder (Y*L) divided by C is less than L. I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure. COMPLEXITY 0 0/1 0.0 ‘c’ COMPLEXITY 1 1/8 0.125 ‘cccccccL’ 1/7 0.142857… ‘ccccccL’ 1/6 0.166666… ‘cccccL’ 1/5 0.2 ‘ccccL’ 1/4 0.25 ‘cccL’
Julian Calendar COMPLEXITY 2 2/7 0.285714… ‘cccLccL’ COMPLEXITY 1 1/3 0.333333… ‘ccL’ COMPLEXITY 2 3/8 0.375 ‘ccLccLcL’
Octaeteris 2/5 0.4 ‘ccLcL’ 3/7 0.428571… ‘ccLcLcL’ COMPLEXITY 1 1/2 0.5 ‘cL’
All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between
a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1. I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap COMPLEXITY 3 5/12 0.416666… ‘ccLcLccLcLcL’ It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.
This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31day months as late as possible). It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B). I reckon the shortest cycles of each level of complexity are 1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two. The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n1). https://en.wikipedia.org/wiki/Pell_number
COMPLEXITY 4 12/29 0.413793… ‘ccLcLccLcLccLcLcLccLcLccLcLcL’ COMPLEXITY 5 29/70 0.4142857… ‘ccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLcL’ Karl 16(01(17 From: Palmen,
Karl (STFC,RAL,ISIS) Dear Calendar People On the May 16 this year in reply to the issue of continued fractions I said: “I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.” I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles. If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar. Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion. I found one measure of structural complexity, which involves Ford circles: The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture. Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are … 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, … These cycles have complexity 1. The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions: 8/33 for solar leap day calendars 2/11 & 3/17 for leap week calendars (but not 5/28) 8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar 3/8 & 4/11 for the leap month years in a lunisolar calendar. You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions: Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128) 11/62 for leap week calendars (also 5/28, 8/45 & 14/79) 26/49 for months in for pure lunar calendars (3yerm cycle) 11/30 for 12month years in pure lunar calendars (but not 29/79) 7/19 for leap month years in a lunisolar calendar Complexity 4 includes the 334year cycle, 353year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions: 52/293 & 69/389 for leap week calendars (also 41/231) 451/850 for months of my yerm calendar or any other oneera cycle 29/79 for 12month years in a pure lunar calendars 123/334 & 130/353 for the leap month years of a lunisolar calendar. Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions: 71/400 & 159/896 for leap week calendars 928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multiera cycle 239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.
Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion. I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Both steps of mixing can be shown as cLccLcLccLccLccLcLccLccLccLcLc Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target. ½ [1]+ 1/3 [1] 3/8 [2]+ 4/11 [2] 11/30 [3] ½ [1] 8/15 [2]+ 9/15 [2] 26/49 [3] So not every iteration increases complexity. Karl 16(01(13 
Dear Karl and Calendar List What would be the complexity level of my 13 month leap month calendar in which every month,,, including the leap month always has 28 days? I'm guessing complexity level 2 Walter Ziobro Sent from AOL Mobile Mail On Tuesday, October 18, 2016 Karl Palmen <[hidden email]> wrote: Dear Calendar People
I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year. I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a Cyear cycle with L leap years is I leap year if and only if Remainder (Y*L) divided by C is less than L. I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.
COMPLEXITY 0 0/1 0.0 ‘c’
COMPLEXITY 1 1/8 0.125 ‘cccccccL’ 1/7 0.142857… ‘ccccccL’ 1/6 0.166666… ‘cccccL’ 1/5 0.2 ‘ccccL’ 1/4 0.25 ‘cccL’ Julian Calendar
COMPLEXITY 2 2/7 0.285714… ‘cccLccL’
COMPLEXITY 1 1/3 0.333333… ‘ccL’
COMPLEXITY 2 3/8 0.375 ‘ccLccLcL’ Octaeteris 2/5 0.4 ‘ccLcL’ 3/7 0.428571… ‘ccLcLcL’
COMPLEXITY 1 1/2 0.5 ‘cL’
All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.
I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap
COMPLEXITY 3 5/12 0.416666… ‘ccLcLccLcLcL’
It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year. This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31day months as late as possible). It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).
I reckon the shortest cycles of each level of complexity are 1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two. The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n1). https://en.wikipedia.org/wiki/Pell_number
COMPLEXITY 4 12/29 0.413793… ‘ccLcLccLcLccLcLcLccLcLccLcLcL’
COMPLEXITY 5 29/70 0.4142857… ‘ccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLcL’
Karl
16(01(17
From: Palmen, Karl (STFC,RAL,ISIS) Sent: 14 October 2016 13:04 To: CALNDRL@... Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible
Dear Calendar People
On the May 16 this year in reply to the issue of continued fractions I said:
“I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”
I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles.
If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.
Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.
I found one measure of structural complexity, which involves Ford circles:
The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.
Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and are … 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, … These cycles have complexity 1.
The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions: 8/33 for solar leap day calendars 2/11 & 3/17 for leap week calendars (but not 5/28) 8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar 3/8 & 4/11 for the leap month years in a lunisolar calendar.
You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11). Another example is the 28year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions: Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128) 11/62 for leap week calendars (also 5/28, 8/45 & 14/79) 26/49 for months in for pure lunar calendars (3yerm cycle) 11/30 for 12month years in pure lunar calendars (but not 29/79) 7/19 for leap month years in a lunisolar calendar
Complexity 4 includes the 334year cycle, 353year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293year leap week cycle used in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions: 52/293 & 69/389 for leap week calendars (also 41/231) 451/850 for months of my yerm calendar or any other oneera cycle 29/79 for 12month years in a pure lunar calendars 123/334 & 130/353 for the leap month years of a lunisolar calendar.
Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions: 71/400 & 159/896 for leap week calendars 928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multiera cycle 239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars
In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.
Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.
I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Both steps of mixing can be shown as cLccLcLccLccLccLcLccLccLccLcLc
Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.
½ [1]+ 1/3 [1] 3/8 [2]+ 4/11 [2] 11/30 [3]
½ [1] 8/15 [2]+ 9/15 [2] 26/49 [3]
So not every iteration increases complexity.
Karl
16(01(13

Dear Walter and Calendar People The complexity of such a calendar would depend on the leap year rule for the leap months and would apply only if the leap years were spread as smoothly as possible. I have forgotten what his leap year rule was, but I guess it is 13 leap years in 293 years. The intervals between the leap years alternate between 22 & 23 years,
except for one pair of consecutive 23s. This I reckon to be complexity 3.
Complexity 2 is when all intervals between years of the minority type (leap years) are equal with ONE exception which is ONE year different. Examples are the
33year cycle of 8 leap years or the 45year cycle of 2 leap years for a 28day month calendar. I think the reason Walter thinks it is complexity 2, may be that he has been looking at the sequence of intervals between the leap years rather than the sequence
common years and leap years. The intervals form two types that are spread as smoothly as possible and so have a complexity and this complexity is 1 less (if the leap years are a minority). This process can be repeated again and again this we have just one
interval and in so doing we measure the complexity. 13/293
ccc…cLc… etc. complexity 3 7/13
23:22:23:22:23:22:23:22:23:22:23:22:23 complexity 2 Taking intervals between the 22s which are the minority type 1/6 3:2:2:2:2:2
complexity 1 Then we have only 1 interval between the 3s (of 6). The complexity can be measured by means of repeated mixing in which each mix has one of the one type and one or more of another type of equal complexity. I show this mixing for the 28day month calendar: 0/1: All years have 13 months, no leap years: Complexity 0. 1/1: All years have 14 months, every year is a leap year: Complexity 0. 1/23: Mix 22 of 0/1 with
1 of 1/1 and we get a 23year cycle with one leap year: complexity 1. 1/22: Mix 21 of 0/1 with
1 of 1/1 and we get a 22year cycle with one leap year: complexity 1 2/45: Mix 1 of 1/23 with
1 of 1/22 and we get 45year cycle with two leap years: complexity 2 3/68: Mix 2 of 1/23 with
1 of 1/22 and we get 68year cycle with three leap years: complexity 2 13/293: Mix 5 of 2/45 with
1 of 3/68 and we get 293year cycle with thirteen leap years: complexity 3. For leap weeks the corresponding cycle is 52/293, which has complexity 4. 0/1: All years have 52 weeks, no leap years: Complexity 0. 1/1: All years have 53 weeks, every year is a leap year: Complexity 0. 1/6: Mix 5 of 0/1 with
1 of 1/1 and we get 6year cycle with one leap year: complexity 1 1/5: Mix 4 of 0/1 with
1 of 1/1 and we get 5year cycle with one leap year: complexity 1 3/17: Mix 2 of 1/6 with
1 of 1/5 and we get 17year cycle with three leap years: complexity 2 2/11: Mix 1 of 1/6 with
1 of 1/5 and we get 11year cycle with two leap years: complexity 2 11/62: Mix 3 of 3/17 with
1 of 1/11 and we get 62year cycle with 11 leap years: complexity 3 8/45: Mix 2 of 3/17 with
1 of 1/11 and we get 45year cycle with 8 leap years: complexity 3 52/293: Mix 4 of 11/62 with
1 of 8/45 and we get 293year cycle with 52 leap years: complexity 4 The Gregorian leap week cycle 71/400 would require one more step,, because it is 5 of 11/62 and
2 of 8/45 and so has complexity 5 as shown in my original note. Walter’s 400leap week cycle based on his modified 33year cycle is an example of this. Note that these mixes are not determined by continued fraction convergents. Sometimes a convergent may be skipped. Instead they are determined by Ford circles
on which this complexity is defined. Karl 16(01(18 Yerm Calendar page updated to show yerm 16 at http://www.hermetic.ch/cal_stud/palmen/yerm1.htm#moon
From: Walter J Ziobro [mailto:[hidden email]]
Dear Karl and Calendar List What would be the complexity level of my 13 month leap month calendar in which every month,,, including the leap month always has 28 days? I'm guessing complexity level 2 Walter Ziobro Sent from AOL Mobile Mail On Tuesday, October 18, 2016 Karl Palmen <[hidden email]>
wrote: Dear Calendar People I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year. I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a Cyear cycle with L leap years is I leap year if and only if
Remainder (Y*L) divided by C is less than L. I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure. COMPLEXITY 0 0/1 0.0 ‘c’ COMPLEXITY 1 1/8 0.125 ‘cccccccL’ 1/7 0.142857… ‘ccccccL’ 1/6 0.166666… ‘cccccL’ 1/5 0.2 ‘ccccL’ 1/4 0.25 ‘cccL’
Julian Calendar COMPLEXITY 2 2/7 0.285714… ‘cccLccL’ COMPLEXITY 1 1/3 0.333333… ‘ccL’ COMPLEXITY 2 3/8 0.375 ‘ccLccLcL’
Octaeteris 2/5 0.4 ‘ccLcL’ 3/7 0.428571… ‘ccLcLcL’ COMPLEXITY 1 1/2 0.5 ‘cL’
All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between
a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1. I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap COMPLEXITY 3 5/12 0.416666… ‘ccLcLccLcLcL’ It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.
This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31day months as late as possible). It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B). I reckon the shortest cycles of each level of complexity are 1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two. The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n1). https://en.wikipedia.org/wiki/Pell_number
COMPLEXITY 4 12/29 0.413793… ‘ccLcLccLcLccLcLcLccLcLccLcLcL’ COMPLEXITY 5 29/70 0.4142857… ‘ccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLcL’ Karl 16(01(17 From: Palmen, Karl (STFC,RAL,ISIS)
Dear Calendar People On the May 16 this year in reply to the issue of continued fractions I said: “I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months
are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.” I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal
line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles. If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are
general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in
which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar. Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern
of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion. I found one measure of structural complexity, which involves Ford circles: The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford
circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture. Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding
Ford circles are those that touch those of complexity 0 and are … 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, … These cycles have complexity 1. The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33year cycle
of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions: 8/33 for solar leap day calendars 2/11 & 3/17 for leap week calendars (but not 5/28) 8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar 3/8 & 4/11 for the leap month years in a lunisolar calendar. You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the
19year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11). Another example is the 28year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions: Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128) 11/62 for leap week calendars (also 5/28, 8/45 & 14/79) 26/49 for months in for pure lunar calendars (3yerm cycle) 11/30 for 12month years in pure lunar calendars (but not 29/79) 7/19 for leap month years in a lunisolar calendar Complexity 4 includes the 334year cycle, 353year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293year leap week cycle used
in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions: 52/293 & 69/389 for leap week calendars (also 41/231) 451/850 for months of my yerm calendar or any other oneera cycle 29/79 for 12month years in a pure lunar calendars 123/334 & 130/353 for the leap month years of a lunisolar calendar. Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions: 71/400 & 159/896 for leap week calendars 928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multiera cycle 239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm
cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.
Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any
cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion. I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Both steps of mixing can be shown as cLccLcLccLccLccLcLccLccLccLcLc Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration
raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target. ½ [1]+ 1/3 [1] 3/8 [2]+ 4/11 [2] 11/30 [3] ½ [1] 8/15 [2]+ 9/15 [2] 26/49 [3] So not every iteration increases complexity. Karl 16(01(13 
In reply to this post by Karl Palmen
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 293year 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
33year 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 3yerm cycle Complexity 4:
399/752, 425/801, 451/850, 477/899, … and all other oneera cycles Complexity 5:
876/1651 & 928/1749 some multiera cycles Complexity 6:
13752/25920 Hebrew Yerm Calendar Lunar Calendars by 12month 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 2^{nd} Goddess Lunar calendar Complexity 7:
2519/6840 MeyerPalmen cycle 28day month Calendar leap month cycles Complexity 3:
13/293 30day month Calendar leap month cycles Complexity 3:
7/40 Julian calendar mean year Complexity 4:
18/103 Karl 16(02(20 
In reply to this post by Karl Palmen
Dear Calendar People In my original note I showed examples grouped according to their complexity. Here I show them according to cycle type. This may better show the relationship between cycles of different complexity. I gain write
each cycle down as the fractional part of its mean year in units of intercalation. The complexity applies only to cycles whose leap years are spread as smoothly as possible. I use a comma to separate mixer cycles (Ford circles touch) and semicolon for nonmixers. Solar Leap Day: Complexity 1: ¼ Complexity 2: 8/33 Complexity 3: 15/62, 23/95, 31/128, 39/161, 47/194, 55/227, 63/260, 71/293; 97/400 Complexity 4: 109/450 Solar Leap Week: 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, 63/355, 52/293, 41/231 Complexity 5: 159/896, 71/400 Lunisolar Leap Month: Complexity 1: 1/3, ½ Complexity 2: 4/11, 3/8 Complexity 3: 7/19 Complexity 4: 116/315, 123/334, 130/353, 137/372, 144/391 Complexity 5: 239/649; 383/1040; 267/725 Complexity 6: 622/1689 Complexity 7: 2519/6840 12Month Lunar Leap Day: Complexity 1: 1/3, ½ Complexity 2: 4/11, 3/8 Complexity 3: 11/30 Complexity 4: 29/79 Lunar: Complexity 1: ½ Complexity 2: 8/17, 7/15 (yerm) Complexity 3: 26/49 (3yerm cycle) Complexity 4: 451/850 (or any other 1era cycle) Complexity 5: 876/1651; 928/1749 Complexity 6: 13753/25920 Yerms hide two levels of the complexity and yermeras hide another two levels. Eclipse Season Lunar (5+fraction months = mean eclipse season) Complexity 1: 6/7, 7/8 (Hepton & Octon) Complexity 2: 13/15, 20/23 (Tzolkinex & Tritos) Complexity 3: 33/38, 53/61 (Saros & Inex) Lunisolar Abundance: Complexity 1: 1/6, 1/5 Complexity 2: 6/31, 7/36 Complexity 3: 13/67, 20/103 Complexity 4 : 97/500; 33/170 Months of 1/13 year as in Victor’s 28/293 and 43/450 calendars: Complexity 1: 1/11, 1/10 Complexity 2: 2/21, 3/31 Complexity 3: 15/157, 13/136 Complexity 4: 43/450; 28/293 Karl 16(04(03 From: Palmen,
Karl (STFC,RAL,ISIS) Dear Calendar People On the May 16 this year in reply to the issue of continued fractions I said: “I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.” I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles. If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar. Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion. I found one measure of structural complexity, which involves Ford circles: The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture. Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are … 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, … These cycles have complexity 1. The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions: 8/33 for solar leap day calendars 2/11 & 3/17 for leap week calendars (but not 5/28) 8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar 3/8 & 4/11 for the leap month years in a lunisolar calendar. You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions: Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128) 11/62 for leap week calendars (also 5/28, 8/45 & 14/79) 26/49 for months in for pure lunar calendars (3yerm cycle) 11/30 for 12month years in pure lunar calendars (but not 29/79) 7/19 for leap month years in a lunisolar calendar Complexity 4 includes the 334year cycle, 353year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions: 52/293 & 69/389 for leap week calendars (also 41/231) 451/850 for months of my yerm calendar or any other oneera cycle 29/79 for 12month years in a pure lunar calendars 123/334 & 130/353 for the leap month years of a lunisolar calendar. Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions: 71/400 & 159/896 for leap week calendars 928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multiera cycle 239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.
Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion. I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3. cLccLcLccLccLccLcLccLccLccLcLc Both steps of mixing can be shown as cLccLcLccLccLccLcLccLccLccLcLc Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target. ½ [1]+ 1/3 [1] 3/8 [2]+ 4/11 [2] 11/30 [3] ½ [1] 8/15 [2]+ 9/15 [2] 26/49 [3] So not every iteration increases complexity. Karl 16(01(13 
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