Regular distribution of 365+ days into fractional year divisions

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Regular distribution of 365+ days into fractional year divisions

Christoph Päper-2
I feel like this had been brought up before, but can't find it in my local archive for this list.

If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most, which layouts are possible and which one should be preferred?

    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  31  30
    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  30  31
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183

If the leap day can be at the start of the 3rd month instead of the end of the 2nd one, we get this instead:

    Twelfths:  31  30  30+ 30  31  30  30  31  31  30  30  31
    Twelfths:  30  31  30+ 30  31  30  30  31  31  30  30  31
    Sixths:      61      60+     61      61      61      61
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Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen - UKRI STFC
Dear Christoph and Calendar People

One could designate 5 days of the year as not being counted and also the leap day as not being counted. Then there would be 360 counted days in each year. These could be divided into the divisions Christoph has mentioned and many more.

To prevent ambiguity, the first day of the year and any division of the year must be a counted day.


One idea is to have five non-counted days occur once every 61 days starting with the 61st day and have the leap day at the end of the year.

Another idea, which fits better with the 7 day week is to have four non-counted days once every 91 days starting with the 91st day and also not count the 365th day and the leap day at the end of the year.

A third idea is to have all then non-counted days in the half of the year when the sun is nearer the aphelion than the perihelion. This fits in better with the ecliptic longitude of the sun.

Karl

16(17(25

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Christoph Päper
Sent: 11 January 2018 17:00
To: [hidden email]
Subject: Regular distribution of 365+ days into fractional year divisions

I feel like this had been brought up before, but can't find it in my local archive for this list.

If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most, which layouts are possible and which one should be preferred?

    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  31  30
    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  30  31
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183

If the leap day can be at the start of the 3rd month instead of the end of the 2nd one, we get this instead:

    Twelfths:  31  30  30+ 30  31  30  30  31  31  30  30  31
    Twelfths:  30  31  30+ 30  31  30  30  31  31  30  30  31
    Sixths:      61      60+     61      61      61      61
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro

Another way is to distribute all the 31 day length months smoothly over an entire Olympiad I did that with my Olympiad calendar
Walter Ziobro

Sent from AOL Mobile Mail




On Thursday, January 11, 2018 Karl Palmen <[hidden email]> wrote:

Dear Christoph and Calendar People

One could designate 5 days of the year as not being counted and also the leap day as not being counted. Then there would be 360 counted days in each year. These could be divided into the divisions Christoph has mentioned and many more.

To prevent ambiguity, the first day of the year and any division of the year must be a counted day.


One idea is to have five non-counted days occur once every 61 days starting with the 61st day and have the leap day at the end of the year.

Another idea, which fits better with the 7 day week is to have four non-counted days once every 91 days starting with the 91st day and also not count the 365th day and the leap day at the end of the year.

A third idea is to have all then non-counted days in the half of the year when the sun is nearer the aphelion than the perihelion. This fits in better with the ecliptic longitude of the sun.

Karl

16(17(25

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Christoph Päper
Sent: 11 January 2018 17:00
To: CALNDR-L@...
Subject: Regular distribution of 365+ days into fractional year divisions

I feel like this had been brought up before, but can't find it in my local archive for this list.

If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most, which layouts are possible and which one should be preferred?

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If the leap day can be at the start of the 3rd month instead of the end of the 2nd one, we get this instead:

Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro
In reply to this post by Karl Palmen - UKRI STFC

Another idea is my Mostly Same Quarter Day Calendar in the format 31-30/1-30-31-30-30-31-30-30-31-30-31 in every common year every weekday in each quarter is exactly 91 days or 13 weeks from the same weekday in the next quarter except for the 365th day Also three quarters of the leap year have the same character
Walter Ziobro

Sent from AOL Mobile Mail




On Thursday, January 11, 2018 Karl Palmen <[hidden email]> wrote:

Dear Christoph and Calendar People

One could designate 5 days of the year as not being counted and also the leap day as not being counted. Then there would be 360 counted days in each year. These could be divided into the divisions Christoph has mentioned and many more.

To prevent ambiguity, the first day of the year and any division of the year must be a counted day.


One idea is to have five non-counted days occur once every 61 days starting with the 61st day and have the leap day at the end of the year.

Another idea, which fits better with the 7 day week is to have four non-counted days once every 91 days starting with the 91st day and also not count the 365th day and the leap day at the end of the year.

A third idea is to have all then non-counted days in the half of the year when the sun is nearer the aphelion than the perihelion. This fits in better with the ecliptic longitude of the sun.

Karl

16(17(25

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Christoph Päper
Sent: 11 January 2018 17:00
To: CALNDR-L@...
Subject: Regular distribution of 365+ days into fractional year divisions

I feel like this had been brought up before, but can't find it in my local archive for this list.

If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most, which layouts are possible and which one should be preferred?

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If the leap day can be at the start of the 3rd month instead of the end of the 2nd one, we get this instead:

Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
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Re: Regular distribution of 365+ days into fractional year divisions

Irv Bromberg
In reply to this post by Christoph Päper-2
If you're going to all of the trouble to carry out such a major calendar reform then why insist that the leap day remain in such an inconvenient place?

Having the leap day at the end of the year very significantly simplifies calendrical calculations because the ordinal day number of every non-leap day of the calendar year remains permanently fixed.

Another popular option that you didn't mention: Divide into 13 months of 28 days = 364 days, plus an extra day in some permanent position to make it 365 days in a non-leap year (this could be the New Year
Day or the Mid-Year Day, or the Year-End Day, for example), then append the leap day after the end of the year.

-- Irv Bromberg, University of Toronto, Canada

http://www.sym454.org/

________________________________________
From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Christoph Päper [[hidden email]]
Sent: Thursday, January 11, 2018 12:00

I feel like this had been brought up before, but can't find it in my local archive for this list.

If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most, which layouts are possible and which one should be preferred?

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If the leap day can be at the start of the 3rd month instead of the end of the 2nd one, we get this instead:

Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro

Dear Irv and Calendar List

With my Olympiad Calendar, you don't even need to designate a leap day Just take the 21 months of 31 days that occur in a 4 year period and distribure them smoothly throughout the 48 months

Walter Ziobro

Sent from AOL Mobile Mail




On Thursday, January 11, 2018 Irv Bromberg <[hidden email]> wrote:

If you're going to all of the trouble to carry out such a major calendar reform then why insist that the leap day remain in such an inconvenient place?

Having the leap day at the end of the year very significantly simplifies calendrical calculations because the ordinal day number of every non-leap day of the calendar year remains permanently fixed.

Another popular option that you didn't mention: Divide into 13 months of 28 days = 364 days, plus an extra day in some permanent position to make it 365 days in a non-leap year (this could be the New Year
Day or the Mid-Year Day, or the Year-End Day, for example), then append the leap day after the end of the year.

-- Irv Bromberg, University of Toronto, Canada

http://www.sym454.org/

________________________________________
From: East Carolina University Calendar discussion List [CALNDR-L@...] on behalf of Christoph Päper [christoph.paeper@...]
Sent: Thursday, January 11, 2018 12:00

I feel like this had been brought up before, but can't find it in my local archive for this list.

If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most, which layouts are possible and which one should be preferred?

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If the leap day can be at the start of the 3rd month instead of the end of the 2nd one, we get this instead:

Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
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Re: Regular distribution of 365+ days into fractional year divisions

Christoph Päper-2
In reply to this post by Irv Bromberg
 Irv Bromberg <[hidden email]>:
>
> If you're going to all of the trouble to carry out such a major
> calendar reform then why insist that the leap day remain in such an
> inconvenient place?

It has to work atop an underlying Gregorian year.

I was trying to find the most logical placement of the remaining 5+ null days for 12 fiscal months of 30 days and when I combined that with other subdivisions like fiscal quarters of either 91 days, I realized that no regular pattern like 30:31 or 30:31:30 can do the job.
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Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen - UKRI STFC
Dear Christoph, Walter and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Christoph Päper
Sent: 12 January 2018 01:05
To: [hidden email]
Subject: Re: Regular distribution of 365+ days into fractional year divisions

 Irv Bromberg <[hidden email]>:
>
> If you're going to all of the trouble to carry out such a major
> calendar reform then why insist that the leap day remain in such an
> inconvenient place?

It has to work atop an underlying Gregorian year.

I was trying to find the most logical placement of the remaining 5+ null days for 12 fiscal months of 30 days and when I combined that with other subdivisions like fiscal quarters of either 91 days, I realized that no regular pattern like 30:31 or 30:31:30 can do the job.

KARL REPLIES: So Christoph was considering my suggestion where each year has 360 counted days and has referred to the non-counted days as null days.

If one had the 5 counted days of a common year once every 73 days, the resulting months of 30 or 31 days would be spread as smoothly as possible like the white and black notes on a piano. Unfortunately this is the shortest cycle of structural complexity 3 and so is not simple.

I also made other suggestions about the placement of these null days.

One could start the year on March 1 and have leap day at end and get a consistent correspondence with the Gregorian calendar for every day of the year. Also Christmas Day would be the 300th day of the year.


I then think about Walter's suggestion: One could spread the 21 null days that normally occur in 4 years as smoothly as possible and this would lead to Walter's suggestion. The drawback is that the number of days in each month would depend on the year, normally repeating every 4 years. One null day would need to dropped occasionally to make the calendar accurate (3 times every 400 years for Gregorian mean year).

The 4 years would have 1440 counted days and normally 21 null days. If these 21 were spread as smoothly as possible, one would get three cycles of 7 null days & 480 counted days (16 months). This cycle would have 3 intervals of 68 counted days = 69 days and 4 intervals of 69 counted days = 70 days. This interval cycle has complexity 2 and is yerm-like (69:68:69:68:69:68:69). The cycle would have 7 months of 31 days in 16 months spread as smoothly as possible and so also has complexity 3.

Karl

16(17(26
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Re: Regular distribution of 365+ days into fractional year divisions

Christoph Päper-2
In reply to this post by Karl Palmen - UKRI STFC
Karl Palmen <[hidden email]>:
>
> One idea is to have five non-counted days occur once every 61 days starting with the 61st day and have the leap day at the end of the year.

That's 30:31 and it does work except it violates the constraint that the leap day should be close to 29 February. In other words, either the second or the third month needs to be 30+ days.

    Months:   30  31  30  31  30  31  30  31  30  31  30  30+
    Sixths:     61      61      61      61      61      60+
    Fourths:      91          92          91          91+
    Thirds:        122             122             121+
    Halves:            183                     182+

You could try to weasel out by moving the start of this year into March, but that violates another constraint. 30:31 with leap day at the end of the second month misses one day in the first quarter-year, but otherwise works. This was basically my starting point.

    Months:   30  30+ 30  31  30  31  30  31  30  31  30  31
    Sixths:     60+     61      61      61      61      61
    Fourths:      90+         92          91          92
    Thirds:        121+            122             122
    Halves:            182+                    183

> Another idea, which fits better with the 7 day week is to have four non-counted days once every 91 days starting with the 91st day and also not count the 365th day and the leap day at the end of the year.

That's 30:30:31 and it does not work because the last month has 32 days in common and 33 days in leap years.

    Months:   30  30  31  30  30  31  30  30  31  30  30  32+
    Sixths:     60      61      61      60      61      61+
    Fourths:      91          91          91          92+
    Thirds:        121             121             123+
    Halves:            182                     183+

30:31:30 and 31:30:30 work better for the last month but still not quite and also not for other divisions. With the constraints, you cannot have the 364+1st day and the leap day next to each other.

    Months:   30  31  30  30  31  30  30  31  30  30  31  31+
    Sixths:     61      60      61      61      60      62+
    Fourths:      91          91          91          92+
    Thirds:        121             122             122+
    Halves:            182                     183+

    Months:   31  30  30  31  30  30  31  30  30  31  30  31+
    Sixths:     61      61      60      61      61      61+
    Fourths:      91          91          91          92+
    Thirds:        122             121             122+
    Halves:            182                     183+

> A third idea is to have all then non-counted days in the half of the year when the sun is nearer the aphelion than the perihelion. This fits in better with the ecliptic longitude of the sun.

Such astronomical considerations are completely irrelevant for a parallel fiscal calendar.
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Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen - UKRI STFC
Dear Christoph and Calendar People

Christoph, please state ALL your constraints.

Karl

16(17(26

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Christoph Päper
Sent: 12 January 2018 15:16
To: [hidden email]
Subject: Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen <[hidden email]>:
>
> One idea is to have five non-counted days occur once every 61 days starting with the 61st day and have the leap day at the end of the year.

That's 30:31 and it does work except it violates the constraint that the leap day should be close to 29 February. In other words, either the second or the third month needs to be 30+ days.

    Months:   30  31  30  31  30  31  30  31  30  31  30  30+
    Sixths:     61      61      61      61      61      60+
    Fourths:      91          92          91          91+
    Thirds:        122             122             121+
    Halves:            183                     182+

You could try to weasel out by moving the start of this year into March, but that violates another constraint. 30:31 with leap day at the end of the second month misses one day in the first quarter-year, but otherwise works. This was basically my starting point.

    Months:   30  30+ 30  31  30  31  30  31  30  31  30  31
    Sixths:     60+     61      61      61      61      61
    Fourths:      90+         92          91          92
    Thirds:        121+            122             122
    Halves:            182+                    183

> Another idea, which fits better with the 7 day week is to have four non-counted days once every 91 days starting with the 91st day and also not count the 365th day and the leap day at the end of the year.

That's 30:30:31 and it does not work because the last month has 32 days in common and 33 days in leap years.

    Months:   30  30  31  30  30  31  30  30  31  30  30  32+
    Sixths:     60      61      61      60      61      61+
    Fourths:      91          91          91          92+
    Thirds:        121             121             123+
    Halves:            182                     183+

30:31:30 and 31:30:30 work better for the last month but still not quite and also not for other divisions. With the constraints, you cannot have the 364+1st day and the leap day next to each other.

    Months:   30  31  30  30  31  30  30  31  30  30  31  31+
    Sixths:     61      60      61      61      60      62+
    Fourths:      91          91          91          92+
    Thirds:        121             122             122+
    Halves:            182                     183+

    Months:   31  30  30  31  30  30  31  30  30  31  30  31+
    Sixths:     61      61      60      61      61      61+
    Fourths:      91          91          91          92+
    Thirds:        122             121             122+
    Halves:            182                     183+

> A third idea is to have all then non-counted days in the half of the year when the sun is nearer the aphelion than the perihelion. This fits in better with the ecliptic longitude of the sun.

Such astronomical considerations are completely irrelevant for a parallel fiscal calendar.
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Re: Regular distribution of 365+ days into fractional year divisions

Christoph Päper-2
Karl Palmen <[hidden email]>:
>
> Christoph, please state ALL your constraints.

Well, I initially wrote this:

>> If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most,

That means:

- runs parallel to Julian or Gregorian calendar
- leap day at 29 February or a nearby boundary of divisions
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

>> which layouts are possible and which one should be preferred?
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro

Dear Chris

I believe my Mostly Same Weekday Quarter Start Calendar meets all of your requirements with 31-30/1-30-31-30-30-31-30-30-31-30-31 days of months

Walter Ziobro

Sent from AOL Mobile Mail




On Friday, January 12, 2018 Christoph Päper <[hidden email]> wrote:

Karl Palmen <karl.palmen@...>:
>
> Christoph, please state ALL your constraints.

Well, I initially wrote this:

>> If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most,

That means:

- runs parallel to Julian or Gregorian calendar
- leap day at 29 February or a nearby boundary of divisions
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

>> which layouts are possible and which one should be preferred?
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro

Dear Chris

Oops

There is one exception: In leap year the 1st sixth has 62 days But that's the only exception I could find IMO you won't do any better

Walter Ziobro

Sent from AOL Mobile Mail




On Saturday, January 13, 2018 Walter J Ziobro <[hidden email]> wrote:

Dear Chris

I believe my Mostly Same Weekday Quarter Start Calendar meets all of your requirements with 31-30/1-30-31-30-30-31-30-30-31-30-31 days of months

Walter Ziobro

Sent from AOL Mobile Mail


On Friday, January 12, 2018 Christoph Päper <christoph.paeper@...> wrote:
Karl Palmen <[hidden email]>:
>
> Christoph, please state ALL your constraints.

Well, I initially wrote this:

>> If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most,

That means:

- runs parallel to Julian or Gregorian calendar
- leap day at 29 February or a nearby boundary of divisions
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

>> which layouts are possible and which one should be preferred?
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Re: Regular distribution of 365+ days into fractional year divisions

Christoph Päper-2
Walter J Ziobro:
>
> There is one exception: In leap year the 1st sixth has 62 days But
> that's the only exception I could find IMO you won't do any better
>
>> I believe my Mostly Same Weekday Quarter Start Calendar meets all of
>> your requirements with 31-30/1-30-31-30-30-31-30-30-31-30-31 days of
>> months

That's 31:30:30 with leap day in "February" and extra day in "December".

Mostly Same Weekday Quarter Start Calendar:

    31  30+ 30  31  30  30  31  30  30  31  30  31
      61+     61      60      61      61      61
        91+         91          91          92
         122+            121             122
             182+                    183

Sixths and Thirds don't work. The leap day must be in the Sixth with 60 days and in the Third with 121 days.

Just for the record, Brij's Earth Calendar also fails:

    Twelfths:  31  29  31  30  31  30+ 30  31  30  31  30  31
    Sixths:      60      61      61+     61      61      61
    Fourths:       91          91+         91          92
    Thirds:         121             122+            122
    Halves:             182+                    183

> IMO you won't do any better

I provided some patterns initially which do not violate any constraint. I thought there were more that I was just failing to see.

    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  31  30
    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  30  31
    Sixths:      60+     61      61      61      61      61  
    Twelfths:  31  30  30+ 30  31  30  30  31  31  30  30  31
    Twelfths:  30  31  30+ 30  31  30  30  31  31  30  30  31
    Sixths:      61      60+     61      61      61      61
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro

Dear Chris

You are correct The 2 formats you made do meet all of your requirements My Mostly Same Weekday Quarter Start Calendar will work in all cases if the leap day is moved to June 31 which would give it the exact same format as the World Calendar but without perrenial weekdays It would also divide the leap year into 2 halves of week cycles and offset the dates from the Gregorian Calendar by 1 day from March 1 to July 1 in leap years

Walter Ziobro

Sent from AOL Mobile Mail




On Sunday, January 14, 2018 Christoph Päper <[hidden email]> wrote:

Walter J Ziobro:
>
> There is one exception: In leap year the 1st sixth has 62 days But
> that's the only exception I could find IMO you won't do any better
>
>> I believe my Mostly Same Weekday Quarter Start Calendar meets all of
>> your requirements with 31-30/1-30-31-30-30-31-30-30-31-30-31 days of
>> months

That's 31:30:30 with leap day in "February" and extra day in "December".

Mostly Same Weekday Quarter Start Calendar:

31 30+ 30 31 30 30 31 30 30 31 30 31
61+ 61 60 61 61 61
91+ 91 91 92
122+ 121 122
182+ 183

Sixths and Thirds don't work. The leap day must be in the Sixth with 60 days and in the Third with 121 days.

Just for the record, Brij's Earth Calendar also fails:

Twelfths: 31 29 31 30 31 30+ 30 31 30 31 30 31
Sixths: 60 61 61+ 61 61 61
Fourths: 91 91+ 91 92
Thirds: 121 122+ 122
Halves: 182+ 183

> IMO you won't do any better

I provided some patterns initially which do not violate any constraint. I thought there were more that I was just failing to see.

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro
In reply to this post by Christoph Päper-2

Dear Chris

I notice that in every case in which your requirements can be met, the leap month is offset from the double 31 day months by 6 months

Walter Ziobro

Sent from AOL Mobile Mail




On Sunday, January 14, 2018 Christoph Päper <[hidden email]> wrote:

Walter J Ziobro:
>
> There is one exception: In leap year the 1st sixth has 62 days But
> that's the only exception I could find IMO you won't do any better
>
>> I believe my Mostly Same Weekday Quarter Start Calendar meets all of
>> your requirements with 31-30/1-30-31-30-30-31-30-30-31-30-31 days of
>> months

That's 31:30:30 with leap day in "February" and extra day in "December".

Mostly Same Weekday Quarter Start Calendar:

31 30+ 30 31 30 30 31 30 30 31 30 31
61+ 61 60 61 61 61
91+ 91 91 92
122+ 121 122
182+ 183

Sixths and Thirds don't work. The leap day must be in the Sixth with 60 days and in the Third with 121 days.

Just for the record, Brij's Earth Calendar also fails:

Twelfths: 31 29 31 30 31 30+ 30 31 30 31 30 31
Sixths: 60 61 61+ 61 61 61
Fourths: 91 91+ 91 92
Thirds: 121 122+ 122
Halves: 182+ 183

> IMO you won't do any better

I provided some patterns initially which do not violate any constraint. I thought there were more that I was just failing to see.

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro
In reply to this post by Christoph Päper-2

Dear Chris

I have also checked my Olympiad Calendar with Smoothly Spread Month Lengths and it appears that all four years of the Olympiad meet all of your requirements, except that it doesn't have a specific leap day in any month

Walter Ziobro

Sent from AOL Mobile Mail




On Sunday, January 14, 2018 Christoph Päper <[hidden email]> wrote:

Walter J Ziobro:
>
> There is one exception: In leap year the 1st sixth has 62 days But
> that's the only exception I could find IMO you won't do any better
>
>> I believe my Mostly Same Weekday Quarter Start Calendar meets all of
>> your requirements with 31-30/1-30-31-30-30-31-30-30-31-30-31 days of
>> months

That's 31:30:30 with leap day in "February" and extra day in "December".

Mostly Same Weekday Quarter Start Calendar:

31 30+ 30 31 30 30 31 30 30 31 30 31
61+ 61 60 61 61 61
91+ 91 91 92
122+ 121 122
182+ 183

Sixths and Thirds don't work. The leap day must be in the Sixth with 60 days and in the Third with 121 days.

Just for the record, Brij's Earth Calendar also fails:

Twelfths: 31 29 31 30 31 30+ 30 31 30 31 30 31
Sixths: 60 61 61+ 61 61 61
Fourths: 91 91+ 91 92
Thirds: 121 122+ 122
Halves: 182+ 183

> IMO you won't do any better

I provided some patterns initially which do not violate any constraint. I thought there were more that I was just failing to see.

Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Sixths: 60+ 61 61 61 61 61
Twelfths: 31 30 30+ 30 31 30 30 31 31 30 30 31
Twelfths: 30 31 30+ 30 31 30 30 31 31 30 30 31
Sixths: 61 60+ 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183
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Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen - UKRI STFC
In reply to this post by Christoph Päper-2
Dear Christoph and Calendar People

- runs parallel to Julian or Gregorian calendar

This is too vague to be meaningful. Any calendar could be run in parallel with the Gregorian calendar.

Perhaps Christoph means that the calendar year coincides with the Gregorian calendar year. That would rule out the March 1 new year, which none of his other conditions do. In this reply, I'll assume this is what he means.

There are 16 solutions.

- 6 divisions of 60 or 61 days each (sixths) requires all sixths have 61 days in a leap year, hence the first sixth must be the only one that has 60 days in a common year. So given
- 12 divisions of 30 or 31 days each (months)
the first month must have 30 days and the second month have 30 days in a common year and 31 days in a leap year.

- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves) are both ensured by this.

The remaining months occur in pairs of 30 days and 31 days, which could occur in either order, but must satisfy
- 4 divisions of 91 or 92 days each (fourths or quarters)
Therefore the third month must have 31 days and the fourth month 30 days. This is necessary to ensure that the first quarter has 91 days in a common year and 92 days in a leap year.

The second quarter may be either 30 31 30 or 30 30 31 and has 91 days.

More arrangements are possible for the remaining two quarters, because they are not constrained by the leap day. There
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
imply
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)

Hence 16 month patterns are possible and they are determined by which of the two months in each of the last four sixths has 31 days. The first four months must be 30 30+ 31 30.

It is possible to make the last 5 months match the last 5 Gregorian months. There are 2 solutions that satisfy this:
30 30+ 31 30 30 31  30 31 30 31 30 31
and
30 30+ 31 30 31 30  30 31 30 31 30 31
In the latter, month 2 begins 1 day early, months 3 to 7 begin 1 day late and the others begin on the same day as in Gregorian.


Karl

16(27(29


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Christoph Päper
Sent: 12 January 2018 23:57
To: [hidden email]
Subject: Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen <[hidden email]>:
>
> Christoph, please state ALL your constraints.

Well, I initially wrote this:

>> If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most,

That means:

- runs parallel to Julian or Gregorian calendar
- leap day at 29 February or a nearby boundary of divisions
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

>> which layouts are possible and which one should be preferred?
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Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen - UKRI STFC
Dear Christoph and Calendar People

- uses same calendar year as Gregorian calendar
- leap day in division of 29 February
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

Here I list the 16 possibilities allowed by the above constraints in Christoph's notation:

    Twelfths:  30  30+ 31  30  30  31  30  31  30  31  30  31
    Twelfths:  30  30+ 31  30  30  31  30  31  30  31  31  30
    Twelfths:  30  30+ 31  30  30  31  31  30  30  31  30  31
    Twelfths:  30  30+ 31  30  30  31  31  30  30  31  31  30
    Twelfths:  30  30+ 31  30  31  30  30  31  30  31  30  31
    Twelfths:  30  30+ 31  30  31  30  30  31  30  31  31  30
    Twelfths:  30  30+ 31  30  31  30  31  30  30  31  30  31
    Twelfths:  30  30+ 31  30  31  30  31  30  30  31  31  30
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          91          92
    Thirds:         121+            122             122
    Halves:             182+                183

    Twelfths:  30  30+ 31  30  30  31  30  31  31  30  30  31
    Twelfths:  30  30+ 31  30  30  31  30  31  31  30  31  30
    Twelfths:  30  30+ 31  30  30  31  31  30  31  30  30  31
    Twelfths:  30  30+ 31  30  30  31  31  30  31  30  31  30
    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  30  31
    Twelfths:  30  30+ 31  30  31  30  30  31  31  30  31  30
    Twelfths:  30  30+ 31  30  31  30  31  30  31  30  30  31
    Twelfths:  30  30+ 31  30  31  30  31  30  31  30  31  30
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183

If you add the constraint that there are no two consecutive months of 31 days, except the 2nd and 3rd month of a leap year we get:

    Twelfths:  30  30+ 31  30  30  31  30  31  30  31  30  31
    Twelfths:  30  30+ 31  30  31  30  30  31  30  31  30  31
    Twelfths:  30  30+ 31  30  31  30  31  30  30  31  30  31
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          91          92
    Thirds:         121+            122             122
    Halves:             182+                183

    Twelfths:  30  30+ 31  30  31  30  31  30  31  30  30  31
    Twelfths:  30  30+ 31  30  31  30  31  30  31  30  31  30
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183

If also you want no two consecutive quarters of 92 days, including around the start of a leap year.
Then you have just have:

    Twelfths:  30  30+ 31  30  31  30  31  30  31  30  30  31
    Twelfths:  30  30+ 31  30  31  30  31  30  31  30  31  30
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183

If you want the last five months to match the Gregorian (maximum possible), then one must suffer two consecutive quarters of 92 days around the start of a leap year.

    Twelfths:  30  30+ 31  30  30  31  30  31  30  31  30  31
    Twelfths:  30  30+ 31  30  31  30  30  31  30  31  30  31
    Sixths:      60+     61      61      61      61      61  
    Fourths:       91+         91          91          92
    Thirds:         121+            122             122
    Halves:             182+                183

I revised the leap day constraint slightly to make it verifiable. I had assumed that the leap day would occur in the division of February 29 in reckoning the 16 possibilities. If this is relaxed, other possibilities occur including:

    Twelfths:  30+  30  31  30  31  30  31  30  31  30  31  30
    Sixths:      60+     61      61      61      61      61
    Twelfths:  31  30  30+  30  31  30  31  30  31  30  31  30
    Sixths:      61     60+      61      61      61      61    
    Fourths:       91+         91          92          91
    Thirds:         121+            122             122
    Halves:             182+                183


Karl

16(17(30

-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 15 January 2018 12:56
To: 'East Carolina University Calendar discussion List'
Subject: RE: Regular distribution of 365+ days into fractional year divisions

Dear Christoph and Calendar People

- runs parallel to Julian or Gregorian calendar

This is too vague to be meaningful. Any calendar could be run in parallel with the Gregorian calendar.

Perhaps Christoph means that the calendar year coincides with the Gregorian calendar year. That would rule out the March 1 new year, which none of his other conditions do. In this reply, I'll assume this is what he means.

There are 16 solutions.

- 6 divisions of 60 or 61 days each (sixths) requires all sixths have 61 days in a leap year, hence the first sixth must be the only one that has 60 days in a common year. So given
- 12 divisions of 30 or 31 days each (months)
the first month must have 30 days and the second month have 30 days in a common year and 31 days in a leap year.

- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves) are both ensured by this.

The remaining months occur in pairs of 30 days and 31 days, which could occur in either order, but must satisfy
- 4 divisions of 91 or 92 days each (fourths or quarters)
Therefore the third month must have 31 days and the fourth month 30 days. This is necessary to ensure that the first quarter has 91 days in a common year and 92 days in a leap year.

The second quarter may be either 30 31 30 or 30 30 31 and has 91 days.

More arrangements are possible for the remaining two quarters, because they are not constrained by the leap day. There
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
imply
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)

Hence 16 month patterns are possible and they are determined by which of the two months in each of the last four sixths has 31 days. The first four months must be 30 30+ 31 30.

It is possible to make the last 5 months match the last 5 Gregorian months. There are 2 solutions that satisfy this:
30 30+ 31 30 30 31  30 31 30 31 30 31
and
30 30+ 31 30 31 30  30 31 30 31 30 31
In the latter, month 2 begins 1 day early, months 3 to 7 begin 1 day late and the others begin on the same day as in Gregorian.


Karl

16(27(29


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Christoph Päper
Sent: 12 January 2018 23:57
To: [hidden email]
Subject: Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen <[hidden email]>:
>
> Christoph, please state ALL your constraints.

Well, I initially wrote this:

>> If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most,

That means:

- runs parallel to Julian or Gregorian calendar
- leap day at 29 February or a nearby boundary of divisions
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

>> which layouts are possible and which one should be preferred?
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Re: Regular distribution of 365+ days into fractional year divisions

Walter J Ziobro

Dear Karl

The 5th option on your list is very close to Brij's proposal Like Brij it takes 1 day from July and adds it to February But it also takes 1 day from January as well In any case, it appears to be the option that requires the fewest changes from the current Gregorian Calendar that meets Christophe's requirements

Walter Ziobro

Sent from AOL Mobile Mail




On Tuesday, January 16, 2018 Karl Palmen <[hidden email]> wrote:

Dear Christoph and Calendar People

- uses same calendar year as Gregorian calendar
- leap day in division of 29 February
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

Here I list the 16 possibilities allowed by the above constraints in Christoph's notation:

Twelfths: 30 30+ 31 30 30 31 30 31 30 31 30 31
Twelfths: 30 30+ 31 30 30 31 30 31 30 31 31 30
Twelfths: 30 30+ 31 30 30 31 31 30 30 31 30 31
Twelfths: 30 30+ 31 30 30 31 31 30 30 31 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 30 31 30 31
Twelfths: 30 30+ 31 30 31 30 30 31 30 31 31 30
Twelfths: 30 30+ 31 30 31 30 31 30 30 31 30 31
Twelfths: 30 30+ 31 30 31 30 31 30 30 31 31 30
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 91 92
Thirds: 121+ 122 122
Halves: 182+ 183

Twelfths: 30 30+ 31 30 30 31 30 31 31 30 30 31
Twelfths: 30 30+ 31 30 30 31 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 30 31 31 30 31 30 30 31
Twelfths: 30 30+ 31 30 30 31 31 30 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 30 31
Twelfths: 30 30+ 31 30 31 30 30 31 31 30 31 30
Twelfths: 30 30+ 31 30 31 30 31 30 31 30 30 31
Twelfths: 30 30+ 31 30 31 30 31 30 31 30 31 30
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If you add the constraint that there are no two consecutive months of 31 days, except the 2nd and 3rd month of a leap year we get:

Twelfths: 30 30+ 31 30 30 31 30 31 30 31 30 31
Twelfths: 30 30+ 31 30 31 30 30 31 30 31 30 31
Twelfths: 30 30+ 31 30 31 30 31 30 30 31 30 31
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 91 92
Thirds: 121+ 122 122
Halves: 182+ 183

Twelfths: 30 30+ 31 30 31 30 31 30 31 30 30 31
Twelfths: 30 30+ 31 30 31 30 31 30 31 30 31 30
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If also you want no two consecutive quarters of 92 days, including around the start of a leap year.
Then you have just have:

Twelfths: 30 30+ 31 30 31 30 31 30 31 30 30 31
Twelfths: 30 30+ 31 30 31 30 31 30 31 30 31 30
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183

If you want the last five months to match the Gregorian (maximum possible), then one must suffer two consecutive quarters of 92 days around the start of a leap year.

Twelfths: 30 30+ 31 30 30 31 30 31 30 31 30 31
Twelfths: 30 30+ 31 30 31 30 30 31 30 31 30 31
Sixths: 60+ 61 61 61 61 61
Fourths: 91+ 91 91 92
Thirds: 121+ 122 122
Halves: 182+ 183

I revised the leap day constraint slightly to make it verifiable. I had assumed that the leap day would occur in the division of February 29 in reckoning the 16 possibilities. If this is relaxed, other possibilities occur including:

Twelfths: 30+ 30 31 30 31 30 31 30 31 30 31 30
Sixths: 60+ 61 61 61 61 61
Twelfths: 31 30 30+ 30 31 30 31 30 31 30 31 30
Sixths: 61 60+ 61 61 61 61
Fourths: 91+ 91 92 91
Thirds: 121+ 122 122
Halves: 182+ 183


Karl

16(17(30

-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 15 January 2018 12:56
To: 'East Carolina University Calendar discussion List'
Subject: RE: Regular distribution of 365+ days into fractional year divisions

Dear Christoph and Calendar People

- runs parallel to Julian or Gregorian calendar

This is too vague to be meaningful. Any calendar could be run in parallel with the Gregorian calendar.

Perhaps Christoph means that the calendar year coincides with the Gregorian calendar year. That would rule out the March 1 new year, which none of his other conditions do. In this reply, I'll assume this is what he means.

There are 16 solutions.

- 6 divisions of 60 or 61 days each (sixths) requires all sixths have 61 days in a leap year, hence the first sixth must be the only one that has 60 days in a common year. So given
- 12 divisions of 30 or 31 days each (months)
the first month must have 30 days and the second month have 30 days in a common year and 31 days in a leap year.

- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves) are both ensured by this.

The remaining months occur in pairs of 30 days and 31 days, which could occur in either order, but must satisfy
- 4 divisions of 91 or 92 days each (fourths or quarters)
Therefore the third month must have 31 days and the fourth month 30 days. This is necessary to ensure that the first quarter has 91 days in a common year and 92 days in a leap year.

The second quarter may be either 30 31 30 or 30 30 31 and has 91 days.

More arrangements are possible for the remaining two quarters, because they are not constrained by the leap day. There
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
imply
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)

Hence 16 month patterns are possible and they are determined by which of the two months in each of the last four sixths has 31 days. The first four months must be 30 30+ 31 30.

It is possible to make the last 5 months match the last 5 Gregorian months. There are 2 solutions that satisfy this:
30 30+ 31 30 30 31 30 31 30 31 30 31
and
30 30+ 31 30 31 30 30 31 30 31 30 31
In the latter, month 2 begins 1 day early, months 3 to 7 begin 1 day late and the others begin on the same day as in Gregorian.


Karl

16(27(29


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Christoph Päper
Sent: 12 January 2018 23:57
To: CALNDR-L@...
Subject: Re: Regular distribution of 365+ days into fractional year divisions

Karl Palmen <karl.palmen@...>:
>
> Christoph, please state ALL your constraints.

Well, I initially wrote this:

>> If one wanted a 12-month calendar with 365 days and a leap day preferably around the same time as in the Julian and Gregorian calendars, but with regular distribution of lengths such that halves, thirds, quarters and sixths of the year also deviate in length by 1 day at most,

That means:

- runs parallel to Julian or Gregorian calendar
- leap day at 29 February or a nearby boundary of divisions
- 12 divisions of 30 or 31 days each (months)
- 6 divisions of 60 or 61 days each (sixths)
- 4 divisions of 91 or 92 days each (fourths or quarters)
- 3 divisions of 121 or 122 days each (thirds)
- 2 divisions of 182 or 183 days each (halves)
- weeks and days of the week do not matter

>> which layouts are possible and which one should be preferred?
12