Simple leap rules with N-day week RE: Leap rule for 366-day year with 6-day week

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Simple leap rules with N-day week RE: Leap rule for 366-day year with 6-day week

Karl Palmen - UKRI STFC

Dear Irv, Christoph and Calendar People


The 6-day cycle is very lucky in having accurate year-length rules that are simple when the long and short years are spread as smoothly as possible. The only other N-day cycles up to 30 days that have such accurate year-length rules are 1-day, 3-day, 17-day & 26-day cycles.


Here I list the cycles of such year-length rules with mean year in the range 365.2416 to 365.2428 days:


N  Days  Years LongYrs  Mean Year 


 1  12053    33       8  365.24242 


3  33237    91      68  365.24176  

 3  34698    95      71  365.24211

 3  36159    99      74  365.24242

3  37620   103      77  365.24272


6  34698    95      83  365.24211

6  37620   103      90  365.24272


17  12053    33      16  365.24242


26 122356   335      16  365.24179

26 114686   314      15  365.24204

26 107016   293      14  365.24232

26  99346   272      13  365.24265



For example, the 293-year cycle for the 26-day cycle can have every 14th year long within each 293-year cycle starting with the 7th year, which gives a symmetrical cycle whose long years are spread as smoothly as possible (i.e. a Helios Cycle). The interval between long years is 14 years within a 293-year cycle, but 13 between 293-year cycles (one year different). Each short year has 14 26-day cycles (364 days) and each long year has 15 26-day cycles (390 days).


Each of the 6-day cycle year-length cycles appears as a 3-day cycle year-length cycle, but with twice as many short years. Also additional year-length cycles appear with an odd number of days.







From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 21 November 2017 13:02
To: 'East Carolina University Calendar discussion List'
Subject: RE: Leap rule for 366-day year with 6-day week


Dear Irv sand Calendar People


I have suggested the 95-year cycle of 12 short years, which is listed in Irv attachment as the first brown cycle. It has a mean year of about  365.2421 days.


The short years form a sequence as simple as the 33-year cycle of 8 leap years and can defined symmetrically (in a Helios cycle) as the

4th, 12th, 20th, 28th, 36th, 44th, 52nd, 60th, 68th, 76th, 84th & 92nd  years of the 95-year cycle. Then the 1st year has an average start. All these numbers are divisible by 4, but not by 8.


The only other cycles listed in the attachment with equal simplicity are

the 103-year cycle of 13 short years (mean year 365.24272 days) and

the 87-year cycle of 11 short years (mean year 365.24138 days).

Both these have a symmetrical cycle with the same short years, except that the 103-year cycle additionally has the 100th year short and the 87-year cycle does not of course have the 92nd year short.


At the end of the attachment, Irv listed various cycles implemented as simple cycles not necessarily spread as smoothly as possible and so may have higher jitter. However in the case of the 87-year, 95-year & 103-year cycle, they are spread as smoothly as possible, so do not have higher jitter. This arises because the interval (of 7) between the cycles is just 1 different from the interval (of 8) within the cycle.


I have thought of a way of extending my idea of structural complexity to some cycles where the two types of year are not spread as smoothly as possible. Then all of Irv’s simple cycles have complexity 2. The 128-year cycle formed by dropping a leap year one every 128 years also has complexity 2. The Gregorian 400-year cycle has complexity 3 and the Revised Julian 900-year cycle has complexity 4. The ISO leap week cycle does not have a value for this complexity, because it has 3 interval lengths.



For “The Julian precision is rather simple to match...” , I’d say  “The Julian mean year is rather simple to match...”.







From: East Carolina University Calendar discussion List [[hidden email]] On Behalf Of Irv Bromberg
Sent: 16 November 2017 18:42
To: [hidden email]
Subject: Re: Leap rule for 366-day year with 6-day week


From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Christoph P├Ąper [[hidden email]]
Sent: Thursday, November 16, 2017 10:27

What is a good, i.e. simple yet accurate, rule to determine when to drop a 6-day week from a 366-day or 61-week calendar? (Alternatively, consider it a 360-day, 60-week calendar with a very frequent 6-day leap week.)

The Julian precision is rather simple to match...

Irv replies: "precision" is the wrong term here, you are talking about matching the mean year, which is mostly concerned with accuracy relative to some specified astronomical target, such as an equinox or solstice. In the present era the most stable astronomical mean years are those of the north solstice and the northward equinox.

Precision may best be considered as reflecting the short-term jitter of the leap cycle.

The ideal leap rule for providing both accuracy and precision (minimum jitter) is a smoothly spread one, and for simplest evaluability of astronomical accuracy the leap cycle ought to be symmetrical also, which makes it valid to evaluate just the first year of each cycle relative to the desired astronomical target (typically mean equinox or solstice "events"), because the first year will always be at the average. The actual astronomical events wobble around so much that you can't match them without using very complex astronomical algorithms for your calendar, including compensating for Delta T.

Your calendar could have twelve 30-day months with five 6-day weeks per month, for a total of 360 days in a short year, and the frequently occurring leap week could be appended to the last month or exist as a stand-alone mini-month at the end of the year. Positioning the leap week at the end of the year simplifies calendrical calculations because all dates in the calendar then have permanent ordinal numbers relative to the start of the year, and fixed month lengths that all contain only a whole number of weeks makes for a very simple perpetual calendar.

All possible candidate leap cycles having a maximum of 1000 years per cycle are listed in the attached PDF. The instructions at the top show how to implement a smoothly spread symmetrical leap cycle. The instructions at the bottom show how to implement some higher-jitter cycle alternatives that you might prefer but I don't recommend.

The rows are coloured according to their suitability to the mean astronomical years of the equinoxes and solstices reckoned for the year 2020 AD. Blue is for the south solstice, with dark blue the most accurate (but keep in mind that the south solstitial year is currently rapidly getting shorter, so no calendar having a fixed mean year can approximate it well). Green is for the northward equinox, with dark green the most accurate (this mean equinoctial year will be quite stable until around the year 6000 AD). Brown is for the southward equinox, with dark brown the most accurate (but keep in mind that the southward equinoctial year is currently rapidly getting shorter, so no calendar having a fixed mean year can approximate it well). Pink is for the north solstice, with red rows indicating the most accurate (this mean solstitial year will be quite stable until around the year 13000 AD). I consider the red rows to contain the "winners", especially the 269-year cycle with 34 short years per cycle because it is the shortest cycle. You may not be as impressed as I am with the long-term stability of the mean north solstitial year -- please allow me to convince you by asking you to study my web page "The Length of the Seasons (On Earth)" at

If you are interested in any lunar associations then various columns show the number of lunar months per cycle, with those close to an integer highlighted with yellow backgrounds.

I generated this list using my freeware Fixed Leap Cycle Finder spreadsheet that is on my web site. You may wish to try different settings:

My freeware Ford Circles spreadsheet generates a similar listing almost instantly but without the lunar information, and without the higher-jitter cycles. You can find it here: and the Ford Circles chart of 6-day leap week cycles that I already have posted on my web site is here:

-- Irv Bromberg, University of Toronto, Canada