Bookmark or cite this page as <http://www.sym454.org/leap/>
by Dr. Irv Bromberg, University of Toronto, Canada
[Click here to go back to the Symmetry454 / Kalendis home page]
This page is a collection of astronomical evaluations of a variety of Earth solar calendar leap rules. The focus is on leap rules that will be reasonably accurate for the next several millennia, either based on astronomical calculations or employing a fixed arithmetic cycle where the number of days per cycle is divisible by 7 for compatibility with both leap day and leap week calendars. Leap week calendars are perpetual, yet preserve the traditional 7-day weekly cycle.
Except for the ISO standard leap rule and its RJiso variant, the arithmetic for these leap rules spreads the leap year intervals as smoothly as possible. All of these leap rules were selected for their ability to closely align calendars with the solar cycle for as long as possible — so don't expect to see large differences between them.
For a leap week calendar, the number of days per cycle is given by DaysPerCycle = (YearsPerCycle × 364) + (LeapWeeksPerCycle × 7).
The calendar mean year for any fixed arithmetic leap cycle = DaysPerCycle / YearsPerCycle.
Alternatively, the calendar mean year for a leap day cycle = 365 + LeapDaysPerCycle / YearsPerCycle, and for a leap week cycle = 364 + 7 × LeapWeeksPerCycle / YearsPerCycle. However, if one calculates the mean year using floating point arithmetic, as with an ordinary programming language or a basic calculator, then one can obtain an extra 3 decimal points for the mean year by calculating the fractional part separately from the number of days in a non-leap year. For a leap day cycle the fractional part of the mean year is simply LeapDaysPerCycle / YearsPerCycle, and for a leap week calendar it is 7 × LeapWeeksPerCycle / YearsPerCycle – 1.
If the fractional part of the calendar mean year is not already known as an exact proper fraction then that can be calculated using a computing engine capable of arbitrary-precision arithmetic, for example Mathematica or the computer programming language "LISP". If the mean year is instead computed using floating point arithmetic then the result may be approximate, even if calculated to double precision. In such cases a continued fraction calculator will probably yield the proper fraction. For further information about continued fractions, see <http://mathworld.wolfram.com/ContinuedFraction.html>. The following are continued fraction calculators that you can freely use on-line:
This one displays the full continued fraction graphically:
<http://www.hostsrv.com/webmaa/app1/MSP/webm1010/continuedfraction>This one shows intermediate values used to compute the continued fraction, and offers a full explanation about continued fraction:
<http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfCALC.html>
The continued fraction calculator may offer a series of convergents that approximate the given decimal fraction with progressively better accuracy. Usually the appropriate one to pick is the most accurate convergent. Confirm that you have picked the correct fraction by checking that its floating point value is exactly the same as that of the calendar mean year.
To express the calendar mean year in terms of the number of hours, minutes, and seconds in excess of 365 days, simply paste its fractional part into one cell of a spreadsheet program such as Microsoft Excel and then format that cell to display time to the second. This works because in such programs the time of day is simply the fraction of the day that has elapsed since midnight. For calendar mean years that are not exact to the second, calculate the leftover fraction of a second as 86400 × (MeanYear – Hours/24 – Minutes/1440) – Seconds.
These leap rules are also applicable to leap day calendars that have smoothly spread leap years, by taking only the middle day of the distribution (centered above and below the green regression line, if shown). For fixed arithmetic leap day calendars the number of leap years per cycle and the number of years per cycle is indicated by the proper fraction in the "Mean Year" column. For example, for the 159/896 leap rule the mean year is shown as 365 days plus the fraction 31/128, which indicates that the leap day equivalent cycle (having the same mean year length) would have 31 leap days in 128 years.
To find fraction that is the leap day equivalent of a given leap week cycle (having exactly the same calendar mean year), LeapDaysPerCycle = DaysPerCycle – (YearsPerCycle × 365). If LeapDaysPerCycle is divisible by 7 then divide LeapDaysPerCycle and YearsPerCycle by 7. For example, the 41/231 leap week cycle has (231 × 364) + (41 × 7) = 84371 days per cycle, so the numerator of the leap day equivalent fraction is 84371 – (231 × 364) = 56. Thus its leap day equivalent fraction is 56/231, but both 56 and 231 are divisible by 7 so it reduces to 8/33 and becomes a 33 year leap cycle with 8 leap days per cycle.
Alternatively, LeapDaysPerCycle = 7 × LeapWeeksPerCycle – YearsPerCycle, and if LeapDaysPerCycle is divisible by 7 then divide LeapDaysPerCycle and YearsPerCycle by 7. For example, for the 159/896 leap week cycle LeapDaysPerCycle = 7 × 159 – 896 = 217, which divides by 7 = 31 so the leap day equivalent cycle has 31 leap days in 896/7 = 128 years.
The number of days per cycle for a leap day calendar is given by DaysPerCycle = (YearsPerCycle × 365) + LeapDaysPerCycle.
To find the leap week cycle that is the equivalent of a given leap day cycle, first check if DaysPerCycle is divisible by 7. If not, then make YearsPerCycle = YearsPerCycle × 7 and DaysPerCycle = DaysPerCycle × 7, so that there will be a whole number of weeks in the cycle. Finally, LeapWeeksPerCycle = [DaysPerCycle – (YearsPerCycle × 364)] / 7. For example, the Revised Julian calendar has 218 leap years per 900-year cycle, so its calendar mean year = 365+218/900 days, which reduces to 365+109/450 days. Next we calculate DaysPerCycle = (450 × 365) + 109 = 164359 days, but that is not divisible by 7 so we set YearsPerCycle = 450 × 7 = 3150 years and DaysPerCycle = 164359 × 7 = 1150513 days. Finally, we obtain LeapWeeksPerCycle = [1150513 – (3150 × 364)] / 7 = 559 leap weeks in 3150 years.
Alternatively, LeapWeeksPerCycle = LeapDaysPerCycle + YearsPerCycle. If LeapWeeksPerCycle is divisible by 7 then LeapWeeksPerCycle = LeapWeeksPerCycle / 7 otherwise YearsPerCycle = 7 × YearsPerCycle.
Calendrical calculations make frequent use of dividing a number and keeping only the remainder, for example, dividing by 7 to determine the weekday, as will be done below. Many programming languages have a MOD operator or function intended for this purpose, but in many languages MOD handles negative or real numbers improperly (the MOD operator of Microsoft Visual Basic is defective on both counts). To avoid the risk of such errors, herein I will use the solution recommended by Dershowitz & Reingold in Calendrical Calculations: 3rd Edition (CC3, see <http://www.calendarists.com/>):
Not being limited to integer division, the CC3 modulus function also works properly with floating point (real number) parameters provided both the x and the y parameter and the function return value are declared as Double Precision.
The modulus function is essential for implementing smoothly spread leap rules, that is where leap years are as uniformly distributed as possible within each leap cycle, thus minimizing equinox or solstice "jitter". A generic leap rule can be stated as:
isLeapYear( TheYear ) = modulus( LeapYearsPerCycle × TheYear + K, YearsPerCycle ) < LeapYearsPerCycle
where the result is boolean (TRUE if TheYear is a leap year, FALSE otherwise), and K is a constant that sets the long-term mean equinox or solstice alignment. In the section entitled "Symmetrical Leap Cycles", below, I will explain how to select K so as to yield a symmetrical distribution of leap years, and the important advantages of that strategy.
I used the calendrical calculation functions and astronomical algorithms described in "Calendrical Calculations" by Nachum Dershowitz and Edward M. Reingold, third edition published in 2008 by Cambridge University Press. I also employed some algorithms from "Astronomical Algorithms" by Jean Meeus, second edition, published in 1998 by Willmann-Bell, Richmond, Virginia, USA.
Limitations: The astronomical algorithm employed for solar longitude ignores Neptune, Pluto and all asteroids, based on Meeus' truncation of the VSOP87 planetary theory. The parabola used to approximate Delta T is based on the assumption that solar days will get 1.75 milliseconds longer each century (for more information about Delta T see this page). Relativistic effects not accounted for.
Method: Various astronomical, mean astronomical, and fixed arithmetic calendar leap rules were implemented in Kalendis and applied to the Symmetry454 calendar. The alignment of the Northward Equinox (March Equinox, Boreal Vernal Equinox, Austral Autumnal Equinox) relative to the 79th calendar day (Sym454 March 16) or of the North Solstice (June Solstice, Boreal Summer Solstice, Austral Winter Solstice) relative to the 171st calendar day (Sym454 June 17), as appropriate to the specific leap rule, was evaluated using the above astronomical algorithms. Each leap rule was normalized for the Prime Meridian, to "level the playing field" for leap rule comparisons. Where a curvilinear relationship was obtained a 3rd or 4th order polynomial was fit to the points by statistical least-squares regression analysis, and plotted as a thick green line which indicates the average long-term alignment of the equinox or solstice. Most of the reasonably short fixed arithmetic leap-week-compatible cycles (total number of days per cycle divisible by 7) were evaluated (see also Karl Palmen's web page on leap week calendars).
These leap rules are introduced in the document "All About the Symmetry454 Leap Week" and their arithmetic and computer implementation is detailed in the document "Symmetry454 Calendar Arithmetic". I omitted evaluations of the Kalendis Mean Orbital Year (MOY) and Rotation-Adjusted Year (RAY) leap rules because they are not intended to align directly with any equinox or solstice, although MOY and RAY maintain an ultra-long-term constant average relationship with respect to ALL of the equinoxes and solstices.
At this time it is not possible to employ any simple fixed arithmetic leap cycle to align any calendar relative to the Southward Equinox (September Equinox, Boreal Autumnal Equinox, Austral Vernal Equinox), because that equinoctial year length is changing too rapidly and will continue to do so for more than twenty thousand years. Likewise the next era of reasonable stability of the South Solstice (December Solstice, Boreal Winter Solstice, Austral Summer Solstice) as well as the Besselian New Year is more than fifty thousand years away. The astronomical basis for these statements is presented on my "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/>, in particular see the charts in rows 3 and 4 on that page.
A leap week calendar must employ some mix of 6- and 5-year intervals between leap years. A constant interval of 5 years between leap years is far too frequent. Such a calendar would have mean year of 365+2/5 ≡ 365.4 days ≡ 365 days 9 hours and 36 minutes per year. That is 3 hours and 47 minutes too long per year, relative to the present era northward equinoctial year of 365 days 5 hours 49 minutes and 0 seconds, corresponding to a drift rate of about another day later than the equinox every 6+1/3 years! Likewise a constant interval of 6 years between leap years is too infrequent. Such a calendar would have a mean year of 365+1/6 ≡ 365.16... ≡ 365 days and 4 hours per year, which is 1 hour and 48 minutes too short per year, corresponding to a drift rate of about another day ahead of the equinox every 13+1/5 years! Therefore to better track that equinox, or any desired point in the solar cycle, a leap week calendar must employ some scheme to alternately use 6- and 5-year intervals between leap years, maintaining a mix to provide a calendar mean year that closely approximates the mean year of the target point in the solar cycle. Since the drift rate of an every-5-years leap cycle is about twice as fast as that of an every-6-years leap cycle, in terms of the number of years per day of drift, for the present era all appropriate mixes must employ a approximately twice as many 6-year intervals as 5-year intervals. The information presented on my "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/> is a useful guide for selecting a calendar mean year and thence an appropriate calendar leap cycle.
For fixed arithmetic leap cycles, the average interval between leap years = YearsPerCycle / LeapYearsPerCycle, or in terms of days = DaysPerCycle / LeapYearsPerCycle. For the 52/293 leap cycle that average is 2058 days = exactly 294 weeks. Other fixed arithmetic cycles may insert the leap week at slightly shorter or longer average intervals.
For the leap week cycles presented here, leap year intervals occur in groups of either ( 6 + 6 + 5 ) = 17 years or ( 6 + 5 ) = 11 years.
These in turn can most commonly be grouped into sub-cycles of ( 3 × 17 + 11 ) = 62 years or ( 2 × 17 + 11 ) = 45 years.
For example, the sub-cycle pattern for the 52/293 leap cycle is ( 4 × 62 + 45 ) = 293 years.
Such leap interval grouping is not a design feature for these cycles. Rather, sub-cycles are a natural outcome of spreading the leap years of a fixed arithmetic leap cycle as smoothly as possible, generating a repeating pattern that is predictable. The existence of sub-cycle patterns does not increase any calendar's equinox or solstice "jitter". The pattern of leap year intervals is simply an observation made after-the-fact. Nevertheless, its recognition does lead to the prediction of other longer and shorter leap cycles.
This Microsoft Excel "Leap Week Cycles" spreadsheet 
44KB shows the grouping of leap year interval sub-cycles for all of the fixed arithmetic leap week cycles presented below, and more. (If you don't have that a compatible program then you can click here to download the free Microsoft Excel Viewer 2003 for Windows.) In particular, compare the ratios of the 62-year sub-cycles and 45-year sub-cycles for various leap cycles.
Similarly, this "Leap Day Cycles" spreadsheet 32KB shows the grouping of leap year interval sub-cycles for the leap day cycles that correspond to the leap week cycles presented here (those that have the same calendar mean year length). Leap day calendar cycles are associated with a greater variety of sub-cycle sizes.
Astronomical leap rules (actual, mean, or approximation) also employ a mix of 6- and 5-year intervals between leap weeks, but that mixture varies over the years, so it is not possible group their leap year intervals into simple repeating patterns. Nevertheless, it is easy to find many instances of 17-, 11-, 62- and 45-year sub-cycles in their leap year lists.
Linear Approximation Progressive Leap RulesIn addition to astronomical and simple fixed arithmetic leap cycles, on this page are presented several fixed arithmetic progressive leap rules that employ linear approximations to the changing year lengths for the mean Northward Equinox (LANEY), mean North Solstice (LANSY), mean Southward Equinox (LASEY), mean South Solstice (LASSY), and mean Besselian Year (LABY), respectively.
With respect to a target equinox or solstice, over the range of years for which such a leap rule is valid linear approximation progressive leap rules provide accuracy approaching that of astronomical algorithms, yet they employ very simple arithmetic. In cases where a simple fixed arithmetic leap rule provides acceptable accuracy, however, there is obviously no point in using linear approximation.
Each progressive leap rule is based on 3 linear segments representing eras that are each typically 10 millennia in duration:
For example, this chart depicts the linear approximation of the LASSY leap rule 
125KB. The "+" symbols shown the duration of the astronomical south solstitial year, with inherent scatter due to variability of the underlying astronomy. The red line plots the 3-segment linear approximation. One can fit the approximation by simply drawing on the chart and judging the goodness of fit by eye, or one could employ linear regression for the diagonal segment. (It is difficult to use linear regression for the first and third segments because they must be constrained to have zero slope.)
Simply by integrating the area under these 3 segments over the range from the calendar epoch backward or forward to the target year number, the linear approximation arithmetic can compute the New Year Moment for any desired year, in terms of the number of days and fraction of a day elapsed relative to the specified epoch.
In the present era the southward equinoctial mean year and the south solstitial mean year are changing too rapidly for use with any simple fixed arithmetic calendar leap cycle, as shown in chart #3 of my "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/>. The LASEY and LASSY linear approximation progressive leap rules, however, offer an excellent yet simple fixed arithmetic fit. In fact, in the present era LASEY and LASSY can parallel the southward equinox and south solstice, respectively, more accurately than any simple fixed arithmetic cycle can parallel the northward equinox or north solstice, even though the latter two presently have almost constant mean years.
One can judge the goodness of the approximation by comparing the linear approximation New Year Moments to the corresponding mean astronomical New Year Moments. For example, this chart shows the New Year Moment differences 94KB between LASSY and the MSS10 leap rule, showing that the New Year Moments from 1000 BC to 12000 AD match to within about 1/10th of a day. This means that a solar calendar of any given structure will have identical dates for almost all days within that range, whether based on LASSY or MSS10 as the leap rule. Optionally, one may tally up statistics within a selected range of years on the proportion of days having identical dates and the proportion that differ when the leap rules disagree on the leap status of a year.
Another way to evaluate the approximation is by plotting when the target equinox or solstice occurs on a calendar that is based on that leap rule. For example, this chart shows the performance of the Symmetry454 calendar with LASSY leap rule, relative to the south solstice 356KB. In this case, the goodness of fit is obviously almost as good as using a mean astronomical leap rule, at least for the range of years shown. There is a small amount of curvature near the ends of the range of years, whereas a mean astronomical leap rule yields a perfectly straight band with monotonously spaced plotted points, see for example the MSS10 (mean south solstice + 10.25 days) leap rule applied to the Symmetry454 calendar as shown in this chart 277KB.
The linear approximation arithmetic inherently distributes leap years at intervals that are as smoothly spread as possible. Its algorithm transparently ensures a smooth transition through the switchover points between the line segments, automatically avoiding exceptionally short or long leap year intervals as well as avoiding "glitches" away from the intended equinox/solstice alignment at the line segment "elbows", without needing explicit logic to check for or handle those transitions.
For a leap day calendar, floor or round the New Year Moment to yield the New Year Day, then that year has a leap day if the next New Year Day is >365 days away (365 or 366 days are the only possible year lengths). Leap years will be at intervals of 4 or 5 years, as smoothly spread as possible.
For a leap week calendar that always starts its calendar year on a certain "start on" weekday, floor or round the New Year Moment and find the nearest "start on" weekday, then that year has a leap week if the next New Year Day is >364 days away (364 or 371 days are the only possible year lengths). Leap years will be at intervals of 6 or 5 years, as smoothly spread as possible.
[ If using viewing with a text editor, don't allow line wrapping, otherwise it will be very hard to read. ]
Belgian astronomer / retired meteorologist Jean Meeus, author of several books about astronomical algorithms, wrote about the futility of designing calendar leap cycles to carry a calendar more than a few millennia into the future, explaining that the uncertainties of the underlying astronomical algorithms and Delta T approximation, and the partial unpredictability of future changes in the Earth rotation rate limit the range of years that can be accurately projected. I have attempted to estimate the uncertainty of Delta T through the following charts:
See also the section "Dynamic Demonstration of Mean Solar Calendar Drift Rates", below, which includes an Excel spreadsheet with macro that dynamically demonstrates the astronomical drift of a variety of fixed arithmetic leap cycles in comparison with these linear approximation leap rules, showing that the latter indeed have minimal astronomical drift over their intended range of years. That spreadsheet allows the user to experiment with a range of Delta T multipliers to see the effect of various Earth rotation slowdown rates.
In much of the second half of this century, the northward equinox will land on Gregorian March 19th. Thus the Gregorian calendar will have drifted 2 days late relative to its design objective of keeping the equinox on March 21st. Obviously a two-day drift is not considered objectionable (yet), otherwise there would be a major international push for reform of the Gregorian calendar. With that in mind, and allowing for up to 2 days of drift due to uncertainty of the future value of Delta T, any of the linear approximation leap rules ought to be acceptable for well in excess of 10 millennia.
Although in theory these Linear Approximation progressive leap rules could carry a calendar as far as 30 millennia into the future, I don't claim that they indeed will be accurate for so long. Rather my purpose in presenting these leap rules is to define a method that could in the future be refined to the necessary accuracy (before the starting and ending years of the Linear Approximation region), when superior astronomical algorithms and Delta T approximation will be available and the changes in Earth rotation rate will be better understood. Nevertheless, even without such refinement the linear approximation leap rules will give a far better fit to future solar cycles than would any fixed arithmetic leap cycle if it were continued for such a long time.
The leap year intervals during the starting horizontal line segment could be made to match any desired calendar that also has smoothly spread leap years, by using the same epoch and starting calendar mean year. For example, the Dee calendar inherently has smoothly spread leap years at 4- or 5-year intervals, so starting LANEY with a mean year of 365+8/33 days could, with the appropriate epoch offset value, be made to match the Dee leap years, until the start of the sloped linear approximation era. By contrast, the Gregorian and Revised Julian calendars have leap years at intervals which are not smoothly spread, that is every 4 years except for certain centurial years. Therefore even if one started with a matching mean year of 365+97/400 days or 365+109/450 days, respectively, the LANEY leap year intervals would not perfectly match those of the Gregorian or Revised Julian calendar.
Every fixed arithmetic leap cycle has a calendar mean year that is calculable as explained in the "Basic Leap Cycle Calculations" section above. In a typical solar calendar application, the leap cycle and hence the calendar mean year is chosen to approximate a selected equinotical or solstitial mean year, which for the present era is usually either the northward equinoctial or the north solstitial mean year, due to their current astronomical stability, as explained on my "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/>. Astronomical evaluations of the long-term drift of a variety of leap cycles are a useful guide to help make the optimal choice.
One way to carry out such a drift analysis is to determine the date and time of day of the target equinox or solstice in each year, and plot that raw data as a function of the calendar year number. A curve can be statistically fitted to the points, for example by using non-linear least-squares polynomial regression, but if the analysis spans many millennia then the trend probably won't parallel a simple monotonic curve, the calculation will take a long time to complete, and the resulting polynomial may not look like a particularly good fit.
Such astronomical analyses can be greatly simplified by choosing a single year for evaluating the timing of the mean equinox or solstice in each cycle, then interpolating from cycle-to-cycle. This always yields the mean equinox or solstice drift relative to its position at the epoch of the calendar (the day upon which calendar year 1 started), but it won't yield the average position in each cycle unless care is taken to choose the year that is always at the average or closest to the average.
If a leap cycle is arranged such that the list of leap years is symmetrical, so that year n of each cycle has the same leap status as the symmetrical year occurring n years before the first year of the next cycle, then the start of the first year of every cycle will always be at the average for that cycle. In other words, such a leap cycle contains an odd number of years C where each pair of mirror image years (1,C), (2,C-1), ... (Y,C-Y+1) has either both years non-leap or both years leap. The key property of such a symmetrical cycle is that the first calendar year starts at the same moment as the first mean calendar new year moment, and thereafter the first year of each cycle starts at the calendar mean new year moment given by epoch + ElapsedYears × CalendarMeanYear. This symmetrical arrangement can simplify choosing the calendar epoch, because it causes the timing of the target mean equinox or solstice to fall at the cycle average in the first year of every cycle. Therefore, to carry out long-term astronomical drift analysis of a symmetrical leap cycle it is only necessary to evaluate the first year of each cycle, then interpolate from cycle-to-cycle.
This symmetrical leap cycle concept, rules, and arithmetic are largely due to K.E.V. (Karl) Palmen of the Rutherford Appleton Laboratory in the United Kingdom, primarily based on correspondence with the CALNDR LISTSERV during 2007-2008, in threads concerned with what he called 'Helios' and 'quasi-Helios' cycles.
Symmetrical leap cycles can be applied to leap day, leap week, or leap month calendars. Hereinafter, the term leap unit refers generically to the leap day, week, or month, as appropriate to the calendar design, where X = the number of days in the leap unit that in leap years is appended to the end of the calendar year or is inserted somewhere after the target equinox or solstice.
If instead the leap unit is inserted at any position that is prior to the target equinox or solstice within the calendar year then the timing of the mean equinox or solstice instead will fall at the cycle average in the last year of every cycle.
This discussion will be limited to fixed arithmetic leap cycles that are not a repetition of a shorter cycle, and in which the intervals between leap years are as smoothly spread as possible so as to minimize the short-term "wobble" of any mean astronomical equinox or solstice that the calendar is intended to target.
Some additional definitions and properties that are relevant to such symmetrical leap cycles include:
See: <http://mathworld.wolfram.com/ModularInverse.html>.
The modular inverse can be computed in Mathematica using the expression PowerMod[ L, -1, C ].
See also the following Wikipedia page, and follow its links to the extended euclidean algorithm: <http://en.wikipedia.org/wiki/Modular_multiplicative_inverse>.
Click here to download a text file containing an iterative Visual Basic implementation of the modular inverse algorithm. This Microsoft Excel "Demonstration of Modular Inverse Function Calls" spreadsheet 62KB uses that algorithm verbatim as a VBA (Visual Basic for Applications) macro to calculate the modular inverse for a variety of leap day, leap week, and 19- or 30-day leap month calendar cycles.
If U doesn't exist then L and C are not mutually coprime and the cycle contains one or more repetitions of a shorter cycle, to which it must be reduced.
A short cycle is convenient to illustrate such a symmetrical leap cycle. For example, the following line shows the leap status for each year of the symmetrical 45-year leap week cycle (D=364, X=7, D+X=371), having leap intervals of 6 or 5 years, using K = (C – 1) / 2 = 22, where the 37 non-leap years are indicated by '0' digits and the 8 leap years are indicated by '1' digits. A space is inserted beside each leap year to make it easier to discern the symmetrical sequence of leap years. This odd-length cycle is fully symmetrical, and because there are an even number of leap years in the cycle it has a non-leap middle year, as highlighted in boldface:
001 000001 000001 00001 00000 10000 100000 100000 100
For all leap week calendars the inter-leap interval is 6 or 5 years, so the shorter inter-leap interval S = 5 is an odd number. Therefore each cycle begins and ends with (S–1)/2 = 2 non-leap years.
The 45-year cycle could have been useful in the past, but its mean year ≡ 365+11/45 days ≡ 365d 5h 52m 0s ≡ 365.2444... days, which is too long for the present era and future. Nevertheless, its short cycle length is useful for illustrating the principles of symmetrical leap cycles. Click here to view 2 graphs depicting the symmetrical accumulator and new year moment variations for the 45-year leap week cycle 25KB. The second-page accumulator chart shows that all leap years have an accumulator <8 and those leap years that have an accumulator <5 mark the 5 leap years that begin a 6-year inter-leap interval, with the other 3 leap years beginning a 5-year inter-leap interval.
If we change this leap cycle to a 45-year leap day cycle (D=365, X=1, D+X=366), then the inter-leap intervals become 4 or 5 years with 11 leap years per cycle, and because of the odd number of leap years the middle year of the cycle will be a leap year. All of the leap years begin 4-year inter-leap intervals except for 5-year interval begun by the last leap year in each cycle, whose non-leap years are split equally 2+2 between the ending cycle and the next cycle:
001 0001 0001 0001 0001 000 1 000 1000 1000 1000 1000 100
For all leap day calendars the inter-leap interval is 4 or 5 years, so the shorter inter-leap interval S = 4 is an even number. Therefore each cycle begins and ends with S/2 = 2 non-leap years.
If there were a ritual or political reason to align the average equinox or solstice relative to a particular meridian in a certain era, K could be adjusted to fine tune that alignment, but the adjusted leap cycle would be non-symmetrical. U is the smallest number such that the accumulator of year Y + U is one greater modulus C than the accumulator of year Y, and hence the new year moment of year Y + U is 1/C of a leap unit later than that of year Y. Each increment of K advances the leap cycle by U years. If C-U has a smaller absolute value than U then that could be more useful for incremental adjustments, for example in the Dee leap day cycle C=33, L=8, U=29 but C-U=4.
Leap cycles that have an even number of years per cycle as well as an even number of leap years per cycle (such as the Revised Julian calendar, which has 218 leap days in 900 years) need to be reduced to a shorter cycle (109 leap days in 450 years for the example given).
Leap cycles that have an even number of years per cycle and an odd number of leap years per cycle can't be perfectly symmetrical, but will be almost symmetrical. If K = C/2 then the sum of any two symmetrical years' accumulators is L (+ C if they are not leap years), year K has the minimum accumulator = 0, year modulus(K – U, C) has the maximum accumulator C-1, the accumulators of the middle two years are 0 and L, respectively, and the only non-symmetry is that year C/2 is leap but year C/2+1 is non-leap. If on the other hand K = C/2–1 then the sum of any two symmetrical years' accumulators is L-2 (+ C if they are not leap years), year U-K-1 has the minimum accumulator = 0, year K+1 has the maximum accumulator C-1, the accumulators of the middle two years are C-1 and L-1, respectively, and the only non-symmetry is that year C/2 is non-leap but year C/2+1 is leap. With either K value the two middle years must differ in leap status and the accumulator of year 0 or C is K. There is no year in which the new year moment or mean equinox or solstice falls exactly at the cycle average, but year 1 of each cycle is one of the two jointly closest years, deviating by only X/(2×C).
For example, the following line shows the leap status for each year of the almost symmetrical 62-year leap week cycle, having leap intervals of 6 or 5 years, using K = C/2 = 31, where the 51 non-leap years are indicated by '0' digits and the 11 leap years are indicated by '1' digits. A space is inserted beside each leap year to make it easier to discern the symmetrical sequence of leap years. Four non-leap years of a 5-year interval are split between the beginning and end of the cycle. This even-length cycle is symmetrical except that its middle two years differ in leap status. The 31st year of every cycle is a leap year, but the 32nd year of every cycle is a non-leap year, as highlighted in boldface:
001 000001 000001 00001 000001 0000 10 0000 100000 10000 100000 100000 100
Changing K to the alternative value C/2-1 = 30 produces an identical sequence of non-leap and leap years except that the middle two years are reversed: the 31st year of every cycle is a non-leap year, but the 32nd year of every cycle is a leap year, as highlighted in boldface:
001 000001 000001 00001 000001 0000 01 0000 100000 10000 100000 100000 100
The following line shows the almost symmetrical 62-year leap day cycle, having leap intervals of 4 or 5 years with K = 31, showing the almost symmetrical arrangement of its 47 non-leap and 15 leap years. Again, four non-leap years of a 5-year interval are split between the beginning and end of the cycle, and the 31st year of every cycle is a leap year, but the 32nd year of every cycle is a non-leap year:
001 0001 0001 0001 0001 0001 0001 000 10 000 1000 1000 1000 1000 1000 1000 100
The detailed tables below give K and U for those fixed arithmetic leap cycles that can be made symmetrical, or almost symmetrical.
I used SOLEX version 9.1 (home page at <http://main.chemistry.unina.it/~alvitagl/solex/>) to calculate numerically integrated Terrestrial Time moments of each equinox and solstice for each century from 4000 BC to 12000 AD, using that data together with polynomial expressions for approximating Delta T as published in January 2007 by Fred Espenak and Jean Meeus at the NASA Eclipses web site at <http://sunearth.gsfc.nasa.gov/eclipse/SEcat5/deltatpoly.html> to create a VBA (Visual Basic for Applications) macro that dynamically calculates the solar calendar leap cycle drift rates of many of the leap rules discussed here, relative to the appropriate equinox or solstice and a user-specified zero reference year (epoch), displaying the results graphically. The user can change the list of leap cycles, and can enter or shift the epoch year independently for each equinox or solstice. The user can also optionally alter a Delta T multiplier to graphically see the effect of the Earth rotation rate tidally slowing down more or less gradually than it has in the recent past.
Click here to download the "Solar Calendar Drift" spreadsheet 353KB
Click here to see the default built-in charts of the "Solar Calendar Drift" spreadsheet 141KB (4 pages)
In order for the macro to run, you must have the full version of Excel, it won't run in the Excel Viewer environment. In addition, your Excel security settings must allow the macro to run, with or without your confirmation, as you prefer. To enable macros, use the Excel "Tools" menu, choose "Macro", then over to "Security...", then choose the desired macro security level.
Given any minimum and maximum calendar mean year, length of short years (non-leap), length of long years (leap), and the maximum allowable number of years per cycle, it is possible to list all compatible fixed arithmetic leap cycles, simply by computing all possible combinations. That is the strategy employed by the Visual Basic macro in this Excel spreadsheet:
Click here to download the "Fixed Leap Cycle Finder" spreadsheet 390KB
Click here to see an example report generated by the "Fixed Leap Cycle Finder" spreadsheet 18KB
Your Excel macro security settings must be enabled to allow the macro to run, as explained at the end of the previous section.
The spreadsheet allows entry of a target year number, for which it calculates the mean equinoctial and solstitial years (and their rate of change in terms of seconds per century), as well as the mean lunar month lengths. While it is generating the list of leap cycles, the macro color-highlights the cycles that best fit each of the mean equinoctial and solstitial year lengths. Since a good fit to a year length that is changing by more than a second per century may not be particularly useful (use the Solar Calendar Drift spreadsheet above to evaluate the long-term calendar drift), where overlaps occur precedence is given to stable equinoctial or solstitial years.
After generating the list of leap cycles the macro sorts the list in descending order by calendar mean year, highlights saros and lunar month counts that are near integers, sets up the first and last listed cycles as short "mixer" cycles that have fewer years per cycle than the shortest cycle within the specified mean year range and having mean years equal to or closest to the specified range, then it calculates the proportion of short mixer cycles that comprise each within-range leap cycle.
Optionally, the user can specify that the number of years per listed cycle must be divisible by a specified number. For example, the user could enter 100 as the "divisible by" value to cause all listed within-range cycles to be whole centuries in length, although obviously for this example only a few cycles will qualify.
Optionally, the user can specify that the cycle mean year must comprise a whole number of a specified fraction of a day. When this option is enabled the maximum number of years per cycle is ignored, so all qualifying cycles will be found. For example:
The user can specify the number of days per week, initially set to 7 days.
A checkbox option allows the user to exclude or allow generation of simple leap cycle repeats. The list will be shorter if repeated cycles are excluded, but in some cases repeated cycles may be useful to obtain near integer counts of lunar months, if desired, or round counts for the number of years per cycle.
A unique feature of this spreadsheet is its ability to calculate leap cycle mean years to exact fractions, both in terms of days and in terms of hours minutes and seconds in excess of 365 days, but this is only possible if the specified year lengths are integers (as is usually the case).
Although preset to find cycles that are compatible with perpetual leap week calendars, the spreadsheet is intended as a general-purpose leap cycle finder for any kind of fixed arithmetic calendar. The number of days per year need not be integers, and can be any valid Excel expressions. The following table lists examples of useful settings for a few calendar types:
| Calendar Type | Short Year Length | Long Year Length | ||
|---|---|---|---|---|
| Days | Months | Days | Months | |
| leap day | 365 | 12 | 366 | 12 |
| leap week (7 days per week, perpetual) | =52*7 | 12 | =53*7 | 12 |
| leap month, all months = 28 days (perpetual) | =13*28 | 13 | =14*28 | 14 |
| leap month, all months = 35 days (perpetual) | =10*35 | 10 | =11*35 | 11 |
| leap month, all months = 30 days | =12*30 | 12 | =13*30 | 13 |
| leap month, all months = 19 days | =19*19 | 19 | =20*19 | 20 |
| leap month, lunisolar | =12*Synodic | 12 | =13*Synodic | 13 |
| leap month, lunisolar, fixed mean month | =12*FMM | 12 | =13*FMM | 13 |
| hex week (long years most common) | 360 | 12 | 366 | 12 |
Buttons are provided to help set up the desired calendar type:
The preset default fixed mean month (FMM) expression is for 29 + 12/24 + 44/1440 + (2+62/89)/86400 days = 29 days 12 hours 44 minutes and 2+62/89 seconds, corresponding to a 49-yerm fixed cycle containing 376 deficient (29-day) and 425 full (30-day) months, which is a good choice for the present millennium. If you prefer a 52-yerm fixed cycle containing 399 deficient and 451 full months, then change the seconds portion of the expression to (2+14/17).
The spreadsheet also contains macro functions for calculation of Continued Fractions (see the "Ratios" worksheet) and calculation of integer Divisors with optional indication as to which divisors are prime numbers (see the "Divisors" worksheet), for further characterization of candidate leap cycles. Where the target numerator and denominator for a continued fraction approximation are both integers, the "Ratios" worksheet shows examples of a novel method for indicating the exact fractional error for each convergent, which can be useful in developing simple leap rules with periodic corrections.
Although the mean year of any particular leap cycle may not exactly match the mean year of an equinox or solstice, each reasonably accurate leap cycle has a mean year that is a stable match to one or two points in the annual solar cycle, which we can call "calendar season(s)" for that leap cycle. In the present era, such calendar seasons are stable for calendar mean years that are as short as about 365 days 5 hours 47 minutes 53 seconds to as long as about 365 days 5 hours 49 minutes 35 seconds, a range of only about 1 minute 42 seconds, corresponding to the shortest and longest available mean years in today's solar cycle:
Expressed as minutes and seconds in excess of 365 days 5 hours (mean solar time).
(click here or on the chart to open a high-resolution PDF version 70 KB)
The trend toward fewer mean solar days per solar year is due to tidal slowing of the Earth rotation rate.
The range of possible calendar seasons is currently getting narrower due to decreasing Earth orbital eccentricity and decreasing Earth axial tilt.
If Earth's orbit were a perfect circle or ellipse and if the Earth axial tilt were constant and its axis not wobbling then the mean length of the solar cycle would be the same no matter where in the solar cycle the measurement were started. In actuality, however, Earth's orbit is not a perfect ellipse (the major axis of the ellipse advances around Sun), and there is a long periodic variation in the Earth axial tilt as well as a long periodic wobble of the axial direction, which collectively cause the solar cycle mean year to vary slightly but significantly, depending on the measurement start point.
In any given era the longest cycle mean year having a stable calendar season will match the mean solar year (in terms of mean solar days) at the ecliptic longitude of the Earth orbital perihelion, and the shortest cycle mean year having a stable calendar season will match the mean solar year at the ecliptic longitude of the Earth orbital aphelion. (These are not the same as the anomalistic year, the time taken for Earth to complete one revolution with respect to perihelion or aphelion, which is presently about 365 days 6 hours 13 minutes and 52 seconds, slightly longer than the sidereal year because of the advance of perihelion and aphelion.) For example, if in the present era the mean perihelion is at an ecliptic longitude of 283° then the longest solar year will be the one measured from that solar longitude until the next time Sun reaches 283°, or if the mean aphelion is at 103° then the shortest solar year will be the one measured from that solar longitude until the next time Sun reaches 103°. (When calculating the mean year for any ecliptic solar longitude, year-to-year variations mainly due to Moon make it necessary to compute averages symmetrically spanning many centuries before and after the target year, or to carry out the calculation for the Earth-Moon center of gravity.) The advance of perihelion (and aphelion, always 180° away) and the tidal slowing of the Earth rotation rate cause calendar seasons to evolve and migrate as the millennia pass.
A leap cycle will not have any calendar seasons in an era in which its mean year is a few seconds shorter than the mean year at the ecliptic longitude of aphelion. With tidal slowing of the Earth rotation rate, however, eventually the mean year at aphelion will equal the cycle mean year, so a calendar season will appear at the ecliptic longitude of aphelion. With progressive tidal rotation slowing that calendar season will split into a more stable season that will migrate in opposite directions. The more stable calendar season migrates to earlier solar longitudes, and is more stable because it moves against the advance of perihelion. The less stable calendar season migrates to later solar longitudes, and is less stable because it moves in the same direction as the advance of aphelion. As tidal rotation slowing continues, eventually the average length of the solar cycle will approximately equal the cycle mean year, and then both calendar seasons will be migrating at slowest rate towards perihelion, and perihelion and aphelion will be situated approximately midway between them, with perihelion having the less stable calendar season behind it (prior solar longitude) and the more stable calendar season ahead of it (later solar longitude). Further tidal rotation slowing will cause the calendar seasons to converge towards perihelion, disappearing in later years when the mean year near the ecliptic longitude region of perihelion becomes longer than the cycle mean year. Migration of the calendar seasons is fastest near the start and end of their existence, and their positions are most stable during the middle several millennia of their existence. Their migration is fastest from the region of about 15° before to about 30° after the ecliptic longitude of perihelion, so both calendar seasons become unstable and disappear before ever reaching perihelion. In total it takes 10-11 millennia for a calendar season to start at aphelion and end at perihelion.
With that description, the reader might think that aphelion should eventually "catch up" to perihelion, but that can never happen because they both advance in unison around Sun, always 180° apart. As aphelion passes through each point of the solar cycle, however, that ecliptic solar longitude has the shortest mean year, and the opposite solar longitude 180° away at perihelion has the longest mean year.
Deeper tidal slowing of the Earth rotation rate, if it persists for many centuries, tends to improve the long-term stability of the solar year lengths for the ecliptic longitudes between perihelion and aphelion, thus stabilizing the most stable calendar season.
In collaboration with K.E.V. (Karl) Palmen of the Rutherford Appleton Laboratory in the United Kingdom, primarily based on correspondence with the CALNDR LISTSERV during early 2009, I developed the following method of finding and characterizing calendar seasons:
Choose an appropriate baseline middle year. Typically that year would be in the present era or recent past for calendars intended to approximate the northward equinox, or a year within the next few millennia for calendars intended to approximate the north solstice. Although calendar seasons exist that could be used to approximate the southward equinox or south solstice, they are only moderately stable in the present era.
Calculate the new year moment (NYM) of the baseline middle year, using the arithmetic presented above in the "Symmetrical Leap Cycles" section. For elapsed each day of the baseline middle year relative to the NYM, calculate the ecliptic solar longitude, then use that to generate the MiddleLine value to be subtracted from other comparison years:
MiddleSoLong = SolarLongitude(NYM(MiddleYear) + ElapsedDaysInYear)
MiddleLine = modulus(ElapsedDaysInYear × 360° / CycleMeanYear – MiddleSoLong, 360°)
where modulus( x, y ) = x – y × floor( x / y ) and where CycleMeanYear = 365 + LeapDaysPerCycle / YearsPerCycle.
On a chart, plot the baseline middle year as a horizontal line, at "zero drift".
For various years appropriately spaced prior to and after the middle year, calculate the their respective NYM and solar longitude at each elapsed day of the year. The spacing between these other evaluated and plotted years could be the length of the leap cycle or an arbitrary step size of 50 to 500 years, as desired. (If the step size is too large, such as 1000 years, then the stable calendar season may get masked by its slow drift to later dates in the calendar year, caused by the advance of perihelion.) For each of these other years, calculate the calendar drift relative to the baseline middle year for each elapsed day of the year relative to the respective NYM, starting with the solar longitude:
SoLong = SolarLongitude(NYM(Year) + ElapsedDaysInYear)
Calculate the solar longitude drift relative to the previously calculated MiddleLine:
Drift = modulus(ElapsedDaysInYear × 360° / CycleMeanYear – SoLong, 360°) – MiddleLine
where the expression ElapsedDaysInYear × 360° / CycleMeanYear is the leap cycle's implied approximation of the solar longitude for that number of elapsed days. Finally, convert the drift to hours of calendar drift relative to the middle year:
HoursOfDrift = Drift × CycleMeanYear / 15
where the divisor 15°/h is 360° divided by 24 hours.
On the resulting chart, the various plotted curves will converge, cross, and reverse their order at the most stable calendar season, if it exists. This is the stable point in the solar cycle that has an astronomical mean year equal to the leap cycle mean year, persisting for at least as long as the interval between the earliest and latest convergent year on the chart. A secondary, less stable calendar season usually exists elsewhere in the solar cycle, but at a point where many of the curves cross each other above or below the horizontal line that represents the baseline middle year. The secondary calendar season will appear better defined if the step size between plotted years is reduced, but occasionally a small step size may make it seem slightly better defined that the primary calendar season (for example 71/293 cycle with 50-year step size and middle year 2500). If the mean year of the leap cycle is too long or too short relative to the era of the baseline middle year then convergence and crossover will be incomplete, or lacking entirely (for example any cycle having a mean year outside the bounds suggested above, with a present era middle year).
One way to find the calendar seasons is to calculate the standard deviation (SD) of the drift for all plotted curves (including the zero baseline) and then find the one or two minima in that list. Another method is to look for curve crossovers, especially where the baseline middle year is crossed. A crossover occurs wherever the drift of a lower year minus the drift of a higher year changes sign. The moment of the crossover can be calculated by linear interpolation as follows:
Let (Ax, Ay) and (Bx, By) be the coordinates of the lower year curve on the day before and after the crossover, respectively, where the x-coordinate is the elapsed day number and the y-coordinate is the drift relative to the middle year, and AB is the line between them. Similarly let (Cx, Cy) and (Dx, Dy) be the coordinates of thei higher year curve on the day before and after the crossover, respectively, and CD is the line between them. The intersection of line AB with CD falls at the moment and drift of the crossover. Calculate the solar longitude for that moment within the lower year (it will be the same solar longitude at the corresponding moment in the higher year, that is why they crossed each other). If the solar longitude is >180° then it may be convenient to subtract 360° to avoid a discontinuity near the northward equinox, so that values near stable calendar seasons will be within the range of ±180°.
If a secondary calendar season is to be used for calendrical purposes then due to its more rapid migration it is best to calculate it for a range of years where the middle year is well into the future, provided that the primary calendar season will still be stable.
Click here to download the "Find Solar Calendar Seasons" spreadsheet 625KB
At the present time the spreadsheet doesn't display its chart nicely in Excel 2007, so for best results use an earlier version of Excel. To see what such chart are supposed to look like, click on any of the following years-per-cycle links to open a PDF version of a calendar seasons chart for that cycle, as generated by the "Find Solar Calendar Seasons" spreadsheet: 293, 524, 33, 400, 327, 389 (each PDF is about 93 KB).
The example chart for the 293-year leap cycle shows that it has a stable calendar season that is presently about 4 days after the northward equinox, which is consistent with its calendar mean year being slightly shorter than the mean northward equinoctial year, and it has a secondary calendar season in the region of the last week of October. By shifting the plot to earlier and also to later millennia, stable calendar seasons can be found for this leap cycle from before 4000 BC until about 5000 AD (>9 millennia), gradually migrating to later dates in the calendar year in parallel with the advance of perihelion. If you set the middle year to 5000 AD then prior plotted years will be drawn with medium thickness lines, indicating that they cross the middle year, but future years will be drawn with thin lines, indicating that they don't cross the middle year (they converge toward the calendar season without crossing the baseline).
In order for the macro to run, you must have the full version of Excel, it won't run in the Excel Viewer environment. In addition, your Excel security settings must allow the macro to run, with or without your confirmation, as you prefer. To enable macros, use the Excel "Tools" menu, choose "Macro", then over to "Security...", then choose the desired macro security level.
To use the spreadsheet, activate the "Setup" worksheet and enter the desired number of leap days per cycle, years per cycle, step size in years (preferably a multiple of the mean synodic month to obtain the smoothest plotted curves, see the continued fraction calculator at the bottom of the Setup worksheet), and middle year number (such as a year near the present era). Observe the calculated cycle mean year as you change these entries, and compare with the shortest, average, and longest solar years for the specified middle year. If your computer system has the Wingdings font installed (a standard TrueType font in Windows) then the spreadsheet indicates if the cycle mean year is within range (happy face icon), too short (down arrows), or too long (up arrows).
If you want help to choose a cycle that has a certain mean year, the middle section of the Setup worksheet offers a continued fraction calculator into which the user can enter a fraction or time or decimal number expression, then a series of progressively more accurate convergents will be listed below. There is also a checkbox to control automatic scaling of the vertical axis of the chart.
After the user clicks the "Update Chart" button the macro will run, automatically switching to the "Chart" worksheet and updating it as the calculation proceeds. The primary (more stable) and secondary (less stable) calendar seasons will be marked with dashed vertical lines and their legend keys at the bottom left show their number of elapsed days, solar longitude, month and day. If the specified middle year number is not divisible by the step size then it will change to the prior year number that is divisible.
The chart also shows the elapsed day positions of the equinoxes, solstices, and mean perihelion and aphelion in the middle year.
Navigation buttons at the top allow you to shift a couple of steps to the past or future.
If you click on one of the "Year ####" curves, then click on the "Shift to Selected Curve" button near the bottom left, the chart will recalculate using that year as the new middle year number.
The "Cross" worksheet shows the detailed list of curve crossover points. If you autofilter on Drift = 0 then the list will display only the baseline crossings.
The "Cycles" worksheet contains a collection of leap cycles that is in turn used for the "Pattern" chart, depicting the approximate mathematical relationship between the calendar mean year fraction (hours minutes and seconds in excess of 365 days) and the stable solar longitude of the calendar season, expressed over the range ±180° relative to the northward equinox. It shows that in the present era there is an approximately linear relationship between the calendar mean year fraction (in excess of 365 days) and the solar longitude of the stable calendar season:
StableSolarLongitude = 28735 – 118580 × MeanYearFraction
where MeanYearFraction = LeapDaysPerCycle / YearsPerCycle and the StableSolarLongitude is expressed as ±180° measured along the ecliptic relative to the northward equinox.
The Leap Key corresponds to the leap rule abbreviation in Kalendis, which can be employed to produce leap year lists for most of these leap rules, as applied to the Symmetry454 or Symmetry010 leap week calendars, or compatible variants. For fixed arithmetic cycles the abbreviation comprises the number of leap years per cycle / number of years per cycle. Where the Mean Year is shown with an equivalency symbol (≡) the decimal value is exactly equivalent to the fraction (with overscored digits and "..." indicating a repeating sequence of decimal digits), otherwise an asymptotic symbol (≈) indicates that the decimal value is rounded but almost equal to the given fraction.
To save space, the Mean Year length, where fixed, is given as the portion of a day in excess of 365 days, expressed as the fraction, hours : minutes : seconds and fraction of a second, the continued fraction equivalent, and as a decimal number. To convert to the actual mean year, just add 365 days.
The fraction in excess of 365 days can only contain a whole number of seconds if the number of seconds per day (24×60×60=86400) is divisible by the denominator, which is usually the number of years per cycle, but may be less if the mean year fraction reduces. The divisors of 86400 are: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 90, 96, 100, 108, 120, 128, 135, 144, 150, 160, 180, 192, 200, 216, 225, 240, 270, 288, 300, 320, 360, 384, 400, 432, 450, 480, 540, 576, 600, 640, 675, 720, 800, 864, 900, 960, 1080, 1152, 1200, 1350, 1440, 1600, 1728, 1800, 1920, 2160, 2400, 2700, 2880, 3200, 3456, 3600, 4320, 4800, 5400, 5760, 7200, 8640, 9600, 10800, 14400, 17280, 21600, 28800, and 43200 (only 2, 3, and 5 are prime numbers). Those that are highlighted in boldface are mentioned in discussion of leap rules below.
For the leap cycles that are astronomical, linear approximations, or in which leap years are not smoothly spread, an astronomical drift chart in PDF format is linked to the Leap Key. To view any chart using a PDF reader such as the freely available Adobe Acrobat Reader, just click on the Leap Key link. Each PDF is 200 to 300 KB in size, as one page in landscape layout. Each will open in a new window so that you won't have to wait for loading.
For any of the simple fixed arithmetic leap cycles the "Dynamic Demonstration of Mean Solar Calendar Drift Rates" spreadsheet, freely available above, offers an excellent way to evaluate long-term astronomical drift.
In several cases equivalent simple lunisolar cycles are mentioned, which could be useful not only for lunisolar calendar purposes, but also for generating an Easter computus having a mean year that is identical to the solar calendar.
| Leap Key | Mean Year (in excess of 365 days) |
Description and Comments (northward equinox = ecliptic solar longitude 0°) |
|---|---|---|
| NE79 | actual equinoctial year | The "gold standard" for the Northward Equinox, taking the New Year moment as 78 days prior to the astronomical equinox. Has small variations (±15 minutes) in the equinoctial year length, mainly due to gravitational interactions with Moon, Venus, and Jupiter. Keeps the average equinox aligned on the 79th calendar day, and has the minimum possible "equinox jitter" (for a leap week calendar) of ±3 days. |
| MNE79 | mean equinoctial year (currently 5:49:00.2) |
The "secondary gold standard" for the Northward Equinox, taking the New Year moment as 78 days prior to the mean equinox. Better regulated than NE79, above, eliminating short-term fluctuations in the equinoctial year length, but the polynomials employed for calculating the mean equinox moment are valid for only ±3 millennia from the present era. Kalendis extends the valid range by using NE79 and averaging ±50 years centered on the target year. Keeps the mean equinox aligned on the 79th calendar day, and has the minimum possible "mean equinox jitter" (for a leap week calendar) of ±3 days. |
| 52/293 | 71/293 ≡ 5h 48m 56+152/293s
≡ ≈ .242320819112628 |
71 leap days or 52 leap weeks per 293-year cycle (293 is a prime number) [cycle has 107016 days, prime divisors are 2, 3, 7, 13]. The primary calendar season for years 0 - 1765 - 3530 is at an ecliptic solar longitude of about 3.1° (about 3 days after the northward equinox), click here to view chart 93 KB.
The 52/293 leap cycle is preferred for the Symmetry454 and Symmetry010 calendars because:
The exact decimal value of the fraction 71/293 has 146 repeating digits: 0.2423208191 1262798634 8122866894 1979522184 3003412969 2832764505 1194539249 1467576791 8088737201 3651877133 1058020477 8156996587 0307167235 4948805460 750853... |
| 93/524 | 127/524 ≡ 5h 49m 60/131s
≡ ≈ .24236641221374 |
127 leap days or 93 leap weeks per 524-year cycle provides slightly tighter equinox alignment than 52/293, but the duration of excellent equinox alignment will be about 500 years shorter (524 is divisible by 2, 4, 131, 262, of which 131 is a full reptend prime number) [cycle has 191387 days, prime divisors are 7,19,1439]. This 524-year cycle contains 6481 synodic or 7005 sidereal lunar months. For an almost symmetrical cycle, K = 261 or 262, U = 293. The primary calendar season for years 706 - 2471 - 4236 is at an ecliptic solar longitude of about 359.2° (at the northward equinox), click here to view chart 93 KB.
For a solar leap month calendar having 19 days per month and 19 regular months per year, an exactly equivalent cycle would have 524 years with 117 leap years and a total of 10073 months per cycle. Because 131 is a full reptend prime divisor of 524, the exact decimal value of the fraction 127/524 equals 0.24 followed by 131-1=130 repeating digits: 0.24 2366412213 7404580152 6717557251 9083969465 6488549618 3206106870 2290076335 8778625954 1984732824 4274809160 3053435114 5038167938 9312977099... |
| LANEY | Linear Approximation to the Northward Equinoctial Year |
For about 10 millennia prior to 5200 AD, the mean Northward Equinoctial year length could have been approximated with a mean year of about 365 days 5 hours 48 minutes 57.5 seconds = 365+71/293 days. From 5200 AD to 14000 AD the equinoctial year will get shorter in almost linear fashion by about 9/5 seconds per century. After 14000 AD the mean Northward Equinoctial year length will level out at about 365 days 5 hours 46 minutes 22 seconds, for about 5 millennia.
Click here for charts depicting the LANEY approximation |
| Leap Key | Mean Year (in excess of 365 days) |
Description and Comments (north solstice = ecliptic solar longitude 90°) |
|---|---|---|
| NS171 | actual solstitial year | The "gold standard" for the North Solstice, taking the New Year moment as 170 days prior to the astronomical solstice. Has small variations (±15 minutes) in the solstitial year length, mainly due to gravitational interactions with Moon, Venus, and Jupiter. Keeps the solstice aligned on the 171st calendar day, and has the minimum possible "solstice jitter" (for a leap week calendar) of ±3 days. |
| MNS171 | mean solstitial year (currently 5:47:55.5) |
The "secondary gold standard" for the North Solstice, taking the New Year moment as 170 days prior to the mean solstice. Better regulated than NS171, above, eliminating short-term fluctuations in the solstitial year length, but the polynomials employed for calculating the mean solstice moment are valid for only ±3 millennia from the present era. Kalendis extends the valid range by using NS171 and averaging ±50 years centered on the target year. Keeps the mean solstice aligned on the 171st calendar day, and has the minimum possible "mean solstice jitter" (for a leap week calendar) of ±3 days. |
| 69/389 | 94/389 ≡ 5h 47m 58+58/389s
≡ ≈ .241645244215938 |
The ideal fixed arithmetic cycle for aligning the Symmetry454 calendar with the North Solstice for the next 10 millennia, with 94 leap days or 69 leap weeks per 389 year cycle (389 is a full reptend prime number) [cycle has 142079 days, prime divisors are 7, 20297]. This cycle has the second shortest mean year of any fixed arithmetic cyle presented here, but for the next 11000 years it will be ideal for alignment with the North Solstice. This 389-year cycle contains almost exactly 4811+1/4 mean synodic months, so 4 such cycles would contain almost exactly 19245 mean synodic months. For a symmetrical cycle, K = 194, U = 327. The primary calendar season for years 1184 - 2664 - 4144 is at an ecliptic solar longitude of about 79.5° (slightly before the north solstice), click here to view chart 93 KB, for years 4440 - 5920 - 7400 it will be at 90.2° (at the north solstice), then it will spend a few millennia slightly after the north solstice.
For a lunisolar leap month calendar an equivalent cycle would have 296 years with 109 leap years, and a total of 3661 months per cycle. Since 389 is a full reptend prime number, the exact decimal value of the fraction 94/389 has 389-1=388 repeating digits: 0.2416452442 1593830334 1902313624 6786632390 7455012853 4704370179 9485861182 5192802056 5552699228 7917737789 2030848329 0488431876 6066838046 2724935732 6478149100 2570694087 4035989717 2236503856 0411311053 9845758354 7557840616 9665809768 6375321336 7609254498 7146529562 9820051413 8817480719 7943444730 0771208226 2210796915 1670951156 8123393316 1953727506 4267352185 0899742930 5912596401 0282776349 6143958868 89460154... |
| 58/327 | 79/327 ≡ 5h 47m 53+43/109s
≡ ≈ .241590214067278 |
An excellent fixed arithmetic cycle for aligning the Symmetry454 calendar with the North Solstice for the next 5 millennia, with 79 leap days or 58 leap weeks per 327 year cycle (327 is divisible by the prime number 3 and the full reptend prime number 109) [cycle has 119434 days, prime divisors are 2, 7, 19, and 449]. This cycle has the shortest mean year of any fixed arithmetic cyle presented here, but for the next 5000 years it will be excellent for alignment with the North Solstice. This 327-year cycle contains 4389 draconic lunar months. For a symmetrical cycle, K = 163, U = 265. The primary calendar season for years 1184 - 2664 - 4144 is at an ecliptic solar longitude of almost 89° (one day before the north solstice), click here to view chart 93 KB, but for years 4440 - 5920 - 7400 it will be at about 95.8° (about 5 days after the north solstice).
For a solar leap month calendar having 19 days per month and 19 regular months per year, an exactly equivalent cycle would have 327 years with 73 leap years, and a total of 6286 months per cycle. Because 109 is a full reptend prime divisor of 327, the exact decimal value of the fraction 79/327 has 109-1=108 repeating digits: 0.2415902140 6727828746 1773700305 8103975535 1681957186 5443425076 4525993883 7920489296 6360856269 1131498470 94801223... |
| LANSY | Linear Approximation to the North Solstitial Year |
For the 10 millennia prior to 10500 AD, the mean North Solstitial year length can be approximated with a mean year of about 365+94/389 days ≈ 365 days 5 hours 47 minutes 58 seconds. From 10500 AD to 18000 AD the solstitial year will get shorter in almost linear fashion by about 5/4 seconds per century. After 17500 AD the mean North Solsticial year length will level out at about 365 days 5 hours 46 minutes 20 seconds, for about 5 millennia.
Click here for charts depicting the LANSY approximation |
| Leap Key | Mean Year (in excess of 365 days) |
Description and Comments (northward equinox = ecliptic solar longitude 0°) |
|---|---|---|
| 71/400 | 97/400 ≡ 5h 49m 12s
≡ ≡ .2425 |
Same mean year as Gregorian calendar, but allocating 97 leap days or 71 leap weeks at intervals that are as smoothly spread as possible per 400-year cycle (400 is divisible by 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, of which 2 and 5 are prime) [cycle has 146097 days, prime divisors are 3, 7, 773]. For an almost symmetrical cycle, K = 199 or 200, U = 231. The primary calendar season for years 0 - 1860 - 3720 is at an ecliptic solar longitude of about 342.5° (nearly 3 weeks before the northward equinox), click here to view chart 93 KB.
Compare with the ragged equinox alignment of the Gregorian calendar (two pages), and the ISO leap rule (next row below), neither of which have smoothly spread leap years. The accuracy of the Gregorian calendar mean year was limited by the sexigesimal numeric notation that was used in the era of the Gregorian reform. The mean spring equinoctial year was expressed to the nearest 1/3600 of a day as 365+14/60+33/3600 days ≡ 365+97/400 days. Each 1/3600 of a day is 24 seconds. Deducting a further 1/3600 of a day would have yielded the mean year that today is used for the Revised Julian calendar. |
| ISO | 97/400 ≡ 5h 49m 12s ≡ .2425 | Same mean year as the Gregorian calendar, with 71 leap weeks per 400-year cycle, but starting the year on the Monday that is closest to Gregorian New Year Day, as per the International Organization for Standardization ISO:8601 leap rule. It is a leap year only if the corresponding Gregorian year starts or ends on a Thursday. Note the ragged edge of the equinox moments distribution, due to the non-uniform spread of Gregorian leap years (intervals of 4 or 8 years, whereas uniform spread would use intervals of 4 or 5 years). |
| 41/231 | 8/33 ≡ 5h 49m 5+5/11s
≡ |
Leap week adaptation of John Dee's leap day calendar cycle (8 leap days per 33-year cycle, note that 231 = 33 × 7 years), with 41 smoothly spread leap years per 231 years (231 is divisible by 3, 7, 11, 21, 33, 77, of which 3, 7 and 11 are prime) [cycle has 84371 days, prime divisors are 7, 17, 709]. This 231-year cycle contains 3062 anomalistic or 3100+1/2 draconic lunar months. For a symmetrical cycle, K = 115, U = 62. The primary calendar season for years 0 - 1765 - 3530 is at an ecliptic solar longitude of about 351.3° (almost 9 days before the northward equinox), click here to view chart 93 KB. |
| 175/986 | 239/986 ≡ 05h 49m 2+394/493s
≡ ≈ .242393509127789 |
239 leap days or 175 leap weeks per 986-year cycle (986 is divisible by 2, 17, 29, 34, 58, and 493, of which 2, 17, and 29 are prime) [cycle has 360129 days, prime divisors are 3, 7, 11, 1559]. For an almost symmetrical cycle, K = 492 or 493, U = 755. The primary calendar season for years 353 - 2118 - 3883 is at an ecliptic solar longitude of about 356.3° (3-4 days before the northward equinox).
The exact decimal value of the fraction 239/986 equals 0.2 followed by 112 repeating digits: |
| 134/755 | 183/755 ≡ 5h 49m 1+149/151s
≡ ≈ .242384105960265 |
183 leap days or 134 leap weeks per 755-year cycle (755 is divisible only by the prime numbers 5 and 151) [cycle has 275758 days, prime divisors are 2, 7, 19697]. This 755-year cycle contains 9338 synodic or 10093 sidereal lunar months. For a symmetrical cycle, K = 377, U = 524. The primary calendar season for years 353 - 2118 - 3883 is at an ecliptic solar longitude of about 357.2° (about 3 days before the northward equinox).
The exact decimal value of the fraction 183/755 equals 0.2 followed by 75 repeating digits: |
| 145/817 | 198/817 ≡ 5h 48m 59+37/817s
≡ ≈ .24235006119951 |
198 leap days or 145 leap weeks per 817-year cycle (817 is divisible only by the prime numbers 19 and 43) [cycle has 298403 days, prime divisors are 7, 47, 907]. For a symmetrical cycle, K = 408, U = 293. The primary calendar season for years 353 - 2118 - 3883 is at an ecliptic solar longitude of about 0.1° (at the northward equinox).
The exact decimal value of the fraction 198/817 has 126 repeating digits: 0.2423500611 9951040391 6768665850 6731946144 4308445532 4357405140 7588739290 0856793145 6548347613 2190942472 4602203182 3745410036 719706... |
| 167/941 | 228/941 ≡ 5h 48m 54+306/941s
≡ ≈ .242295430393199 |
228 leap days or 167 leap weeks per 941-year cycle (941 is a full reptend prime number) [cycle has 343693 days, prime divisors are 7, 37, 1327]. For a symmetrical cycle, K = 470, U = 648. The primary calendar season for years 353 - 2118 - 3883 is at an ecliptic solar longitude of about 7° (about a week after the northward equinox).
Since 941 is a full reptend prime number, the exact decimal value of the fraction 228/941 has 941-1=940 repeating digits: 0.2422954303 9319872476 0892667375 1328374070 1381509032 9436769394 2614240170 0318809776 8331562167 9064824654 6227417640 8076514346 4399574920 2975557917 1094580233 7938363443 1455897980 8714133900 1062699256 1105207226 3549415515 4091392136 0255047821 4665249734 3251859723 6981934112 6461211477 1519659936 2380446333 6875664187 0350690754 5164718384 6971307120 0850159404 8884165781 0839532412 3273113708 8204038257 1732199787 4601487778 9585547290 1168969181 7215727948 9904357066 9500531349 6280552603 6131774707 7577045696 0680127523 9107332624 8671625929 8618490967 0563230605 7385759829 9681190223 1668437832 0935175345 3772582359 1923485653 5600425079 7024442082 8905419766 2061636556 8544102019 1285866099 8937300743 8894792773 6450584484 5908607863 9744952178 5334750265 6748140276 3018065887 3538788522 8480340063 7619553666 3124335812 9649309245 4835281615 3028692879 9149840595 1115834218 9160467587 6726886291 1795961742 8267800212 5398512221 0414452709 8831030818 2784272051 0095642933 0499468650 3719447396 3868225292... |
| 115/648 | 157/648 ≡ 5h 48m 53+1/3s
≡ ≡ .242 283950617... |
157 leap days or 115 leap weeks per 648-year cycle (648 is divisible by 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, of which only 2 and 3 are prime numbers) [cycle has 236677 days, prime divisors are 7, 33811]. For an almost symmetrical cycle, K = 321 or 324, U = 355. The primary calendar season for years 353 - 2118 - 3883 is at an ecliptic solar longitude of about 7.9° (about 8 days after the northward equinox). |
| 63/355 | 86/355 ≡ 5h 48m 50+50/71s
≡ ≈ .24225352112676 |
86 leap days or 63 leap weeks per 355-year cycle (355 is divisible only by the prime numbers 5 and 71) [cycle has 129661 days, prime divisors are 7, 18523]. For a symmetrical cycle, K = 177, U = 62. The primary calendar season for years 353 - 2118 - 3883 is at an ecliptic solar longitude of about 10.8° (almost 11 days after the northward equinox). The exact decimal value of the fraction 86/355 equals 0.2 followed by 35 repeating digits: |
| 137/772 | 187/772 ≡ 5h 48m 48+96/193s
≡ ≈ .242227979274611 |
187 leap days or 137 leap weeks per 772-year cycle (772 is divisible by 2, 4, 193, and 386, of which 193 is a full reptend prime number) [cycle has 281967 days, prime divisors are 3, 7, 29, 463]. This 772-year cycle contains 10233 anomalistic lunar months. For an almost symmetrical cycle, K = 385 or 386, U = 417. The primary calendar season for years 334 - 2004 - 3674 is at an ecliptic solar longitude of about 13.8° (almost 2 weeks after the northward equinox).
Because 193 is a full reptend prime divisor of 772, the exact decimal value of the fraction 187/772 equals 0.24 followed by 193-1=192 repeating digits: 0.24 2227979274 6113989637 3056994818 6528497409 3264248704 6632124352 3316062176 1658031088 0829015544 0414507772 0207253886 0103626943 0051813471 5025906735 7512953367 8756476683 9378238341 9689119170 9844559585 49... |
| 559/3150 | 109/450 ≡ 5h 48m 48s
≡ ≡ .242... |
Leap week adaptation of the "Revised" Julian (New Orthodox) mean year, with 559 smoothly spread leap weeks per 3150 years = 7 × 450 years, which is half of the New Orthodox cycle that has 118 leap days per 900 years (3150 is divisible by 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 25, 30, 35, 42, 45, 50, 63, 70, 75, 90, 105, 126, 150, 175, 210, 225, 315, 350, 450, 525, 630, 1050, and 1575, of which only 2, 3, 5 and 7 are prime) [cycle has 1150513 days, prime divisors are 7, 13, 47, 269]. Compare with the ragged equinox alignment of the Revised Julian calendar (two pages), and the RJiso leap rule (next row below), neither of which have smoothly spread leap years. The primary calendar season for years 334 - 2004 - 3674 is at an ecliptic solar longitude of about 13.7° (almost 2 weeks after the northward equinox). |
| RJiso | 109/450 ≡ 5h 48m 48s ≡ .242... | Same mean year as the Revised Julian calendar shown in the row above, with 559 leap weeks per 3150-year cycle, but starting the year on the Monday that is closest to Revised Julian New Year Day. It is a leap year only if the corresponding Revised Julian year starts or ends on a Thursday. Note the ragged edge of the equinox moments distribution, due to the non-uniform spread of Revised Julian leap years (intervals of 4 or 8 years, whereas uniform spread would use intervals of 4 or 5 years). Also note that due to its shorter mean year length the RJiso leap rule will retain satisfactory equinox alignment for about two thousand years longer than the ISO standard leap rule. RJiso Sym454 dates match ISO Sym454 dates from the 13th century until the year 2809. |
| 74/417 | 101/417 ≡ 5h 48m 46+86/139s
≡ ≈ .24220623501199 |
101 leap days or 74 leap weeks per 417-year cycle = half of the Brij Vij 834-year cycle (417 is divisible only by the prime numbers 3 and 139) [cycle has 152306 days, prime divisors are 2, 7, 11, 23, 43]. This 417-year cycle contains 5597 draconic lunar months. For a symmetrical cycle, K = 208, U = 62. The primary calendar season for years 334 - 2004 - 3674 is at an ecliptic solar longitude of about 15.7° (almost 16 days after the northward equinox).
The exact decimal value of the fraction 101/417 has 46 repeating digits: |
| 159/896 | 31/128 ≡ 5h 48m 45s
≡ |
Same mean year as a leap day calendar having 31 leap days per 128-year cycle and as the Bonavian leap week calendar, using 159 smoothly spread leap weeks per 896-year cycle = 7 × 128 years (896 is divisible by 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224 and 448, of which only 2 and 7 are prime) [cycle has 327257 days, prime divisors are 7, 46751]. This 896-year cycle contains 11082 synodic lunar months. For an almost symmetrical cycle, K = 447 or 448, U = 479. The primary calendar season for years 1670 - 3340 - 5010 is at an ecliptic solar longitude of about 20.6° (almost 3 weeks after the northward equinox). |
| Leap Key | Mean Year | Description and Comments (southward equinox = ecliptic solar longitude 180°) |
|---|---|---|
| SE100 | actual equinoctial year | The "gold standard" for the Southward Equinox, taking the New Year moment as 100+1/4 days after the astronomical equinox. Has small variations (±15 minutes) in the equinoctial year length, mainly due to gravitational interactions with Moon, Venus, and Jupiter. Keeps the equinox aligned on the 266th calendar day, and has the minimum possible "equinox jitter" (for a leap week calendar) of ±3 days. Without the extra +1/4 day offset, the equinox would align on the 266th day after 06:00h in about 3/4 of years, and on the 267th day before 06:00h in about 1/4 of years. |
| MSE100 | mean equinoctial year | The "secondary gold standard" for the Southward Equinox, taking the New Year moment as 100+1/4 days after the mean equinox. Better regulated than SE100, above, eliminating short-term fluctuations in the equinoctial year length, but the polynomials employed for calculating the mean equinox moment are valid for only ±3 millennia from the present era. Kalendis extends the valid range by using SE100 and averaging ±50 years centered on the target year. Keeps the mean equinox aligned on the 266th calendar day, and has the minimum possible "mean equinox jitter" (for a leap week calendar) of ±3 days. Without the extra +1/4 day offset the mean equinox, would align on the 266th day after 06:00h in about 3/4 of years, and on the 267th day before 06:00h in about 1/4 of years. |
| LASEY | Linear Approximation to the Southward Equinoctial Year |
For about 10 millennia prior to 3000 BC, the mean Southward Equinoctial year length could have been approximated with a mean year of about 365 days 5 hours 50 minutes 35 seconds. From 3000 BC to 5200 AD the equinoctial year has been getting shorter and will continue to get shorter in almost linear fashion by almost 5/2 seconds per century. After 5200 AD the mean Southward Equinoctial year length will level out at about 365 days 5 hours 47 minutes 13 seconds (±15 seconds), again for about 10 millennia.
Click here for charts depicting the LASEY approximation |
| 11/62 | 15/62 ≡ 5h 48m 23+7/31s
≡ ≡ .2 4193548387 09677... |
15 leap days or 11 leap weeks per 62-year cycle (62 is divisible only by the prime numbers 2 and 31) [cycle has 22645 days, prime divisors are 5, 7, 647]. Selected by Josef Šuráň to approximate what he called the "mean tropical year" for his leap week calendar reform proposal, as published in Vistas in Astronomy 1998; 41(4): 493-506.
Such a 62-year cycle can serve as an excellent approximation to the Southward Equinoctial mean year from an epoch of 1800 AD until at least 2800 AD. With 15 leap days per cycle it would have been an optimal and simple choice for the French Republican calendar. For an almost symmetrical cycle, K = 30 or 31, U = 17. The secondary calendar season for years 630 - 2205 - 3780 is at an ecliptic solar longitude of about 178° (about 2 days before the southward equinox). [The primary calendar season for this cycle is almost mid-way between the northward equinox and north solstice, at an ecliptic longitude of about 42.5°.] Six serial repeats of this cycle (62×6=372 years) contain 4601 synodic or 4973 sidereal lunar months. |
| Leap Key | Mean Year | Description and Comments (south solstice = ecliptic solar longitude 270°) |
|---|---|---|
| SS10 | actual solstitial year | The "gold standard" for the South Solstice, taking the New Year moment as 10+1/4 days after the astronomical solstice. Has small variations (±15 minutes) in the solstitial year length, mainly due to gravitational interactions with Moon, Venus, and Jupiter. Keeps the solstice aligned on the 356th calendar day, and has the minimum possible "solstice jitter" (for a leap week calendar) of ±3 days. Without the extra +1/4 day offset, the solstice would align on the 356th day after 06:00h in about 3/4 of years, and on the 357th day before 06:00h in about 1/4 of years. The 10+1/4 day offset could be omitted if it were acceptable to take the moment of the solstice as the New Year moment. |
| MSS10 | mean solstitial year | The "secondary gold standard" for the South Solstice, taking the New Year moment as 10+1/4 days after the mean solstice. Better regulated than SS10, above, eliminating short-term fluctuations in the solstitial year length, but the polynomials employed for calculating the mean solstice moment are valid for only ±3 millennia from the present era. Kalendis extends the valid range by using SS10 and averaging ±50 years centered on the target year. Keeps the mean solstice aligned on the 356th calendar day, and has the minimum possible "mean solstice jitter" (for a leap week calendar) of ±3 days. Without the extra +1/4 day offset, the mean solstice would align on the 356th day after 06:00h in about 3/4 of years, and on the 357th day before 06:00h in about 1/4 of years. The 10+1/4 day offset could be omitted if it were acceptable to take the moment of the mean solstice as the New Year moment. |
| LASSY | Linear Approximation to the South Solstitial Year |
For about 10 millennia prior to 1200 AD, the mean South Solsticial year length could have been approximated with a mean year of about 365 days 5 hours 49 minutes 47 seconds. From 1200 AD to 9200 AD the solstitial year has been getting shorter and will continue to get shorter in almost linear fashion by about 9/4 seconds per century. After 9200 AD the mean South Solsticial year length will level out at about 365 days 5 hours 46 minutes 47 seconds (±20 seconds), again for about 10 millennia.
Click here for charts depicting the LASSY approximation |
| 30/169 | 41/169 ≡ 5h 49m 20+160/169s
≡ ≈ .24260355 |
41 leap days or 30 leap weeks per 169-year cycle (169 is divisible only by the prime number 13) [cycle has 61726 days, prime divisors are 2, 7, 4409]. For a symmetrical cycle, K = 84, U = 62. Although its mean year is too short for the present era, the mean South Solstitial year is getting progressively shorter, so this cycle could serve as an excellent approximation to the South Solstice from now until beyond the year 4500 AD. The secondary calendar season for years 372 BC - 1488 - 3348 is at an ecliptic solar longitude of about 230° (about mid-way between the southward equinox and the south solstice).
The exact decimal value of the fraction 41/169 has 78 repeating digits: |
| 71/400 | 97/400 ≡ 5h 49m 12s
≡ |
Same mean year as Gregorian calendar, but allocating 97 leap days or 71 leap weeks at intervals that are as smoothly spread as possible per 400-year cycle (400 is divisible by 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, of which 2 and 5 are prime) [cycle has 146097 days, prime divisors are 3, 7, 773]. For an almost symmetrical cycle, K = 199 or 200, U = 231. Although its mean year is too short for the present era, the mean South Solstitial year is getting progressively shorter, so this cycle could serve as a fairly good approximation to the South Solstice from now until beyond the year 6000 AD. The secondary calendar season for years 372 - 2232 - 4092 is at an ecliptic solar longitude of about 243° (almost 4 weeks before the south solstice). |
| Leap Key | Mean Year | Description and Comments (Besselian New Year moment = ecliptic solar longitude 280°) |
|---|---|---|
| BNY | actual Besselian year | The "gold standard" for the Besselian Year, taking the New Year moment as the moment when Sun reaches an ecliptic longitude of 280°. Has small variations (±15 minutes) in the Besselian year length, mainly due to gravitational interactions with Moon, Venus, and Jupiter. Keeps the average Besselian New Year moment aligned on the calendar New Year Day. The BNY moment is closely related to the South Solstice, occurring about 10 days after the solstice. |
| LABY | Linear Approximation to the Besselian Year |
For the 10 millennia prior to 1600 AD, the mean Besselian year length could have been approximated with a mean year of about 365 days 5 hours 49 minutes 44 seconds. From 1600 AD to 9700 AD the mean Besselian year will get shorter in almost linear fashion by about 12/5 seconds per century. After 9700 AD the mean Besselian year length will level out at about 365 days 5 hours 46 minutes 46 seconds (±18 seconds), again for about 10 millennia.
Click here for charts depicting the LABY approximation |
| 71/400 | 97/400 ≡ 5h 49m 12s
≡ |
Same mean year as Gregorian calendar, but allocating 97 leap days or 71 leap weeks at intervals that are as smoothly spread as possible per 400-year cycle (400 is divisible by 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, of which 2 and 5 are prime) [cycle has 146097 days, prime divisors are 3, 7, 773]. For an almost symmetrical cycle, K = 199 or 200, U = 231. Although its mean year is too short for the present era, the mean Besselian year is getting progressively shorter, so this cycle could serve quite well from the present era until beyond the year 5000 AD. The secondary calendar season for years 372 - 2232 - 4092 is at an ecliptic solar longitude of about 243° (almost 4 weeks before the south solstice). |
| Leap Key | Mean Year | Description and Comments |
|---|---|---|
| 7/39 | 10/39 ≡ 6h 9m 13+11/13s
≡ |
10 leap days or 7 leap weeks per 39-year cycle (39 is divisible by 3 and 13) [cycle has 14245 days, prime divisors are 5, 7, 11, 37]. The mean year of this cycle is an excellent approximation to the mean sidereal year, useful for sidereal calendars, for example calendars that are intended to maintain average long-term alignment relative to the positions of the constellations of the zodiac. This cycle contains 517 anomalistic lunar months. For a symmetrical cycle, K = 19, U = 28. |
This page updated January 22, 2010 (Symmetry454) = January 25, 2010 (Gregorian)
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