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The Extended-Range Secular Calendar:
A (Semi)Formal Specification

"If you fear the number 13, or the number 666, your fear is rational. But fearing the number π (pi) is just irrational."
~ Popular math joke; author unknown

This page presents a more-or-less formal specification of the Extended-Range Secular Calendar, with commentary in a light blue font. To keep this page from running on at length, this specification describes in only a general way the epoch of the calendar, the ranges in which the calendar is effective, and its complex leap-year system, without expending much effort to justify these features, or to elucidate their origins. Readers who seek a deeper explanation of the aforementioned features should also turn to the webpage titled Ranges, Epoch, and Leap Years in the XRS Calendar, as soon as these features are made available to the public.

Epoch; Year Zero and Day Zero; Eras

1. The XRS Calendar Epoch is defined as midnight UTC, Saturday, 4967 BCE April 30 of the Proleptic Julian Calendar. (The year 4967 BCE is equivalent to the astronomical year -4966.) In the XRS Calendar this date is equivalent to Monday, March 1 of the XRS year 0, and is also deemed day 0 of the calendar. Unlike the Julian and Gregorian calendars that preceded it, the XRS Calendar incorporates a zero day and a zero year by design, and is fully and explicitly mindful of astronomical years.

2. The XRS Calendar divides the continuum of time into two eras: the Post-epochal Era, and the Pre-epochal Era. The Post-epochal Era begins at the instant 0 March 1.00000 (or XRS day 0.00000), while the Pre-epochal Era concludes with the instant -1 Intercalary 2.99999 (or XRS day -0.00001). (For those who haven't yet experimented with the XRS Calendar demo app, the meanings of these phenomena should become clearer once you've perused more of this specification.)

Comments: It's an astronomical fact that solar longitudes can be determined fairly accurately as far back as the astronomical year -5000, with an uncertainty estimated to be only about 18 arcminutes in either direction. Accordingly, setting the XRS Calendar epoch at right around -5000 is advantageous because it predates the epochs of virtually every calendar still in use today, including the Hebrew Calendar, and even predates the epoch of the Julian Period by more than 250 years. With the epoch set as far back as this, we can also reference nearly every significant event in human history using positive year-numbering. As we do so, we can be quite confident that an actual occurrence of the northward equinox took place on 4967 BCE April 30.

It cannot be emphasized too strongly that the XRS Calendar epoch does not depend in any way on historical, human-made events. Rather, the choice of epoch is determined by the calendar's leap-year system, which awaits further explanation on this webpage. In consequence, the XRS Calendar pays no regard whatsoever to the "Anno Domini" year-numbering system (i.e., the system that has culminated in A.D. year 2016 as of this writing). This is in keeping with the concept of making a fresh start, and designing a calendar without unnecessary presuppositions and biases, particularly those of a religious or cultural nature. Additionally, the decision to incorporate a zero year is very deliberate: it's intended to undo the confusion caused by the "Anno Domini" calendars' immediate transition from 1 B.C. to 1 A.D., without a zero year—a terrible oversight that has forced astronomers to assign 0 to 1 B.C., -1 to 2 B.C., -2 to 3 B.C., and so on.

Julian and Epochal Day Numbers

1. The XRS Calendar is built upon the Julian day-numbering system used by astronomers. At the same time, the XRS Calendar establishes its own system of day-counting called Epochal days, which is simply an arithmetic offset of the set of Julian days. By definition, Epochal day 0 is equivalent to Julian day -92654.5, which coincides with midnight UTC, 4967 BCE April 30—the epoch of the XRS Calendar. Converting Julian day numbers to Epochal day numbers, and vice versa, is simple and straightforward: given an instant in time t, and expressing both the Julian day and Epochal day as functions of this instant, we have Epochal day(t) - Julian day(t) = 92654.5

Comments: Despite their apparent similarity, it's important to note the conceptual and philosophical differences between these systems of day-numbering. In the XRS Calendar, day-counting consists very simply of real numbers arranged on a line extending infinitely in both directions. There is no concept of periodicity involved, as in Joseph Justus Scaliger's conception of the Julian day-numbering system in 1583, with its periods of 7,980 years. As such, Epochal day-numbering does not depend on human-made cycles such as the Roman indiction cycle of 15 years, nor does it depend on preconceived notions such as the seven-day week, and so on. Moreover, zero in the Epochal day-numbering system is made to coincide with the epoch of the XRS calendar. This is quite unlike the epoch of the Julian Period, which has nothing to do with the epoch of the Julian Calendar, but which is instead based on the time when the first day of each of the Roman indiction, Metonic, and solar cycles last coincided, in 4713 BCE.

Note also—and this is no small matter indeed—that Epochal day-numbering establishes the beginning of each day at midnight UTC, not noon. This is intended to eliminate the ubiquitous—and consummately irritating—half-day offset imposed by Scaliger's system.

Lastly, as is historically the case for Julian days, the XRS Calendar's day-numbering system lends itself to special variants resulting either from truncating Epochal day numbers, or exploiting rounded XRS years—either of which may suit various technical purposes. See the section near the bottom of this page, Epochal Day Variants, for more information.

Differences and similarities aside, both day-numbering systems are indispensable for converting Julian and Gregorian calendar dates into their XRS calendar equivalents, and vice versa. First, the Julian or Gregorian date is converted to its equivalent Julian day number; then the Julian day number is converted to its equivalent Epochal day number; finally, the Epochal day number is translated into the equivalent date on the XRS Calendar. The process of converting XRS calendar dates into their Julian or Gregorian equivalents is exactly the reverse of the above.


1. The beginning of each XRS Calendar year approximates to an occurrence of the northward equinox such that this occurrence will fall on either the first or second day of the year1—that is, either on Monday, March 1 or Tuesday, March 2. In the period from -4999 through +8000 (astronomical years as measured on the Proleptic Julian and Gregorian calendars), special care has been taken to ensure that no occurrence of a northward equinox regresses to the end of the previous year—which, if it did, would result unfortunately in the previous year spanning two consecutive occurrences of the northward equinox.

Comments: In an age where electronic, digital computing holds sway, and complex astronomical calculations can be performed in milliseconds, the time is ripe for a new world calendar that anchors the year (albeit somewhat loosely) to a definite point determined astronomically in space, and to which the entire world would have good reason to give its agreement. The XRS Calendar joins a very small group of extant (and extinct) calendars2 that define the year with reference to one of the equinoctial or solsticial points. It must be emphasized, however, that unlike most of the calendars in this group, the XRS Calendar is not an astronomical calendar, but rather, an arithmetical one that approximates to the northward equinox through pure arithmetic rather than through strictly astronomical formulae.

The proximity of the northward equinox to the beginning of the XRS year necessitates an especially low tolerance for seasonal drift over periods spanning centuries and millennia, and for seasonal wobble from one calendar year to the next. This is one of the reasons why the XRS Calendar deploys a new and sophisticated leap-year formula that constitutes a radical departure from that of the Gregorian Calendar. For more information about the new formula, see the section below titled Leap Years and Common Years, and also the Ranges, Epoch, and Leap Years page of this site.

Months; Intercalation

1. The XRS Calendar is an intercalated calendar comprising 13 identical months of 28 days each. Because these months add up to only 364 days, there is an intercalary period at the end of each year, called simply Intercalary, that lasts one day during common years, and two days during leap years.

2. The 13 months, in order of their appearance throughout the year, are named March, April, May, June, July, August, September, October, November, December, Undecimber, Duodecimber, and Tredecimber. Intercalary comes at the end of each year, right after Tredecimber and just before the start of the following year on March 1. Although it is not a month per se, Intercalary is represented internally by the calendar as a brief "month" lasting one or two days. These days are named Intercalary 1 and (in leap years) Intercalary 2, and are not associated with any month or any day of the week.

Comments: As stated in the set of design goals for this calendar, only years, days, and seasonal points have any basis in reality. Everything else—months, weeks, and weekdays in particular—is just a social construct at best, and pure fiction at worst. The decision to intercalate the XRS Calendar, and divide the bulk of the year into 13 equal segments, is a political one that readily affords a way to make the months of the calendar not only permanent, but identical to one another. Obviously, the most controversial consequence of this decision is that the seven-day sabbatical cycle, as spelled out in the Fourth Commandment of the Old Testament, will be preserved only during the first 51 weeks of the year, while in the 52nd and final week the cycle will be broken by the intercalary period. This issue warrants more extended discussion than is either feasible or desirable here, so we will defer discussion to the Objections and Replies page of this website.

Second, note that in the XRS scheme of things, March is no longer the third month of the year, but the first; April is no longer the fourth month, but the second; and so on. As a result, the calendar restores the months of September, October, November, and December to their original places as established in the Roman Calendar: September really is the seventh month, October really is the eighth month, November the ninth, and December the tenth month of the year. Gone are the months of January and February; in their place are three new months: Undecimber, Duodecimber, and Tredecimber. Why these names? Well, it happens that in the Latin language, the words septem, octo, novem, decem, undecim, duodecim, and tredecim translate into the numbers seven through thirteen. The months of Undecimber, Duodecimber, and Tredecimber merely follow the trend already established by the months of September through December. (But see the section near the bottom, Semantic Freedom in the XRS System.)

Third, even though it contains 13 months of 28 days each, the XRS Calendar is not a lunar calendar, nor should it be regarded as such. The decision to make each month 28 days long is not intended to follow the lunar archetype. It is instead the result of applying pure arithmetic: 13 months/year ✕ 28 days/month = 364 days/year, which approximates nicely to the length of a natural solar year—currently about 365 days, five hours and 49 minutes—and makes the intercalary period as brief as possible mathematically. (As we'll see in time, the XRS Calendar has several precursors that follow the same model, albeit in different ways. For want of a better term, we shall henceforth refer to this class of calendars as 13 ✕ 28 calendars.)

One final comment about the way in which the XRS Calendar is intercalated. When the decision was made to incorporate leap years in the old Julian Calendar—a decision dating back to 46 BCE—the intercalary day was interjected within the month of February. This day was called the bissextile, and was taken to occur on February 24, chiefly because the Julian Calendar had a system of weeks and weekdays that did not resemble in any wise the seven-day weekly cycle that we're familiar with today. Over time the leap-year day evolved to become February 29. At that time it made sense to append the leap-year day to February's end, because February was the last month of the year in the old Julian Calendar. Then, at some succedent point in the calendar's history, the beginning of the year was changed from March to January. This shunted the leap-year day awkwardly to a point just some 59 days in front of the beginning of the year.

The only way the calendarists of that time could have done worse than this would have been to make February the first month of the year. Why? As Dr. Bromberg implies in his fourth criticism of the Gregorian Calendar, when calendars are implemented in a computer program, the days of the year are usually numbered consecutively from 1 to 365 or 366. In a common year of the Gregorian Calendar, March 1 would be reckoned as day number 60 of that year. However, during leap years March 1 becomes day 61. Even worse, in a leap year every day from March 2 through December 31 must be incremented accordingly. This causes massive and unnecessary calendrical complexity—the kind of complexity that must be dealt with by the program that implements the calendar on a computer. When the Gregorian Calendar was first proposed in the 16th century CE, no measure was even considered to correct this defect (which in turn illustrates the danger of hewing too closely to a prevailing calendar design while proposing a new one). We can forgive the counsel of Caesar's calendrical circle, and similar experts within the 16th-century Vatican Council, for not thinking too much about digital computing, but on the other hand this defect in both the Julian and Gregorian calendars is just one of many unfortunate circumstances that cry out for solution by way of a modern computer and a fresh, new calendar design. The XRS Calendar avoids this circumstance entirely, by placing its intercalary days at the end of every year, not before.

Weeks and Weekdays

1. The XRS Calendar retains the seven-day week, as well as the names of the days of the week, of the Gregorian and Julian calendars. (Again, please see the section near the bottom, Semantic Freedom in the XRS System, in this regard.) However, the days of the week are arranged somewhat differently. Each week in the XRS Calendar begins on a Monday, and ends on a Sunday, effectively putting the two "weekend days"—Saturday and Sunday—together consecutively, at the end of each week.

2. Each of the 13 months begins on Monday the 1st, and ends on Sunday the 28th. By applying the function f(x) = x modulo 7 to each day of the month, we see that Mondays always have a (modulo 7) of 1; Tuesdays always have a (modulo 7) of 2; Wednesdays always have a (modulo 7) of 3; and so on through Sunday, which always has a (modulo 7) of 0. To put the point in somewhat less technical terms, any date numbered 1, 8, 15, or 22 will always fall on a Monday; any date numbered 2, 9, 16, or 23 will always fall on a Tuesday; any date numbered 3, 10, 17, or 24 will always fall on a Wednesday; and so on through Sunday, which is never assigned to any date except the 7th, 14th, 21st, and 28th.

3. Intercalary 1 is always the 365th day of any year—common or leap—and always occurs the day after Sunday, Tredecimber 28; Intercalary 2 occurs only during leap years, and is always the 366th day of the leap year in question. After the intercalary period of one or two days, the new year begins on Monday, March 1.

Comments: The relation of weeks to months in the XRS Calendar, specified in Item 2 above, makes this calendar not only fixed, or permanent. It also makes every month immutable, and identical to every other except in name only—the latter a feat that neither the Hanke-Henry nor the Symmetry leap-week calendars, nor the World Calendar, can accomplish. Note too that every week of the year is identical to every other week, and every week is integral: there is no month that contains only part of a week. By making every month identical to every other month, and every week identical to every other week, the XRS Calendar, if adopted on a wide scale, stands to save billions of dollars in scheduling costs, as well as untold natural resources consumed by all manner of paraphernalia relating to a calendar that is not permanent and immutable—for instance, the Gregorian Calendar.

Secondly, the days labeled Intercalary 1 and Intercalary 2 are never to be understood as belonging to any regular month (not even Tredecimber), nor are they to be understood as having a "day of the week," whether it's the Monday or Tuesday following the month of Tredecimber. They are intercalary dates, meaning they stand completely outside the regular monthly and weekly cycles. More than any other factor, it is the intercalary dates that achieve perpetuity within the XRS system.

Note that retention of the seven-day week is not motivated in the slightest by the sabbatical cycle spelled out in the Fourth Commandment of the Old Testament. Once again, the decision is based on pure arithmetic: seven is an even divisor of 364. This is very likely the real reason why so many calendars around the world are founded on a seven-day weekly cycle. There is no reason in principle why the XRS Calendar could not have been built upon weeks of other lengths. For instance, the weeks could have been made five days long, or six days, or even ten days, and apportioned into twelve identical months of 30 days each. It's fairly clear that such solutions would have been far less than optimal, for the intercalary period would have been five or six days long; a week of five days, or even six days, is rather dizzying; and a ten-day week might have made for a very tedious workweek indeed!3


1. The XRS Calendar year is divisible into quarters as follows. The first quarter begins Monday, March 1 and ends Sunday, June 7; the second quarter runs Monday, June 8 through Sunday, September 14; the third quarter, from Monday, September 15 through Sunday, December 21; and the fourth quarter from Monday, December 22 through Intercalary 1 in common years, and through Intercalary 2 in leap years.

Comments: Detractors of 13-month calendars commonly insist that these calendars do not easily or conve- niently admit of division into quarter-years. This belief is pure hogwash, born largely of the unreasoning desire to keep monthly periods integral within each quarter. As it happens, the quarters in the XRS Calendar are of more uniform length than even those in the Gregorian Calendar! When one assays the lengths of XRS quarters, from the first through the fourth, one will find them to be 91, 91, 91, and 92 days respectively in common years, and 91, 91, 91, and 93 days in leap years. Contrast with the Gregorian Calendar, which produces quarters of 90, 91, 92, and 92 days in common years, and 91, 91, 92, and 92 days in leap years.

Moreover, in the Gregorian Calendar the quarters can begin and end on any day of the week, yielding unequal ratios of workdays to the number of days in the quarter overall (that is, before non-working holidays are accounted for). For example, in a typical common Gregorian year the ratios for the four quarters amount to 64/90, 65/91, 66/92, and 66/92—or roughly 0.7111, 0.7143, 0.7174, and 0.7174. In contrast, since every week is integral throughout the XRS calendar year, these ratios hold steady at 0.7143 through the first three quarters, while the ratio differs only slightly in the fourth quarter: about 0.7065 in common years, and 0.6989 in leap years. Even more important, every XRS quarter has the same number of workdays as every other: 65 (before non-working holidays are taken into account).

Thus would one hope—with respect to the XRS Calendar at least—that we've put an end to this obnoxious dispute over quarterly division of the year.

Seasonal Points

1. If adoption of the XRS Calendar were to become widespread in the future, many people—astronomers and agronomists especially—would be concerned to know when the seasonal points fall in terms of XRS Calendar dates. Over the next 600 years or so, the northward equinox falls on March 1 nearly 99 percent of the time. Only in the years 7171, 7365, 7398, 7431, 7464, 7530, and 7563 is the northward equinox anticipated to occur just after midnight UTC on March 2 instead. The remaining seasonal points fall as follows: the northern solstice, on June 9 or 10; the southward equinox, on September 19 or 20; and the southern solstice, on December 25 or 26.

Comments: As a handy bonus, this website comes with a 600-year almanac 4 that presents the dates and times of the equinoxes and solstices for all XRS years from 6971 through 7570 (2005 through 2604 Gregorian). Each timing is presented in two formats: a decimal format, rounded to the nearest thousandth of a day (86.4 seconds); and the conventional 24-hour clock format, rounded to the nearest minute. The estimated uncertainty of these timings ranges from zero in the year 6971 to about ±13.7 minutes out to 7570. (You may have noticed that the range of timings for each seasonal point is much narrower than the corresponding range in the Gregorian Calendar. This phenomenon is explained in the section just below, on leap years in the XRS Calendar.)

Leap-Year Formulas

The Extended-Range Secular Calendar is arguably the most sophisticated and complex arithmetical calendar ever proposed or implemented, and even if it's not, it is definitely the world's most sophisticated and complex arithmetical solar calendar. Because its leap-year system is so complex, we will discuss that system only in a general way here, and provide some examples of the leap years that it generates.

Unlike virtually every other calendar known to humankind, the XRS Calendar does not employ a single, static leap-year rule that doesn't change over time. On the contrary, the XRS leap-year system is dynamic, changing in stepwise intervals to accommodate the changing length of the mean northward equinoctial year (MNEY) as the calendar traverses the period from -94167 through 104045 in XRS years [-99132 through 99079 in the Proleptic Julian and Gregorian calendars; henceforth we shall just enclose these years in square brackets to distinguish them from the equivalent XRS years]. The XRS system divides this period into 88 segments, and deploys an aggregation of 74 different leap-year rules, one for each segment. (Some rules are deployed more than once.)

In the XRS system, leap years occur four calendar years apart in the majority of cases, while in the remaining cases the leap years occur five calendar years apart. In the distant past end of the spectrum, the "anomalous" five-year intervals seldom appear, because at that time the length of the MNEY is estimated to be quite close to 365.25 days (and actually exceeds 365.25 days from about -85352 to -80712 [-90317 to -85677]). However, as the MNEY shortens over time, the five-year intervals tend to appear more and more frequently. By the time we arrive at the distant future end of the spectrum, the five-year intervals are often designed to appear in one out of three instances, with leap years apportioned in cycles of 13 years as follows: < 4, 4, 5 >. The objective of the XRS Calendar's leap-year system is to distribute the five-year intervals as smoothly as possible throughout the entire spectrum, while holding as closely as possible to the projected length of the MNEY for any given segment within that spectrum.

Comments: The first and most important thing to remember about leap years is this: In nature there is no such thing as a leap year! All leap-year systems are human devices5 intended to reconcile the irrational length of the natural yearly cycle with the human desire for integral relations between days and years. As human artifice goes, the XRS Calendar's leap-year schema certainly is complex—at least sufficiently complex that it cannot be reduced to one of those enviable one-line rules enjoyed by the Julian and Gregorian calendars, and by some leap-week reform calendars as well.6 Yet it has never been a goal of this calendar to make its leap-year rule(s) easy to articulate, or to remember. For one thing, there is nothing particularly easy about the workings of celestial mechanics. For another, it has always been one of the paramount operative principles of the calendar to embrace complexity only when there is no better alternative. This schema is the best alternative available for keeping the beginnings of calendar years closely aligned with natural occurrences of the northward equinox.

But why would this leap-year schema be the best alternative, if it's so complex? One quick-and-vague reply is that this schema applies its corrective measures more frequently, more efficiently, and more smoothly, than does the static Gregorian leap-year rule. In the region circumscribing the present day, for instance, it takes only 33 years (and occasionally, only 29 years) for the XRS Calendar to apply virtually the same amount of corrective force that the Gregorian Calendar requires 400 years to achieve! Another is that, in the Gregorian Calendar, the northward equinox can fall within a range of about 53.2 hours, which further entails its occurrence on any of three calendar dates in the present era: March 19, 20, or 21. In contrast, the XRS system narrows this range to just under 24 hours, making it possible for the northward equinox to fall on March 1 in roughly 98.8 percent of all XRS years from 5966 through 7966 [1000 through 3000 CE]. Outside this period, the equinox will fall on either March 1 or March 2 for the years 254 through 5965 [-4712 through 999] and 7967 through 12886 [3001 through 7920].7 Most important of all, the multi-rule XRS leap-year system is designed with an eye to preventing seasonal drift over periods lasting many thousands of years—a highly important consideration inasmuch as we seek to prevent the northward equinox from creeping backward in the calendar, causing two occurrences of the equinox in the same year.

With all that being said, it must be emphasized that no one needs to understand the theory behind the XRS Calendar's leap-year system in order to use that system. (Would you expect someone to understand fully the engineering principles behind the internal-combustion engine as a prerequisite for obtaining a driver's license?) Even the most hideously complex software systems imaginable—say, a program for managing a world banking system, which can easily absorb millions of lines of code—could be adapted to the XRS leap-year system through a simple array containing only 20 to 40 integers, and/or a small amount of accessory code. There is no need for the program to "understand" or assimilate the entire XRS system. Similarly, for most lay persons it suffices merely to enumerate some XRS leap years in the more-or-less-immediate past and future. Just below is such an enumeration: it's a tabulation of leap years circumscribing the present day, from 6812 through 7105 [equivalent to the period from 1846 CE March 20 through 2139 CE March 19]. Note as well how this table groups the leap years into subcycles of 33 and 29 years:

                                                               6812,    ----------
       6816,   6820,   6824,   6828,   6832,   6836,   6840,   6845,    <33 years>
       6849,   6853,   6857,   6861,   6865,   6869,   6873,   6878,    <33 years>
       6882,   6886,   6890,   6894,   6898,   6902,   6906,   6911,    <33 years>
       6915,   6919,   6923,   6927,   6931,   6935,   6939,   6944,    <33 years>
       6948,   6952,   6956,   6960,   6964,   6968,   6972,   6977,    <33 years>
       6981,   6985,   6989,   6993,   6997,   7001,   7005,   7010,    <33 years>
       7014,   7018,   7022,   7026,   7030,   7034,   7038,   7043,    <33 years>
       7047,   7051,   7055,   7059,   7063,   7067,   7071,   7076,    <33 years>
       ----    7080,   7084,   7088,   7092,   7096,   7100,   7105     <29 years>

Conventions for Naming Dates and Times

1. XRS Calendar dates are typically written in big-endian format, meaning the most significant terms are spelled out first, and the least significant last. This entails that the following be specified in this order: (a) the sign of the year - or + (for positive years, the plus sign is usually omitted); (b) the year; (c) the month; and (d) the day of the month.

2. Options include prepending the date with the day of the week (where applicable—remember, intercalary dates are not associated with any day of the week!), and appending the date with a decimal time (about which, much more on this webpage). For example, "-0031 Tredecimber 17" and "Wednesday -0031 Tredecimber 17.875" are acceptable alternatives, as are "6007 July 7", "Sunday +6007 July 7", and "Sunday 6007 July 7.97153". (On this website, decimal times are almost always distinguished by being rendered in a slightly different color from the rest of the date.) Keep in mind, however, that intercalary dates must never be prepended with any day of the week; in other words, "7033 Intercalary 1" and "-0043 Intercalary 2.33333" are valid, whereas "Monday 7033 Intercalary 1" and "Tuesday -0043 Intercalary 2" are absolutely not (for there are no such dates, by definition).

3. Another option for naming dates in the XRS Calendar consists of something called an ISO 8601 semblance. This is a date format that more or less resembles the format established by ISO 8601. But it is only a semblance, because it has not (yet) been sanctioned by the ISO. Examples of ISO 8601 semblances include "-0046.04.13" (equivalent to Saturday -0046 June 13) and "+7107.13.04.75" (equivalent to Thursday 7107 Tredecimber 4.75).

4. Writing an intercalary date as an ISO 8601 semblance yields a brief, simple result. Examples include "+6997.i1", "-1045.i2", "-0018.i1.375", "+8017.i2.86734", and the like.

Comments: Note that, unlike ISO 8601 dates, the years, months, and days of an ISO 8601 semblance are not separated by dashes, but by periods instead. This is done in part to disambiguate separators from minus signs prepended to dates in the Pre-epochal Era (although such ambiguity can certainly be resolved in context—just as context will distinguish the periods used to separate date components from the decimal point preceding decimal time). Therefore, use of periods as separators really amounts to nothing more than a personal preference of this author.

Readers can view many more examples of dates written in various XRS Calendar formats by experimenting with the XRS Calendar demo app and the date conversion demo; see the final section of this page, Explore this Calendar!, for more information.

Semantic Freedom in the XRS System

1. In the XRS Calendar the names of months, and days of the week, have been chosen rather expediently for the most part, chiefly because there already exist translations of these names in many other languages. Nevertheless, the calendar shall remain flexible with regard to the names of calendar properties that are constructed socially, not astronomically. Any culture shall be free to substitute its own preferred names for the months and weekdays, just so long as the names convey the cardinal and ordinal relations among themselves as clearly as do the original names.

2. The XRS Calendar also permits options for arraying weeks vertically rather than horizontally, and for writing and reading months and weeks from right to left.

Comments: Although the XRS Calendar strives to be free of cultural bias in principle, it is impossible to achieve this goal perfectly in practice. It may be argued, justifiably, that the XRS Calendar cannot help but be infected by its own peculiar biases: for instance, its nod-of-the-head to the scientific community rather than to the mythical; and its reliance upon Latinate, Eurocentric nomenclature for the better part of it. (Along this vein, it would have been a grotesque mistake to follow the lead of the French philosopher Auguste Comte (1798-1857) and his Positivist Calendar (an important 13 ✕ 28 precursor to this one, devised in 1849), by assigning to the months of the XRS Calendar the names of famous persons in Western European history in the fields of science, religion, philosophy, industry and literature, as Comte had done.) As an antidote to inevitable built-in cultural bias, the real point of permitting semantic freedom in the XRS Calendar is that it's the combination of astronomy and arithmetic underpinning this calendar that matters, not the window-dressing provided by nominatives of one sort or another.

Epochal Day Variants

Although they would not constitute part of the XRS Calendar's design per se, certain variants of the XRS day-numbering system can be created to suit various technical purposes. These variants can result either from reducing Epochal day numbers, or exploiting rounded XRS years. Just below is a table showing some possible variants. The first four are reductions, and may be useful wherever lengthy integers are deemed undesirable; the latter five mark the onsets of rounded XRS years. The entries labeled Year 7000* are variants of a variant, falling not only on a rounded XRS year, but on a somewhat rounded Epochal day as well.

     Variant:        Calculation:      Epoch:                    Equivalent to:

     Day 2400000     EDN - 2400000     Tue 6570 Tredecimber 23   Sun 1605 March 13
     Day 2500000     EDN - 2500000     Wed 6844 Undecimber 3     Fri 1878 December 27
     Day 2550000     EDN - 2550000     Sun 6981 November 21      Thu 2015 November 19
     Day 2600000     EDN - 2600000     Wed 7118 October 10       Wed 2152 October 11
     Year 6000       EDN - 2191454     Mon 6000 March 1          Fri 1034 March 15
     Year 6500       EDN - 2374075     Mon 6500 March 1          Wed 1534 March 11
     Year 7000       EDN - 2556696     Mon 7000 March 1          Mon 2034 March 20
     Year 7000*      EDN - 2556700     Fri 7000 March 5          Fri 2034 March 24
     Year 7000*      EDN - 2557000     Thu 7000 Undecimber 25    Thu 2035 January 18

Prolepsis (and Eventual Adjustment) of the Calendar

The (semi)formal specification of the XRS Calendar is now just about complete. From the above specification, you can readily infer that the XRS Calendar is not only proleptic; it was designed from the get-go to be a proleptic calendar, meaning that it extends just as easily back in time as it does moving forward, pretty much as though the calendar had already existed from Day One. (Well, actually, Epochal day 0! (;-}= ).

Just as for any other proleptic calendar, however, it does not follow strictly that the pattern established by future dates will coincide exactly with the pattern established for dates that extend into the past. For one thing, extrapola- tions into the distant past are always speculative to a greater or lesser degree. We cannot make amends for the crudity of the calendrical systems that held sway back then, nor can we literally travel backward in time to rectify the astronomical observations made in the distant past. Much the same can be said for our extrapolations into the distant future as well. For even though (and because) our astronomical methods are considerably more sophisticated and accurate, we now recognize slight but persistent changes in certain astronomical cycles over time, particularly the period of Earth's rotation about its own axis. Due to tidal friction near the earth's equator, the mean solar day is currently lengthening at a rate of about 1.45 milliseconds per 100 tropical years. Consequently, the length of the tropical year (measured in Terrestrial Time) is decreasing at a rate of approximately 0.53 seconds per 100 tropical years. As if to compound the uncertainty, these rates of change are themselves changing, such that astronomical effects will not be precisely predictable over periods spanning thousands of years.

However the XRS Calendar errs for dates in the distant past, at least the stakes are either very low, or nonexistent. We cannot say the same for dates extending into the distant future. For as long as the human species persists on Earth, there will always be a need to adjust our calendrical calculations to accord with newly-observed phenomena in outer space whenever these conflict with predictions we had made thousands of years prior. As it currently stands, the XRS Calendar's leap-year system embodies several sources of possible error that will make adjusting the calendar for future dates a virtual certainty. This does not mean, however, that the theory underlying the system is necessarily and inherently flawed, for we are simply working with the best information we have on hand at the present time. Those who criticize the calendar's leap-year system on that account are missing the real point of this system: to provide future custodians of the calendar with a blueprint and a methodology for revising its arithmetic to agree with future astronomical observations.

In light of the above, those in future generations who will become custodians of the XRS Calendar should be accorded the freedom to devise new arithmetical leap-year formulas where necessary to avoid such accidents as having two consecutive occurrences of the northward equinox falling within the same year—or even the freedom to convert this calendar to an astronomical one, or a hybrid arithmetical/astronomical calendar, if the situation warrants such a conversion.

Explore this Calendar!

Now that you've studied this specification, you may find the XRS Calendar to be quite foreign to your existing "calendrical mindset"—a mindset that is locked in place through the idiosyncracies, vagaries, and confusions of the Gregorian Calendar. Never fear, for this website provides tools that actually implement the XRS Calendar, and help you better understand how it works! Your next step is to tinker with the XRS Calendar demo app, with which you can actually "step through" the calendar, going day-to-day, or month-to-month, stepping forward and backward in time through the years, centuries, and even millennia. Go ahead, click around, go nuts—you won't break anything! Afterward, play around with the XRS Calendar's surprisingly versatile and full-featured date conversion demo, with which you can convert many Gregorian dates into XRS Calendar dates, and vice versa. Lastly, experiment with the day-of-week conversion demo, a handy tool that lets you see how XRS days of the week change from year to year in relation to days of the week in the Gregorian Calendar.

I am quite confident that once you have explored these three tools, you will no longer find the XRS Calendar to be a strange, foreign, and exotic device, but rather, a simple, orderly, and logical replacement for the Gregorian Calendar. In contrast, the Gregorian Calendar will soon strike you as not only completely illogical, but messy, crazy, senseless, inaccurate, and expensive as well!


1. Nevertheless, after the XRS year 12880, when the length of the mean northward equinoctial year begins to shorten precipitously, it's possible that in some years the northward equinox will fall on March 3 instead. While these possibilities lie safely far into the future, and do relatively little harm from a practical standpoint, they do portend eventual adjustments to the XRS Calendar's leap-year algorithm in the event that occurrences of the northward equinox begin falling much more frequently on March 2 or 3 than on March 1. Cf. the section above, Prolepsis (and Eventual Adjustment) of the Calendar; also, Ranges, Epoch, and Leap Years for illustrative tables and graphs that await publication in the coming months.

2. Two calendars in particular that begin the year at the northward equinox are the modern Iranian Solar Hijri Calendar—one of the most elegant and sophisticated calendars ever devised by humankind—and the Bahá'í Calendar. The Solar Hijri Calendar, established in 1925 (Gregorian), is the descendant of the Jalali Calendar, an astronomical calendar dating back to 11th century Persia; the Bahá'í Calendar was established in 1844 (Gregorian). The French Republican Calendar was another, striking example of an astronomical calendar, the epoch of which was the southward equinox of 1792 (Gregorian). Alas, it didn't last very long; Napoléon abolished the calendar, effectively from January 1, 1806 (Gregorian), scarcely more than twelve years after its official adoption on October 24, 1793 by the Jacobin-controlled National Convention.

3. These are not idle speculations; history bears these claims out. Workers in post-revolutionary France objected vehemently to the aforementioned French Republican Calendar because each 30-day month was divided into three weeks of ten days each, called décades. The fact that these workers had just one day off in ten was a major factor that led to the calendar's downfall. Between 1929 and 1940, the Soviet Union implemented a series of disastrous calendars, beginning with one containing five-day weeks with work schedules rotated to maintain continuous, round-the-clock production. After just two years, the measure failed because family members and friends had different days off work, and because it was nearly impossible to maintain and repair equipment that was in constant use. What's more, machines actually broke down more frequently because they were used by workers who were rotated constantly, and so were not fully familiar with their machinery. So in 1932 a six-day week was implemented, with one common day off. By 1940 the Soviets reverted to the seven-day week, but only to increase productivity, as workers were now laboring 85.7 percent of the time rather than a mere 83.3 percent! (It's not difficult to imagine this presumed 2.86-percent increase in productivity being easily wiped out, and probably shown to be counterproductive in fact, owing to workers' exhaustion, lack of attention, increased injury rates on the job, and so on.)

4. Please see the section titled XRS Calendar Seasonal Almanac: Fair Use and Disclaimer of the Copyrights, Protections, and Terms of Use page for important limitations on fair use of this almanac.

5. Some of you may be thinking, "Aha! Surely there are counterexamples to your notion that leap years are contrived. What about astronomical calendars, where leap years are occasioned naturally through a span of 366 calendar days between successive occurrences of some annual orbital phenomenon—for example, the northward equinox?" But believe it or not, your argument would be unsound. Not because your premise isn't true, but because your premise doesn't acknowledge that the very meaning of "calendar day" is up for grabs! Definitions of "calendar day" are political decisions that vary from culture to culture. Moreover, these decisions depend in no small way on the choice of seasonal point with which to anchor the calendar year—a choice that is in no way mandated by nature. For example, the Iranian Solar Hijri Calendar defines the date of occurrence of the northward equinox as follows: If the equinox falls before noon (Tehran true time) on a particular day, then that day is deemed the first day of the Solar Hijri calendar year. If the equinox falls after noon (Tehran time), then the following day is the first day of the year. Another example is the French Republican Calendar, the establishing decree of which stated: "Each year starts at midnight, with the day when the true autumnal equinox falls for the observatory of Paris." For that matter, an early prototype of the XRS Calendar was actually a hybrid of arithmetical and astronomical calendars; according to that prototype, if the northward equinox occurs between midnight UTC and 23:59:59 UTC of a given date, that date shall mark the first day of the XRS year.

6. But the concision of a leap-year rule does not necessarily make it any easier to wrap one's head around that rule. For example, the leap-year rule of the Symmetry454 Calendar reads: "It is a leap year only if the remainder of (52 × Year + 146) / 293 is less than 52." This rule masks the underlying complexity of Bromberg's leap-year system, which is based on interesting symmetries of leap-year arrangements within spans of 45 and 79 years, and a composite of these spans totaling 293 years.  See this webpage for a more thoroughgoing illustration.

On the other hand, the algorithm for determining leap years in the Hanke-Henry Permanent Calendar turns out to be strangely congruent with another single-sentence rule that states: "It is a leap year only if the corresponding year in the Gregorian Calendar either begins or ends on a Thursday." But this spawns the odd (and, ultimately, undesirable) consequence that one can make quick sense of the Hanke-Henry rule only by referring to the very calendar that it seeks to overthrow! (By the way, although the Symmetry and Hanke-Henry are both leap-week reform calendars, they produce different sets of leap years.)

7. Cf. note 1 above.