# Fixed Arithmetic Calendar Cycle Jitter

Created by Dr. Irv Bromberg, University of Toronto, Canada

## Introduction to Calendar Cycle Jitter

The jitter of a fixed arithmetic calendar cycle is short- to medium-term variation that is due to the inherent arrangement of the cycle, isolated from long term drift (caused by inaccuracy of the cycle mean year or mean month) and from astronomical variations (which span about 10 minutes for a solar calendar or about 28 hours for a lunar calendar).

Many of the concepts and calculations of calendar cycle jitter are due to Karl Palmen of the UK, who is now retired but used to work for the Science & Technology Facilities Council (STFC) at the Rutherford Appleton Laboratory in Oxfordshire, England.

## Solar calendar jitter relative to cycle mean year

An interval of consecutive years contains a whole number of days if it is a leap day calendar, otherwise a whole number of weeks if it is a leap week calendar, or a whole number of months if it is a leap month calendar. This normally differs from the number of days (or weeks or months) in the same number of mean calendar years and so causes the calendar to jitter by a deficit or surplus over the interval. An interval that starts just after an intercalated year and ends just before an intercalated year will have a deficit, especially if it contains intercalated years that are further apart than the average, whereas any interval that starts and ends with an intercalated year will have a surplus, especially if it contains intercalated years that are closer together than the average. Jitter analysis seeks to calculate and/or plot the jitter and thereby find the interval over which the deficit or surplus is largest. If a selected interval has the maximum surplus or deficit then the interval containing the complementary years (the years of the cycle that don’t belong to the selected interval) will have the complementary deficit or surplus, respectively. Exaggerated jitter is unavoidable when a cycle is very long and contains corrective adjustments at long intervals.

Minimum possible solar calendar jitter, which is slightly less than the length of the intercalation unit (day, week, or month), occurs when leap years are distributed at intervals that are as smoothly spread as possible. For a C-year cycle of L leap years, this is simply and easily obtained by having year Y be leap year if and only if the remainder of (LY + K) / C is less than L, where K is any non-negative integer less than C. There are advantages to choosing K so as to distribute the leap years symmetrically within the cycle — for further information, please see my discussion about Smoothly Spread Symmetrical Leap Cycles.

Since the jitter range is the difference between maximum and minimum jitter, the initial value doesn’t matter and for consistency has been chosen to be zero on the y-axis. This causes each repetition of the cycle to start again at zero, regardless of the cycle length (odd or even), cycle symmetry, or smoothness of the distribution of leap years.

Almost all cycles that aren’t symmetrical will have one year other than the first at the mean (odd length cycles) or closest to the mean (even length cycles), where the mean is the average of the maximum and the minimum jitter deviations, recurring for each repetition of the cycle. In extreme cases all of the points except the first per cycle may be above zero, in which case the mean will be close to one-half of the jitter maximum, or all of the points except the first per cycle may be below zero, in which case the mean will be close to one-half of the jitter minimum. The mean and the arithmetic jitter average for symmetrical cycles will be zero.

If the ratio of leap years to the number of years per cycle is reducible then the non-reduced cycle will have more than one year at the average, depending on the reduction that is possible, but the jitter range will be the same. For example, the 900-year Revised Julian leap cycle could be reduced to a 450-year cycle having the same mean year and jitter range. The 900-year cycle has two years at the average, whereas the 450-year reduced cycle has only one year at the average. Similarly, the Alpo Balognian 9000-year cycle has 3 years at the average per cycle, but reduces to a 3000-year cycle having the same mean year and jitter range but only one year at the average. The denominators of the exact mixed fractions for the cycle mean year and jitter range always equal the number of years in the reduced cycle.

If a calendar is intended to approximate an equinoctial or solstitial mean year then it might be preferable to arrange the calendrical calculations to keep that average close to the target, so that the deviations before and after the event will be similar.

### The following table offers links to jitter charts for leap day calendar cycles:

Jitter (days) Leap Day Cycle Description and Comments (mostly sorted by ascending jitter range)
0.75 Julian Calendar (leap every year divisible by 4, uniformly spread, least possible jitter but rather long mean year)
The same chart is also available as an interactive Excel spreadsheet that is compatible with Excel 2007 or later for Windows, Excel 2011 or later for macOS, and LibreOffice CALC. Enable execution of macros. Use the up/down arrow control to change K to any value from 0 to 3.
0.96 Smoothly Spread Symmetrical 6/25 (minimal jitter)
1.68 (6/25)×4=24/100 Centurial (out of jitter sequence, to show that just these modifications increase the jitter range by 1.75×. Both of these 6/25 leap cycles have a rather short mean year of only 365+6/25 days ≡ 365.24 days ≡ 365 d 5h 45m 36s, which is and will be too short for any solstitial or equinoctial year until around the year 33000 AD)
0.96 Smoothly Spread 8/33 (K=8 is the same as the Dee Calendar, but K=16 is perfectly symmetrical)
The same chart is also available as an interactive Excel spreadsheet that is compatible with Excel 2007 or later for Windows, Excel 2011 or later for macOS, and LibreOffice CALC. Enable execution of macros. Use the up/down arrow control to change K to any value from 0 to 32, or directly type into the yellow K value field on the Data worksheet.
1.69 Georgian 132-year cycle, leap every 4 years except last year of each cycle (out of jitter sequence, same mean year as the 8/33 cycle above, but these modifications likewise increase the jitter range 1.75×) [published 1745, see web site]
≈ 1.02 Arithmetic Persian (2820 years, Birashk) Although this jitter is almost the minimum possible, it was silly for Birashk to propose such an absurdly long leap cycle for a leap day calendar, and his extreme accuracy claim was misguided.
Comparable short cycles include the 31/128 cycle (mean year almost one second shorter) or the 101/417 cycle (mean year nearly 2/3 second longer). One of each of these cycles could be combined into a 132/545 cycle having an intermediate mean year that is only about 1/4 second longer than the 2820-year cycle. Alternatively, combine two 128-year cycles with one 417-year cycle to make a 163/673 cycle that has a mean year differing by less than 50 milliseconds from Birashk’s mean year.
1.6953125 Mädler 128-Year Cycle (leap every year divisible by 4 except years divisible by 128) [published 1864, see web site]
Compare with this 128-year minimal jitter smoothly spread almost symmetrical leap cycle.
1.742 Dual uncorrelated cycles (500 years, an educational example from Karl Palmen)
2.036 Gregorian-Like 300-Years
2.1975 Gregorian (400 years)
2.294 Gregorian-Like 500-Years
2.3583 Gregorian-Like 5×120-Years
2.362 Revised Julian (900 years)
3.04575 Herschel-Modified Gregorian (4000 years, John Herschel, 1876)
3.2554 Newton-Modified Gregorian (5000 years, unpublished, from Sir Isaac Newton’s personal notes, see Belenkiy manuscript)
3.2296 Palmen-Modified Alpo Balognian [see next row] (rules tweaked by Karl Palmen to reduce jitter by 66/125 day)
3.7576 Alpo Balognian [web site] (reduced from 9000 to 3000 years to reduce chart clutter, same jitter range, but 3× faster)

### For the following leap week cycles click on a cycle description to be taken directly to its jitter chart:

Jitter (days) Leap Week Cycle Description and Comments
≈ 6.8871 Smoothly Spread Almost Symmetrical 11/62
6.96 ISO-Dee
≈ 6.976 Smoothly Spread Symmetrical 52/293 (used for the Symmetry454 and Symmetry010 calendars)
6.9825 Smoothly Spread Almost Symmetrical 71/400
6.9825 Pragmatic Civil Calendar (400 years)
7.9275 International Organization for Standardization (ISO, 400 years, 1988)
Note that the ISO leap cycle has leap years at intervals of 6 or 5 years with a single exception:
a 7-year interval before the 303rd year of each cycle (hence it is asymmetrical).
≈ 8.0 11/62 Even-Numbered
8.2075 71/400 Even-Numbered
9.75 28-year cycle with leap week only allowed in Julian leap years
≈ 10.3 11/62 Divide-by-Six
11.8825 71/400 Divide-by-Five
11.9875 400-year cycle with leap week allowed only in Gregorian leap years
12.495 Weekdate Calendar (400 years, Rick McCarty)
≈ 12.607 Brij Vij 834-Year Divide-by-Six
12.703125 Palmen Modified Brij Vij 896-Year Divide-by-Seven
12.7225 Brij Vij 1200-Year Divide-by-Six
13.4453125 Bonavian Civil Calendar (896 years, Chris Carrier, 1975, web site)
14.1328125 Brij Vij 6 & 45 896-Year: append leap week if (Year within 896-year cycle) is divisible by 6 or by 45.
16.0475 Another 400-year cycle with leap week only allowed in Gregorian leap years
16.7825 Bob McClenon’s Reformed weekly Calendar (400 years)
17.3075 Jubilee Calendar (400 years, Professor Cecil L. Woods, 1955)
17.3075 Searle Calendar (400 years, Rev. George M. Searle, 1905)
17.625 Brij Vij 896-Year Divide-by-Seven
18.2875 Pax Calendar (400 years, James A. Colligan, 1930)

### For the following leap month cycles click on a cycle description to be taken directly to its jitter chart:

Jitter (days) Leap Month Cycle Description Comments
≈ 27.9 days Jitter of 293-year leap month solar cycle, relative to cycle mean year Intended for a 13-month calendar with 28 days per month plus a 28-day leap month appended to end-of-year at long intervals (23 or 22 years). Like all smoothly spread symmetrical leap cycles it is vertically centered on the zero jitter line. For the intended calendar type it has the minimum possible jitter (just under 28 days).

## Lunar calendar jitter relative to cycle mean month

Fixed arithmetic lunar calendar cycles contain only full (30-day) and deficient (29-day) month lengths, mostly arranged in an alternating sequence.

A yerm, as defined by Karl Palmen, has an odd number of months starting with a full month, then alternating month lengths, and ending with a full month. For minimum jitter and reasonable accuracy for the present era lunar cycle the longest allowable yerm length is 17 months and the shortest allowable is 15 months. The use of longer or shorter yerms will exaggerate the jitter and usually causes the mean month to be too short or too long, respectively.

An interval of consecutive months contains a whole number of days. This normally differs from the same number of mean calendar months and so causes the calendar to jitter by a deficit or surplus over the interval. Any interval that starts and ends with a deficient month will have a deficit, whereas yerms that are 15-months or shorter will have a surplus. Jitter analysis seeks to calculate and/or plot the jitter and thereby find the interval over which the deficit or surplus is largest. If a selected interval has the maximum surplus or deficit then the interval containing the complementary months (the months of the cycle that don’t belong to the selected interval) will have the complementary deficit or surplus, respectively.

The minimum possible lunar calendar jitter occurs when full and deficient months are distributed as smoothly as possible, such that deficient months never occur consecutively and there are never more than two consecutive full months.

The original developer’s motivation for several of the lunar calendars listed below was to simplify the calendar arithmetic, but that comes at the cost of exaggerated jitter, and in reality calendrical calculations are actually made more complex by multiple rules and exceptions. The arithmetic for calendars with minimum possible jitter is simple enough. Various lunar calendar cycles are listed below in order of increasing jitter range.

The 850-month 52-yerm, 801-month 49-yerm, 360-month 22-yerm, and 49-month 3-yerm cycles have minimum possible jitter, and jitter decreases slightly with progressively shorter yerm cycles.

The following tables contain links to interactive Microsoft Excel spreadsheet files in XML macro-enabled format (.xlsm file extension). They contain data, charts, and macro code written in Visual Basic for Applications (VBA). The interactivity requires Excel 2007 or later for Windows, or Excel 2011 or later for macOS. Upon launching each file the user must enable execution of VBA macros.

These files can be opened by LibreOffice CALC on any platform they support, and the user should enable macros, but none of the control buttons or checkboxes will work properly, due to CALC using a chart object model that differs from Excel.

≈ 23.51 49-month, 3-yerm cycle least jitter of this collection of cycles, but its mean month of 29 days 12 hours and 4+44/49 seconds ≡ 29+26/49 days is rather long for the present era, however it would have been ideal around the year 7000 BC
23.93 360-month, 22-yerm cycle its mean month of 29 days 12 hours and exactly 44 minutes ≡ 29+191/360 days is rather short for the present era, although it will be ideal soon after the year 9300 AD
≈ 23.97 801-month, 49-yerm cycle minimum possible jitter, my personal favorite for general-purpose present era use, with intentionally slightly short mean month
≈ 23.972 850-month, 52-yerm cycle minimum possible jitter, Karl Palmen’s choice for his Yerm Lunar Calendar and present era use, with accurate mean month
≈ 41.723 Terence McKenna’s Goddess Lunar Calendar 130-month cycle [web site]
≈ 43.95 Palmen-McKenna Lunar Calendar 2418-month cycle [web site]
≈ 65.12 McKenna-Meyer Goddess Lunar Calendar 3055-month cycle [web site]
≈ 68.425 Palmen Seventeen Month Lunar Calendar 850-month cycle [web site]
≈ 78.946 Meyer Goddess Lunar Calendar 850-month cycle

I haven’t included longer cycles because the file size would be inconveniently large. For example, the 25-saros cycle contains 25 × 223 = 5575 months, the Tibetan Phugpa cycle contains 5656 months, the full molad cycle of the traditional Hebrew calendar contains 25920 months (same as the number of parts or chalakim per day), the modern Hindu Surya cycle contains 13358334 months, the Gregorian Easter computus contains 70499183 months, the old Hindu Arya calendar cycle contains 53433336 months, and the progressive molad of my rectified Hebrew calendar contains an infinite number of months because its mean month slowly gets progressively shorter to closely approximate the actual astronomical mean synodic month.

## Lunisolar calendar jitter relative to the lunar cycle mean month

≈ 47.77 Orthodox Easter computus jitter of paschal moon relative to mean month 4 × 7 × 19 = 532-year cycle
≈ 191.77 Orthodox Easter computus jitter of Easter Sunday relative to mean month 4 × 7 × 19 = 532-year cycle, has 6 days more jitter than paschal moon