Fixed Arithmetic Calendar Cycle Jitter

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

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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 works 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.

Click here to open a leap day jitter chart collection, sorted by ascending jitter, starting at the minimum possible. 470KB

Click here to open a leap week jitter chart collection, sorted by ascending jitter, starting at the minimum possible. 1.4MB

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 following are compressed ZIP archives each containing an interactive Microsoft Excel for Windows spreadsheet file. Download and extract the original spreadsheet files, and upon launching each file the user must enable execution of VBA (Visual Basic for Applications) macros. The charts are designed to have equal jitter y-axes for direct inter-comparison. 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 ascending order of number of months per cycle:

(Each ZIP archive is roughly 1/3 the size of the original spreadsheet file.)

Lunisolar calendar jitter relative to the lunar cycle mean month

The following compressed ZIP archives each contain an interactive Microsoft Excel for Windows spreadsheet file as explained above.

The following PDF files each plot the jitter of the traditional fixed arithmetic Hebrew calendar molad cycle relative to the calendar mean year (235 molad intervals per 19 years):

For further information about the traditional and variant Hebrew calendar leap rules, and definition of terms used in these charts, please see my web page: The Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar.


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" 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.


CALNDR-L Updated 7 Iyar 5776 (Traditional) = 8 Sivan 5776 (Rectified) = May 14, 2016 (Symmetry454) = May 12, 2016 (Symmetry010) = May 15, 2016 (Gregorian)