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Dates are Gregorian (YYYY.MM.DD) unless stated otherwise.

- 2017.11.18
- The Length of the Lunar Cycle, Periodic Earth-Moon Distance Variations section: improved all of the images and their PDFs. Minor tweaks to the discussions.
- 2017.11.16
- The Length of the Lunar Cycle, Periodic Earth-Moon Distance Variations section: image 3 of 4 is new. Improved the discussion and moved academic sources to the end of the section.
- 2017.11.15
- The Length of the Lunar Cycle, Periodic Earth-Moon Distance Variations section: improved the discussion, added citations for sources. Minor chart improvements, and in the first PDF the first two charts are shown with and without colored curves connecting the lunar events.
- 2017.11.14
- The Length of the Lunar Cycle, Periodic Earth-Moon Distance Variations section: improved the charts and edited the discussion to suit the changes. The lunar daily distance charts now have faint green grid lines at 59-day intervals (about 2 lunar cycles), and the 21st century cumulative centiles chart of lunar quarterly distances now has two first derivative curves showing that the most extreme distances have the highest frequencies.
- 2017.11.12
- The Length of the Lunar Cycle: Added new section about Periodic Earth-Moon Distance Variations.
- 2017.10.23
- Replaced
*moladot*example report with this year's update for Jerusalem. - Added to discussion of
*Shabbat Rosh Chodesh*on the Weekly Torah Portions and Special Sabbaths web page, explaining that*Rosh Chodesh*dates that**never**fall on*Shabbat*are caused by the*Rosh Hashanah*postponement rules. - Minor revisions to the Rectified Hebrew Calendar web page indicating the current period of perfect agreement with the traditional Hebrew calendar.
- 2016.10.09
- Updated information about the SOLEX web site and its programmer on the following web pages here:

The Lengths of the Seasons (on Earth), Length of the Lunar Cycle, Solar Calendar Leap Rules, The Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar, Moon and the*Molad*of the Hebrew Calendar,*Rambam*and the Seasons, and Request for Rain (*Sh'ela*). - 2016.09.30
- Why Divide Hours into 1080 Parts?: Added the distance that light travels per
*chelek*of time. - 2016.05.15
- Fixed Arithmetic Calendar Cycle Jitter: Added a symmetrical 6/25 cycle and a centurial (6/25)x4 = 24/100 leap cycle chart to the solar calendar leap day calendar jitter charts collection.

Rectified Hebrew Calendar: This spring marked the end of a 7-year block of perfect agreement between the traditional and rectified Hebrew calendars. The next 6+ year block of perfect agreement will start next spring from 1*Nisan*5777 until 29*Cheshvan*5784. - 2015.12.07
- Fixed Arithmetic Calendar Cycle Jitter: Updated the leap week solar calendar jitter charts by adding a mirror version of the lower jitter chart having leap week only in Gregorian leap years.
- 2015.11.29
- Fixed Arithmetic Calendar Cycle Jitter: Updated the leap week solar calendar jitter charts by revising the chart having leap week only in Gregorian leap years and adding a new version that has substantially less jitter.
- 2015.11.09
- Fixed Arithmetic Calendar Cycle Jitter: Updated the solar calendar jitter charts by splitting the collection into separate leap day and leap week collections, then added an 896-year cycle having a leap week if the year number within the cycle is divisible by 6 or by 45.

Request for Rain: fixed broken links of external references and added another external reference. - 2015.10.14
- Fixed Arithmetic Calendar Cycle Jitter: Updated the solar calendar jitter charts collection after adding two 400-year leap week cycles — one with leap weeks allowed only in years when a smoothly spread 400-year leap day calendar has a leap year (leap years can be at intervals of 4, 5, 8, or 9 years), the second with leap weeks allowed only in years when the Gregorian calendar has a leap year (leap years can be at intervals of 4 or 8 years).
- 2015.09.16
- Calendar Seasons: Updated the second chart, "Solar Year Length Variations on Earth", so that the x-axis goes from 180° to 360° (0°) then back to 180° to better illustrate the calendar season trends, with the northward equinox in the middle.
- 2015.04.16
- Symmetry454 and Symmetry010 calendars: Added statement that in leap years the 4th quarter has 14 weeks, but this occurs only once per 24 or 20 quarters.
- 2015.04.11
- Symmetry454 and Symmetry010 calendars: Added statement that the year before the epoch was year zero, to be consistent with the astronomical convention. (This statement has been long present in the basic calendar arithmetic documentation.)
- 2015.03.21
- Refined instructions for setting up
*Kalendis*to run under WINE in Linux.

Withdrew web page about the Bahá'í Calendar, and my proposed variants. - 2015.01.04
- Arithmetic of the Bahá'í Calendar, and Variants: Further revisions of the introductory section.
- 2015.01.01
- Arithmetic of the Bahá'í Calendar, and Variants: Updated introduction to discuss the World Bahá'í Centre announcement of the switch to a worldwide unified astronomically reckoned Bahá'í calendar next spring.
- 2014.12.23
- The Duration of Twilight: Ooops, had to fix crashing DST function in the Twilight-Charter spreadsheet, but also added Daytime Duration to "Twilight" chart, removed Dusk Tam (remains on list) and added 3 Bright Stars to "Zmanim" chart. Rearranged charts legends so that equinoxes and solstices at the top, then legend entries proceed in a top-down fashion like the curves on the chart.
- Jitter charts: New version of solar jitter charts with some aesthetic chart improvements.
- 2014.12.21
- The Duration of Twilight: Updated Israel DST rule in the Twilight-Charter spreadsheet.
- 2014.12.20
- Jitter charts: Lunisolar section two variants of the Orthodox Easter
*computus*jitter — I had neglected to upload the two updated Orthodox Easter files, sorry about that, now fixed. Added Arithmetic Persian (Birashk) leap day solar 2820-year cycle to solar calendar jitter charts PDF. - 2014.12.04
- Jitter charts: Added 4 PDF charts to the lunisolar section, depicting the jitter of the traditional Hebrew calendar
*molad*cycle relative to the calendar mean year. - 2014.12.04
- Jitter charts: Lunisolar section now has two variants of the Orthodox Easter
*computus*jitter — paschal moon relative to mean month, and Easter Sunday relative to mean month, the latter having 6 days' greater jitter. - 2014.11.28
- Solar Calendar Leap Rules: every occurrence of the word "jitter" now links to my new web page that explains jitter and offers charts showing the jitter of various solar, lunar, and lunisolar calendar cycles.
- 2014.11.21
- Posted new version of Symmetry 454/010 FAQs with minor change, and new version of Ford-Circles-of-Leap-Cycles.xls spreadsheet file with minor changes.
- 2014.11.20
**Released new version of Kalendis**: To learn what is changed, see the Release Notes.- Minor updates to WINE web page (for installing Kalendis under Linux operating systems).
- Solar Calendar Leap Rules: Added a sentence explaining that the average calendar season mean year corresponds to the two points where Earth's orbital elliptical minor axis intersects the orbit. Added a sentence about Euler, giving an example of a relationship between fractions and circles.
- Updated links to CALNDR-L LISTSERV.
- 2014.06.02
**Released new version of Kalendis**: To learn what is changed, see the Release Notes.- Updated Locales web page, fixing a broken USA TIGER link and adding a link to KosherJava web site.
- 2013.07.25
- Lengths of the Seasons: Minor edits to explain that Earth spends more time moving slowly near aphelion as its orbital eccentricity increases, and that may cause globally decreased temperatures.
- 2013.05.19
- Why Divide Hours into 1080 Parts? In the complete traditional
*molad*cycle of 25920 months there are 793 more full months than one half cycle, exactly corresponding to the 793 parts excess over 29.5 days of the traditional*molad*interval. Added the duration of a "part" for the 1144-, 1095-, 850-, and 801-month lunar cycles, to illustrate that the traditional choice of 1080 parts per hour or 25920 parts per day has nothing to do with divisors nor with time resolution precision. Corrected a minor grammar error. - 2013.03.31
- Symmetry454 and Symmetry010 calendars: Posted a regenerated basic arithmetic PDF file. The old version was generated with a glitch that causes the symbols for CEILING to get lopped off, so that they looked like absolute value bars instead. No actual changes to the document, but the file size more than doubled as a consequence of the different generation method.
- 2013.01.21
- Lengths of the Seasons: Expanded discussion of astronomical seasons and the lag between them and surface weather seasons.
- 2013.01.17
- Torah Portions and Special Sabbaths: Added an analysis of
*Shabbat Rosh Chodesh*frequencies. - 2012.12.30
- Torah Portions and Special Sabbaths: Added tables showing the 14 unique Hebrew calendar year types with their exact frequencies, and the number of combined portions for each year type in Israel and in the diaspora.
- 2012.11.27
- Installing
*Kalendis*under WINE or CrossOverMac: Minor revisions, most significantly that the display glitches of CrossOverMac are gone since the release of version 11. - 2012.05.17
- Torah Portions and Special Sabbaths: Mentioned that
*Shabbat Shekalim*and*Shabbat HaChodesh*can coincide with*Rosh Chodesh*. Also mentioned*Shabbat Channukah*, and mentioned that it can occasionally occur on the both the first and last day of*Channukah*. - Why March 21st? : Added equinox jitter paragraphs to the Gregorian and Revised Julian calendar cycle comments.
- Solar Calendar Leap Rules: Added comment explaining excess equinox jitter for the ISO leap cycle.
- 2012.02.29
- Bahá'í Calendar: Added a list of alternate translations for the Arabic word
*Badí*. - 2012.02.27
- Why Divide Hours into 1080 Parts? In the discussion of exact calculation, corrected a recurring error which inadvertently omitted 12 hours from the arithmetic expressions. (Thanks to Friedrich Reincke for bringing this error to my attention.) Also, in the discussion on that page about the progressively shorter mean synodic month, added links from there to where I show the differences between the molad interval and the mean synodic month.
- 2012.01.15
- Mediant Fractions, Farey Pairs, and Ford Circles:

Released a newer version of the**Ford Circles of Leap Cycles**spreadsheet with VBA macro. Changed "Selected LS Cycles" and "Computus" cycles to use Excel formula expressions for decimal values. Computus feature is now prevented from running if the solar list is already lunisolar. Now including default lunar cycle list so that computus feature has a list to work with. Cycle Patterns: Fixed a problem that prevented some cells from getting the intended borders all around in Excel 2010. - 2012.01.18
- Mediant Fractions, Farey Pairs, and Ford Circles:

Updated the discussion of the**Ford Circles of Leap Cycles**spreadsheet to cover the new features (computus, cycle patterns, exact expressions for decimal columns), as well as a workaround for the case of leap day cycles with mixers 1/4 and 0/1. - 2012.01.15
- Mediant Fractions, Farey Pairs, and Ford Circles:

Released a newer version of the**Ford Circles of Leap Cycles**spreadsheet with VBA macro. Changed "Selected LS Cycles" and "Computus" cycles to use Excel formula expressions for decimal values. Computus feature is now prevented from running if the solar list is already lunisolar. Now including default lunar cycle list so that computus feature has a list to work with. Cycle Patterns: Fixed a problem that prevented some cells from getting the intended borders all around in Excel 2010. - Later updated the same spreadsheet yet again to further improve the pattern borders in Excel 2010, and reduced download file size from 2.8 MB to 1.1 MB.
- 2012.01.13
- Mediant Fractions, Farey Pairs, and Ford Circles:

Released a new version of the**Ford Circles of Leap Cycles**spreadsheet with VBA macro. Changes:

- Added Computus feature (select a solar cycle, click "Use selection to generate computus cycles" command button, uses list of lunar cycles to generate computus cycles having same solar mean year.
- Show Cycle Pattern (shows the symmetrical leap cycle pattern for the selected non-lunar cycle).
- Decimal fraction of year, decimal seconds: now saved as Excel formula expressions, which Excel displays as decimals values to the full precision supported by the application.
- Minor bug fixes and improvements.

- 2012.01.10
**The Lengths of the Seasons:**Added a new section near the top concerning Chinese Major Solar Terms.- 2012.01.08
**Symmetry454 and Symmetry010 calendars**: Added a new section at the top,*Credo*: What We Believe.- 2012.01.07
**Symmetry454 and Symmetry010 calendars**: Revised the Events PDF to repair a broken link to the WCC web site concerning a Common Date for Easter, and added quotations from that document, and to repair broken links to the Vatican web site concerning a papal ruling on calendar reform.- 2012.01.04
**Symmetry454 and Symmetry010 calendars**: Revised the FAQs PDF and the Basic Arithmetic PDF. Both now discuss the on-going 4-week cycle that can be used for regular payments such as payroll or rental. The FAQs has a new topic heading discussing the impact of the leap week on the duration of contracts.**Why March 21st?**: In the table at the end of the page, revised the discussion of the Gregorian calendar, and added a row to depict the March equinox on the Dee calendar.- 2012.01.03
**Why March 21st?**: Added a new paragraph ("Religious sources...") explaining the novel insight that arranging the Gregorian calendar so that the spring equinox lands on March 21st ensures that the*entire*month of March falls within the spring*weather*season in Rome.- 2012.01.01
**Released new version of Kalendis**: To learn what is changed, see the Release Notes.- 2011.12.24
**Seasonal Drift of the Traditional Hebrew Calendar**: Added a new paragraph ("A major difficulty...") explaining why our traditional criteria actually imply that the*entire*month of*Nisan*must be within the spring*weather*season.- Symmetry454 and Symmetry010 calendar pages: Added Click here to see the detailed 293-year leap cycle pattern 29 KB.
- 2011.12.14
**Using**: This page was previously based on correspondence with other people who used*Kalendis*under Linux using WINE*Kalendis*under WINE, but now that I actually have WINE running under Ubuntu the text is revised to provide specific instructions and tips.- 2011.12.12
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Further revision of the introductory text, using simpler direct arithmetic to calculate the full cycle length in years. Added "the month before the month before*Nisan*[*Shevat*or*Adar Rishon*], which is also calculated by checking**only 4 years**.- Repaired or removed broken external links throughout this site.
- 2011.12.04
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Added some helpful shading to the first table, and revised the introductory text. In particular note that the exact weekday frequencies of all dates from the month before*Nisan*through to the 29th of*Cheshvan*can be calculated by checking**only 4 years**.- 2011.11.12
**Lengths of the Seasons**and**Seasonal Drift of the Traditional Hebrew Calendar**and**Request for Rain**and: Improved the discussion of sunrise and sunset direction in relation to the astronomical equinoxes.*Rambam*and the Seasons- Irvember: Corrected the last paragraph, because the stand-alone leap week at the end of the year was restored as the Symmetry010 calendar default in
*Kalendis*. - 2011.07.20
**Solar Calendar Leap Rules**: Added sub-cycle information to all of the listed fixed arithmetic leap cycles. Every*almost*symmetrical cycle (even number of years per cycle with odd number of leap years per cycle) is identical to some combination of perfectly symmetrical sub-cycles.- 2011.06.27
**Lunar Cycle Variations**: Major new section to demonstrate that periodic variations of the lunar cycle are least at lunar conjunctions and greatest at the first quarter.- 2011.06.21
**Solar, Lunar, Indiction cycle arithmetic**: revised to incorporate corrections from Jean Meeus (solar, lunar, and indiction cycles all count from one, not zero), also now it warns when the user enters an invalid solar, lunar, or indiction number, and the warning shows the valid range for each field.- 2011.06.13
: Added a final paragraph explaining how to use this equinox reckoning to reproduce traditional Hebrew calendar dates.*Tekufat Nisan*of*Rav Adda*- 2011.05.16
**Basic Leap Cycle Calculations**: Added two example calculations to the new subsection detailing exact mean year calculations for**lunisolar**calendars. In the course of doing so, discovered a formula error which is now corrected, and also made the formula easier to understand.- 2011.05.10
- Mediant Fractions, Farey Pairs, and Ford Circles: Added a PDF list of lunar eclipse cycles. Posted new version of Ford Circles spreadsheet (minor bug fix: wasn't clearing mean month info on Ford chart if lunar cycles were drawn after lunisolar cycles).
- 2011.05.09
**Seasonal Drift of the Traditional Hebrew Calendar**: Added a paragraph the end section "**Proposed More Accurate Leap Rule for the Hebrew Calendar**" to explain that truncating a 19-year cycle to only 8 years (instead of 11 years) would have the opposite effect, making the equinox drift worse.- Why 1080 Parts?: HTML syntax improvements to pass W3C validation. Added a menu of topics, and added two paragraphs explaining the implied lunar cycle of the traditional
*molad*interval, and which shorter cycles can yield comparable accuracy both for the past and for the present era. - 2011.05.06
**Leap Rules for Aligning Calendars Relative to the North Solstice**: Moved the 296-year lunisolar cycle to the description cell belonging to the 327-year leap week cycle, and added the 611-year lunisolar cycle to the description cell for the 389-year leap week cycle.- 2011.05.03
**Leap Rules for Sidereal Calendars (Zodiac)**: Added 160-year**lunisidereal**cycle. Also added a paragraph to the discussion of the Ford Circles spreadsheet explaining how to use it to generate a list of lunisidereal cycles.- 2011.04.28
**Mediant Fractions, Farey Pairs, and Ford Circles**: New version of the Ford Circles spreadsheet, now including lunar eclipse cycles.- 2011.04.27
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Added to the year types table the exact number of weeks for each year type. Only perfect leap years contain a whole number of weeks = 55.- 2011.04.22
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Reworded comments below the year types table.- 2011.04.17
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Added totals by weekday and by year length to the year types table.- 2011.04.15
**Drift of Fixed Arithmetic Lunar Cycles**: Revised spreadsheet to replace some unimportant cycles with the 49-yerm and 25-saros cycles, and added details to all cycle comments on the "Lunar_Cycles" worksheet.

On the web page also updated table of fixed arithmetic lunar cycles.- Later also updated the
**Solar Calendar Drift spreadsheet**to include the interesting 100-saros cycle. - 2011.04.11
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Revised to include a printable version that has a page break before each table, so that the print-optimized version can fit each table on a single A4 or Legal page. The print-optimized and view-optimized versions are interlinked at the top of the report. The view-optimized version has links for all months ahead of each table, but these are omitted in the print-optimized version so as not to waste space on the printout. Also corrected a recurring empty table cell HTML syntax error.- 2011.04.07
**The Duration of Twilight**: Added menu of topics, plus a new topic, offering a freeware Excel spreadsheet that calculates and plots twilight moments and*zmanim*for an entire year for any specified locale.- 2011.04.05
**Exact Hebrew Calendar Full Cycle Weekday Frequencies**: Minor typographical corrections and rewording of the introductory text. Index of links to months now is two lines with separate links to each*Adar*table.- 2011.04.02
**Basic Leap Cycle Calculations**: Inserted a new subsection detailing exact mean year calculations for**lunisolar**calendars.- 2011.04.01
**Mediant Fractions, Farey Pairs, and Ford Circles**: Major rewrite of the Ford Circles spreadsheet to introduce exact mean months for exact**lunisolar**leap cycles, and posted revised PDF charts including 4 variants of lunisolar cycles.- 2011.04.01
**Solar, Lunar, Indiction cycle arithmetic**: small Excel spreadsheet posted, accessed directly from home page, to show how to interconvert year numbers with the**Solar, Lunar, Indiction**cycles.

Older records deleted 2011.12.12