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

- 2023.10.29
- Special Sabbaths: Added a detailed analysis of
**the frequencies of**.*Shabbat Machar Chodesh* - 2023.10.22, 2023.10.26
- Special Sabbaths: Improved the discussions of
*Shabbat Machar Chodesh*and*Shabbat Rosh Chodesh*. - 2023.07.30
- The Duration of Twilight: Updated external URLs to use secure connections (https:) and to point to the correct current locations, mostly in the section entitled
Calculations, and the Shortest and Longest Twilight

.*Kalendis*seems to be also compatible with Windows 11. - 2023.01.30
*Kalendis*: updated year-specific example reports.- 2022.12.04
- Request for Rain (
*Sh'ela*): updated links to external resources. - 2022.07.31
**Jitter Charts**: Completed updating all leapcalendar cycle jitter charts as follows:*week*- each jitter chart is now in a stand-alone file, either PDF or an interactive Excel workbook
- all had an elapsed days arithmetic error that is now fixed
- each chart shows the jitter range as both a decimal number and an exact fraction
- each chart shows the cycle mean year as an exact fraction, exact time, and as a decimal number
- improved the appearance, layout, and consistency
- added some citations and comments
- each chart starts with a zero jitter value at year 1
- each chart ends at the year after at least 2 complete cycles, returning to the same zero jitter value as at year 1
- many charts that are related to the Gregorian calendar are now plotted on two pages, the first showing the jitter against the calendar mean year, and the second showing the jitter against the Gregorian calendar.

- 2022.07.26
**Jitter Charts**: Completed updating all leapcalendar cycle jitter charts as follows:*day*- each jitter chart is now in a stand-alone file, either PDF or an interactive Excel workbook
- many had an elapsed days arithmetic error that is now fixed
- each chart shows the jitter range as both a decimal number and an exact fraction
- each chart shows the cycle mean year as an exact fraction, exact time, and as a decimal number
- improved the appearance, layout, and consistency
- added some citations and comments
- each chart starts with a zero jitter value at year 1
- each chart ends at the year after at least 2 complete cycles, returning to the same zero jitter value as at year 1

- 2022.07.11
**Fixed Leap Cycle Finder**: New version replaces all ActiveX controls with normal form controls, for compatibility with Excel for macOS while retaining compatibility with Excel for Windows.

Minor changes to the**Lunar Calendar Drift**and**Solar Calendar Drift**workbooks for Excel.- 2022.07.10
**Calendar Leap Cycles**: moved my**Mediant Fractions Algorithm**to a subordinate web page.- 2022.07.09
- Posted a minor update to the
**Twilight-Charter**workbook: changedTemporal Time (sundial time)

toTemporal Time (seasonal hours)

, and changedLocal Apparent Time

toLocal Apparent Time (sundial time)

. Temporal Time and sundial time are not even close, except near equinoxes. - 2022.07.08
**The Lengths of the Seasons (on Earth)**: Full review with minor changes (last previous changes October 2016). Chinese Major Solar Terms section: Introduced new terms for seasons__beginning__at equinoxes and solstices = thermal seasons (western astronomical seasons), as opposed to equinoxes and solstices marking the__middle__of seasons = sunlight seasons (solar insolation seasons). Added new chart showing the lengths of the solar insolation seasons from 8000 BC to 15000 AD.

**Inverting the**(given a weekday and*molad**molad*time, find the corresponding Hebrew year and month): removed reference to Maimonides. Added to end of section external links to on-line*molad*inversion calculators written by Prof. Nachum Dershowitz.

**Moon and the Molad of the Hebrew Calendar**: Started revising the section discussing Temporal Time (seasonal hours). Also detailed what*Rambam*wrote about the invisibility of Moon near its conjunction with Sun.

**Hebrew Calendar**: Detailed what*Rosh Hashanah*Postponements*Rambam*wrote about the visibility of the new lunar crescent on*Rosh Hashanah*.

**Why Divide Hours into 1080 Parts?**: Detailed what*Rambam*wrote about the divisors of 1080.- 2022.06.23
**Ford-Circles-of-Leap-Cycles**: minor workbook changes to add support for Martian 28-sol skip month calendar cycles.- 2022.06.20
**Freeware Excel Workbook Demonstrating Annual Twilight Variations**: Posted a new version of the Twilight-Charter (with Directions) workbook, with open-source Visual Basic for Applications (VBA) macros, which is now compatible with Excel 2007 or later for Windows as well as Excel 2011 for macOS.- 2022.06.08
**Inverting the**(given a weekday and*molad**molad*time, find the corresponding Hebrew year and month): updated the downloadable open-source demonstration Excel workbook to make minor improvements and to make it compatible with LibreOffice CALC. A week later another update added two worksheet cells to theInvert

worksheet with formulae for direct inversion of the*molad*, and some extra VBA functions such as*ModularInverse( )*.- 2022.06.01
**Ford-Circles-of-Leap-Cycles**: major new version (replaced ActiveX controls with normal form controls, allowing this workbook to also work in Excel for Macintosh 2011 or later).- 2022.05.12
- Fixed-Leap-Cycle-Finder: new version, with minor improvements.

Lunar Calendar Drift: Replaced file, which was damaged somehow. - 2022.05.02
**Jitter Charts**: added individual leap day jitter chart for dual uncorrelated cycles as proposed by Karl Palmen as an educational example.

**Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar**: added a new section before the concluding section entitled**Hebrew Calendar**.Equinox Jitter

- 2022.04.05
**Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar**: Thorough review from beginning to end, revised and updated as appropriate, grammatical and layout improvements, and added a final**Conclusions and**.Executive Summary

- 2022.03.25
- Improved and extended the discussion of
.Calendar Seasons: Stable Points in the Solar Cycle

**Hebrew Calendar Studies**: added links to theweb site.*Sanhedrin* - 2022.03.16
- New
chart.Solar Year Length Variations With Cycles

- 2022.03.15
**Jitter Charts**: added individual jitter chart for Gregorian-like 5×120-year leap day cycle.

**Calendar Leap Cycles**: added a comment about the excellent calendar season of the 11/62 cycle positioned mid-way between the northward equinox and the north solstice.- 2022.03.14
- Significant update to the
workbook (added markers for the beginnings of the astronomical solar insolation seasons, midway between the equinoxes and solstices), and updated all of the PDF example charts to show this new feature. Also added a newFind Solar Calendar Seasons

Solar-Year-Length-Variations

chart for year 2000 and 4000 AD, with horizontal lines marking the mean year of various selected leap day calendar cycles. - 2022.03.12
- Minor updates to the
workbook, and updated all of the PDF example charts. Equinox and solstice drift extenders are now always drawn from the most extreme drift curves. Extender points discontinued for perihelion and aphelion.Find Solar Calendar Seasons

Shift to Selected Curve

no longer leaves the original chart series selected. - 2022.03.11
- Major update to the
workbook, and updated all of the PDF example charts (and added 2 more). Formerly the charts used vertical lines to show the positions of perihelion, aphelion, and the equinoxes and solstices, but those all applied to the middle year (horizontal black line at zero drift). Now the macro usesFind Solar Calendar Seasons

*curves*to plot them in relation to all of the plotted curves, and then extends vertically to the top and bottom of the chart. - 2022.03.10
: Added my empirical*Rambam*and the SeasonsHalf-Cosine Curve Fit

toSecond Latitude amount to set aside

on theTables

worksheet of the Excel workbook file.

**Jitter Charts**: Added a

at the top of the page.**Menu of Topics**- 2022.03.09
**Jitter Charts**: Replaced ActiveX controls with ordinary form controls in all interactive lunar and lunisolar jitter charts, so now they are compatible with Excel for macOS 2011 or later yet retain compatibility with Excel for Windows 2007 or later. An annoying limitation is that the font and text style of ordinary form checkboxes can’t be edited, so they look rather small and inconspicuous.- 2022.03.08
- Updated the
workbook, now compatible with Excel for Windows 2007 or laterFind Solar Calendar Seasons

__and__compatible with Excel for Mac 2011 or later (removed ActiveX controls). - 2022.03.07
**Mixed Radix Conversions**: Fixed a bug that caused the**ApplyUnits**( ) function tomangle

the results when the radix point was leftmost. Added a**ContinuedFraction**( ) function and used it in several of the calendar examples. Added a Lunisolar Cycle calculation example.**ReduceFractStr**( ) function accepts a text fraction and then calls**ReduceFraction**( ) to perform the reduction. Improved compatibility with LibreOffice CALC.- 2022.03.03
**Mixed Radix Conversions**: minor web page improvements.- 2022.03.02
**Mixed Radix Conversions**: updated workbook, adding Roman coins mixed radix example, and the**ApplyUnits**( ) VBA function now has an option to skip zeros.- 2022.03.01
**Mixed Radix Conversions**: updated workbook with improved and additional examples and some minor VBA function bug fixes.- 2022.02.28
**Mixed Radix Conversions**: new version of the workbook — now the**ToRadix**( ) function can optionally reduce the least fractional radix value, returning a fraction, the**ApplyUnits**( ) function recognizes#

as a special symbol, and I added many Tibetan calendar__arithmetic__examples from Henning's book.- 2022.02.25
**Jitter Charts**: added individual jitter charts for Gregorian-like 300- and 500-year leap day cycles. Added simple equivalent leap rule to Pragmatic jitter chart.- 2022.02.24
**Jitter Charts**: added individual jitter charts for the Georgian 132-year leap day cycle and for the Pragmatic Civil Calendar leap week cycle.- 2022.02.23
**Jitter Charts**: Reorganized the jitter charts again, into tabular layouts.- 2022.02.22
**Jitter Charts**: Reorganized the**Leap Day**and**Leap Week**jitter charts so that they are in order of increasing jitter [shown in days in square brackets].- 2022.02.21
**Jitter Charts**: Reorganized the**Lunar**and**Lunisolar**jitter charts sections so that they are in order of increasing jitter [shown in hours in square brackets], and added the 49-month 3-yerm cycle as the least jitter lunar cycle. Corrected 360-month cycle description by changing 21 yerms to 22.- 2022.02.19
**Mixed Radix Conversions**: new version of the workbook — now the**ToRadix**( ) function can also accept mixed, proper, or improper fractions (formerly it only accepted decimal numbers with optional decimal points).- 2022.02.14
**Mixed Radix Conversions**: new version of the workbook — now the**FromRadix**( ) function is able to return fractions, which are used in about 80 mixed radix Tibetan Calendar examples added to theCalendar

worksheet.- 2022.02.09
**Jitter Charts**: Substantial rewrite of the**Lunar**and**Lunisolar**jitter charts sections, with all of the interactive jitter spreadsheets now directly downloadable in macro-enabled XML format (.xlsm file extension).

Fixed Leap Cycle Finder: posted a new version with a bug fixed (no longer crashes when pre-deleting the worksheet that it is about to generate if that worksheet doesn't exist).- 2022.02.06
**Jitter Charts**: moved to a newjitter

subfolder of theintervalmath

directory, which is now reserved for theInterval Math Demo

project. For a while, surfing to the old path will get automatically redirected to the new subfolder.

Mixed Radix Conversions: added more calendar-related examples, some new functions, and improved LibreOffice Calc compatibility.

*Rambam*and the Seasons: improved LibreOffice Calc compatibility.- 2022.02.04
- Mixed Radix Conversions: collected the calendar-related examples onto the new
Calendar

worksheet. - 2022.02.01
- Mixed Radix Conversions: added an important new feature to the web page and the workbook — the
**ApplyUnits**( ) function. - 2022.01.31
- Mixed Radix Conversions: added more applications to web page and examples to workbook.
- 2022.01.30
: added a link to the*Rambam*and the Seasons*Chabad*web site for*Kiddush HaChodesh*, and added a link for downloading my Excel spreadsheet demonstrating all of the calculations in this book, along with instructions for using the spreadsheet in Excel and in LibreOffice Calc.

**Mixed Radix Conversions**: Integer rounding of the least radix value is now the default, instead of truncation, and there is improved control of decimal point rounding of the least radix value. Later added syntax and aesthetic improvements to this web page, and similar changes to theSyntax

worksheet of the conversions workbook. Later expanded the web page's introductory comments and described some typical applications of mixed radix conversions.- 2022.01.27
**Mixed Radix Conversions**: I got tired of LibreOffice Calc harassing me with notifications of the Nth( ) item number being out-of-range, so I changed the function to gracefully return zero in such cases. Added arithmetic to show that Karl Palmen's reduced-jitter modification of the 3000-year cycle retains the same calendar mean year. Also changed all worksheet zoom settings from 200% to 150% to fit better on laptop displays.- 2022.01.26
**Mixed Radix Conversions**: added some documention of the mixed-radix VBA functions to this web page.- 2022.01.26
**Jitter Charts**: added a new chart for the higher-jitter Modified Apo-Balognian 3000-year leap cycle, and made minor improvements to the unpublished Issac Newton cycle chart, see the last two pages of solar leap day jitter charts (PDF).- 2022.01.25
- Added a new web page:
**Mixed Radix Conversions** - 2021.11.27
- The Length of the Lunar Cycle and also Moon and the
*Molad*of the Hebrew Calendar: Added a new paragraph and histogram chart showing the durations of the lunar cycle that occur most frequently. - 2021.11.03
- Moon and the
*Molad*of the Hebrew Calendar: Added to lunar crescent discussions that no visible crescent is ever due to Earth casting any shadow on Moon. - 2021.11.02
- Hebrew Calendar
*Rosh Hashanah*Postponements: Edited the main introductory paragraph and inserted table showing the lengths of the months of*Cheshvan*and*Kislev*, and deleted the*molad*interval as a reason for postponements. Expanded the discussion regarding*when*the Hebrew calendar started using postponements. Added*shofar*blowing at the end as another reason why*Yom Kippur*can't be on Friday. Fixed broken calendar mean year links in the*Rectified*Hebrew Calendar web page, and appended a chart to that section depicting its progressively shorter mean year in comparison with several other calendars and the mean equinoctial and solstitial mean years. - 2021.09.19
- Corrected long-standing embarrassing mistakes in the leap year intervals discussion of the traditional Hebrew calendar in the paragraph starting with "In each cycle of the traditional Hebrew calendar" on the Seasonal Drift of the Traditional Hebrew Calendar and
*Rectified*Hebrew Calendar web pages and made minor improvements to the mean year paragraphs that follow. Other paragraphs that discuss the leap year intervals at this web site were alright. Thanks to Dr. Israel Hanukoglu, Professor of Biochemistry and Molecular Biology, Ariel University, Israel, for bringing these mistakes to my attention. - 2021.08.31
- Updated the
**Locales**web page to fix broken links and removed sites that are no longer available. - 2021.04.02
- Expanded discussion of Gauss’ proposed modification to the traditional
*molad*epoch, which would eliminate all*molad zakein*postponements of*Rosh Hashanah*. - 2021.02.21
- Hebrew Calendar
*Rosh Hashanah*Postponements: Inserted new section after the introductory section — When did the Hebrew calendar start using postponement rules? - 2021.01.12
*Rectified*Hebrew Calendar: Corrected another long-standing mistake, in the first point of the Executive Summary, the omitted 8 years contain 3 leap months, not 2. Thanks to actuary Ethan E. Kra for bringing this mistake to my attention. This web site’s other references to the 8-year subcycle (*octaeteris*) correctly specify the 3 leap months.- 2020.12.10
*Rectified*Hebrew Calendar:- Corrected a long-standing mistake in the
*NisanFirstDay*function (the+1

had been omitted, only from that web page, in the given arithmetic expression). - Added a paragraph in the section entitled
The Locale in Jerusalem

, highlighted there as blue text, explaining that the Nile-Euphrates Midpoint is a more appropriate reference locale than Jerusalem. - Added a paragraph in the section entitled
The Number of Prior Elapsed Months — the

, highlighted there as blue text, explaining why Δ=139 is preferred.*ElapsedMonths*Function - Corrected a mistake in the discussion of The
*MoladMoment*function: previously stated that the mean synodic month was equal to the traditional*molad*interval in the era of Hillel ben Yehuda, changed to the correct earlier era of the Maccabees.

- Corrected a long-standing mistake in the
- 2020.10.08
- Released new
*Kalendis*version**11.157(1678)**. For a brief summary of changes please see the Release Notes. The actual programming changes were completed almost 2 months ago, but release was delayed for thorough testing. - 2020.05.01
- Symmetry Calendar FAQs: corrected error, pointed out today by Landis Markley, about the ratio of 6-year to 5-year leap intervals for the 293-year leap cycle (it's not 2 to 1, it's 33 to 19).
- 2020.04.06
- Counting the
*Omer*for Hebrew year 5780 (Gregorian 2020) - 2020.03.06
- The Duration of Twilight: Full review and revision (last edited more than 5 years ago), in particular correcting seriously incorrect information about the parallactic angle. Posted new experimental version of the Twilight-Charter spreadsheet, compatible with Excel 2007 and later.
- 2019.12.22
- Added The Hebrew Calendar "Y6K" to the menu of Hebrew calendar studies, after fully reviewing and revising that page as appropriate.
- 2019.12.05
- Request for Rain: fixed broken link to Lasker's article at seforimblog.com.

Released new*Kalendis*version**11.156(1677)**, for a brief summary of changes please see the Release Notes. - 2019.11.23
- Released new
*Kalendis*version**11.155(1676)**, for a brief summary of changes please see the Release Notes. - 2019.09.03
- Complete rewrite of
**instructions for using**, in particular detailing new steps for using the current stable version of*Kalendis*under macOS or Linux**WINE**for macOS or Linux. - 2019.08.27
- Appended new section to the Hebrew calendar
*molad*web page, about**inverting the**(given a weekday and*molad**molad*time, find the corresponding Hebrew year and month), with downloadable open-source demonstration Excel workbook. - 2019.08.23
- Released minor new
*Kalendis*version**11.153(1674)**, with a significant old/new lunar crescent Daylight Savings bug fix, and some enhancements, see the Release Notes.

Minor update to Sym454/Sym010 basic arithmetic documentation. - 2019.08.15
- Released minor new
*Kalendis*version**11.150(1671)**. - 2019.08.08
- Released major new
*Kalendis*version**11.150(1670)**. - 2019.05.20
- Fixed Arithmetic Calendar Cycle Jitter: Added a chart of Newton's 5000-year modified Gregorian cycle as the last page (worst jitter) of the solar calendar leap day calendar jitter charts collection. Also modified the 2nd-last page (Herschel's 4000-year modified Gregorian cycle) to include the cycle's exact mean year at the bottom of the page.
- 2019.05.14
*Torah*Portions: revised the final "Verification" section and modified the "Torah-Portions.xls" spreadsheet file to include more columns and to use the same Hebrew calendar year type character codes as used elsewhere on this site, as explained in the spreadsheet.- 2019.04.03
*Rosh Hashanah*postponements: revised headings for rules #3 and #4 with corresponding changes in the bodies of their respective discussions, and added a weekday note to start of the Introduction. Revised 500-year example postponements list accordingly.- 2019.04.01
*Rosh Hashanah*postponements: reviewed, made minor changes (last revision of this page 7 years ago!), most notably expanded the discussion of rule #3. Added to end of introductory section a link to 500-year list of*Rosh Hashanah*dates and postponements.- 2019.01.07
- Symmetry454 Calendar and Symmetry010 Calendar: corrected an error in the FAQs document regarding matching of months between Gregorian and Sym454 calendar, and added table of matching months for Sym010 calendar. The former FAQs version, revised 5 years ago, claimed that any Gregorian month that starts on Monday matches the corresponding Sym454 month, but that was incorrect for March in a Gregorian non-leap year (Gregorian 7 days ahead) and for November in a Gregorian leap year (Gregorian 7 days behind). Thanks to Karl Palmen of the UK for catching this error. The total number of matching months is greater for the Sym010 calendar, but there are 6 types of Gregorian year that don't contain any months matching the Sym010 calendar.
- 2018.12.05
- Request for Rain: fixed broken link to Professor Sokolow's article at The Lookstein Center.
- 2018.10.07
- The Hebrew Calendar: added a link to a lunar full rotation movie.
- The
*Kalendis*Calendar Calculator: updated example reports for omer counts, twilight, zmanim, moladot. - 2018.07.11
- Hebrew Prayer for Maintenance of Good Health: minor improvements to English and French translations.
- Moon and the
*Molad*of the Hebrew Calendar: updated the USNO web site links. - 2018.05.05
- Hebrew Prayer for Maintenance of Good Health: added French translation as third page.
- 2018.04.02
- Kalendis freeware calendrical calculator: major new version 10.118(1599) released.
- 2018.03.20
- Hebrew Prayer for Maintenance of Good Health: added English translation as second page.
- 2018.01.24
- The Length of the Lunar Cycle: minor revisions regarding 111-lunation period, which is actually 109.5 lunations over the longer term.
- 2017.12.19
- Symmetry Calendar Date Conversion Sheets: removed all past years, added years 2018 through 2026.
- 2017.12.03
- The Length of the Lunar Cycle, Periodic Earth-Moon Distance Variations section: minor text edits.
- 2017.11.22
- The Length of the Lunar Cycle, Periodic Earth-Moon Distance Variations section: added new charts depicting the distributions of lunar distances when Moon is actually at perigee or apogee. Later made minor edits to this section and added sources for mean perigee.
- 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