366-date solar calendar regulated by lunar calendar and Primary Lunisolar Cycles

Dear Amos and Calendar People

 

About the idea of a solar calendar with 366 dates in a year with one date omitted each new yerm in a lunar yerm calendar;

 

I’ve already mentioned that if the lunar calendar were a simple 49-month 3-yerm cycle calendar you get a 725-year cycle. If you modify the lunar calendar to increase the mean month, you reduce the mean year and if you reduced the mean month, you increase the mean year.

 

If you take 85 of these 3-yerm cycles and remove two full yerms of 17 months from these you get a 4131-month cycle of 121991 days and 253 yerms. For 334-years to have the same number of days the number of leap years needs to be 81 and 81+253=334. Hence one can construct the solar calendar by having 366 dates per year and omit one date each new yerm in this 253-yerm cycle. The resulting cycle is the primary 334-year cycle.

 

Another example is got by taking 98 3-yerm cycles and adding 2 full yerms of 17 months to them (so creating 2 yerm-eras). You get a 4836-month cycle of 142810 days and 296 yerms. For 391 years to have the same number of days one needs 95 leap years and 296+95=391. Hence one can construct the solar calendar by having 366 dates per year and omit one date each new yerm in this 296-yerm cycle. The resulting cycle is the primary 391-year cycle.

 

All cycle that can be constructed from these two primary cycles can have such a 366-date solar calendar with one date omitted each new yerm. This includes the 725-year cycle which is simply the sum of the primary 334-year and 391-year cycles.

 

 

With any of these cycles either the mean lunar month or the mean solar year is at least a little too long. For a more accurate calendar, one needs to occasionally omit an additional date from the solar year of 366 dates. Any lunisolar cycle can be constructed by omitting the appropriate number of solar dates in addition to those omitted at a new yerm. For the primary 315-year cycle just one extra omission is needed (because 76+238+1=315). Therefore in any cycle the number of additional omissions is equal to the number of 315-year primary cycles in the mix of primary cycles that make the said cycle.

 

The spreadsheet Lunisolar_391_334_315.xls in http://www.the-light.com/cal/kp_Lunisolar_xls.html calculates the mix of primary cycles in any given cycle, specified as having A years, B leap months and C days in excess of 354 per 12-month year and 384 per 13-month year. Also for any cycle so specified, the number X of additional date omissions is given by

 

X = 12A – 31B – 3C.

 

This formula is used in the aforementioned spreadsheet to work out the number of 315-year primary cycles in the mix.

 

The spreadsheet of http://www.the-light.com/cal/Lunisolar_333.html tells us that the cycles of 649, 706 and 1040 years need one extra omission, cycles of 1689 and 1803 years need two extra omissions, the 3234-year cycle of 98 33-year cycles needs 3 extra omissions and the MPSLC cycle of 6840-years needs 7 extra omissions.

 

See my lunisolar spreadsheets at http://www.the-light.com/cal/kp_Lunisolar_xls.html  for more details.

 

Karl

 

10(08(12

 

 

 

 

Dear Amos And Calendar people

 

You may have heard from me before of the idea of a solar calendar with 366 dates in a year with one date omitted each new yerm in a lunar yerm calendar.

 

If the yerm calendar is a simple 3-yerm cycle with 49 lunar months lasting 1447 days. Then these 3 yerms have 1447+3=1450 dates and so 183 of these 3-yerm cycles have 183*1450 = 366*725 dates and so 725 years, giving rise to the 725-year cycle that Irv recently mentioned.

 

The 3 yerm cycle is always 14 dates short of 4 years and this is useful in working out which lunar date occurs at  the start of each year. One has to add a lunar date at the end of each yerm for the solar date that has no day. For a calendar where year 1 begins on the first day of yerm 1 of the 3-yerm cycle, I reckon that the lunar yerm dates at the start of each year are as follows for the first 80 years of the 725 year cycle are:

 

0001 1(01(01  0002 1(13(13  0003 2(08(24  0004 3(04(05

0005 1(01(15  0006 1(13(27  0007 2(09(09  0008 3(04(19

0009 1(01(29  0010 1(14(11  0011 2(09(23  0012 3(05(04

0013 1(02(13  0014 1(14(25  0015 2(10(07  0016 3(05(18

0017 1(02(27  0018 1(15(09  0019 2(10(21

 

0020 3(06(02  0021 1(03(12  0022 1(15(23  0023 2(11(06

0024 3(06(16  0025 1(03(26  0026 1(16(07  0027 2(11(20

0028 3(07(01  0029 1(04(10  0030 1(16(21  0031 2(12(04

0032 3(07(15  0033 1(04(24  0034 1(17(06  0035 2(12(18

0036 3(07(29  0037 1(05(09  0038 1(17(20

 

0039 2(13(03  0040 3(08(13  0041 1(05(23  0042 2(01(03

0043 2(13(17  0044 3(08(27  0045 1(06(07  0046 2(01(17

0047 2(14(01  0048 3(09(12  0049 1(06(21  0050 2(02(01

0051 2(14(15  0052 3(09(26  0053 1(07(06  0054 2(02(15

0055 2(15(29  0056 3(10(10  0057 1(07(20

 

0058 2(02(29  0059 2(16(13  0060 3(10(24  0061 1(08(04

0062 2(03(14  0063 2(16(27  0064 3(11(09  0065 1(08(18

0066 2(03(28  0067 2(17(12  0068 3(11(23  0069 1(09(03

0070 2(04(12  0071 2(17(26  0072 3(12(07  0073 1(09(17

0074 2(04(26  0075 3(01(09  0076 3(12(21

 

0077 1(10(01  0078 2(05(11  0079 3(01(23  0080 3(13(06

 

If this table were extended, one would eventually reach a year beginning on the extra date added to the end of a yerm 1, this indicates that the year omits its first date and so begins with its second date, which is also the first day of a yerm 2. This won’t happen with the extra dates of  other yerms, because they are even-numbered dates of the 3-yerm cycle, which occur only on even-numbered dates of the year (never new year’s date).

 

NB: Yerm 1 and 2  each have 17 lunar months of 502 days = 503 dates and yerm 3 has 15 lunar months of 443 days  = 444 dates.

 

Years 0001, 0005, 0009, 0013, 0017, 0021, 0025, 0029, 0033, 0037, 0042, 0046, 0050, 0054, 0058, 0062, 0066, 0070, 0075 and 0079 have 366 days because they begin and end in the same yerm.

 

 

Karl

 

10(05(05

 

 

How about the opposite type of solar calendars, those which have more dates in a year than days?  (I can think of useful calendars of at least 366 and 368 dates schemes).  Then, instead of "phantom days" which are date-less days, we'd have "phantom dates", which are day-less dates; that is, a date is skipped every N days.
 
Using the same method, I get the results:
 
366 482.977 483.105 483(365.242236)
367 208.784 208.807 209(365.244019)
368 133.440 133.449 133(365.233083)
369  98.196  98.201  98(365.234694)
370  77.767  77.770  78(365.256410)

Amos Shapir
 

Date: Thu, 8 Jan 2009 13:10:53 +0000
From: karl.palmen@...
Subject: Phantom Day Ratios
To: CALNDR-L@...

Dear Calendar People

There have been on this list a few examples of a solar calendar where a year has a fixed number of ordinary days between which are occasionally  inserted a phantom day.

Here I have a table that shows the ratio of ordinary days to phantom days for various numbers of ordinary days per year (column 1) for a mean year of 365.2422 days (column 2) and a mean year of 365.2424 days (column 3). You can get the ratio of days to phantom days by adding one to the ratio of ordinary days to phantom days.

Subsequent columns have a suggested approximation of the ratio followed by the resulting mean year in days enclosed in().

365  1507.019  1505.776    1507 (365.242203)    1506 (365.242364)

364   293.028   292.981     293 (365.242321)

363   161.895   161.880     162 (365.240740)  1457/9 (365.242278)

362   111.653   111.646   335/3 (365.241791)

361    85.097    85.093   85.1  (365.242068)  936/11 (365.242521)

360    68.673    68.671   206/3 (365.242718)

Karl

10(04(12

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