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Phantom Day Ratios

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	Phantom Day Ratios by Karl Palmen :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Phantom Day Ratios 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 -- Scanned by iCritical.
	Re: Phantom Day Ratios by Amos Shapir :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Some parts of this message have been removed. Learn more about Nabble's security policy. 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 -- Scanned by iCritical. check out the rest of the Windows Live™. More than mail–Windows Live™ goes way beyond your inbox. More than messages
	Re: Phantom Day Ratios by Karl Palmen :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Some parts of this message have been removed. Learn more about Nabble's security policy. Dear Amos and Calendar People Thank you Amos for giving this alternative, which I considered for a later note. In general the calendar will repeat in as many years as the numerator of the ratio (e.g. 293 years for the 364 date example). However, if the number of dates per year has a common divisor with this numerator, the numerator divided by this common divisor would give the number of years. This happens in Amos’s example of 483 for 366 dates, which repeats once every 161 years and was first mentioned by Victor. Amos has not considered any fractional approximations. Here are some: 366 482.977 483.105 483(365.242236) 367 208.784 208.807 209(365.244019) 208.8(365.242337) 368 133.440 133.449 133(365.233083) 400/3(365.24) 1201/9(365.242298) 369 98.196 98.201 98(365.234694) 98.2(365.2423625) 370 77.767 77.770 78(365.256410) 700/9(365.242857) 311/4(365.241158) 1011/13(365.242334) Karl 10(04(12 From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir Sent: 08 January 2009 15:31 To: CALNDR-L@... Subject: Re: Phantom Day Ratios 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 -- Scanned by iCritical.
	Re: Phantom Day Ratios by Brillig :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Some parts of this message have been removed. Learn more about Nabble's security policy. Dear Karl and Calendar People, My favorite calendar involving the number 161 is the one I mentioned a few years ago that has alternating months of 1 and 2 days. This pattern of alternating months repeats for 161 days, then starts over. Each year has 244 months. Karl then modified this calendar, making months generally twice as long, having usually 3 days each, and occasionally 2. These are discussed in the document http://the-light.com/cal/ve244.txt Additionally, I came up with a lunisolar scheme of some sort. I’ve lost my notes, so I’ll have to reconstruct it, unless someone has a copy of the conversation where I mentioned it originally. If you arrange the 1 and 2 day months described, above, into groups of 20, you have what I’ll call now a standard lunation. A lunation consists of 20 of these very short months. The average length of a lunation with no adjustment, is 241/16120 = 29.938 days. However, in http://the-light.com/cal/ve161m.txt I show how the pattern can be shifted either every 3 or every 4 lunations to get a better value for the mean lunation length. I don’t recall what the pattern of shifts was. With some rough calculations, it looks like a shift should occur approximately every 11/3 lunations. I’ll see if I can find the original emails where I discussed this. I also crocheted a Metonic cycle using this scheme. See http://the-light.com/cal/vecrochet0.jpg for an illustration. The crochet pattern consists of these 1 and 2 day months. A 1 day month is simply a double crochet. A 2 day month is two double crochets placed in the same spot, with the top loop being drawn through both stitches. The color changes every 161 days. Each row is a new year, so the Metonic cycle is given by 19 rows. Victor From:* East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Palmen, KEV (Karl) Sent: Thursday, January 08, 2009 9:57 AM To: CALNDR-L@... Subject: Re: Phantom Day Ratios Dear Amos and Calendar People Thank you Amos for giving this alternative, which I considered for a later note. In general the calendar will repeat in as many years as the numerator of the ratio (e.g. 293 years for the 364 date example). However, if the number of dates per year has a common divisor with this numerator, the numerator divided by this common divisor would give the number of years. This happens in Amos’s example of 483 for 366 dates, which repeats once every 161 years and was first mentioned by Victor. Amos has not considered any fractional approximations. Here are some: 366 482.977 483.105 483(365.242236) 367 208.784 208.807 209(365.244019) 208.8(365.242337) 368 133.440 133.449 133(365.233083) 400/3(365.24) 1201/9(365.242298) 369 98.196 98.201 98(365.234694) 98.2(365.2423625) 370 77.767 77.770 78(365.256410) 700/9(365.242857) 311/4(365.241158) 1011/13(365.242334) Karl 10(04(12 From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir Sent: 08 January 2009 15:31 To: CALNDR-L@... Subject: Re: Phantom Day Ratios 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 -- Scanned by iCritical.
	161-year calendar etc. RE: Phantom Day Ratios by Karl Palmen :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Some parts of this message have been removed. Learn more about Nabble's security policy. Dear Victor and Calendar People My suggested calendar where most of the months have three days and some have two days, could be realised by having a year of 366 dates, where every 483^rd date is omitted. The months would then have 3 dates. This causes every 161^st month to be short. This does not work with Victor’s months of 1 or 2 days. One could time months alternating between 1 and 2 dates, so that only a 2-date month has a date omitted, but this would produce three consecutive 1-day months, rather than two as in Victor’s calendar. One could instead have months alternating between 30 and 31 dates, in this cases a date would be omitted in months of either length, so some months would have 29 days. One can get an accurate lunar calendar cycle by omitting 11 dates every 703 dates from months of 30 dates. I also note that the 482-day cycle of date omission in the solar calendar is approximately one mean yerm long. So one could have a lunar calendar where months alternate between 29 and 30 dates, except after each omission of a date in the solar calendar when an additional month of 30 days is inserted. The mean lunar month would be (59482)/(2482-1) = 29.530633 days. Also I invented a lunisolar calendar where each solar year has 372 dates, but a date is omitted at the end of each 29-day lunar month and another date is omitted once every 13^th lunar month. See http://www.hermetic.ch/cal_stud/palmen/lunar13.htm . Karl 10(03(13 From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Engel,Victor Sent: 08 January 2009 17:02 To: CALNDR-L@... Subject: Re: Phantom Day Ratios Dear Karl and Calendar People, My favorite calendar involving the number 161 is the one I mentioned a few years ago that has alternating months of 1 and 2 days. This pattern of alternating months repeats for 161 days, then starts over. Each year has 244 months. Karl then modified this calendar, making months generally twice as long, having usually 3 days each, and occasionally 2. These are discussed in the document http://the-light.com/cal/ve244.txt Additionally, I came up with a lunisolar scheme of some sort. I’ve lost my notes, so I’ll have to reconstruct it, unless someone has a copy of the conversation where I mentioned it originally. If you arrange the 1 and 2 day months described, above, into groups of 20, you have what I’ll call now a standard lunation. A lunation consists of 20 of these very short months. The average length of a lunation with no adjustment, is 241/16120 = 29.938 days. However, in http://the-light.com/cal/ve161m.txt I show how the pattern can be shifted either every 3 or every 4 lunations to get a better value for the mean lunation length. I don’t recall what the pattern of shifts was. With some rough calculations, it looks like a shift should occur approximately every 11/3 lunations. I’ll see if I can find the original emails where I discussed this. I also crocheted a Metonic cycle using this scheme. See http://the-light.com/cal/vecrochet0.jpg for an illustration. The crochet pattern consists of these 1 and 2 day months. A 1 day month is simply a double crochet. A 2 day month is two double crochets placed in the same spot, with the top loop being drawn through both stitches. The color changes every 161 days. Each row is a new year, so the Metonic cycle is given by 19 rows. Victor From:* East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Palmen, KEV (Karl) Sent: Thursday, January 08, 2009 9:57 AM To: CALNDR-L@... Subject: Re: Phantom Day Ratios Dear Amos and Calendar People Thank you Amos for giving this alternative, which I considered for a later note. In general the calendar will repeat in as many years as the numerator of the ratio (e.g. 293 years for the 364 date example). However, if the number of dates per year has a common divisor with this numerator, the numerator divided by this common divisor would give the number of years. This happens in Amos’s example of 483 for 366 dates, which repeats once every 161 years and was first mentioned by Victor. Amos has not considered any fractional approximations. Here are some: 366 482.977 483.105 483(365.242236) 367 208.784 208.807 209(365.244019) 208.8(365.242337) 368 133.440 133.449 133(365.233083) 400/3(365.24) 1201/9(365.242298) 369 98.196 98.201 98(365.234694) 98.2(365.2423625) 370 77.767 77.770 78(365.256410) 700/9(365.242857) 311/4(365.241158) 1011/13(365.242334) Karl 10(04(12 From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir Sent: 08 January 2009 15:31 To: CALNDR-L@... Subject: Re: Phantom Day Ratios 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 -- Scanned by iCritical. -- Scanned by iCritical.
	Re: 161-year calendar etc. RE: Phantom Day Ratios by Brillig :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Dear Karl and Calendar People, On 1/9/09, Palmen, KEV (Karl) <karl.palmen@...> wrote: > My suggested calendar where most of the months have three days and some have > two days, could be realised by having a year of 366 dates, where every 483rd > date is omitted. The months would then have 3 dates. This causes every 161st > month to be short. This does not work with Victor's months of 1 or 2 days. I'm not sure what Karl means here. It's my recollection that Karl's first formulation of this scheme was to simply merge each two consecutive month of my 1 and 2 day alternating months to form his 2 and 3 day months. The result would be 80 months of 3 days for the first cycle of 161 days, leaving 1 day left over. This remainder is combined with the first day of the next 161 day cycle to form a 2 day month. The remainder of the second 161 day cycle forms another string of 80 3-day months. The pattern is thus 80 3-day months, 1 2-day month, 80 3-day months. Rearranging the months, it's simply 160 3-day months followed by a single 2-day month. In other words, a single cycle of the 2/3-day month scheme corresponds to two cycles of the 1/2-day month scheme. Reviewing the archives for January, 2007, I need to also clarify that dividing the 1/2-day months into groups of twenty months to approximate lunations appears to ahve been Karl's idea. Victor
	Re: 161-year calendar etc. RE: Phantom Day Ratios by Karl Palmen :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Dear Victor and Calendar People -----Original Message----- From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Victor Engel Sent: 09 January 2009 16:13 To: CALNDR-L@... Subject: Re: 161-year calendar etc. RE: Phantom Day Ratios Dear Karl and Calendar People, On 1/9/09, Palmen, KEV (Karl) <karl.palmen@...> wrote: > My suggested calendar where most of the months have three days and some have > two days, could be realised by having a year of 366 dates, where every 483rd > date is omitted. The months would then have 3 dates. This causes every 161st > month to be short. This does not work with Victor's months of 1 or 2 days. I'm not sure what Karl means here. It's my recollection that Karl's first formulation of this scheme was to simply merge each two consecutive month of my 1 and 2 day alternating months to form his 2 and 3 day months. The result would be 80 months of 3 days for the first cycle of 161 days, leaving 1 day left over. This remainder is combined with the first day of the next 161 day cycle to form a 2 day month. The remainder of the second 161 day cycle forms another string of 80 3-day months. The pattern is thus 80 3-day months, 1 2-day month, 80 3-day months. Rearranging the months, it's simply 160 3-day months followed by a single 2-day month. In other words, a single cycle of the 2/3-day month scheme corresponds to two cycles of the 1/2-day month scheme. KARL SAYS: The 160 3-day months followed by a single 2-day month can be constructed out of 161 3-date months if the every 483rd date is omitted. This causes every 161st month to have 2 days instead of the usual 3 days. Karl 10(04(15 till noon -- Scanned by iCritical.
	366-date solar calendar regulated by lunar calendar RE: Phantom Day Ratios by Karl Palmen :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Some parts of this message have been removed. Learn more about Nabble's security policy. 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 1831450 = 366725 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 From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir Sent: 08 January 2009 15:31 To: CALNDR-L@... Subject: Re: Phantom Day Ratios 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 -- Scanned by iCritical. check out the rest of the Windows Live™. More than mail–Windows Live™ goes way beyond your inbox. More than messages Scanned by iCritical.
	366-date solar calendar regulated by lunar calendar and Primary Lunisolar Cycles by Karl Palmen :: Rate this Message: Reply to Author \| View Threaded \| Show Only this Message Some parts of this message have been removed. Learn more about Nabble's security policy. 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 From: Palmen, KEV (Karl) Sent: 30 January 2009 15:42 To: 'East Carolina University Calendar discussion List' Subject: 366-date solar calendar regulated by lunar calendar RE: Phantom Day Ratios 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 1831450 = 366725 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 From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir Sent: 08 January 2009 15:31 To: CALNDR-L@... Subject: Re: Phantom Day Ratios 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 -- Scanned by iCritical. check out the rest of the Windows Live™. More than mail–Windows Live™ goes way beyond your inbox. More than messages Scanned by iCritical.