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Re: Phantom Day Ratios
by Brillig
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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 241/161*20 = 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:
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:
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
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