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

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