Dear Irv and Calendar People In
http://www.individual.utoronto.ca/kalendis/hebrew/chelek.htm Irv said about the Hebrew lunar month of a whole numbers of parts (with 1080 parts per hour): “A shorter lunar cycle (smaller denominator, hence fewer parts per day) with a similar mean month could work, but most don't have an integer number of parts per hour (that is, the denominator isn't divisible
by 24).” I thought I’d check this out. The Hebrew calendar has 1080 parts per hour and 18 parts per minute. Could we get an accurate lunar month with fewer parts per hour? If there are also a whole number of parts per minute, then 29 days 12 hours and 44 minutes has a whole number of parts and so any shorter cycle whose mean month is a whole number of parts, cannot have a mean month between 29 days 12 hours
and 44 minutes and the 29 days 12 hours 44 minutes and 1 part of the Hebrew calendar. The shortest mean month more than 29 days 12 hours 44 minutes is one part more than that amount.
For a part of 1/17 minute, that mean month would be just over 29 days 12 hours 44 minutes and 3.5 seconds.
For a part of 1/16 minute, that mean month would be 29 days 12 hours 44 minutes and 3.75 seconds. For a part of 1/15 minute, that mean month would be 29 days 12 hours 44 minutes and 4 seconds. There are plenty of examples that have a whole number of parts per hour, but not per minute.
One example is the 70-yerm era of the Hebrew yerm calendar, which has 1144 months and mean month is 29 days 12 105/143 hours. The 105/143 hours = 44 minutes 3.356643… seconds.
Another example is the 67-yerm era of the Hebrew yerm calendar, which has 1095 months and mean month of 29 days 268/365 hours. The 268/365 hours = 44 minutes 3.287671… seconds. In both cases, the denominator is not divisible by 24, but each has a common divisor with 24 and when divided by this common divisor is less than 1080. Also,
any cycle of less than 1080 months would do, if its mean month were acceptable. Karl 16(11(06 |
Dear Calendar People I generalise this. If you want the mean month or year to be a whole number of parts, so 1/M of a day has a whole number of parts , which is less than N, then this will occur if and only if the number C of months or years in the
cycle is less than N when divided by the highest common divisor of M & C. For example M=24, N=1080 is satisfied by C=1144, because HCD(24,1144) = 8 and 1144/8 = 143 < 1080. But suppose you want a whole number of parts in quarter of an hour that is fewer than 1080/4 = 270, then M=96, N=270 is also satisfied by C=1144, because HCD(96,1144) = 8 and 1144/8 = 143 < 270. If you want a whole number of parts in a minute then M=1440 and N=18. C=1144 months would then fail, because HCD(1440,1140) = 8 and 1144/8 = 143 > 18. Karl 16(11(07 From: Palmen,
Karl (STFC,RAL,ISIS) Dear Irv and Calendar People In
http://www.individual.utoronto.ca/kalendis/hebrew/chelek.htm Irv said about the Hebrew lunar month of a whole numbers of parts (with 1080 parts per hour): “A shorter lunar cycle (smaller denominator, hence fewer parts per day) with a similar mean month could work, but most don't have an integer number of parts per hour (that is, the denominator isn't divisible
by 24).” I thought I’d check this out. The Hebrew calendar has 1080 parts per hour and 18 parts per minute. Could we get an accurate lunar month with fewer parts per hour? If there are also a whole number of parts per minute, then 29 days 12 hours and 44 minutes has a whole number of parts and so any shorter cycle whose mean month is a whole number of parts, cannot have a mean month between 29 days 12 hours
and 44 minutes and the 29 days 12 hours 44 minutes and 1 part of the Hebrew calendar. The shortest mean month more than 29 days 12 hours 44 minutes is one part more than that amount.
For a part of 1/17 minute, that mean month would be just over 29 days 12 hours 44 minutes and 3.5 seconds.
For a part of 1/16 minute, that mean month would be 29 days 12 hours 44 minutes and 3.75 seconds. For a part of 1/15 minute, that mean month would be 29 days 12 hours 44 minutes and 4 seconds. There are plenty of examples that have a whole number of parts per hour, but not per minute.
One example is the 70-yerm era of the Hebrew yerm calendar, which has 1144 months and mean month is 29 days 12 105/143 hours. The 105/143 hours = 44 minutes 3.356643… seconds.
Another example is the 67-yerm era of the Hebrew yerm calendar, which has 1095 months and mean month of 29 days 268/365 hours. The 268/365 hours = 44 minutes 3.287671… seconds. In both cases, the denominator is not divisible by 24, but each has a common divisor with 24 and when divided by this common divisor is less than 1080. Also,
any cycle of less than 1080 months would do, if its mean month were acceptable. Karl 16(11(06 |
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