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« Return to Thread: 13 x 353 = 4589-year lunisolar cycle
13 x 353 = 4589-year lunisolar cycle
by Irv Bromberg
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Dear Calendar People:
I was unable to find a reasonably accurate intentionally slightly
short mean month and mean year that yielded an exact integer number of
days in the 353-year, 4366 lunar month cycle.
Thirteen repeats of that cycle = 4589 years or 56758 lunar months,
however, can contain exactly 1676097 days (not divisible by 7),
yielding a mean month of 1676097/56758 = 29+30115/56758 days or 29d
12h 44m 2+17882/28379s (the fraction is just under 2/3 second) and a
calendar mean year of 1676097/4589 = 365+1112/4589 days = 365d 5h 48m
56+1496/4589s (the fraction is just under 1/3 second), or about
365.242318588 days.
The stable calendar season in the present era is at an ecliptic
longitude of about 3°, very close to the target northward equinox.
With that mean month, 4366 lunar months (353 years) contains
128930+7/13 days, so 13 repeats are necessary to make the fraction go
away (adding up to one week). In other words, in the full 4589-year
cycle, 353-year subcycles having 128931 days alternate with 353-year
subcycles having 128930 days, for a total of 7 + 6 = 13 of the 353-
year subcycles, respectively.
The mean month fraction numerator 30115 indicates it has that many
full lunar months per cycle, and 56758-30115=26643 deficient lunar
months. The difference 30115-26643=3472 is the number of yerms per
4589-year cycle. The full 4589-year cycle contains a mix of 61 eras
of 40 yerms with 24 eras of 43 yerms.
The time unit 1/56758 of a day is just a tad more than 1+1/2 seconds.
Compare with the 1/25920 of day "part" used for the traditional Hebrew
calendar molad, which is exactly 3+1/3 seconds.
If this 4589-year cycle were used for reform of the traditional Hebrew
calendar, with a fixed revised molad interval of 29+30115/56758 days,
then the cycle would nicely approximate the mean northward equinox for
at least a millennium longer than the 353-year cycle with traditional
molad interval, but not as long as the 353-year cycle with
progressively shorter molad interval. Correspondence with the mean
lunar cycle would be improved compared to the traditional molad while
retaining similar arithmetic simplicity, but the progressively shorter
molad would have much less drift.
-- Irv Bromberg, Toronto, Canada
<http://www.sym454.org/>
I was unable to find a reasonably accurate intentionally slightly
short mean month and mean year that yielded an exact integer number of
days in the 353-year, 4366 lunar month cycle.
Thirteen repeats of that cycle = 4589 years or 56758 lunar months,
however, can contain exactly 1676097 days (not divisible by 7),
yielding a mean month of 1676097/56758 = 29+30115/56758 days or 29d
12h 44m 2+17882/28379s (the fraction is just under 2/3 second) and a
calendar mean year of 1676097/4589 = 365+1112/4589 days = 365d 5h 48m
56+1496/4589s (the fraction is just under 1/3 second), or about
365.242318588 days.
The stable calendar season in the present era is at an ecliptic
longitude of about 3°, very close to the target northward equinox.
With that mean month, 4366 lunar months (353 years) contains
128930+7/13 days, so 13 repeats are necessary to make the fraction go
away (adding up to one week). In other words, in the full 4589-year
cycle, 353-year subcycles having 128931 days alternate with 353-year
subcycles having 128930 days, for a total of 7 + 6 = 13 of the 353-
year subcycles, respectively.
The mean month fraction numerator 30115 indicates it has that many
full lunar months per cycle, and 56758-30115=26643 deficient lunar
months. The difference 30115-26643=3472 is the number of yerms per
4589-year cycle. The full 4589-year cycle contains a mix of 61 eras
of 40 yerms with 24 eras of 43 yerms.
The time unit 1/56758 of a day is just a tad more than 1+1/2 seconds.
Compare with the 1/25920 of day "part" used for the traditional Hebrew
calendar molad, which is exactly 3+1/3 seconds.
If this 4589-year cycle were used for reform of the traditional Hebrew
calendar, with a fixed revised molad interval of 29+30115/56758 days,
then the cycle would nicely approximate the mean northward equinox for
at least a millennium longer than the 353-year cycle with traditional
molad interval, but not as long as the 353-year cycle with
progressively shorter molad interval. Correspondence with the mean
lunar cycle would be improved compared to the traditional molad while
retaining similar arithmetic simplicity, but the progressively shorter
molad would have much less drift.
-- Irv Bromberg, Toronto, Canada
<http://www.sym454.org/>
« Return to Thread: 13 x 353 = 4589-year lunisolar cycle
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