Hexaseasons

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Hexaseasons

Helios
Dear Calendar People,

The sixth-of-year could be called a hexad or a hexaseason, perhaps good for a desk cube calendar. Here are three things I found.

Three incontestable laws of  hexaseasons

1) Should we elect to choose the simplest and most immediate solar hexaseason cycle, we find [ 95 hexaseasons = 5783 days ]. This is ineluctable because the shorter [ 87 hexaseasons = 5296 days ]  is insufficient.

2) Should we elect to choose the simplest and most immediate  lunar hexaseason cycle, we find [ 1377 months = 668 hexaseasons ] . This is ineluctable because the shorter [ 806 months = 391 hexaseasons ] is insufficient.

3) By coincidence, the mean value of the previous two are nearly exact to one another.
 
--------------------------------------------------------------------

61 61 61 60 61 61 61 61 61 61 61 60 61 61 61 61 61 61 61
60 61 61 61 61 61 61 61 60 61 61 61 61 61 61 61 60 61 61
61 61 61 61 61 60 61 61 61 61 61 61 61 60 61 61 61 61 61
61 61 60 61 61 61 61 61 61 61 60 61 61 61 61 61 61 61 60
61 61 61 61 61 61 61 60 61 61 61 61 61 61 61 60 61 61 61

Let a solar hexaseason be described as deficient if it has 60 days.

The quantity of days hitherto hexaseason H, is
61*( H - 1 ) - FLOOR[ H*( 12 / 95 ) ]

--------------------------------------------------------------------

The Lunar Hexaseason  most usually contains 2 lunar months, but sometimes 3 months. There are many models one could suggest here.

( 30 days ) + ( 29 & 48 / 1483 days ) + ( 41 / 668 )*( 30 days )

Let the 2nd month of the lunar hexaseason, usually 29 days, be described as abundant if it has 30 days, or the entire hexaseason also described as abundant if it contains an abundant 2nd month.

The previous should transfer over to a Pontisso method using the same stem.

( 30 days ) + ( 29 & 48 / 1483 days ) + ( unknown )*( 30 days )

By the third law of hexaseasons, we can expect the leap hexaseasons to follow the 41 / 668 regime.

As Karl was exploring, there is the displacement between the greater solar heap and the lesser lunar heap. This difference tells us when the solar new season's day falls after the lunar new season's day.

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Re: Hexaseasons

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

I thought of a hexaseason calendar and put it at
http://www.hermetic.ch/cal_stud/palmen/rainbow.htm#double .
It is a double Rainbow calendar with a 6-day rainbow week and a rainbow year of 6 seasons.
Five of these seasons begin on the first day of a Gregorian month and the exception (blue) begins on January 31.
Each season has 61 days, unless it contains the end of a 28-day February when it has 60 days.

This can be generalised to any solar leap day calendar. Divide the leap year into six equal hexaseasons of 61 days. In a common year the hexaseason that would have the leap day has one day less so is deficient. If the solar calendar has its leap years spread as smoothly as possible (e.g. 33-year cycle), the deficient hexaseasons can also be spread as smoothly as possible (25 in 198 for the 33-year cycle) without changing the number of days in any year.


The Genius of Helios is to convert these solar hexaseasons into lunar hexaseasons by means of a Pontisso-like method, which he explains below. Using hexaseasons instead of years does not change the number of abundant ones needed nor the number of leap months needed.
However, the rules for abundance become simpler.

Note that once the solar calendar is selected one only needs to select the abundance rule.


I have suggested a Pontisso like calendar in which 20 in 103 years is abundant. This translates to 10 in 309 hexaseasons abundant.
An alternative is 33 in 170 years abundant, which translates to 11 in 340 hexaseasons abundant.

This suggests normally having 1 in 31 hexaseasons abundant and this alone would work for the Gregoriana.
This can then be corrected occasionally by reducing the interval from 31 to 30.
Correcting once 1 in 10 abundant hexaseasons gives the 10 in 309.
Correcting 1 in 11 abundant hexaseasons gives the 11 in 340.
Correcting 1 in 9 abundant hexaseasons gives 9 in 278.

I had also suggested 233 abundant years in 1200 years for the Gregorian calendar and this leads to the 18,000-year cycle. This translates to 233 abundant hexaseasons in 7200 hexaseasons. 23 abundant hexaseasons would be 30 hexaseasons after the previous one and the other 210 would be 31 hexaseasons after the previous one.

A simple (but not very accurate) example can be generated from a solar calendar with a 62-year cycle of 15 leap days, which translates into 372 hexaseasons of which 47 are deficient. Now make every 31st lunar hexaseason abundant. This I believe will give you a Gregoriana of 372 years = 4601 months = 135870 days = 19410 weeks.


I have more comments below, but the most important part of this note has passed.

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 01 June 2015 17:13
To: [hidden email]
Subject: Hexaseasons

Dear Calendar People,

The sixth-of-year could be called a hexad or a hexaseason, perhaps good for
a desk cube calendar. Here are three things I found.

Three incontestable laws of  hexaseasons

1) Should we elect to choose the simplest and most immediate solar
hexaseason cycle, we find [ 95 hexaseasons = 5783 days ]. This is
ineluctable because the shorter [ 87 hexaseasons = 5296 days ]  is
insufficient.

KARL REPLIES:
Also there is [103 hexaseasons = 6270 days]. The mean year is a little too long, but it can be mixed with the 95-hexaseason cycle to form a 33-year cycle [ 198 hexaseasons = 12053 days]. Also 6270 = 3x10x11x19.


HELIOS CONTINUED:
2) Should we elect to choose the simplest and most immediate  lunar
hexaseason cycle, we find [ 1377 months = 668 hexaseasons ] . This is
ineluctable because the shorter [ 806 months = 391 hexaseasons ] is
insufficient.

KARL REPLIES:
We could look for a lunisolar cycle whose number months and number of days is divisible by six. Then divide it by six.
So I look for such a cycle in http://the-light.com/cal/LunisolarA.htm. I do not find any in a quick search.
I decide to look at those cycles that are a whole number of lunar cycles. If that whole number is divisible by six we have one!
So I look in http://the-light.com/cal/LunisolarML.htm . There are none for which the whole number is divisible by six but quite a few for which the whole number is divisible by three. Such a cycle can be doubled to get one for which the whole number is divisible by six.

The simplest is a double 725-year cycle of 1450-years and 366 lunar 49-month cycles. The mean month is 29.530612... days and mean year is 365.2476... days.
Divided by six, we get 1450 hexaseasons = 2989 months = 61 49-month cycles.
The occurrence of 366 and 61 may lead to a simple calendar. I'll leave it to Helios to exploit this.

Others are
Double 3014-year cycle = 6028 years of 114 654-month cycles
Double 3511-year cycle = 7022 years of 90 965-month cycles
Double 5515-year cycle = 11030 years of 78 lunar 1749-month cycles
Double 8464-year cycle = 16928 years of 210 lunar 997-month cycles

However, if one applies the Pontisso method (not yet mentioned by Helios), one does not need to concern oneself with the number of days in the lunisolar cycle, because the solar calendar will take care of that. I can then look for cycles for which just number of months is divisible by six.
In http://the-light.com/cal/Lunisolar4.html I find the 391-year cycle and this leads to the [ 806 months = 391 hexaseasons ] that Helios found. Also two 334-year cycles lead to the [ 1377 months = 668 hexaseasons ] that Helios found.


HELIOS CONTINUED:
3) By coincidence, the mean value of the previous two are nearly exact to
one another.
 
--------------------------------------------------------------------

61 61 61 60 61 61 61 61 61 61 61 60 61 61 61 61 61 61 61
60 61 61 61 61 61 61 61 60 61 61 61 61 61 61 61 60 61 61
61 61 61 61 61 60 61 61 61 61 61 61 61 60 61 61 61 61 61
61 61 60 61 61 61 61 61 61 61 60 61 61 61 61 61 61 61 60
61 61 61 61 61 61 61 60 61 61 61 61 61 61 61 60 61 61 61

Let a solar hexaseason be described as deficient if it has 60 days.

The quantity of days hitherto hexaseason H, is
61*( H - 1 ) - FLOOR[ H*( 12 / 95 ) ]

--------------------------------------------------------------------

The Lunar Hexaseason  most usually contains 2 lunar months, but sometimes 3
months. There are many models one could suggest here.

( 30 days ) + ( 29 & 48 / 1483 days ) + ( 41 / 668 )*( 30 days )

Let the 2nd month of the lunar hexaseason, usually 29 days, be described as
abundant if it has 30 days, or the entire hexaseason also described as
abundant if it contains an abundant 2nd month.

The previous should transfer over to a Pontisso method using the same stem.

( 30 days ) + ( 29 & 48 / 1483 days ) + ( unknown )*( 30 days )

By the third law of hexaseasons, we can expect the leap hexaseasons to
follow the 41 / 668 regime.

As Karl was exploring, there is the displacement between the greater solar
heap and the lesser lunar heap. This difference tells us when the solar new
season's day falls after the lunar new season's day.

KARL REPLIES:
I don't understand what Helios means by "the greater solar heap and the lesser lunar heap".

He appears to be suggesting that 41 in 668 lunar hexaseasons  have a leap month. Three cycles of 688 lunar hexaseasons then make a 334-year cycle of 123 leap months.
Also he appears to suggest that 48 in 1483 lunar hexaseasons are abundant. I have no idea why he chose 48/1483. My suggestions have been 1/31, 9/278, 10/309, 11/340 & 233/1200.


The correct approach is

(61 - deficient/hexaseasons days) = ( 30 days ) +( 29 & abundant/hexaseasons days ) + ( unknown )*( 30 days )

Then 'unknown' can be calculated, but is not needed in the calendar rules.


Karl Palmen

14(17(16

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Re: Hexaseasons

Helios
Dear Karl and Calendar People,

We can equate the solar and lunar hexaseasons and solve for the abundance x.

mean hexaseason = 5783 / 95 = 59 + x + 30*{ [ ( 5783 / 95 ) / M ] - 2 }

M = 30 / [ 1 + ( 95 / 5783 )*( 1 - x ) ]
x = 1 + ( 5783 / 95 )*( 1 - 30 / M )

x = .0323664923557 or examining continued fractions,

x = 10 / 309 ( M = 29.53058691 )
x = 19 / 587 ( M = 29.53058954 )
x = 29 / 896 ( M = 29.53058863 )
 x = 48 / 1483 ( M = 29.53058899 )
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Re: Hexaseasons

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

Thank you Helios for giving explanation of 48/1483.

I consider 10/309 (first approximation) sufficiently accurate, taking account of the slow decrease of the number of days in the mean synodic month.


My formula for the mean month of a Pontisso calendar

M = 30*(365 + s)/(371 + x - a)

can be converted to use hexaseasons.

Let x be the proportion of abundant hexaseasons (as used by Helios below) and d the proportion of deficient hexaseasons. Then

a = 6*x
s = 1 - 6*d

We then get

M = 30*(366 - 6*d)/(372 - 6*d - 6*x)

Noting that both 366 and 372 are divisible by six, we can cancel out the sixes to get

M = 30*(61 - 12/95)/(62 - d - x)


Helios's 95-hexaseason cycle of 5783 days has 61*95 - 5783 = 12 deficient hexaseasons and so we then get
M = 30*(61 - 12/95)/(62 - (12/95) - x)
This I believe is identical to
M = 30 / [ 1 + ( 95 / 5783 )*( 1 - x ) ]
But I prefer to deal with smaller numbers.

Note that 61 - d is the mean number of days in a hexaseason and 62 - d - x is the mean number of tithis (1/30 month) in a hexaseason.


The x = 29 / 896 approximation is interesting, because it suggests a 896-year cycle of seven 128-year cycles, for which d = 97/768. Then
M = 30 * ( 61 - 97/768 )/( 62 - 97/768 - 29/896 ) = 29.530588734.... days and follows the 268800-year cycle listed at
http://the-light.com/cal/LunisolarA.htm equal to 210 128-year cycles.
Also the 29 is the number of additional leap weeks required the Divide-By-Six triple 896-year cycle.

Note that 97/768 can be made from seven of 12/95 and one of 13/103.
Helios must have used a unusually small mean year to miss the 103-hexaseason cycle of 6270 days, which has d = 13/103.


Karl

14(17(17


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 03 June 2015 08:00
To: [hidden email]
Subject: Re: Hexaseasons

Dear Karl and Calendar People,

We can equate the solar and lunar hexaseasons and solve for the abundance x.

mean hexaseason = 5783 / 95 = 59 + x + 30*{ [ ( 5783 / 95 ) / M ] - 2 }

M = 30 / [ 1 + ( 95 / 5783 )*( 1 - x ) ]
x = 1 + ( 5783 / 95 )*( 1 - 30 / M )

x = .0323664923557 or examining continued fractions,

x = 10 / 309 ( M = 29.53058691 )
x = 19 / 587 ( M = 29.53058954 )
x = 29 / 896 ( M = 29.53058863 )
 x = 48 / 1483 ( M = 29.53058899 )




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Re: Hexaseasons

Karl Palmen - UKRI STFC
In reply to this post by Helios
Dear Helios and Calendar People

I think Helios missed what I think is the biggest virtue of the lunar hexaseason system.

This is the approximation x = 1 / 31.

This means that most abundant hexaseasons occur 31 hexaseasons = 5 years & 1 hexaseason after the previous abundant hexaseason. There are a few exceptional abundant hexaseasons that occur 30 hexaseaons = 5 years after the previous. I refer to such an abundant hexaseason as an early abundant hexaseason.

Helios listed his approximations:
x = 10 / 309 ( M = 29.53058691 )  -> ( 1 / 10 ) abundant hexaseasons are early
x = 19 / 587 ( M = 29.53058954 )  -> ( 2 / 19 ) abundant hexaseasons are early
x = 29 / 896 ( M = 29.53058863 )  -> ( 3 / 29 ) abundant hexaseasons are early
x = 48 / 1483 ( M = 29.53058899 ) -> ( 5 / 48 ) abundant hexaseasons are early

My opinion is that the following is sufficiently accurate:
x = 10 / 309 ( M = 29.53058691 )-> ( 1 / 10 ) abundant hexaseasons are early
and there is no point in adding complication to make the mean month more accurate than about five decimal places, because such additional accuracy would perish, from change in the number of days in the synodic month, before becoming worth the complication.


A similar situation exists with the deficient hexaseasons. Most deficient hexaseasons occur 8 hexaseasons after the previous deficient hexaseason. If they all occurred so, the Julian mean year of 365.25 days would result. To correct this there is an occasional early deficient hexaseason and that occurs just once in Helios's 95-hexaseason cycle.
d = 12 / 95 ( Y = 365.2410526... ) -> ( 1 / 12 ) deficient hexaseasons are early.

Also of interest
d = 25 / 198 ( Y = 365.242424... ) -> ( 2 / 25 ) deficient hexaseasons are early (= 33-year cycle),
which has
x = 10 / 309 ( M = 29.5305873... )-> ( 1 / 10 ) abundant hexaseasons are early.

Karl

14(17(18

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 03 June 2015 08:00
To: [hidden email]
Subject: Re: Hexaseasons

Dear Karl and Calendar People,

We can equate the solar and lunar hexaseasons and solve for the abundance x.

mean hexaseason = 5783 / 95 = 59 + x + 30*{ [ ( 5783 / 95 ) / M ] - 2 }

M = 30 / [ 1 + ( 95 / 5783 )*( 1 - x ) ]
x = 1 + ( 5783 / 95 )*( 1 - 30 / M )

x = .0323664923557 or examining continued fractions,

x = 10 / 309 ( M = 29.53058691 )
x = 19 / 587 ( M = 29.53058954 )
x = 29 / 896 ( M = 29.53058863 )
 x = 48 / 1483 ( M = 29.53058899 )




--
View this message in context: http://calndr-l.10958.n7.nabble.com/Hexaseasons-tp15872p15881.html
Sent from the Calndr-L mailing list archive at Nabble.com.