tithi as 1 / 29th month

[Helios]

Suppose we have a tithi which is newly defined as 1 / 29th month.

T = 1 & 32 / 1749 days = 1781 / 1749 days

With a sundial on the moon using 29 "hours" and the moon divided into 29 meridians, the tithi resembles a lunar hour. A lunar observer would note that

1749 lunar hours ( tithis ) = 1781 terrestrial days

Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).

1 / S = 1 - 1 / T

S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
T = 1 + 1 / ( S - 1 )  = 1 & 32 / 1749 days

The skip day is to have the same tithi value ( 1 - 29 ) as that of the previous day. If the mathermatics is correct, the tithi value should repeat on the skip day.

Should skip days be distributed evenly spaced and symmetrically throughout the 1781 period, then they are found by

( 32*D + 890 )MOD( 1781 ) < 32

The skip days are then

28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808, 863,
919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531, 1587, 1642, 1698, 1754

This leaves only to formulate the "Z number" that converts days into tithis by subtraction.

Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) / 1781 ]

Finally,

tithi count = day count - Z

moon age = tithi count MOD( 29 )

[Amos Shapir]

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A better approximation may use the fact that the difference between a lunar month and a solar month (defined as 1/12 of a solar year; yes I know this could vary), is very close to 29/32 solar day.  So using a unit of 1/32 day (45 mins) we can define a solar month of 974 units and a lunar month of 945 units.  For an even better approximation, these may be shortened by one unit every 48 or 47 months, respectively.

Amos Shapir
 



 

> Date: Fri, 11 Sep 2009 05:01:20 -0700
> From: suntheorem@...
> Subject: tithi as 1 / 29th month
> To: CALNDR-L@...
>
> Suppose we have a tithi which is newly defined as 1 / 29th month.
>
> T = 1 & 32 / 1749 days = 1781 / 1749 days
>
> With a sundial on the moon using 29 "hours" and the moon divided into 29
> meridians, the tithi resembles a lunar hour. A lunar observer would note
> that
>
> 1749 lunar hours ( tithis ) = 1781 terrestrial days
>
> Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).
>
> 1 / S = 1 - 1 / T
>
> S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
> T = 1 + 1 / ( S - 1 ) = 1 & 32 / 1749 days
>
> The skip day is to have the same tithi value ( 1 - 29 ) as that of the
> previous day. If the mathermatics is correct, the tithi value should repeat
> on the skip day.
>
> Should skip days be distributed evenly spaced and symmetrically throughout
> the 1781 period, then they are found by
>
> ( 32*D + 890 )MOD( 1781 ) < 32
>
> The skip days are then
>
> 28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808,
> 863,
> 919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531, 1587,
> 1642, 1698, 1754
>
> This leaves only to formulate the "Z number" that converts days into tithis
> by subtraction.
>
> Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) / 1781 ]
>
> Finally,
>
> tithi count = day count - Z
>
> moon age = tithi count MOD( 29 )
>
> --
> View this message in context: http://www.nabble.com/tithi-as-1---29th-month-tp25399679p25399679.html
> Sent from the Calndr-L mailing list archive at Nabble.com.
>

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[Karl Palmen]

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Dear Amos, Helios and Calendar People

 

Amos’s lunar approximation of 945/32 days is much poorer than Helios’s 29*1781/1749.

 

Amos’s approximation can be expressed as 945/32 = 30*63/64 and so is got by skipping a tithi of 1/30 lunar month once every 63 days (7 weeks). This can be corrected by having every 11^th skip occur 62 days after the previous skip instead of the usual 63 days. Then we have a 692-day cycle of 703 tithis making up a 703-month cycle of 692*30=20760 days with a mean month of 29.5305832 days. This sequence of skips is considerably simpler than Helios’s suggested skips.

 

 I’ve stated that 118 of these 703-month cycle can be approximated to 6707 years.

Also 39 of Helios’s 1749-month cycles can be approximated to 5515 years.

Amos approximates a solar year to 974*12/32 = 365.25 days.

 

Amos’s approximation creates a lunisolar 1260-year cycle with

1260*12*974 = 945*16*974 units =

16*974= 15584 lunar months.

This cycle is made up of four 315-year cycles with 116 leaps months each. This more accurate than  the 19-year cycle.

Consequently, shortening the mean  unit slightly would improve accuracy significantly. Perhaps, this could be done as suggested by skipping a unit once every 48 lunar months or 47 solar months (45360 or  45778 units respectively).

 

Karl

 

10(13(03

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir
Sent: 11 September 2009 22:44
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

 

A better approximation may use the fact that the difference between a lunar month and a solar month (defined as 1/12 of a solar year; yes I know this could vary), is very close to 29/32 solar day.  So using a unit of 1/32 day (45 mins) we can define a solar month of 974 units and a lunar month of 945 units.  For an even better approximation, these may be shortened by one unit every 48 or 47 months, respectively.

Amos Shapir
 



 
> Date: Fri, 11 Sep 2009 05:01:20 -0700
> From: suntheorem@...
> Subject: tithi as 1 / 29th month
> To: CALNDR-L@...
>
> Suppose we have a tithi which is newly defined as 1 / 29th month.
>
> T = 1 & 32 / 1749 days = 1781 / 1749 days
>
> With a sundial on the moon using 29 "hours" and the moon divided into 29
> meridians, the tithi resembles a lunar hour. A lunar observer would note
> that
>
> 1749 lunar hours ( tithis ) = 1781 terrestrial days
>
> Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).
>
> 1 / S = 1 - 1 / T
>
> S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
> T = 1 + 1 / ( S - 1 ) = 1 & 32 / 1749 days
>
> The skip day is to have the same tithi value ( 1 - 29 ) as that of the
> previous day. If the mathermatics is correct, the tithi value should repeat
> on the skip day.
>
> Should skip days be distributed evenly spaced and symmetrically throughout
> the 1781 period, then they are found by
>
> ( 32*D + 890 )MOD( 1781 ) < 32
>
> The skip days are then
>
> 28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808,
> 863,
> 919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531, 1587,
> 1642, 1698, 1754
>
> This leaves only to formulate the "Z number" that converts days into tithis
> by subtraction.
>
> Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) / 1781 ]
>
> Finally,
>
> tithi count = day count - Z
>
> moon age = tithi count MOD( 29 )
>
> --
> View this message in context: http://www.nabble.com/tithi-as-1---29th-month-tp25399679p25399679.html
> Sent from the Calndr-L mailing list archive at Nabble.com.
>

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[Karl Palmen]

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Dear Amos and Calendar People

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir
Sent: 11 September 2009 22:44
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

 

A better approximation may use the fact that the difference between a lunar month and a solar month (defined as 1/12 of a solar year; yes I know this could vary), is very close to 29/32 solar day.  So using a unit of 1/32 day (45 mins) we can define a solar month of 974 units and a lunar month of 945 units.  For an even better approximation, these may be shortened by one unit every 48 or 47 months, respectively.

If the correction were done exactly once every 49 lunar months (49*945 = 46305 = 1447*32+1 units). The lunar months would follow the 49-month cycle of 1447 days giving a mean lunar month of 29.530612 days and the mean year would have 365.242112 days.

Karl

10(13(03

Amos Shapir
 

 

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[Karl Palmen]

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Dear Amos and Calendar People

 

Here are some mean year and months for different rates of correction of the unit of 1/945 lunar month and 1/974 solar month. The unit is normally 1/32 day (45 minutes). A correction consists of skipping a unit so making the skipped unit have zero duration. This is equivalent to removing one unit of 1/32 day from a month (lunar or solar).

 

Correction             Units per Correction   Mean Lunar Month   Mean Year

None                   Infinite               29.53125 days      365.25 days

1 per 4 years          46752                  29.530618… days    365.2421875 days

1 per 49 lunar months  46305                  29.530612… days    365.242112… days

1 per 48 lunar months  45360                  29.530599… days    365.2419775… days

1 per 47 lunar months  44415                  29.530585… days    365.241776… days

 

If correction is done by removing one unit from a month, then it can be done at a different rate for solar and lunar months. Amos’s suggestion of a solar month correction once every 48 months (4 years) and a lunar month correction once every 47 months is very accurate.

 

There is an advantage in doing both corrections at the same rate. Then one can ignore these corrections when reckoning lunar months in relation to solar months (ignoring days). A 315-year cycle then occurs for solar and lunar months ignoring days.

 

 

Karl

 

10(13(03 till noon

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir
Sent: 11 September 2009 22:44
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

 

A better approximation may use the fact that the difference between a lunar month and a solar month (defined as 1/12 of a solar year; yes I know this could vary), is very close to 29/32 solar day.  So using a unit of 1/32 day (45 mins) we can define a solar month of 974 units and a lunar month of 945 units.  For an even better approximation, these may be shortened by one unit every 48 or 47 months, respectively.

Amos Shapir
 



 
> Date: Fri, 11 Sep 2009 05:01:20 -0700
> From: suntheorem@...
> Subject: tithi as 1 / 29th month
> To: CALNDR-L@...
>
> Suppose we have a tithi which is newly defined as 1 / 29th month.
>
> T = 1 & 32 / 1749 days = 1781 / 1749 days
>
> With a sundial on the moon using 29 "hours" and the moon divided into 29
> meridians, the tithi resembles a lunar hour. A lunar observer would note
> that
>
> 1749 lunar hours ( tithis ) = 1781 terrestrial days
>
> Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).
>
> 1 / S = 1 - 1 / T
>
> S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
> T = 1 + 1 / ( S - 1 ) = 1 & 32 / 1749 days
>
> The skip day is to have the same tithi value ( 1 - 29 ) as that of the
> previous day. If the mathermatics is correct, the tithi value should repeat
> on the skip day.
>
> Should skip days be distributed evenly spaced and symmetrically throughout
> the 1781 period, then they are found by
>
> ( 32*D + 890 )MOD( 1781 ) < 32
>
> The skip days are then
>
> 28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808,
> 863,
> 919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531, 1587,
> 1642, 1698, 1754
>
> This leaves only to formulate the "Z number" that converts days into tithis
> by subtraction.
>
> Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) / 1781 ]
>
> Finally,
>
> tithi count = day count - Z
>
> moon age = tithi count MOD( 29 )
>
> --
> View this message in context: http://www.nabble.com/tithi-as-1---29th-month-tp25399679p25399679.html
> Sent from the Calndr-L mailing list archive at Nabble.com.
>

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[Karl Palmen]

Dear Helios and Calendar People

Some time ago I investigated the idea of a tithi equal to 1/29 lunar
month.

I found that a year of 365.2422 days has 358.67972 tithis for a lunar
month of 29.53059 days.

This suggests approximating the year to 358.68 tithis. This can be done
with a 25-year cycle with 8 short years of 358 tithis and the other
years of 359 tithis. The eight short years can be arranged into a Helios
cycle, in which case they'd be years 02, 05, 08, 11, 14, 17, 20, 23 of
the 25-year cycle and so be spaced once every three years within each
25-year cycle.

A 359-tithi year gives an epact increment of 359-(29*12)= 11 mod 29,
while a short years of 358 tithis give an epact increment of 10 mod 29.
This gives rise the a 725-year cycle of 267 leap months with annual
epacts beginning thus:

00 11  21 03 14  24 06 17  27 09 20  01 12 23  04 15 26  07 18 00  10 21
03  13 24
06 17  27 09 20  01 12 23  04 15 26  07 18 00  10 21 03  13 24 06  16 27
09  19 01
12 23  04 15 26 etc...

These epacts repeat once every 19 years, except that two epacts are
lowered by 1 mod 29. The following rearrangement of the above epacts
shows this.

00 11  21 03 14  24 06 17  27 09 20  01 12 23  04 15 26  07 18
00  10 21 03  13 24 06 17  27 09 20  01 12 23  04 15 26  07 18
00  10 21 03  13 24 06  16 27 09  19 01 12 23  04 15 26 etc...

Karl

10(13(05

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 11 September 2009 13:01
To: CALNDR-L@...
Subject: tithi as 1 / 29th month

Suppose we have a tithi which is newly defined as 1 / 29th month.

T = 1 & 32 / 1749 days = 1781 / 1749 days

With a sundial on the moon using 29 "hours" and the moon divided into 29
meridians, the tithi resembles a lunar hour. A lunar observer would note
that

1749 lunar hours ( tithis ) = 1781 terrestrial days

Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).

1 / S = 1 - 1 / T

S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
T = 1 + 1 / ( S - 1 )  = 1 & 32 / 1749 days

The skip day is to have the same tithi value ( 1 - 29 ) as that of the
previous day. If the mathermatics is correct, the tithi value should
repeat
on the skip day.

Should skip days be distributed evenly spaced and symmetrically
throughout
the 1781 period, then they are found by

( 32*D + 890 )MOD( 1781 ) < 32

The skip days are then

28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808,
863,
919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531,
1587,
1642, 1698, 1754

This leaves only to formulate the "Z number" that converts days into
tithis
by subtraction.

Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) /
1781 ]

Finally,

tithi count = day count - Z

moon age = tithi count MOD( 29 )

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[Karl Palmen]

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Dear Calendar People

 

Amos suggested having 945 units in a lunar month and 947 units in a solar month and each unit is close to 1/32 day.

This approximates the solar month to 974/945 lunar months. This approximation leads to a 315-year cycle of 116 leap months.

 

Another possibility is to have 391 units in a lunar month and 403 units in a solar month (or 372 units in 13-month calendar solar month).

This approximate the solar month to 403/391 lunar months. The approximation leads to a 391-year cycle of 144 leap months.

 

Since 945*403-974*391=1, there is no simpler approximation whose ratio lies in between and all approximations whose ratio does lie in between can be made up of a whole number of these two (by adding numerators and denominators separately).

 

Adding them gives 1377/1336, which gives rise to the 334-year cycle of 123 leap months.

Adding two 403/391 to this gives 2183/2118, which gives the 353-year cycle of 130 leap months.

 

Karl

 

10(13(05

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir
Sent: 11 September 2009 22:44
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

 

A better approximation may use the fact that the difference between a lunar month and a solar month (defined as 1/12 of a solar year; yes I know this could vary), is very close to 29/32 solar day.  So using a unit of 1/32 day (45 mins) we can define a solar month of 974 units and a lunar month of 945 units.  For an even better approximation, these may be shortened by one unit every 48 or 47 months, respectively.

Amos Shapir
 



 
> Date: Fri, 11 Sep 2009 05:01:20 -0700
> From: suntheorem@...
> Subject: tithi as 1 / 29th month
> To: CALNDR-L@...
>
> Suppose we have a tithi which is newly defined as 1 / 29th month.
>
> T = 1 & 32 / 1749 days = 1781 / 1749 days
>
> With a sundial on the moon using 29 "hours" and the moon divided into 29
> meridians, the tithi resembles a lunar hour. A lunar observer would note
> that
>
> 1749 lunar hours ( tithis ) = 1781 terrestrial days
>
> Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).
>
> 1 / S = 1 - 1 / T
>
> S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
> T = 1 + 1 / ( S - 1 ) = 1 & 32 / 1749 days
>
> The skip day is to have the same tithi value ( 1 - 29 ) as that of the
> previous day. If the mathermatics is correct, the tithi value should repeat
> on the skip day.
>
> Should skip days be distributed evenly spaced and symmetrically throughout
> the 1781 period, then they are found by
>
> ( 32*D + 890 )MOD( 1781 ) < 32
>
> The skip days are then
>
> 28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808,
> 863,
> 919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531, 1587,
> 1642, 1698, 1754
>
> This leaves only to formulate the "Z number" that converts days into tithis
> by subtraction.
>
> Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) / 1781 ]
>
> Finally,
>
> tithi count = day count - Z
>
> moon age = tithi count MOD( 29 )
>
> --
> View this message in context: http://www.nabble.com/tithi-as-1---29th-month-tp25399679p25399679.html
> Sent from the Calndr-L mailing list archive at Nabble.com.
>

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[Karl Palmen]

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Dear Amos Helios and Calendar People

 

A tithi of 1/30 lunar month hence has 31.5 units. A year of 12*974 units hence has 12*974/31.5 = 8*974 /21 = 371 1/21 tithis. So it can be realised by having a saltus lunae (372 tithi year) once every 21 years (other years have 371 tithis). This gives rise to the 315-year cycle of 116 leap years as already stated.

 

For a tithi of 1/29 lunar months, a solar month of 2989/2900 lunar months could be used, which gives rise to the 725-year cycle of 267 leap months (and 49*183 months in total). Then a year has 12*2989/100 = 358.68 tithis as previously suggested.

 

Karl

 

10(13(05

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Amos Shapir
Sent: 11 September 2009 22:44
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

 

A better approximation may use the fact that the difference between a lunar month and a solar month (defined as 1/12 of a solar year; yes I know this could vary), is very close to 29/32 solar day.  So using a unit of 1/32 day (45 mins) we can define a solar month of 974 units and a lunar month of 945 units.  For an even better approximation, these may be shortened by one unit every 48 or 47 months, respectively.

Amos Shapir
 



 
> Date: Fri, 11 Sep 2009 05:01:20 -0700
> From: suntheorem@...
> Subject: tithi as 1 / 29th month
> To: CALNDR-L@...
>
> Suppose we have a tithi which is newly defined as 1 / 29th month.
>
> T = 1 & 32 / 1749 days = 1781 / 1749 days
>
> With a sundial on the moon using 29 "hours" and the moon divided into 29
> meridians, the tithi resembles a lunar hour. A lunar observer would note
> that
>
> 1749 lunar hours ( tithis ) = 1781 terrestrial days
>
> Now the day takes a lap on the tithi ( T ) every skip day cycle ( S ).
>
> 1 / S = 1 - 1 / T
>
> S = 1 + 1 / ( T - 1 ) = 55 & 21 / 32 days
> T = 1 + 1 / ( S - 1 ) = 1 & 32 / 1749 days
>
> The skip day is to have the same tithi value ( 1 - 29 ) as that of the
> previous day. If the mathermatics is correct, the tithi value should repeat
> on the skip day.
>
> Should skip days be distributed evenly spaced and symmetrically throughout
> the 1781 period, then they are found by
>
> ( 32*D + 890 )MOD( 1781 ) < 32
>
> The skip days are then
>
> 28, 84, 140, 195, 251, 307, 362, 418, 474, 529, 585, 641, 696, 752, 808,
> 863,
> 919, 974, 1030, 1086, 1141, 1197, 1253, 1308, 1364, 1420, 1475, 1531, 1587,
> 1642, 1698, 1754
>
> This leaves only to formulate the "Z number" that converts days into tithis
> by subtraction.
>
> Z = skip days prior to and including day D = FLOOR[ ( D*32 + 890 ) / 1781 ]
>
> Finally,
>
> tithi count = day count - Z
>
> moon age = tithi count MOD( 29 )
>
> --
> View this message in context: http://www.nabble.com/tithi-as-1---29th-month-tp25399679p25399679.html
> Sent from the Calndr-L mailing list archive at Nabble.com.
>

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[Helios]

Dear Karl, Amos, et al.

I suppose that one could refer to

mean yerm of 12ths = W = 1 / [ ( 732 / Y ) - 2 ]

and set this to a top-heavy fraction of months. This would automatically lead to a luni-solar cycle.

suppose W = 8 & 9 / 56 months = 457 / 56 months

Y = 732 / [ 2 + ( 1 / W ) ]

and this is close to

Y = 365 & 101 / 417 days

W = 240 &  313 / 316

M = W*( 56 / 457 ) = 29.5305653

but I suppose one would have to fish around to find a short lunar cycle.

[Karl Palmen]

Dear Helios and Calendar People

If Helios wants a short lunar cycle, I suggest he try one of the
lunisolar cycles that is a multiple of a short lunar cycle.
I'll show it for the 725-year and 5515-year cycles.


725-YEAR CYCLE

The number of days in the 725-year cycle is 183*1447=264801 days.
If the mean solar month were 30.5 days, the number of days would be
12*725*30.5=265350
The number of solar yerms (for the solar months) is twice the difference
which is 1098.
The mean solar yerm is 241 1/15 days.

If all these solar yerms were to alternate between seven and nine
months, then the 725-year cycle would have 8*1098=8784 months, but the
725-year cycle actually has 12*725=8700 solar months and we have 42 more
yerms with seven months and 42 fewer with nine months. This gives 591
yerms of seven months and 507 yerms of nine months. This breaks into
three solar yerm cycles of 197 seven-month yerms and 169 nine-month
yerms. Each of these three cycles would have 14 super-yerms of either 27
or 25 yerms alternating between nine and seven months.

However, I'd prefer a solar calendar where each year has 359 dates,
except years 02, 05, 08, 11, 15, 18, 21, 24 of a 25-year cycle which
have one less date. The 30th day of each lunar month (in the 49-month
cycle) does not have a date so every lunar month has 29 dates (or
tithis).


5515-YEAR CYCLE

I expect Helios would like to find a way of using solar yerms for his
5515-year lunisolar cycle, which is made up of 39 lunar 1749-month
cycles. The 5515-year cycle has 39*51649=2014311 days. If the solar
months had an average of 30.5 days this would be 12*5515*30.5=2018490
days. The number of solar yerms is twice the difference, which is 8358.
The mean solar yerm is 241 11/2786 days.

However, the 5515-year cycle is suited to a 13-month solar year rather
than a 12-month solar year, because three lunar 1749-month cycles can be
made up of a whole number of these solar months, which is 5515 months,
of which 527 would have 29 days and the rest have 28 days. The mean
interval between long solar months of 29 days is 294 9/527 days.


Karl

10(13(07


-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 25 September 2009 09:31
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

Dear Karl, Amos, et al.

I suppose that one could refer to

mean yerm of 12ths = W = 1 / [ ( 732 / Y ) - 2 ]

and set this to a top-heavy fraction of months. This would automatically
lead to a luni-solar cycle.

suppose W = 8 & 9 / 56 months = 457 / 56 months

Y = 732 / [ 2 + ( 1 / W ) ]

and this is close to

Y = 365 & 101 / 417 days

W = 240 &  313 / 316

M = W*( 56 / 457 ) = 29.5305653

but I suppose one would have to fish around to find a short lunar cycle.

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[Karl Palmen]

Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 25 September 2009 09:31
To: CALNDR-L@...
Subject: Re: tithi as 1 / 29th month

Dear Karl, Amos, et al.

I suppose that one could refer to

mean yerm of 12ths = W = 1 / [ ( 732 / Y ) - 2 ]

and set this to a top-heavy fraction of months. This would automatically
lead to a luni-solar cycle.

suppose W = 8 & 9 / 56 months = 457 / 56 months

KARL SAYS: I don't see what use this defining the solar yerm in terms of
lunar months has in designing a calendar.

Y = 732 / [ 2 + ( 1 / W ) ]

and this is close to

Y = 365 & 101 / 417 days

KARL SAYS: This is the same cycle the Brij has used in his 834-year
cycle leap week calendar.

W = 240 &  313 / 316

M = W*( 56 / 457 ) = 29.5305653

but I suppose one would have to fish around to find a short lunar cycle.

KARL SAYS:
The resulting lunar and lunisolar cycle has 2919 years, 36103 months and
1066142 days = 152306 weeks. It is equal to seven solar 417-year cycles
(each of 101 leap days or 74 leap weeks). It is also equal to 79 of
Helios's 457-month cycles of 56 solar yerms. Consequently the 2919-year
cycle should have 4424 solar yerms and 2*(12*2919*30.5-1066142)=4424, so
it does have that many solar yerms.
The parameters for my lunisolar spreadsheets
http://www.the-light.com/cal/kp_Lunisolar_xls.html  are A=2919 B=1075
C=566.
It might have appeared in http://www.the-light.com/cal/Lunisolar7.html
were the mean lunar month more accurate.

So Helios has discovered a lunisolar cycle that is a multiple of a solar
calendar cycle. I've list others in
http://www.the-light.com/cal/Lunisolar33.html (multiples of 33-year
cycle) and
http://www.the-light.com/cal/Lunisolar400.html (multiples of Gregorian
400-year cycle).

If you take one short Metonic cycle of 6939 days from Helios's 2919-year
cycle, you get the 29-century cycle I listed in
http://www.the-light.com/cal/Lunisolar4.html which has a mean month of
29.5305844 days and a mean year of 365.24241 days.
This is a multiple of the 725-year cycle that I have mentioned.


Karl

10(13(07

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[Helios]

Here's a comparison of lunar and solar formulae with 6707 year luni-solar cycle for example. Lower case for lunar, upper case for solar,
-----------------------------
Yerms

m = 59 / [ 2 - ( 1 / w ) ]
M = 61 / [ 2 + ( 1 / W ) ]

w = 482 & 34 / 43 days
W = 241 & 13 / 847 days
-----------------------------
Skip cycles

m = 30*[ 1 - 1 / ( s + 1 ) ]
M = 30*[ 1 + 1 / ( S - 1 ) ]

s = 62 & 10 / 11 days
S = 69 & 197 / 293 days
-----------------------------
Tithis

t = [ 1 - 1 / ( s + 1 ) ]
T = [ 1 + 1 / ( S - 1 ) ]

t = 692 / 703 days
T = 20414 / 20121 days
-----------------------------

[Karl Palmen]

Dear Helios and Calendar People

Helios does not explain his variables, so I put in my explanations.

I also suggest a simpler solar calendar that has a 353-year cycle of 17
leap years equalling 4366 lunar months.

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 28 September 2009 00:51
To: CALNDR-L@...
Subject: solar skip cycle and tithi

Here's a comparison of lunar and solar formulae with 6707 year
luni-solar
cycle for example. Lower case for lunar, upper case for solar,
-----------------------------
Yerms

m = 59 / [ 2 - ( 1 / w ) ]
M = 61 / [ 2 + ( 1 / W ) ]

w = 482 & 34 / 43 days
W = 241 & 13 / 847 days

The variable w is the length of the mean (lunar) yerm and W is the
length of the mean solar yerm.
The variable m is the mean lunar month in days and M is the mean solar
month in days.
-----------------------------
Skip cycles

m = 30*[ 1 - 1 / ( s + 1 ) ]
M = 30*[ 1 + 1 / ( S - 1 ) ]

s = 62 & 10 / 11 days
S = 69 & 197 / 293 days

The variable s is the mean number of days between skipped dates from
lunar months of 30 dates.
The variable S is the mean number of days between skipped days not
counted as dates to solar months of 30 dates.
-----------------------------
Tithis

t = [ 1 - 1 / ( s + 1 ) ]
T = [ 1 + 1 / ( S - 1 ) ]

t = 692 / 703 days
T = 20414 / 20121 days

The variable t is the ratio of days to lunar dates and so is the length
of the tithi associated with the lunar date.
The variable T is the ratio of days to solar dates and so is the length
of the solar tithi associated with the solar date.

Helios has missed out

t = m/30
T = M/30

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

Unless the fractions
S = 69 & 197 / 293 days and T = 20414 / 20121 days can be expressed as a
short continued fraction, the solar skip rule would be structurally very
complicated. Likewise with W = 241 & 13 / 847 days for solar yerms. Also
this solar calendar does not to exploit the fact that the 6707-year
cycle is a multiple of the 353-year cycle.

Instead, I suggest the following for a solar calendar:

Common year of 371 lunar dates (tithis) counting skipped dates and a
leap year of 372 lunar dates (tithis).
Each solar month has 31 lunar dates (= lunar month + 1 date), except for
the last month of a common year, which has 30 lunar dates (= lunar
month).
The calendar has a 353-year cycle of 17 leap years (and exactly 4366
lunar months).

Nineteen 353-year cycles make a complete 6707-year cycle. Each of these
353-year cycles is identical, except for the skipped dates.
The 17 leap years of each 353-year sub-cycle can be arranged into a
Helios cycle. It would have four intervals of 20 years and 13 intervals
of 21 years. One 21-year interval (with 20 common years) would span the
end of the cycle, so the first leap year is year 11.

Karl

10(13(10

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