Mathematics of the Kasshi Calendar

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This page is for those who want to know more details about the mathematics behind the Kasshi calendar.

Segments

The basic unit of the calendar is the segment, of which there are 1,152. Each segment represents an equal angular division of the orbit, equal to .3125 degrees. Segment 1 starts at perihelion. The length of time from the beginning of the 1st segment to the beginning of each individual second is a fixed value, measured down to the daythird. For example, the 2nd segment begins 11 daymins, 9 daysecs, and 26 daythirds after the start of the 1st segment. The segments vary in length from 11:9:26[1] for the 1st and 1152nd segments to 14:30:55 for the 576th and 577th segments. To these fixed values, an offset is added which is determined for each sidereal year by multiplying the number of the year minus one by the length of the sidereal year - calculated at 244 days, 52:13:26 (accurate to within a fraction of a daythird), adding the standard offset of 22:30:42. This is used to determine which segments are in a given day in a continuous count of days. For example, for 965, the offset is 236,055 days 3:42:42 making segments 1-6 part of day 236,055, segments 7-11 day 236,056, 12-16 day 236,057, 17-22 day 236,058, and so on. Each month is assigned to a particular range of segments. Early Spring, for example, is segments 756-851, Mid-Spring is 852-947, etc. Because the tropical, and thus calendar, year does not begin on segment 1 of the orbit, one must calculate offsets for two orbits. The later part of the year thus uses an onset calculated as above, but adding one to the year-number, as that part of the tropical year is included in the next sideral year. For example, Early Spring through the first part of Mid-Summer of 965 is included in sidereal year 965 (segments 756-1152) while the later part of Mid-Summer through Late Winter of 965 are in sidereal year 966 (segments 1-755).

Months

To calculate the months for a particular year, one only needs to note the days on which 12 segments occur, namely, at present, 756 for Early Spring, 850 for Mid-Spring, etc. The fixed values for the key segments are:

Month Segment Day Daymin Daysec Daythird
Late Summer 84 15 29 56 10
Early Fall 180 33 55 23 29
Mid-Fall 276 53 22 7 0
Late Fall 372 74 9 27 8
Early Winter 468 96 16 39 12
Mid-Winter 564 119 17 25 55
Late Winter 660 142 25 12 42
Early Spring 756 164 50 34 41
Mid-Spring 852 186 0 38 54
Late Spring 948 205 47 58 45
Early Summer 1,044 224 27 19 8
Mid-Summer 1,140 242 27 9 50

The fixed values for Early Spring through Mid-Summer are added to the offset for the sidereal year with the same number as the Calendar Year, while Late Summer through Late Winter are added to the offset for the following sidereal year. For Sidereal Year 965, the offset is 236,055 days 26:20:25 while for Sidereal Year 966, the offset is 236,300 days 18:33:52 making the values for Calendar Year 965 as follows:

Sid. Year Month Segment Day Daymin Daysec Daythird
965 Early Spring 756 236,220 16 55 7
Mid-Spring 852 236,241 26 59 20
Late Spring 948 236,261 14 19 11
Early Summer 1,044 236,279 53 39 34
Mid-Summer 1,140 236,297 53 30 16
966 Late Summer 84 236,315 48 30 2
Early Fall 180 236,334 13 57 21
Mid-Fall 276 236,353 40 40 52
Late Fall 372 236,374 28 1 0
Early Winter 468 236,396 35 13 4
Mid-Winter 564 236,419 35 59 47
Late Winter 660 236,442 43 46 34

Each month begins on the day that starts after the first segment occurs. Thus, Early Spring starts on day 236,221. Note, by way of example, that Early Summer starts on day 236,280 and Mid-Summer on day 236,298. If the offset for this year were 6 daymins 20 daysecs, 26 daythirds longer longer, then Early Summer would start on day 236,281 while Mid-Summer would still start on day 236,298. Thus, Early Summer would only have 17 days. But if the offset were 6:29:44 longer, then Mid-Summer would start on day 236,299, giving Early Summer 18 days. This shows why short Early Summers are so rare[2]. It requires a very narrow range of offsets to cause that. As the length of each segment is fixed, the time between any two segments is constant. At present, the lengths of the months are:

Month Day Daymin Daysec Daythird
Early Spring 21 10 4 12
Mid-Spring 19 47 19 50
Late Spring 18 39 20 23
Early Summer 17 59 50 42
Mid-Summer 17 54 59 45
Late Summer 18 25 27 18
Early Fall 19 26 43 31
Mid-Fall 20 47 20 8
Late Fall 22 7 12 3
Early Winter 23 0 46 43
Mid-Winter 23 7 46 47
Late Winter 22 25 21 59

Calendar types

Explanation

For any given shift, there are 13 possible calendars, as defined by combinations of long and short days, though not all will actually be realized. This follows logically from the following consideration: the day a month starts is determined by adding the fixed value to the annual offset. The first day of each month can thus fall on one of two dates, depending on that year's offset. For each month, there is a particular offset that will be necessary to shift it to the later date, in addition to the Early Spring of the following year (necessary to determine the length of Late Winter). It is easiest to explain when measuring relative to the segment at the start of spring, 756 at present, if that segment begins at 0:0:0, then the year would appear as follows:

Month Day Daymin Daysec Daythird
Early Spring 0 0 0 0
Mid-Spring 21 10 4 12
Late Spring 40 57 24 12
Early Summer 59 36 44 8
Mid-Summer 77 36 35 8
Late Summer 95 31 34 54
Early Fall 113 57 2 13
Mid-Fall 133 23 45 44
Late Fall 154 11 5 52
Early Winter 176 18 17 56
Mid-Winter 199 19 4 39
Late Winter 222 26 51 26
(Next Year) 244 52 13 25

Thus, each month's first day and length is:

Month First Day Length
Early Spring 1 21
Mid-Spring 22 19
Late Spring 41 19
Early Summer 60 18
Mid-Summer 78 18
Late Summer 96 18
Early Fall 114 20
Mid-Fall 134 21
Late Fall 155 22
Early Winter 177 23
Mid-Winter 200 23
Late Winter 223 22
(Next Year) 245 N/A

Start dates

The total length of the year is thus 244 days, a short year. With an offset of 2 daymins, 35 daysecs, 56 day thirds, Late Spring's start becomes 41 days, 0:0:0 making its first day day number 42, and making Mid-Spring 20 days and Late Spring 18 days, in effect, what had formerly been the first day of Late Spring becomes the last day of Mid-Spring. A slightly higher offset of 0:2:57:46 pushes Early Fall to day 115, shortening it to 19 days and lengthening Late Summer to 19, etc. Thus, each of 12 different offsets pushes the start of a different month or the next year ahead one day, 13 calendars in total, including those that result when the offset is less than the smallest offset. At present, the thirteen possible calendars with the shifted day highlighted in red, are:

Month Minimum Offset (Daymin:Daysec:Daythird)
0:0:0 2:35:56 2:57:46 7:46:35 23:15:33 23:24:51 28:25:5 33:8:33 36:14:16 40:55:20 41:42:3 48:54:7 49:55:47
Early Spring 1 1 1 1 1 1 1 1 1 1 1 1 1
Mid-Spring 22 22 22 22 22 22 22 22 22 22 22 22 23
Late Spring 41 42 42 42 42 42 42 42 42 42 42 42 42
Early Summer 60 60 60 60 61 61 61 61 61 61 61 61 61
Mid-Summer 78 78 78 78 78 79 79 79 79 79 79 79 79
Late Summer 96 96 96 96 96 96 97 97 97 97 97 97 97
Early Fall 114 114 115 115 115 115 115 115 115 115 115 115 115
Mid-Fall 134 134 134 134 134 134 134 134 135 135 135 135 135
Late Fall 155 155 155 155 155 155 155 155 155 155 155 156 156
Early Winter 177 177 177 177 177 177 177 177 177 177 178 178 178
Mid-Winter 200 200 200 200 200 200 200 200 200 201 201 201 201
Late Winter 223 223 223 223 223 223 223 224 224 224 224 224 224
Next Year 245 245 245 246 246 246 246 246 246 246 246 246 246

Lengths

As a result, the lengths of each month in the above calendars is (typical months in white, long months in green, short months in blue)

Month Minimum Offset (Daymin:Daysec:Daythird)
Calendar type
0:0:0
1
2:35:56
2
2:57:46
3
7:46:35
4
23:15:33
5
23:24:51
6
28:25:5
7
33:8:33
8
36:14:16
9
40:55:20
10
41:42:3
11
48:54:7
12
49:55:47
13
Early Spring 21 21 21 21 21 21 21 21 21 21 21 21 22
Mid-Spring 19 20 20 20 20 20 20 20 20 20 20 20 19
Late Spring 19 18 18 18 19 19 19 19 19 19 19 19 19
Early Summer 18 18 18 18 17 18 18 18 18 18 18 18 18
Mid-Summer 18 18 18 18 18 17 18 18 18 18 18 18 18
Late Summer 18 18 19 19 19 19 18 18 18 18 18 18 18
Early Fall 20 20 19 19 19 19 19 19 20 20 20 20 20
Mid-Fall 21 21 21 21 21 21 21 21 20 20 20 21 21
Late Fall 22 22 22 22 22 22 22 22 22 22 23 22 22
Early Winter 23 23 23 23 23 23 23 23 23 24 23 23 23
Mid-Winter 23 23 23 23 23 23 23 24 24 23 23 23 23
Late Winter 22 22 22 23 23 23 23 22 22 22 22 22 22

Note how the months change in length effectively as the transfer of days from one month to another. For example, between calendar 1 and calendar 2, one day is shifted from Late Spring to Mid-Spring, and in calendar 3 a day is shifted from Early Fall to Late Summer, in calendar 4 an extra day is added to the calendar itself (the first three calendars are short years). These calendar-types change over time as the calendar shifts. Not every shift changes the calendar-types available, but they do at least change the relative frequencies

Weekdays

The day of the week of a particular day can be found by adding 3 to its day number and then finding the modulo 9 of that number, 0 representing Opening and so forth. Upper and Lower can be found by taking the day number + 3 and dividing that by 9, and then taking the integral portion. If it is an odd number, then you have an Upper-Week day, if even, it is a Lower-Week day.

Precession and Calendar Shifts

Above, it was noted that Early Spring begins in segment 756. That is true at present, but due to axial precession, it is not always true. Thus, 756 is not a fixed value. The first segment of Early Spring is determined by the formula 8089/75 -4/75

  • YEAR and dropping the fractional part. It can be seen from this formula that the first segment of Early Spring will move back 4 segments in 75 years, or approximately once every 19 years. When this happens, it is referred to as a shift. Consequently, 4 times every 75 years, there is a year that is only 1,151 segments long. Calendar year 977, for example, will run from segment 756 of Sidereal Year 977 to segment 754 of SY 978, with calendar year 978 starting on segment 755. Each of the other months will likewise move back one segment, so that Mid-Spring will start on 851, and so forth, altering the lengths of each month slightly. Over time, these shifts can make for considerably different calendars, though there can never be months shorter than 17 days or longer than 24. The shortest possible month is 17 d 52:36:20 and the longest possible is 23 d, 11:16:1. Both the longest possible and the shortest possible would occur in the same year.

For example, after the next shift, in the year 978, the fixed values will be

Month Segment Day Daymin Daysec Daythird
Late Summer 83 15 18 38 26
Early Fall 179 33 43 36 25
Mid-Fall 275 53 9 34 1
Late Fall 371 73 56 1 29
Early Winter 467 96 2 29 5
Mid-Winter 563 119 2 29 5
Late Winter 659 142 10 53 50
Early Spring 755 164 36 54 59
Mid-Spring 851 185 47 51 13
Late Spring 947 205 36 0 7
Early Summer 1,043 224 15 55 11
Mid-Summer 1,139 242 16 0 11

Giving month-lengths of:

Month Day Daymin Daysec Daythird
Early Spring 21 10 56 14
Mid-Spring 19 48 8 53
Late Spring 18 39 55 3
Early Summer 18 0 5 0
Mid-Summer 17 54 51 41
Late Summer 18 24 57 58
Early Fall 19 25 57 36
Mid-Fall 20 46 27 27
Late Fall 22 6 27 35
Early Winter 23 0 26 13
Mid-Winter 23 7 58 31
Late Winter 22 26 1 8

The months of Mid-Winter through Early Summer become slightly longer, by up to 52 daysecs 2 daythirds (for Early Spring) while Mid-Summer through Early Winter become slightly shorter, by up to 52 daysecs 40 daythirds (for Mid-Fall). In the case of Early Summer, the extra time is enough to change it from just slightly under 18 days to slightly over 18 days, meaning that 17-day Early Summers will become impossible, while 19-day Early Summers will be possible, though they will remain rare for quite some time[3]

By way of comparison, in the year 721 when the modern Empire was established, the month lengths were:

Month First segment Day Daymin Daysec Daythird
Early Spring 769 20 58 44 8
Mid-Spring 865 19 36 51 50
Late Spring 961 18 32 7 10
Early Summer 1,057 17 57 5 44
Mid-Summer 1 17 57 5 44
Late Summer 97 18 32 7 10
Early Fall 193 19 36 51 50
Mid-Fall 289 20 58 44 8
Late Fall 385 22 16 33 38
Early Winter 481 23 4 44 10
Mid-Winter 577 23 4 44 10
Late Winter 673 22 16 33 38

Note the perfect symmetry. This only happens when a month starts on segment 1. Comparing the year 721 to 965, and for a more dramatic comparison, 1 OE, noting the typical length (that is, the most common length) with the length with which it alternates:

Month 1 721 965
Early Spring 20 (21) 21 (20) 21 (22)
Mid-Spring 19 (20) 19 (20) 20 (19)
Late Spring 18 (19) 18 (19) 19 (18)
Early Summer 18 (17) 18 (17) 18 (17)
Mid-Summer 18 (19) 18 (17) 18 (17)
Late Summer 19 (18) 18 (19) 18 (19)
Early Fall 20 (21) 19 (20) 19 (20)
Mid-Fall 21 (22) 21 (20) 21 (20)
Late Fall 23 (22) 22 (23) 22 (23)
Early winter 23 (24) 23 (24) 23 (24)
Mid-Winter 22 (23) 23 (24) 23 (24)
Late Winter 22 (21) 22 (23) 22 (23)

It is clear that only small differences exist between 721 and 965. Mostly frequencies of the month-lengths, though Early Spring has changed from being usually 21 days with some 20-day months to being usually 21 days with some 22-day months. Thus, 20-day Early Springs occurred in the early days of the Empire, but no longer occur, while 22-day Early Springs now occur but never did in the early days. However, in comparison with the calendar of 1 OE, greater shifts have occurred. 19-day Mid-Summers occurred, which never occur today, likewise, 21-day Early Falls and 22-day Late Falls could occur. Mid-Winter could never have 24 days, and most often had 22 days, while Late Winter never had 23 days, and sometimes had only 21.

Possible future refinements

At present, the calendar assumes that Galhaf's orbit never changes. In reality, due to gravitational influences from other planets, especially Zheftakh, the planet's eccentricity does change gradually. At present, the eccentricity is slowly decreasing. It has been suggested that the orbital segments themselves be periodically recalculated to maintain accuracy. However, it is difficult to calculate exactly what the future eccentricity will be, making it impossible to project future calendars. In addition, the changes in eccentricity are very small over time spans relevant to humans.

Notes

  1. Colon divides daymins, daysecs, and daythirds, thus, this is 11 daymins, 9 daysecs, 26 daythirds
  2. In fact, the last short Early Summer was in 792
  3. The first occurrence of a long Early Summer will be in the year 1051