Mathematics of the Kasshi Calendar

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 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. 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

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.