| Index |
|---|
| Introduction |
| Invalid Dates |
| Variables |
| Tzolk’in Position |
| Haab Position |
| Number of Haabs |
| CR Position |
| Relative Computions |
| References |
The complete calendar round (CR) comprises 18980 days. You might think that it would start on 1 ’Imix 0 Pohp, but that leads rather swiftly to the conclusion that the day 4 ’Ahaw 8 Kumk’u could never occur in such a system.
In fact, it turns out that if you require such a combination in the calendar round, you must set the starting point of the haab at 0 and the starting point of the tzolk’in at 156, thereby setting the starting date to 1 Kaban 0 Pohp, CR 0 and the ending date to 13 K’ib 4 Wayeb, CR 18979.
We could also view the calendar round as beginning on day 0 set to 4 ’Ahaw 8 Kumk’u, and ending with day 18979 set to 3 Kawak 7 Kumk’u, giving us the same total of 18980 days. This is exactly what we do when we calculate the interval, or distance, between 4 ’Ahaw 8 Kumk’u and any other date in the CR. For most purposes, dealing with the “absolute” position in the CR is preferable, unless we are specifically working with distance numbers.
The CR is made up of four components:
The trecena, or “13”,
a cycle of
13 numbers from 1-13;
any modulo function yielding a position in the trecena must
replace a 0 result with 13.
You can see various number glyphs in the number page.
The veintena,
or “20”, a cycle of
20 day names from “’Imix” to “’Ahaw”;
you can view the glyphs for the days on the Days page. Mathematically, the day names may be treated as a sequence
of numbers beginning with ’Imix and ending with ’Ahaw. We can
assign ’Imix to either 1 or 0, as required. If we assume
that ’Imix is 0, then ’Ahaw is 19; on the other
hand, if we make ’Imix equal to 1, then 0 is ’Ahaw.
On this page, we are treating ’Ahaw as 0.
The haab month, a cycle of 18
20-day months followed
by a 5-day intercalary period called “Wayeb”;
the haab months are named Pohp-Wayeb, or,
numerically, 0-18.
You can see these
glyphs on the Months page.
The haab day, a day number
from 0-19 (0-4 for Wayeb); these
are the same number glyphs as
used for the trecena, although very often the zero glyph
is replaced with 
,
meaning the seating of., as in “the seating
of Pohp”
instead of “0 Pohp”
.
Infrequently, you will see a day that
would normally be shown as day 0 of a month displayed as, instead,
the last day
of the previous month. For example, instead of
“0 Pohp,”
you might see “5 Wayeb.”
The day following 5 Wayeb would still be, then, 1 Pohp.
The trecena and the veintena together comprise the tzolk’in, a 260-day interlocking cycle (13 * 20 = 260); to view a complete tzolk’in, click here.
The haab month and haab day make up a 365-day linear cycle called the haab; this is a sequence of the 18 20-day months and the 5-day wayeb ((18 * 20) + 5 = 365), and the day number in each month. To view a complete haab, click here.
If you have a date that consists of just a CR, you can determine possible long count dates for the given CR using the Calendar Rounds to Long Counts tool.
Not all CR dates that you will see on monuments or in the codicies are valid; the veintena and the haab day are intimately related, and, as shown in the following table, veintena days can occur only on certain haab days. Most of the invalid dates seen are the result of mistakes in calculation, which the Mayans may have seen as the work of the gods. There are inscriptions where it is plain that the scribe realized that a calendar round date was in error, and attempted to fudge later calculations in order to make the ending date come out right. The reason that this is so fascinating is that stelae were not, could not have been, carved on the dates they were set in place; and there’s just no way that the stonecutters could have just started in carving—someone had to lay out the panel well in advance. And it is highly unlikely that the scribes would have drawn the plan directly on the stone, since they did have paper on which to work out the dimensions and locations of all the glyphs. So why didn’t they just erase the mathematical errors and re-do the calculations, and re-draw the glyphs, before ever committing chisel to stone? Maybe Mayanists are too limited, but the best explanation so far seems to be a sacred version of “The Devil made me do it.”
Some weird dates, however, are the result of differing systems, as described by Prouskouriakoff and Thompson (1947), where the haab day is reduced by one from what would normally be expected, as in 9 ’Ahaw 17 Mol instead of the expected 9 ’Ahaw 18 Mol. There are not many dates using this system (termed “Puuc” style, referring to the geographical area where most of these dates are found), however; Prouskouriakoff and Thompson list only seven. See also Thompson (1952). To achieve internal consistency in such a modified system, you must set the starting point of the tzolk’in in the CR cycle back one, from 156 to 155.
| Veintena | Legal Haab day coefficients | |||
|---|---|---|---|---|
’Ahaw
|
8 | 13 | 18 | 3 |
’Imix
|
4 | 9 | 14 | 19 |
’Ik’
|
5 | 10 | 15 | 0 |
Ak’bal
|
6 | 11 | 16 | 1 |
K’an
|
7 | 12 | 17 | 2 |
Chik’chan
|
8 | 13 | 18 | 3 |
Kimi
|
4 | 9 | 14 | 19 |
Manik’
|
5 | 10 | 15 | 0 |
Lamat
|
6 | 11 | 16 | 1 |
Muluk
|
7 | 12 | 17 | 2 |
Ok
|
8 | 13 | 18 | 3 |
Chuwen
|
4 | 9 | 14 | 19 |
’Eb
|
5 | 10 | 15 | 0 |
Ben
|
6 | 11 | 16 | 1 |
’Ix
|
7 | 12 | 17 | 2 |
Men
|
8 | 13 | 18 | 3 |
K’ib
|
4 | 9 | 14 | 19 |
Kaban
|
5 | 10 | 15 | 0 |
’Etz’nab
|
6 | 11 | 16 | 1 |
Kawak
|
7 | 12 | 17 | 2 |
A CR date such as 11 Ix 12 K’ank’in has a numerical position, or pCR, which can be calculated by some simple formulae. Variables required are:
| Variable | Meaning |
|---|---|
| tr | Day of the trecena, any of the numbers 1-13 |
| v | Day of the veintena, any of the named Mayan days ’Imix-’Ahaw, as a number (0-19), where ’Ahaw is 0; ’Imix=1, ’Ik=2, Ak’bal=3, . |
| tz or Δtz | Day of the tzolk’in, the numerical position in the 260-day cycle sometimes known as the Sacred Almanac; this number varies from 0-259 |
| m | “Month”: any of the named Mayan “months,” Pohp-Wayeb; numbered 0-18. The haab month |
| d | Day of the “month”: any of the numbers 0-19, except for Wayeb, where the range is 0-4. The haab day |
| h or Δh | Position in the haab: any of the numbers 0-364; equivalent to the so-called “Julian Day” printed on business calendars in the US. Don’t confuse this with the “Julian Period Day,” which is a different entity entirely. |
For our example, 11 Ix 12 K’ank’in, the four known variables are:
| tr | 11 |
|---|---|
| v | Ix, or 14 |
| d | 12 |
| m | K’ank’in, or 13 |
To find Δtz (position in the tzolk’in) from tr and v, you need to know the minimum number of days between 1 ’Imix and your desired day. To find this value, apply the formula:
Δtz = 40[(tr2 - tr1) - (v2 - v1)] + (v2 - v1) % 260
which is not nearly as complicated as it looks. tr1 simply means the day we start counting from, and tr2 is the day we end on. So (tr2 - tr1) just translates (for our example) into 11 - 1, i.e., our tr (11) minus 1 (for 1 ’Imix).
Similarly, v1 is the day name we start counting from, and v2 where we want to stop. Thus, (v2 - v1) translates (again, for our example) into 14 - 1 (’Ix - ’Imix). Substituting, we get:
Δtz =
40[(11 - 1) - (14 - 1)] + (14 - 1) % 260
Δtz =
40[(10) - (13)] + (13) % 260
Δtz = 40[-3] + (13) % 260
Δtz = -120 + (13) % 260
Δtz = -107 % 260
Δtz = 153
To determine the tzolk’in position from any trecena
and veintena, click here 
.
With the tzolk’in position (Δtz), we can easily recover both the trecena and veintena. To find the trecena, we just apply this formula:
tr = (Δtz + 1) % 13
We add 1 because day 0 in the tzolk’in is 1 ’Imix; and, since the range of the trecena is 1-13, when our modulo function returns 0 we replace the 0 with 13.
tr = (153 + 1) % 13
tr = (154) % 13
tr = 11
Finding the veintena from the position in the tzolk’in is equally straightforward:
v = (Δtz + 1) % 20
However, this time when our modulo function returns 0, we use it directly, to signify the day ’Ahaw:
v = (153 + 1) % 20
v = 154 % 20
v = 14
v = ’Ix
To find Δh (position in the haab) from m and d, you need to know the minimum number of days between 0 Pohp and your desired day. To find this value, apply the formula:
Δh = (d2 - d1) + 20(m2 - m1) % 365
which is, again, very simple. d1 is just the day of the month we start on, and d2 is where we stop. In fact, since d1 is always 0 (for 0 Pohp) and m1 is also always 0 (Pohp, month 0), we can convert the formula for absolute position in the haab to:
Δh = (d + 20(m)) % 365
Substituting the values from our example, we get:
Δh = (12 + 20(13)) % 365
Δh = (12 + 260) % 365
Δh = (272) % 365
Δh = 272
To determine the haab position from any month
and day, click here 
.
Recovering the haab day and haab month values from Δh is straightforward. For the haab day, apply the formula:
d = Δh % 20
For our example, this is:
d = 272 % 20
d = 12
And for the haab month, the formula is:
m = (int)(Δh / 20)
which is:
m = (int)(272 / 20)
m = (int)(13.6)
m = 13
m = K’ank’in
In order to calculate pCR, the next variable we need to find is the number of whole haabs (n(H)) contained in the interval between 4 ’Ahaw 8 Kumk’u and the desired day (in our example, this is 11 Ix 12 K’ank’in). This is found by applying the formula:
n(H) = (Δtz - Δh) % 52
Using the values derived above, we have:
n(H) = (153 - 272) % 52
n(H) = -119 % 52
n(H) = 37
Finally, we are able to calculate pCR with the following formula:
pCR = 365(n(H)) + Δh
Applying the values we determined above, we get:
pCR = 365(37) + 272
pCR = 13505 + 272
pCR = 13777
This value is referenced to 1 Kaban 0 Pohp (the day in the CR when Δh is set to 0 and Δtz is set to 156), or what you might call the “absolute position” in the CR. In order to find the position relative to 4 ’Ahaw 8 Kumk’u, we must subtract 7283 (the position found by applying the above sequence of formulae to the CR co-ordinates 4 ’Ahaw 8 Kumk’u) from pCR, and modulo the answer 18980:
pCR0 = (pCR - 7283) % 18980
pCR0 = (13777 - 7283) % 18980
pCR0 = (6494) % 18980
pCR0 = 6494
To determine the Calendar Round position from any trecena,
veintena, haab day and haab month,
click here 
.
Recovering the haab and tzolk’in positions from the position in the CR is possible, although not particularly direct. We have a choice of two methods, depending on whether we have pCR or pCR0:
| pCR | pCR0 |
|---|---|
| Δh = pCR % 365 | Δh = (pCR0 + 348) % 365 |
| Δtz = (pCR + 156) % 260 | Δtz = (pCR0 + 159) % 260 |
(Remember that the starting point for pCR is defined as the co-ordinates tz = 156 and h = 0, while the starting point for pCR0 is tz = 159 and h = 348.)
Substituting the values determined for our examples, then, we have:
| pCR | pCR0 |
|---|---|
|
Δh = 13777 % 365 Δh = 272 |
Δh = (6494 + 348) % 365 Δh = (6842) % 365 Δh = 272 |
|
Δtz = (13777 + 156) % 260 Δtz = (13933) % 260 Δtz = 153 |
Δtz = (6494 + 159) % 260 Δtz = (6653) % 260 Δtz = 153 |
Applying the same formulae, we can determine the minimum interval between any two CR dates. For example, if we wish to determine how many days between 8 ’Ahaw 13 Pohp and 6 Etz’nab 11 Yax, we use the four formulae given above:
| Formula 1, Position in the Tzolk’in, Δtz |
|---|
| Δtz = 40[(tr2 - tr1) - (v2 - v1)] + (v2 - v1) % 260 |
|
Δtz = 40[(6 - 8) - (18 - 0)] + (18 - 0) % 260 Δtz = 40[(-2) - (18)] + (18) % 260 Δtz = 40[-20] + 18 % 260 Δtz = -800 + 18 % 260 Δtz = -782 % 260 Δtz = 258 |
| Formula 2, Position in the Haab, Δh |
| Δh = (d2 - d1) + 20(m2 - m1) % 365 |
|
Δh = (11 - 13) + 20(9 - 0) % 365 Δh = (-2) + 20(9) % 365 Δh = (-2) + 180 % 365 Δh = 178 % 365 Δh = 178 |
| Formula 3, Number of whole Haabs, n(H) |
| n(H) = (Δtz - Δh) % 52 |
|
n(H) = (258 - 178) % 52 n(H) = (80) % 52 n(H) = 28 |
| Formula 4, Position in the Calendar Round, pCR |
| pCR = 365(n(H)) + Δh |
|
pCR = 365(28) + 178 pCR = 10220 + 178 pCR = 10398 which is 1.8.15.18 in Mayan notation, a touch less than 30 years |
To calculate intervals between any two CR dates,
click here 
.
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Main web site: http://www.pauahtun.org