# The Calendar Round ## Introduction

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.

## Mathematics

### Invalid Calendar Round Dates

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.

Haab day coefficients possible for given Veintena Days
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

### Variables

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 Ix, or 14 12 K’ank’in, or 13

### Tzolk’in Position

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

### Haab Position

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

### Number of Haabs

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

### Position in the Calendar Round

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

### It’s All Relative

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 .

## References

• Lounsbury, F. G., “Formulae for Maya Calendrical Computations,” n.d., Maya File 124.
• Lounsbury, Floyd G., “Maya Numeration, Computation, and Calendrical Astronomy,” in Dictionary of Scientific Biography, ed., Charles Coulston Gillespie, Vol. 15, Supplement 1 (1978), Scribners, New York, 1978 (Maya File 316e).
• Prouskouriakoff, Tatiana and J. Eric S. Thompson, “Maya Calendar Round Dates Such as 9 Ahau 17 Mol,” Notes on Middle American Archaeology and Ethnology No. 79, January 27, 1947.
• Thompson, J. Eric S., “The Introduction of Puuc Style of Dating at Yaxchilan,” Notes on Middle American Archaeology and Ethnology No. 110, May 15, 1952.

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