4 Ahaw 3 K'ank'in Numerical Factors and Periods Important in Maya Calendrics
by Mark Van Stone adapted from John B. Carlson1
This table contains a listing of many of the numerical factors and periods of time that were important in Maya calendric numerology. An asterisk in the first column indicates that the number was probably inspired at least indirectly by some astronomical considerations. A question mark indicates those periods or numbers for which there is only tentative support regarding their use by the Maya. (A ° indicates intervals of apparent numerological significance; all footnotes mine.) The numbers in brackets are the modern values for the astronomical periods to at least two decimal places. The "Maya equivalent" column contains the number of days for each period of time in Maya Long Count notation.
  Name of Period # of days/units Maya Equivalent Factor Decompositions Prime Factors
° (prime-3) 3 .3 3 3
° Four World-directions 4 .4 2*2 22
° Five World-directions (+center) 5 .5 5 5
° Seven Classical Planets(?) 7 .7 7 7
* Venus invisibility at Inferior Conjunction 8 .8 2*4 23
° Glyph G Cycle (Bolon-ti-k'u) Lords of the Night 9 .9 3*3 33
° Trecena (Oxlajun-ti-k'u) 13 .13 13 13
° Veintena / Uinal 20 1.0 13 13
° (Lounsbury) prime-232 23 1.3 23 23
? Two Trecenas (Oxlajun-ti-k'u) 26 1.6 2*13 2*13
° Half Calendar Round 26 (years) 1.6 2*13*365 2*13(*5*73)
* 29-day Lunation 29 1.9  [29.530588] 29 29
* 30-day Lunation 30 1.10  [29.530588] 30 30
° Calendar Round 52 (years) 2.12 (years) 4*13=2*26 22*13(*5*73)
° Five Trecena Cycle 65 3.5 5*13 5*13
*° (Prime-73) 73 3.13 73 73
* Palenque Moon Ratio
(81 Lunations = 2392 days)		 81 (lunations) 4.1 (*29.530588) 9*9 34
* Venus invisibility at
Superior Conjunction		 90 4.10 2*5*9 2*32*5
* (Mercury Synodic Period) 116 5.16 [115.88] 4*29 22*29
°
*?		 Cycle associated w/ Rain God
(Mercury Synodic Period?)		 117 5.17 [11.88] 9*13 33*13
* Venus Visibility as Morning Star 236 11.16 4*59 22*59
* Venus Visibility as Evening Star 250 12.10 4*59 2*53
° Sacred Almanac = Tzolk'in 260 13.0 (9*29)-1=4*65=13*20 22*5*13
* Chronological Year = Tun 360 1.0.0 18*20 23*32*5
*° Computing Year 364 1.0.4 4*91=13*28=7*52 22*7*13
* Calendar Year = Haab
"Vague Year"		 365 1.0.5 [365.2422] (4*91)+1=(18*20)+5=5*73 5*73
*? (Saturn Synodic Period) 378 1.0.18 [378.09] 2*3*7*9 2*33*7
*? (Jupiter Synodic Period) 399 1.1.19 [398.88] 2*3*7*9 3*7*19
° Twenty Squared 400 1.2.0 20*20 24*52
* Maya Eclipse Cycle 405 (lunations) 1.2.5 (*29.530588) 5*81=3*135 34*5 (moons)
* Double Sacred Almanac =
Triple Lunar Half-Node Cycle		 520 1.8.0 [3*173.30906] 3*173.3=2*260 23*5*13
* (Venus Synodic Period) 584 1.11.4 [583.92] 8*73 23*73
° 585-day Cycle 585 1.11.5 584+1=5*9*13=5*117 32*5*13
*? Half Year-Drift Cycle 754 (years) 2.1.14 (yrs) ½(1508 Calendar years)
=½(1507*365.2422 days)
=14.5*52 yrs		 2*13*29(*365)
*° Triple Sacred Almanac
(Mars Synodic Period)		 780 2.3.0 [779.94] 3*260 22*3*5*13
° 819-day Cycle 819 2.4.19 3*273=13*63=9*91=7*117 32*7*13
° 1/8 K'atun3 900 2.9.0 5*9*20=2.5*360 22*32*52
*? Year Drift Cycle 1507 (solar yr) 4.3.7 (*365.2422 days) 11*137 11*137
*° 1507*365.422 days =
ca. 1508*365
Year Drift Cycle		 1508 (Calendar yr) 4.3.8 (*365 days) 2*754=29*52 2*13*29
° 1/4 K'atun4 1800 5.0.0 2*5*9*20=5*360 22*32*52
° Seven Sacred Almanac =
Five Computing Year Cycle		 1820 5.1.0 7*260=5*364 22*5*7*13
°? Nine Sacred Alamanac =
Twenty 117-day Cycle5		 2340 6.9.0 4*585=9*260=20*117=20*9*13 22*32*5*13
* Palenque Moon Ratio
(81 lunations = 2392 days)		 2392 6.11.12 (=2391.977628)
81*29.530588		 23*13*23
* Five Venus Years =
Eight Calendar Year Cycle		 2920 8.2.0 5*584=8*365 23*5*73
° Four 819-day Cycles 3276 9.1.16 9*364=4*819 22*32*7*13
° 1/2 K'atun6 3600 10.0.0 9*20*20=10*360 24*32*52
* Copan Moon Ratio7
(149 lunations=4400 days)		 4400 12.4.0 149*29.530588=4400.057612
4*5*11*20		 (149)
24*52*11
  Thompson's "mystic 13 tuns"
(Venus correction factor)8		 4680 13.0.0 13*360=18*260=40*17 23*32*5*13
° K'atun (Winikhaab) 7200 1.0.0.0 20*20*18=20*360 25*32*52
*° "Micro 11-Bak'tun" 8660 1.4.1.0 433*20 22*5*433
* Dresden Venus Factor9 9100 1.5.5.0 5*20*7*13 2252*7*13
* Dresden Eclipse Cycle
(405 lunations)		 11,960 1.13.4.0 46*260=5*2392=5*81*29.530588 23*5*13*23
°? Twenty 819-day Cycles 16,380 2.5.9.0 45*364=5*3276=20*819 22*32*5*7*13
° 52 Tun/72 Almanac Cycle 18,720 2.12.0.0 53*360=72*260 24*32*5*13
° Calendar Round
52 Haab/73 Almanac Cycle		 18,980 2.12.13.0 52*365=73*260 22*5*13*73
*° "Micro 3-11 Bak'tun" 25,980 3.12.3.0 3*433*20 22*3*5*433
*° Great Venus Cycle correction10
=104-year Cycle—13.0.0		 33,280 4.12.8.0 2CR-(13*360)
20*32*52=128*260		 29*5*13
*° Great Venus Cycle
"unabrideged"=104-year Cycle11		 37,960 5.5.8.0 2*18980=65*584=2*52*365
=104*365=2*73*260=146*260		 23*5*13*73
° Bak'tun (Pik) 144,000 1.0.0.0.0 20*20*20*18=400*360 27*32*53
*? Half Calendar-Year
Drift Cycle		 275,210
(754 Haab)		 1.18.4.8.10 365*754=365*26*29 2*5*13*29*73
*? Calendar Year Drift Cycle =
29 Calendar Rounds		 550,420
(1508 Haab)		 3.16.8.17.0 1508*365=1507*365.2422=
2117*260=942.5*584=29*18980		 22*5*13*29*73
*? Double Calendar Year
Drift Cycle		 1,110,840
(2*1508 Haab)		 7.12.17.16.0 2*1508*365=2*1507*365.2422=
4234*260=1885*584=29*37960		 23*5*13*29*73
° PAL Cross Interval12 1,359,540 9.8.16.9.0 7*9*13*20*83 22*32*5*7*13*83
° Dresden Venus DN13 1,366,560 9.9.16.0.0 9*260*584 25*32*5*13*73
° 80 Calendar Rounds
(Palenque TI Interval)		 1,518,400 10.10.17.14.0 4*20*52*365 26*52*13*73
° Thirteen Bak'tun 1,872,000 13.0.0.0.0 13*20*20*360 27*32*53*13
° Piktun 2,880,000 1.0.0.0.0.0 204*18=8000*360 28*32*54
* "3-11-Bak'tun" 4,752,000 1.13.0.0.0.0 3*11*202*360 27*33*53*11
*? 65 Bak'tun
5*13 Bak'tun		 9,360,000 3.5.0.0.0.0 5*13*202*360 27*32*54*13
* 66 Bak'tun
(double 3-11-Pik)		 9,504,000 3.6.0.0.0.0 6*11*20*20*360 28*33*53*11

Notes

  1. Adapted from John B. Carlson, "Numerology and the Astronomy of the Maya," publ. in Williamson, Ray A., ed., Archaeoastronomy in the Americas. Bean, Lowell John, and Blackburn, Thomas C., series eds., Ballena Press Anthropological Papers No. 22, 1981.
  1. Lounsbury (1976), etc.
  1. Discovered by David Stuart in Temple XIX at Palenque (Stuart 2005 pp.34-35).
  1. The quarter-k'atun and half-k'atun (5 years and 10 years) period-endings were commonly celebrated by the erection of stelae.
  1. Lounsbury (1978, Table 1, p. 780) reports that the Dresden Venus Pages have three (out of six) intervals based on multiples of 6.9.0 = 2340 days: once, thrice (19.9.0 = 7020) and four times (1.6.0.0 = 9360 days).
  1. The quarter-k'atun and half-k'atun (5 years and 10 years) period-endings were commonly celebrated by the erection of stelae.
  1. Coe 1992, p. 132, quoting Teeple.
  1. Lounsbury 1978 p. 787; Barb MacLeod, private communication.
  1. Dresden p. 24, the introduction to the Venus Tables, second row (Lounsbury 1978 pp. 784-5).
  1. Dresden p. 24, the introduction to the Venus Tables, Lounsbury (1978 pp. 784-5) tells us that subtracting 13 tuns from the 65-Venus-year/104-Calendar-year run of the Table on Dresden pp. 46-50 compensates for the Venus Drift, while retaining the beginning on a 1 Ajaw.
  1. Dresden p. 24, the introduction to the Venus Tables, has in its top row multiples of 5.5.8.0.
  1. Lounsbury (1976) tells us that this interval includes every "significant Maya number" except 23 and 73, and "only the 83 is without significance in its own right." (p. 215).
  1. Lounsbury 1978, pp. 786-7, "The … ring-number base and the historical date (to which this DN counts, i.e., the beginning and end dates) occupy the same positions in the 260-day almanac, the 365-day year, the 584-day canonical Venus year, a 2,340-day cycle that unites that of the almanac with that of the 9 Lords of the Night, and with a 117-day cycle pertaining to the rain god (besides being also the corrective foreshortening of the Venus great cycle), and various higher-order cycles deriving from these, such as the Calendar Round (18,980 days), the unabridged great cycle of Venus (37,980 days), and so forth. Thus 9.9.16.0.0 = 1,366,560 days = 5256*260 = 3744*365 = 584*2340 = 468*2920 = 72*18,980 = 36*37,960."

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