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Q17 Lunivenusian Calendar v3.01.008


The Q17 calendar is a daycounting mechanism which tracks the mean synodic moon. It grew from research this author did into the worlds calendars to see whether there were some elegant way to track the real moon via a coefficient related to the gregorian calendar. The issue rapidly reduces to the establishment & synchronisation of an independent robust lunar metric which I came to call the Qbundle. The Qbundle is a period of 502 days which is startlingly close to 17 lunar months.

Although at first seemingly somewhat arcane, a lunar calendar based on the Qbundle has the following key advantages:

The symmetric structure of the fixed cyclical almanac allows it to be tracked much like the Maya tzolkin insofar as it comprises an invariant daycount which proceeds for 50 cycles (68+ years) before requiring adjustment by a single intercalary day. & this latter adjustment is the only requisite to give the calendar itself a theoretical accuracy of a day in some 131,000 years

This high degree of accuracy restores the moon to its rightful role as TimeKeeper rather than TimeKept. By tracking the mean synodic moon, Q17 avoids the artifactual & idiosyncratic monthlength calculations of the worlds astronomical, observational, & empirical lunar calendars, & allows this function to be performed by ordinary daycounters such as Java applets

My personal view is that of all the mechanisms which distort & obscure our universe the calendar should be the least. It should be able to elevate astronomical day-counting from a civil necessity to a sacred mantra, & that compellingly so. & based on a highly-accurate cyclical schema it should serve as a tool to enhance perception of real astronomical events.

a brief Introduction

Something is up with the moon. Solar eclipses themselves are actually a bit hard to believe. & why does the worlds dominant solar calendar maintain a single yearly purely lunar observance? Further, why do millions of disparate people around the world assiduously observe a lunation every 29.53 days or so & celebrate a lunar newyear? My theory is that whales can count to seven. As immensely old tidal beings I believe they feel the 4 phases of the moon & they see 7 flashes of light between each one as the days spin by. Although the correspondence of the weekdays with the planets is the gift of the kings, the week itself is not lost on the whales.

All of us are the products of the cosmic cycles around us, & when we become calendar designers there are certain numbers we cannot escape. Like the whales our primary tool is -- of course -- the best integral approximation in days to some other intrinsic coelestial cycle. Todays historically-derived calendars are as good as theyre going to be because they can approximate years or months in only so many ways. But the existence of the 17mo Qbundle & its 50fold evolute the Q-Era -- although widely known in astronomical circles -- provides an approximation so accurate it is difficult to imagine not using it calendrically. We are in fact at all moments enveloped by it & may as well become aware of it.

Table 1: Calendrical cycles - time to accrue 1 day discrepancy

cycle                  equivalence  yearly   1 day
julian year       1461/4   =  d/y  11.2min    128y
gregorian year  146097/400 =  d/y  26.8sec   3226y
Metonic cycle      235/19  = mo/y   6.6min    219y
metameton         4131/334 = mo/y   7.4sec  11554y
Qbundle (vague)    502/17  =  d/mo 21.0min     69y
Q-Era            25101/850 =  d/mo  0.7sec 130870y

As can readily be seen, 17 months is within a half an hour of 502 days (which in itself is phenomenal) but adjusted with an almost absurdly infrequent leap day the effective approximation of 502.02/17 to the mean synodic month is essentially perfect.

so how does it work?

The structure of the Qbundle arises directly from the length in days of the mean synodic month:

29.5305888531 days / month

This value (known with great accuracy for thousands of years) is readily (& optimally) approximated as 502/17 or 25101/850 (day/mo). A 17-month almanac (Qbundle) would comprise 502.0200105027 days, so the discrepancy from bundling days in 502-day groups would not add to a full day for 50 cycles (850mo / 68+ years). A leap day added every 50th Qbundle to lengthen it to 503 days equates 25101 days with 850 months, an approximation accurate to a day in more than 130,000 years.

The Qbundle is a fixed cyclical almanac comprising these 502 days divided into an alternating sequence of 9 30-day & 8 29-day months. Monikers of the months abbreviate to A thru Q & the lengths are invariant except every 50th Qbundle when a 30th day is added to month P. A group of 50 Qbundles comprises a 25101-day "Q-Era".

Table 2: Simple diagram of the Qbundle

                Dragon     Hourglass     Love        Peace
           Bird       Flower      Justice     Nothing
  01  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  29
          | Brd | Dra | Flo | Hrg | Jus | Lov | Not | Pax |
       | Ani | Cha | Egg | Glx | Inf | Kar | Mus | Ord | Qny |
  01 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 30
        Anima        Egg       Infinity     Music      Quanyin
             Chalice     Galaxy       Karma       Order

The only further issue is adopting a particular synchronisation. I have taken the first day of the first Qbundle to be ad0554mar22 (ns / gregorian), & formally observed the first Anima Moonday (1st day of Qbundle 1056) on 2004apr20ns. Although much of the theoretical work which follows was sketched out prior to that time, it is what I have learned in using the system for 3 years that has motivated my rewrite of the original Q17 document.

qbundle grid
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ani
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  brd
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o cha
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  dra
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o egg
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  flo
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o glx
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  hrg
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o inf
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  jus
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o kar
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  lov
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o mus
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  not
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ord
  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  pax
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o qny

fundamental cycles: weekly / foxfold / Q-Era

The Q17 user is immediately struck by the viability of strictly alternating 30 & 29 day months. The short (vacant) months begin & end on the same weekday & the long (robust) months are simply a day longer. Each incipient month thus starts either 1 or 2 weekdays after the previous, & when a month "seats" it is only necessary to note its moniker & weekday (the "planetary character" or PC day) to establish its attributes as they relate to the coming 4 weeks. This is indicated diagramatically in the following rune by the cyan & green rings & their attachments to the Q17 months:

Above: Q17 fundamental mandala.

This diagram also calls attention to what I call "foxfold partitioning" or, more simply, the foxfold, which is the pattern of the cardinal lunar phases one obtains via binary partitioning of the 17 months into 16 equal parts. The Qbundle begins & ends on a newmoon, thus its midpoint is the full Infinity moon; its quarter points are quarter moons &c. This symmetrical effect proves exceedingly handy in certain calculations.

The periphery of the diagram attests to the lengthening of the penultimate month via a "Pax30" day every Qbundle divisible by 50. At the beginning of each Q-Era the newmoons fall generally on the day prior to the 1st, then as the 50 Qbundles progress newmoons drift towards falling generally on the 1st itself, whereupon the cycle is reset at the end of the Q-Era for another 68+ years. At this writing we are not yet 10 Qbundles into the 22nd Q-Era.

parametonic bundling strategies

Here begins the slightly more mathematical aspect of the analysis. Briefly, the best approximations of well-known calendrical values are achieved by decomposition via continued fractions (qv presented below) into integral ratios; the 17mo & 850mo periods in Q17 arise directly from this process. & just as the Qbundle effects optimal rule-based bundling of days into months, so arises the question of bundling whole synodic months into solar years (ie the "lunisolar" calendar) or even into venusian years (ie "lunivenusian") or other periods. The Q17 approach to the Metonic & related cycles is "non-embolismic" & results in nothing more exotic than lunisolar years starting with different months.

The table below summarises the Q17 lunar (& Venusian) intervals along with the most useful integral month/year ratios of lunisolar commensurability (136/11, 235/19, & 4131/334; conceivably called the inframetonic, Metonic, & metametonic cycles). The months comprising the meton itself are not divisible by 17, but (notably) the others contain an integral number of Qbundles.

Table 3: Q17 lunisolar & lunivenusian commensurabilities

cycle     abbr    days  months    qb  Q-Eras years  notes
week      w          7
month     mo        29+      1
Qbundle   Q/qb     502      17     1          11/8-
Q-Era     E      25101     850    50    1       68+ Q17 IC lunar
IMC       i       4016     136     8    0.16    11- endecaeteris
meton     m       6939+    235    14-           19+ enneadecaeteris
metameton M     121991-   4131   243    4.86   334+ Q17 IC lunisolar
M50       M50  6099543  206550 12150  243    16700  hypermetameton
Vmeton    vm     28615+    969    57    1.14    78+ venusian meton
Vround    Vr    193278-   6545   385    7.70   529+ venus almanac

the endecaeteris & the Metonic cycle

The InfraMetonic Cycle (IMC or endecaeteris) is a fundamental subcycle in the meton & posits an equivalence between 136 months (8 Qbundles) & 11 years (which is almost exactly 1.5 days longer). This means the length of the Qbundle is quite accurately approximated by 11/8 of a year. What proves to be the most noticeable effect of this is that at a given time the newmoon of one of the Q17 months falls closer to the equinoxes, solstices, & crossquarter points than any other, serving to anchor the Qbundle with respect to the sun. The "Anchor Moons" shift in bundle-bearer sequence (qv below) roughly every 20yrs. At this writing it is month Q whose incipience traces out this slowly-rotating 8-pointed star in the solar tropic each 11yr period, to wit:

 2004 spring equinox
 2005 lammas
 2006 winter solstice
 2008 beltane
 2009 fall equinox
 2011 imbolc
 2012 summer solstice
 2013 samhain

Curiously, these 136 synodic months of the endecaeteris are precisely the 132 gregorian months of the solar activity cycle, a synchrony whose significance is beyond the scope of this document.

The legacy meton allocates 235 months neatly among 19 years & is composed of an 11y/8Q IMC (the Q17 "leap-meton" qv below) followed by an octaeteris (ie an IMC truncated to 8y). Here are two views of the prototypical Q17 meton (beginning with month A): the classical diagram beside a rendering into Qbundles (12/13 month years are in lower/upper case with year boundaries indicated by colons):

Table 4: the classical Metonic cycle mapped by Q17

                                   qb  + - + - + - + - + - + - + - + - +  mo
                                   01 :a b c d e f g h i j k l:m n o p q 017
       length   235-month  Q17mA   02  a b c d e f g:H I J K L M N O P Q 034
years (months)    index    month   03  A B C:d e f g h i j k l m n o:p q 051
===== ======== =========== =====   04  a b c d e f g h i j:K L M N O P Q 068
01-03 12 12 13 001 013 025 A M H   05  A B C D E F:g h i j k l m n o p q 085
04-06 12 12 13 038 050 062 D P K   06  a:B C D E F G H I J K L M N:o p q 102
07-08    12 13     075 087   G B   07  a b c d e f g h i:j k l m n o p q 119
09-11 12 12 13 100 112 124 O J E   08  a b c d:E F G H I J K L M N O P Q 136
12-14 12 12 13 137 149 161 A M H   09 :a b c d e f g h i j k l:m n o p q 153
15-17 12 12 13 174 186 198 D P K   10  a b c d e f g:H I J K L M N O P Q 170
18-19    12 13     211 223   G B   11  A B C:d e f g h i j k l m n o:p q 187
                                   12  a b c d e f g h i j:K L M N O P Q 204
                                   13  A B C D E F:g h i j k l m n o p q 221
                                   14  a:B C D E F G H I J K L M N       235

As 235mod17 = 14, each Q17 meton begins with a different month in the 17-fold progression AOLIFCQNKHEBPMJGD. These are the "meton-bearers": the months which determine which 11 of the Q17 months begin years in each meton. The meton number determines the meton-bearer via the expression ((m?-1)*14)mod17+1.

the metameton

With a robust & symmetrical lunar synodic framework in place, it is actually trivial to introduce lunisolar capability. This is merely the ability to specify the beginning of a solar year coincident with the beginning of a month. Although this question of reckoning is often addressed via the Metonic cycle, it is not inconvenient to take advantage of the significantly greater accuracy of its successor the "metameton":

Table 5: overview of the metameton

<----------------------- 4131 months ----------------------> 4131 = 243*17
<------------------------ 243 Qbundles --------------------> 243 = 235+08
<-- Q-Era --><-- Q-Era --><-- Q-Era --><-- Q-Era --><35><08> 4.86 Q-Eras
<- mA mB mC mD mE mF mG mH mI mJ mK mL mM mN mO mP mQ -><11> 17+ metons
<------------------------ 334 years -----------------------> 334 = 323+11
....+....1....+....2....+....3....+....4....+....5....+....6 not to scale

The 4131 months of the metameton (just short of 5 Q-Eras) comprise an integral number of both Qbundles (243) & solar years (334). This period can also be seen to be composed of 17 complete 19-year Metonic cycles (equalling precisely 235 Qbundles) plus a "leap-meton" shortened to an 11y/8Q IMC. The following table shows the beginning months for the complete cycle of 334 years in the metameton (the last year in each column consists of 13 months):

Table 6: the metameton mapped by Q17

m?   003 006 08 011 014 017 19    y   mo
mA : AMH DPK GB OJE AMH DPK GB : 019 0235
mB : OJE AMH DP LGB OJE AMH DP : 038 0470
mC : LGB OJE AM IDP LGB OJE AM : 057 0705
mD : IDP LGB OJ FAM IDP LGB OJ : 076 0940
mE : FAM IDP LG COJ FAM IDP LG : 095 1175
mF : COJ FAM ID QLG COJ FAM ID : 114 1410
mG : QLG COJ FA NID QLG COJ FA : 133 1745
mH : NID QLG CO KFA NID QLG CO : 152 1880
mI : KFA NID QL HCO KFA NID QL : 171 2115
mJ : HCO KFA NI EQL HCO KFA NI : 190 2350
mK : EQL HCO KF BNI EQL HCO KF : 209 2585
mL : BNI EQL HC PKF BNI EQL HC : 228 2820
mM : PKF BNI EQ MHC PKF BNI EQ : 247 3055
mN : MHC PKF BN JEQ MHC PKF BN : 266 3290
mO : JEQ MHC PK GBN JEQ MHC PK : 285 3525
mP : GBN JEQ MH DPK GBN JEQ MH : 304 3760
mQ : DPK GBN JE AMH DPK GBN JE : 323 3995
mZ : AMH DPK GB OJE            : 334 4131

The progression of metametonic bundling against a fixed framework over a long period of time is clearly exhibited here by the Q17 paradigm. The 17-fold symmetry (including 17 complete metons) is quite fortuitous; this synchrony means the simplest possible commensurability is exactly realised between 17 metons & the 235 Qbundles they comprise. The metametonic rule is merely that the classical Metonic cycle progress uninterrupted thru 334 complete years & then start over. This serves to truncate mZ to 8 Qbundles; thus the leap-meton is itself an endecaeteris which leaves the incipient monthly cycles completely undisturbed.

The meton has an interesting relationship with the 136mo/11y of the IMC. As 19 & 11 are relatively prime, a question of commensurability arises after 11 metons (ie mK), putatively 19 IMCs. 19i*136 = 2584mo whereas 11m*235 = 2585mo, ie 209 years is the time required for the IMC to be a full month slow vs the meton. An 11-fold symmetry is clearly illustrated by the Q17 months, which begin years 136 months apart in 11 parallel cycles. With each new meton 10 of these months carry over, but the interval from year 17 to year 09 is 137 months so one month is incremented (ie disappears / debuts). Each 11-year cycle begins with the same month 19 times (ie 209y) & then begins with the subsequent month. With the adoption of the leap-meton the 11-year cycles repeat 20 times for the 11 Q17 months it contains, & the commensurability question arises after 11 metametons (ie 334i*136 = 45424mo vs 11M*4131 = 45441mo). The metameton effectively adds 17 (!) months during this 3674-year period; 216.1 years for a full month of IMC drift (this is far closer to the actual value of 216.0y).

In short, these tables provide a way unambiguously to determine the beginning month of any Q17 lunisolar year, proleptic or not, & determine its number in the Q17 scheme. Although the Venus cycle (below) obviously needs its own numbering, Im ambivalent as to whether Q17 solar years should just use gregorian numbers anyway; but I now view an actual solar or astronomical calendar as an unnecessary part of Q17. Whats laborious is naming the years by initial month, & this is what the Java applet does, as well as track the Qbundle number which overall is more intrinsic to the Q17 date itself.

the lunivenusian calendar

The Venusian aspect of Q17 is simply a synthesis of the preceding techniques. Of course a sufficiently accurate daycount can be calendrified. The Venus cycle of 583 47/51 days implies each group of 51 584-day periods should be lightened by 4; as 584 itself is divisible by 4, these days can be subtracted symmetrically from the 146-day quarters 51, 102, 153 & 204.

The 19.77345 month Venus cycle can also be bundled into whole months like the tropical year. The basic Venus year is 20 months which are shortened to 19 every 4th or 5th cycle. The fundamental intercalary building blocks are composed of complete tetrads & pentads (19:20:20:20 / 19:20:20:20:20):

Table 7: venusian parametonic approximations

 V                -mo   mo
004:            4 - 1   79
009:           45 - 2  178
013:          445 - 3  257
018:         4545 - 4  356
022:        45445 - 5  435
026:       445445 - 6  514
031:      4545445 - 7  613
035:     45445445 - 8  692
039:    445445445 - 9  771
044:   4544545445 -10  870
049:  45454545445 -11  969
053: 454454545445 -12 1048

As the coefficients of the continued fraction for the Venusian period remain small for many iterations, it is not difficult to find ratios of comparable accuracy other than the principal convergents. Q17 achieves a 6545/331 mo/V ratio by repeated occurrences of a Venusian meton (vm) with a 969/49 ratio. Like its solar counterpart, the Venus meton becomes more accurate by periodically shortening a cycle (ie each 7th cycle becomes 731/37). As all 3 of these cycles comprise a multiple of 17 months (ie 385Q = 6*57Q + 43Q), successive passes thru the almanac begin on the same month (the first year in each column consists of 19 months):

Table 8: venus round (venusian metametonic cycle)

v?   0004 00009 0013 00018 0022 00027 0031 00036 0040 0044 00049    V   mo
v7 : CEHK NPBEH KMPB EGJMP BDGJ MOADG JLOA DFILO A               : 331 6545

The discrepancy is 9 hours every (529-year) cycle or a day in 1432 years.

the jovisat

The Saturn / Jupiter conjunction or "jovisat" (also called the Crono-Zeusian Grouping or CZG) is the longest planetary synodic period in classical calendrics. The 7253.445 days of the mean cycle are amazingly close to 1965/8 months (245.6248mo); thus with each conjunction the moons phase advances by (exactly) 5/8 resulting in an 8-fold jovisat cycle correlated with the 8 phases of the moon. This is a plot of the newmoon jovisats from just before the kali yuga to the present:

Table 9: the jovisat

czg  gregorian     Q17      czg  gregorian     Q17
000 -3123/03/25 6625ord01 | 136 -0422/02/03 8590ord01
008 -2964/02/09 6741hrg01 | 144 -0264/12/19 8706glx30
016 -2806/12/25 6857ani01 | 152 -0105/11/03 8821qny30
024 -2647/11/08 6972kar01 | 160 +0054/09/18 8937kar01
032 -2488/09/24 7088dra01 | 168 +0213/08/03 9053cha29
040 -2329/08/08 7203mus30 | 176 +0372/06/18 9168mus30
048 -2170/06/24 7319glx01 | 184 +0531/05/04 9284glx01
056 -2011/05/08 7434qny01 | 192 +0690/03/18 0099qny01
064 -1852/03/23 7550jus01 | 200 +0849/01/31 0215inf30
072 -1693/02/07 7666cha01 | 208 +1007/12/17 0331brd29
080 -1535/12/21 7781mus01 | 216 +1166/11/01 0446mus01
088 -1376/11/06 7897flo01 | 224 +1325/09/15 0562egg29
096 -1217/09/21 8012ord30 | 232 +1484/07/31 0677ord30
104 -1058/08/06 8128inf01 | 240 +1643/06/15 0793hrg29
112 -0899/06/21 8244ani30 | 248 +1802/05/01 0909ani29
120 -0740/05/05 8359kar30 | 256 +1961/03/16 1024kar30
128 -0581/03/21 8475dra29 | 264 +2120/01/29 1140dra29

As this table shows, each 8th mean jovisat is tied so closely to the newmoon that more than 21 cycles are required to realise a full day drift. This is because 1965 months is 58027.61 days while 8 jovisats is 58027.56 (58027 13:36:24) days; ie *1.1 hours* every 158+ years.

The regression of the jovisats thru 1/8 of the lunar cycle can be taken to be 9300 Qbundles (186 Q-Eras; 4668786 days; nearly 2.5 Maya long counts). This is the modulus for the Q17 cyclical Qbundle count & is referred to as a "Q-Aeon". The cycle of 8 Q-Aeons represents the complete drift thru the 8 lunar phases: some 102300 years, far slower than the precession of the equinoxes.

synchronisation & intercalation

The Q17 Qbundle count is cyclical (rather than using zero or negative numbers), repeating after each Q-Aeon of Qbundles numbered 0001-9300. The Qbundle 0001 closest to our era is synchronised with ad0554mar19 (julian), also known as ad0554mar21 (ns / gregorian) or (Maya long count).

This equinoctial day witnessed a total solar eclipse & is the Q17 zero day of Anima, the last day of Qbundle 9300. The following day is the Anima Moonday, the beginning of the first month of Qbundle 0001. Anima 01 is the beginning of metametonic year 1, preceded by a zero year preceded by negative years.

The Venus round is synchronised with ad0554may24ns, the inferior conjunction closest to the beginning of Qbundle 0001 (this can generally be approximated as 2 months). Venus years are cyclically numbered 0001-8000 in vague proportion to the Q-Aeon.

continued fractions
lunar synodic (days) = 29.5305888531 (as of 2000jan01) =
29 + ----------------------------------------------------
     1 + ------------------------------------------------
         1 + --------------------------------------------
             7 + ----------------------------------------
                 1 + ------------------------------------
                     2 + --------------------------------
                        17 + ----------------------------
                             2  + -----------------------
[29;1;1;7;1;2;17;2;5;2;9;4; ... ]
29 + 1/1 = 30
29 + 1/(1+1/1) = 29 1/2 = 59/2
29 + 1/(1+1/(1+1/7)) = 29 8/15 = 443/15
29 + 1/(1+1/(1+1/(7+1/1))) = 29 9/17 = 502/17
29 + 1/(1+1/(1+1/(7+1/(1+1/2)))) = 29 26/49 = 1447/49
29 + 1/(1+1/(1+1/(7+1/(1+1/(2+1/17))))) = 29 451/850 = 25101/850
tropical year (days) = 365.242203052422 (1507 denominator) =
365 + 1/4 = 365 1/4 = 1461/4
365 + 1/(4+1/7) = 365 7/29 = 10592/29
365 + 1/(4+1/(7+1/1)) = 365 8/33 = 12053/33
365 + 1/(4+1/(7+1/(1+1/3))) = 365 31/128 = 46751/128
365 + 1/(4+1/(7+1/(1+1/(3+1/3)))) = 365 101/417 = 152306/417
365 + 1/(4+1/(7+1/(1+1/(3+1/(3+1/1))))) = 365 132/545 = 199057/545
365 + 1/(4+1/(7+1/(1+1/(3+1/(3+1/(1+1/2)))))) = 365 365/1507 = 550420/1507
tropical year (months) = 12.368267 =
[12;2;1;2;1;1;17;2;2;15; ... ]
12 + 1/2 = 12 1/2 = 25/2
12 + 1/(2+1/1) = 12 1/3 = 37/3
12 + 1/(2+1/(1+1/2)) = 12 3/8 = 99/8 (octaeteris)
12 + 1/(2+1/(1+1/(2+1/1))) = 12 4/11 = 136/11 (endecaeteris)
12 + 1/(2+1/(1+1/(2+1/(1+1/1)))) = 12 7/19 = 235/19 (Metonic cycle)
12 + 1/(2+1/(1+1/(2+1/(1+1/(1+1/17))))) = 12 123/334 = 4131/334
years/Qbundle = 1.374485 = 
[1;2;1;2;29;1;46;1;4;3; ... ]
1 + 1/2 = 1 1/2 = 3/2
1 + 1/(2+1/1) = 1 1/3 = 4/3
1 + 1/(2+1/(1+1/2)) 1 3/8 = 11/8 (endecaeteris)
1 + 1/(2+1/(1+1/(2+1/29))) = 1 88/235 = 323/235 (17 metons)
1 + 1/(2+1/(1+1/(2+1/(29+1/1)))) = 1 91/243 = 334/243 (metameton)
venus synodic (days) = 583.92166 =
[583;1;11;1;3;3;1;20; ... ]
583 + 1/1 = 584
583 + 1/(1+1/11) = 583 11/12 = 7007/12
583 + 1/(1+1/(11+1/1)) = 583 12/13 = 7591/13
583 + 1/(1+1/(11+1/(1+1/3))) = 583 47/51 = 29780/51
583 + 1/(1+1/(11+1/(1+1/(3+1/3)))) = 583 153/166 = 96931/166
venus synodic (months) = 19.77345 =
[19;1;3;2;2;2;2;4;67; ... ]
19 + 1/1 = 20
19 + 1/(1+1/3) = 19 3/4 = 157/4
19 + 1/(1+1/(3+1/2)) = 19 7/9 = 178/9
19 + 1/(1+1/(3+1/(2+1/2))) = 19 17/22 = 435/22
19 + 1/(1+1/(3+1/(2+1/(2+1/2)))) = 19 41/53 = 1048/53
19 + 1/(1+1/(3+1/(2+1/(2+1/(2+1/2))))) = 19 99/128 = 2531/128
jovisat synodic (months) = 245.6248 =
[245;1;1;1;2;1;77; ... ]
245 + 1/1 = 246
245 + 1/(1+1/1) = 245 1/2 = 491/2
245 + 1/(1+1/(1+1/1)) = 245 3/4 = 983/4
245 + 1/(1+1/(1+1/(1+1/2))) = 245 5/8 = 1965/8
jovisat synodic (Qbundles) = 14.448518 =
14 + 1/2 = 29/2
14 + 1/(2+1/4) = 14 4/9 = 130/9
14 + 1/(2+1/(4+1/2)) = 14 9/20 = 289/20
14 + 1/(2+1/(4+1/(2+1/1))) = 14 13/29 = 419/29
14 + 1/(2+1/(4+1/(2+1/(1+1/4)))) = 14 61/136 = 1965/136

weekdays & the planetary character

The Planetary Character day of the Anima Moonday (the first day of the first month) determines the PCs of the months in its Qbundle & is thus called the "bundle-bearer". This weekday (like that of every date in the 502-day Qbundle) increments by 5 (ie regresses by 2) with each reoccurrence resulting in a repeating sequence of Qbundles characterised by their initial day:

Table 10: 7fold qbundle planetary character

Ani Brd Cha Dra Egg Flo Glx Hrg Inf Jus Kar Lov Mus Not Ord Pax Qny
Sun Tue Wed Fri Sat Mon Tue Thu Fri Sun Mon Wed Thu Sat Sun Tue Wed
Fri Sun Mon Wed Thu Sat Sun Tue Wed Fri Sat Mon Tue Thu Fri Sun Mon
Wed Fri Sat Mon Tue Thu Fri Sun Mon Wed Thu Sat Sun Tue Wed Fri Sat
Mon Wed Thu Sat Sun Tue Wed Fri Sat Mon Tue Thu Fri Sun Mon Wed Thu
Sat Mon Tue Thu Fri Sun Mon Wed Thu Sat Sun Tue Wed Fri Sat Mon Tue
Thu Sat Sun Tue Wed Fri Sat Mon Tue Thu Fri Sun Mon Wed Thu Sat Sun
Tue Thu Fri Sun Mon Wed Thu Sat Sun Tue Wed Fri Sat Mon Tue Thu Fri

This 7-fold cycle of weekday / month combinations consists of 7*17 = 119 months (9.62 years) occurring 7 times in a Q-Era, comprising the first 49 Qbundles (a "Qubilee"). The 50th Qbundle shares the PC of the Q-Era & its subordinate cycles. With the occurrence of each Pax30 day the beginning of the new Q-Era is delayed 1 weekday, so the "Era-bearer" weekday advances by 6 instead of 5 (ie regresses by 1 with each new Q-Era). This results in a 7-fold repeating cycle of Q-Eras comprising 350 Qbundles / 5950 months (481.07y):

Table 11: 7fold Q-Era planetary character

E?   0000007 0000014 0000021 0000028 0000035 0000042 0000049 50   Q   mo

These periodic structures completely determine the relationship of the Q17 dates to the weekdays, facilitating numerical calculations.

working with the greg

Between month turnovers the Q17 & gregorian daycounts are related to one another by a fixed offset (the "Qdelta") which it is useful to note. Each new Qdelta is (obviously) the date in the non-exiting month of the last day of the exiting month. A little thought shows that the upcoming Qdelta is also the current Qdelta subtracted from the length of the exiting month.

The Qdelta arises from the discrepancy between the mean lengths of the Q17 vs gregorian months (29.5305888531d vs 30.43685d), roughly 0.9 days. The "real" Qdelta reaches zero every 992 days as this discrepancy adds to a full Q17 month. Put another way, 33.585 Q17 months equals 32.585 gregorian months, & thus (barring the "jitters" in the gregorian layout) the months begin on the same day every 2.7 years or so. In whole months, this interval is conveniently close to 2 complete Qbundles, & because of the gregorian irregularity the Q17 alignment month remains fairly stable even while drifting slowly one month backward (ie 67 Q17 months is 65 greg months). A tabulated examination of these events is useful:

Table 12: Q17 vs gregorian Qdelta alignments (* = exact)

Q17       gregorian *   Q17       gregorian *
1055glx01 2003may31   | 1071drg01 2025feb28  
1057egg01 2005dec31   | 1073brd01 2027oct01 *
1059drg01 2008sep01 * | 1075ani01 2030jun01 *
1061egg01 2011jul01 * | 1077ani01 2033mar01 *
1063egg01 2014mar31   | 1079ani01 2035nov30  
1065drg01 2016nov30   | 1080qny01 2038jul31  
1067drg01 2019aug31   | 1082pax01 2041apr01 *
1069cha01 2022may01 * | 1084ord01 2043dec01 *

Using a quartz watch to keep the Q17 calendar is easier than using a mechanical watch with a (31) day-only display to keep any of the historical lunar calendars (where it is necessary to adjust every month). At the start of the (non-Q) 30-day months a quartz watch can be set to 01Jun & left unadjusted for 59 days; when it reaches 30Jul the process is repeated.

the lunar tzolkin

The Maya calendar is primarily planetary rather than tropical, with the Tzolkin crafted from astrometrically-dictated divisors rather than seeking to approximate any single coelestial cycle. The motivation for the Qbundle is no less astrometric; it arises from a coincidence of commensurability & its 17 months correspond to nothing but themselves. Indeed, as the Qbundle & the Tzolkin are both fixed cyclical almanacs, conversion between them is amazingly painless despite the differences in internal structure.

Unlike the Tzolkin, the Qbundle (as a sum of monthlengths) has no factors other than 2*251. Half a Qbundle is thus 9 days short of a Tzolkin; 2 Tzolkins being 18 days longer than a Qbundle. The oft-noted 325-day period (approximating the 324.836 days of 11 months) is exactly 5/4 of a Tzolkin; ie 260+65 days. Intervals of 11 full months in Q17 are 325 days 14/17 of the time (324d beginning with B:D:F); a single Tzolkin is thus (with proper adjustments) 11 months -59d -06d, quite straightforward via Q17. The double Tzolkin is reached by adding 1Q:18d.

Of course finding the next Qbundle in the Tzolkin calendar is even easier via the Maya modular technique. 502mod20 is 2 while 502mod13 is 8; thus if a Qbundle begins on 9 Ben the next one will begin on 4 Men.

Daniel 9 & the 71 weeks

Exegetes have long discussed the prophecy outlined in Daniel 9:24-27. It becomes most clear when it is noted that the 502-day Qbundle consists of 70 weeks, a week, & a portion of a week, & that the 25101-day Q-Era consists of 49 ordinary Qbundles followed by a 503-day Qbundle incorporating the only leapday required by the Q17 system. An invariant & religiously habitual daycounting process is thus in effect for some 68 years whereupon intercalary necessity calls for a singular & venerable event & the princely wisdom required to effect an apocalyptic adjustment.

9:24 Seventy weeks are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy.

Indeed, the Q17 Qbundle is nothing other than the New Jerusalem itself, remedying the errors of history, its 502 days understood by the people as 70 weeks followed 98% of the time by a 12 day coda.

9:25 Recognise wisely that from the time the word goes forth that Jerusalem should be restored and rebuilt, seven weeks shall pass till the appearance of one anointed, a prince; reprising for sixty-two weeks justly & meaningfully to temper the age.

Here the weeks refer to "sevens" of ordinary Qbundles, 49 in all, which progress from the incipience of each Q-Era until the 50th one restores & rebuilds the integrity via an intercalary day. The leap-Qbundle is the messianic anointee, proceeding as usual for 434 days, ie thru the 21st day of Order, the completion of 3/4 of the 15th moon. The subsequent 99 days (69 remaining in the Qbundle plus the inaugural first month of the new Q-Era) witness the closings of 4 consecutive 30-day months.

9:26 And after threescore and two weeks shall Messiah be cut off, but not for himself: and the people of the prince that shall come shall destroy the city and the sanctuary; and the end thereof shall be with a flood, and unto the end of the war desolations are decreed.

The remainder of this Qbundle is different; the regularity of daycounting observed for nearly 70 years is suddenly interrupted by an extra day in the 16th month. No matter how it is presented by those who wisely insist upon it, it may seem to be an abomination, & the people will need to get over it.

9:27 And he shall confirm the covenant with many for one week: and in the partial week he shall cause the sacrifice and the oblation to cease, and for the overspreading of abominations he shall make it desolate, even until the consummation, and that determined shall be poured upon the desolate.

This refers simply to the conclusion of any Qbundle, when after 490 days there is a 71st week which confirms the covenant, followed by an incomplete week of 5 or (rarely) 6 days. Those ill prepared for the sophistication of the arrangement shall be regarded as idolators, & shall thereby suffer from the drift of their inaccurate calendar.

The Q17 calendar is not (at this time) some syncretistic replacement for the worlds great calendars, recouping their incredible achievements. It is foremost a bundling strategy for the days used to count the revolutions of the earths nearest coelestial neighbour, & little more. Its advantage over historical schemes is that it begins with a finely-tuned purely lunar time grid recognising the identities of 17 distinct months whose monikers & lengths are determined entirely from their positions in the Q17 sequence. Thus although the mathematical relationships to the true solar year &/or a system of metametonic lunisolar bundling can be tracked, the months themselves are not "(inserted) leapmonths" & their lengths are invariant. & because of the accuracy of its fundamental layout the intercalary mechanisms used by Q17 can be benign & almost absurdly infrequent.

the myth (!)

From an inconceivably infinite time in the past the Bird God materialised at the edge of a beach on an ephemeral island between Africa & Antarctica. At precisely the same instant, from an inconceivably infinite time in the future, the Peace Goddess appeared as well. It was Thursday 19 March 554 AD. "Havent I seen you somewhere before?" the Bird God asked. The Peace Goddess smiled.

"Its clear," she said, "that what this planet needs is a robust lunar calendar based on the mean synodic period." The Bird God walked to the shore & returned with 270 smooth spheroidal rocks which he proceeded to lay out on the sand in a 9 * 30 rectangle. When he was finished, the Peace Goddess produced a basket containing 232 shells, then placed one in each recess between the groups of four rocks, creating an 8 * 29 rectangle on the surface.

"Of course you realise well have to add a day every 850 lunations to produce a true 25101 day count," the Bird God observed. The Peace Goddess merely looked at him with an expression embodying the omnipresent poignancy of sentient awareness throughout creation. "Intercalate *this* dood," she finally said, glancing upward.

At that moment there was a total eclipse of the sun; Saturn was visible in broad daylight briefly near the corona. When it was over the Peace Goddess sat down & meditated; the Bird God performed a brief ceremonial dance (or it might have been the other way around). "The equinox is right about now," the Bird God suggested. "Whatever ..." the Peace Goddess replied. Then they both disappeared into the eternal cosmic wind.


Last update: 2023jun24pm1824
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