Mathetic Year Beginning Mismatch, and a Revised Grammatēmerologion

Much like how I recently encountered one devil of an author having put something out for public use (though it turned out to be a complete non-issue), now I’m facing another one, this time a lot more serious for me.

So, here’s the issue I face.  I have this thing called the Grammatēmerologion, a lunisolar calendar system that allots the letters of the Greek alphabet to the days, months, and years in a regular, systematized way.  I developed this system of keeping track of lunar months and days for my Mathesis work, a system of theurgy based on Neoplatonic and Neopythagorean philosophy and practices in a Hermetic and loosely Hellenic framework largely centered on the use of the Greek alphabet as its main vehicle for understanding and exploring spirituality.  Not only can the Grammatēmerologion be used as a system of calendrical divination a la Mayan day sign astrology (or tzolk’in), but also for arranging for rituals, festivals, and worship dates in a regular way according to the ruling letter of the day, month, and (rarely) year.  Sounds pretty solid, right?  I even put out a free ebook for people to use and reference, should they so choose, just for their convenience in case they were curious about the Grammatēmerologion for their own needs.

However, this isn’t the only system of time and timing that I need to reference.  In reality, I’m dealing with two cycles: one is the calendrical cycle of the Grammatēmerologion, which starts a new year roughly at the first New Moon after the summer solstice, and the zodiacal cycle that starts at the spring equinox.  The fact that they don’t line up is something that I noted rather early on, yet, passed off easily as “well, whatever, not a big deal”.  However, the more I think about it and how I want to arrange my own system of rituals and ritual timing, the more I realize that this is actually a big deal.

Let’s dig into this a bit more.  Why does the Grammatēmerologion start at the first New Moon after the summer solstice?  This is because the Grammatēmerologion is loosely based on the old Attic calendar, which had the same practice; for the Attics and Athenians, the new year started with summer.  Why did I bother with that?  Honestly, because the system seemed easy enough to apply more-or-less out of the box, and there is a rather convenient solar eclipse on the summer solstice in 576 BCE that would serve as a useful epoch date, this also being the first time the Noumenia coincided with the summer solstice since the stateman Solon reformed Athenian government and laws in 594 BCE.  I figured that this was a pleasant way to tie the Grammatēmerologion into a culturally Greek current as well as tying it to an astronomical event to give it extra spiritual weight.

However, by linking it to the summer solstice, I end up with two notions of “new cycles”, one based on this lunisolar system and one based on the passage of the Sun through the signs of the Zodiac.  The zodiacal stuff is huge for me, and only stands to become even bigger.  While there can truly be no full, exact match between a lunisolar calendar (Grammatēmerologic months) and a strictly solar one (Zodiacal ingresses), having them synced at least every once in a while is still a benefit, because I can better link the Noumēnia (the first day of the lunar month) to an actual zodiac sign.  This would give the months themselves extra magical weight, because now they can officially overlap.  Technically, this could still be done with the Grammatēmerologion as it is, except “the beginning of a cycle” ends up having two separate meanings: one that is strictly zodiacal based, and one that is lunisolar and slapped-on starting a full season later.

The issue arises in how I plan to explore the Tetractys with the letter-paths according to my previous development:

The plan was to traverse the 10 realms described by the Tetractys according to the letters of the Greek alphabet, using twelve paths associated with the signs of the Zodiac, starting with Bēta (for Aries).  This would be “the first step”, and would indicate a new cycle, just as Aries is the first sign of the Zodiac and, thus, the astrological solar year.  Pretty solid, if you ask me, and the cosmological implications line up nicely.  Except, of course, with the notion of when to start the year.  If I really want my Grammatēmerologion system to match well as a lunisolar calendar for my needs, then I’d really need to make it sync up more with the Zodiac more than it does, at least in terms of when to start the year.  So long as the Grammatēmerologion calendar has its Prōtokhronia (New Years) within the sign Aries, this would be perfect, because then I could give, at minimum, the first day of the first month of the year to the first sign of the Zodiac.

So, there are several solutions that I can see for this:

  1. Set the Prōtokhronia (New Year) of the Grammatēmerologion to be the first New Moon after the spring equinox, using the first occurrence of this time after the original epoch date of June 29, 576 BCE.  This would put the first Noumenia of the most recent cycle 69 on April 15, 2010, though the epoch date would remain the same; we’d simply shift what letters would be given to what months.  This would be the least change-intensive option, but it causes all significance to the epoch year to vanish and seems like a giant kluge to me.
  2. Set the Prōtokhronia of the Grammatēmerologion to be the first New Moon after the spring equinox, using a new epoch date where a solar eclipse occurred up to two days before the spring equinox so that the Noumenia coincides with the equinox, hopefully in a year wherein something meaningful happened or which fell within a 19-year period (one Metonic cycle) after a moment where something meaningful happened.  There are very few such dates that satisfy the astronomical side of things.
  3. Reconfigure my own understanding of the flow of the Zodiac to start with Cancer (starting at the summer solstice) instead of with Aries (spring equinox).  This…yikes.  It would leave the Grammatēmerologion system intact as it is—even if at the expense of my own understanding of the nature of the Zodiac (which bothers me terribly and would go against much of well-established education and understanding on the subject) as well as the letter-to-path assignment on the mathetic Tetractys (which doesn’t bother me terribly much, since I still admit that it’s still liable to change, even if it does have a neat and clean assignment to it all).  This is the least labor-intensive, but probably the worst option there is.
  4. Leave both the Grammatēmerologion and zodiacal cycles as they are: leave the Grammatēmerologion to continue starting at summer and the zodiac to start in spring, and just deal with the mismatch of cycles.  This just screams “no” to me; after all, why would I tolerate something that causes me anguish as it is without any good reason or explanation for it, especially in a system that I’m designing of my own free will and for my own needs?  That would be ridiculous.

Based on my options above, I’m tempted to go with establishing a new epoch for the Grammatēmerologion to be set at a solar eclipse just before the spring equinox, with the Prōtokhronia set to coincide with the spring equinox itself.  If I want a reasonable epoch date that goes back to classical times or before…well, it’s not like I have many options, and comparing ephemerides for spring equinoxes and solar eclipses (especially when having to deal with Julian/Gregorian calendar conversions) is difficult at the best of times.  Here are such a few dates between 1000 BCE and 1 BCE, all of which use the Julian calendar, so conversion would be needed for the proleptic Gregorian calendar:

  1. March 30, 1000 BCE
  2. March 30, 935 BCE
  3. March 28, 647 BCE
  4. March 27, 628 BCE
  5. March 27, 609 BCE
  6. March 27, 563 BCE
  7. March 27, 544 BCE
  8. March 25, 294 BCE
  9. March 25, 275 BCE
  10. March 24, 256 BCE
  11. March 24, 237 BCE

As said before, the Attic-style summer-starting Grammatēmerologion has its epoch in 576 BCE, the first time that the Noumenia coincided with the summer solstice (and immediately after a solar eclipse), and the first such time either happened following Solon’s reforms in Athens.  The date that would most closely resemble this for a Mathetic spring-starting Grammatēmerologion would have its epoch in 563 BCE, only a handful of years later.  In the proleptic Gregorian calendar, this would mean that we’d start the epoch on March 21, 563 BCE, with the Noumēnia falling on the day after, the first day the New Moon can be seen and the first full day of spring.

On its face, this would seem to be an easy change to make; just change the epoch date and recalculate everything from there, right?  After all, I have all the programs and scripts ready to go to calculate everything I need, and since we know that a full grammatēmerologic cycle is 38 years which would get us to basically the next time the New Moon happens just after the spring equinox, we know that we’d currently be in cycle 68 (starts in 1984 CE).  Except…the spring equinox in 1984 occurs on March 20, and the New Moon occurs on…April 1.  That’s quite a large drift, much larger than I’d expect.  So I investigated that out and…yeah, as it turns out, there’s an increasing number of days’ difference between the spring equinox and the following New Moon over successive cycles.  I forgot that the Metonic cycle isn’t exact; there is a small amount of error where the lunar cycle shifts forward one day every 219 years, and between 1984 CE and 563 BCE, there’re 2550 years, which means a difference of just over 11 days…which is the number of days between March 20 and April 1, 1984.

And on top of that, I had originally calculated my original epoch date for the Attic-style summer-starting calendar incorrectly: the New Moon should have been on June 17, 576 BCE, not June 29; as it turns out, I had misconverted 576 BCE for year -576, when it should have been -575 (because 1 BCE is reckoned as year 0, 2 BCE as year -1, and so forth).  I majorly screwed myself over there; not only is my epoch system not working for how the revised Grammatēmerologion should work, but the epoch for the original Grammatēmerologion was wrong, anyway.  Splendid.

So much for having a long-term classically-timed epoch, then.  Without periodically fixing the calendar alignment or using a more precise cycle, such as the Callipic or Hipparchic cycle which still have their own inaccuracies, there’s still going to be some drift that won’t allow for establishing long-term cycles how I originally envisioned.  I still want to use the 38-year dual Metonic cycle, but since there’s no real need to tie it to any historical period except for my own wistfulness, I suppose I could use a much more recent epoch.  The most recent time that a solar eclipse happened just before the spring equinox, then, would have been March 20, 1643 CE, putting us in cycle 10 that starts in 1985 CE (which would start on March 22, since the New Moon is on March 21, just after the spring equinox on March 20, which is acceptable), making 2018 CE year 33 in the cycle.  The next cycle would start on March 22, just after the New Moon on March 21, just after the spring equinox (again) on March 20.  Again, this would be acceptable.  The issue of drift would be more evident later on, say, in year 3277 CE, which would start on March 27, which is definitely several days too late.  We start seeing a stable drift of more than two days starting in 2213 CE, but looking ahead a few years, we can see that 2216 CE would have a Prōtokhronia start perfectly on March 20, the day of that year’s spring equinox.

So, here’s my method for applying corrections to the Grammatēmerologion:

  1. Establish an epoch where the Prōtokhronia starts on the day of or the day after the spring equinox.
  2. Grammatēmerologic cycles are to be grouped in sets of seven, which would last 266 years, after which the drift between the dual Metonic cycle and the solar year becomes intolerable.  (We could use six cycles, getting us to 228 years, but seven is a nicer number and the error isn’t always completely stable at that point just yet due to the mismatch between lunations and equinoxes.)
  3. After the end of the seventh grammatēmerologic cycle, start up a “false” cycle to keep track of full and hollow months, until such a year arrives such that the Prōtokhronia of that year starts on the day of or the day after the spring equinox.
  4. That year is to mark the new epoch, and a new set of cycles is established on that day.  (This leads to a “false” cycle of only a few years, none of which should be lettered as usual.)

Let’s just make this simple, then: forget about aligning the beginning cycles with a spring equinox tied to a solar eclispe, and just settle for when the Noumēnia is either on or the day after the spring equinox.  The most recent time a New Moon coincided with the spring equinox was in 2015 CE.  Knowing that the New Moon coincided with the spring equinox on March 20 that year, this makes the epoch date for this cycle March 21, 2015.  This means that we’re currently in year four of the first cycle.  While I’m not entirely thrilled about losing the whole equinox eclipse significance thing, setting 2015 as a cycle start epoch makes sense; after all, the whole system of Mathesis really could be considered to start around then.

However, there’s one extra wrench thrown into the works for this; I want to make sure that the Prōtokhronia always falls while the Sun is in the sign of Aries, so the Noumēnia of the first month of the year must fall when the Sun has already crossed the spring equinox point.  Because twelve lunar months isn’t long enough to ensure that, we’d need to ensure that certain years are full (13 lunar months) and other years are hollow (12 lunar months), and it turns out that the regular Metonic scheme that the old Attic-style Grammatēmerologion doesn’t ensure that.  For instance, the first year of a cycle, according to the Metonic scheme, is supposed to be hollow; if we start the first year off immediately after the spring equinox, then the second year will start off about two weeks before the spring equinox, so we’d need to change how the years are allocated to be full or hollow.  And, to follow up with that, tweaks also need to be made to the scheme of figuring out which months are full (30 days) or hollow (29 days) to make sure they stay properly aligned with the dates of the New Moon, while also not going over the Metonic count of 235 lunar months consisting of 6940 days.

So.  After a day or so of hastily plotting out lunar phases, equinox dates, and eclipse times, I reconfigured my scripts and programs to calculate everything for me to account for all the changes to the Grammatēmerologion, rewrote my ebook to document said changes, and now have a revised Grammatēmerologion for the period between March 2015 and March 2053.  In addition, I took the opportunity to explore a useful extension of the Grammatēmerologion system and the seven-day week to account for days of planetary strength or weakness, as well, and documented them in the ebook, too.  (Normally, there would be no interaction, but this is one that actually makes sense in how the powers of the letters of the day are channeled.)

Download the revised Grammatēmerologion (March 2015 — March 2053) here!

I apologize for the confusion, guys.  Even though I know few people are ever going to take this little pet project of mine seriously, I regret having put something out that was so broken without realizing it.  I’m taking down the old version from my site, and only keeping the new revised version up; if anyone is interested in the old copy (even with its flaws), I can send it to them upon request, but I’d rather it not be so freely available as it was.

10 responses

  1. Pingback: Grammatēmerologion Calendar for Cycle 69, available for download now! « The Digital Ambler

  2. Also, as an addendum, I just want to boast/clarify that churning out a 450 page calendar with a 120 page directory of events that happen in that same span isn’t actually as difficult as it appears to be. I essentially have three three programs that do all the heavy lifting for me:

    – one to calculate astronomical phenomena (ecliptic position of the Sun, lunar phases, eclipses)
    – one to calculate the Metonic cycle and associate the letters of the Greek alphabet with the days, months, and years of the cycle, as well as ferret out other information like when the letter of the day matches up with the month or when the stoicheia of the letter matches up with the weekday
    – one to compile all the data/content files together and emit a PDF to read

    Of these, probably the most laborious program to code was the astronomical phenomena program because of all the astronomical factors that needed to be accounted for, but the most intensive one was the calendar calculation one, because so much needs to be validated, vetted, and tested for it to ensure that it works right.

    With those done, all I really had to do was write the actual introduction and set the formatting, which was trivial. It may seem weird to go through all the work to automate much of the book production, but for a project of this scale, it absolutely paid off in spades, especially since I had to redo the entire damn thing basically with the discovery and reconciliation of the errors mentioned in this post. Without these programs at my beck and call, I’d’ve worked for a year straight on getting the first calendar set up, and more than that on getting the second one done; instead, it took me about a week to get the first calendar done, and a day to revise it and put out the second one (because I didn’t have to do so much coding).

    Knowing how to program is truly a blessing, y’all. I highly encourage you to learn it if you have the opportunity to.

  3. One of the things that really impresses me about this massive project of yours, is that you’re taking on work that has occupied some of the most sophisticated mathematicians of several different cultures on several different continents over a period of several thousand years. The Maya and the Aztecs, the Chinese, the Egyptians (and the Romans through borrowing), the Gregorian-era reformers, have all sorted this calendar problem in a variety of ways at various times; Islam and Judaism each made decisions of their own, as did several branches of Hinduism.

    It’s a knotty problem, designing a calendar. And here you are, tackling that problem as a blogger instead of as some sort of Astronomer Royal or Imperial Keeper of the Calendar, or as part of a Hierophantic Commission. If I ever find myself ruling as a philosopher-king, I’ll be sure to grant you the royal warrant for this sort of work.

    But your point about learning to program and code is well-taken. In the event that someone DID want to design a calendar from scratch, would you consider writing a post about what resources you consulted, what you got from each, and some example code that helps translate the astronomical calculations into machine language? I think that kind of “under the hood” tinkering and explanation would be at least as informative as your learning process for designing a mathetic calendar…

  4. I wanted to thank you for all the work you’ve put into this. You’ve inspired me to finally get around to settling a ritual calendar system for myself. I did have one question though. In table 13 on page 24 where you discuss 7-day week correspondences, you list gamma as an Astremera of Thursday and Phugemera of Tuesday. I expected delta in these positions and I’m wondering if I’m just misunderstanding part of the system.

    • Delta is the letter associated with the zodiac sign Gemini, which has the domicile of Mercury, so its Astrēmera should be Wednesday and the exile of Jupiter, so its Phugēmera should be Thursday. Both of these are in the table.

      Gamma is the letter for the sign of Taurus, domicile of Venus (Ast. Friday), exaltation of the Moon (Dox. Monday), and exile of Mars (Phu. Tuesday). It has no fall, so it doesn’t have a Phthorēmera. All these are in the table, as well.

      I think you misread the Astrēmerai of Thursday wrong; it’s Pi and Tau (Sagittarius and Pisces, respectively). Would that be the case?

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