I’ve had this idea in my head for prayer practice that revolves around the notion of cycles.  For instance, as part of my daily prayer practice, I’ve written a set of seven prayers, one for each of the seven days of the week, which I recite on an ongoing cycle.  They’re not necessarily planetary prayers, like you might find in the Hygromanteia or Heptameron, but they do have some planetary allusions and hints thrown into them.  The seven-day week, which is fundamentally a Mesopotamian invention, makes for a simple cycle of prayers, but I’ve been thinking about ways I could incorporate more cycles into my prayers.  For instance, a simple and short invocation for each of the days of a lunar month—with my Grammatēmerologion, my oracular Greek letter lunisolar calendar—based around the powers and potencies of each of the letters of the Greek alphabet, along with their spirits or gods, could be something fun to toy around with.  There’s lots of opportunities for this sort of practice:

• the four turns of the Sun each day, a la Liber Resh (sunrise, noon, sunset, midnight)
• the seven days of the week
• the 24 planetary hours of a given day
• the four (or eight) phases of the Moon (new, crescent, first quarter, gibbous, full, disseminating, third quarter, balsamic)
• the 29/30 days of a synodic lunar month
• the 28 days of a sidereal lunar month (a la the 28 lunar mansions)
• the 30/31 days of a solar month (a la the 12 signs of the Zodiac)
• the four seasons (solstices and equinoxes), perhaps also with the four cross-quarter days (midpoints between the solstices and equinoxes)
• the 10 days of a decan
• when a planet stations retrograde or direct
• when eclipses occur
• when a planet or star is seen at its heliacal rising or setting

There are lots of opportunities to engage in prayers linked to or with the natural cycles of the cosmos, many of which are fundamentally astrological in nature.  The idea of coming up with a large-scale overarching prayer practice that engages in such cycles, to me, would be a fantastic way to recognize these natural cycles, bring oneself into alignment with them, and tap ever more greatly into the power of these cycles, especially when certain cycles interact or sync up with each other.  By aligning ourselves with these cycles, we can not just make use of χρονος khronos “time” generally, but also καιρος kairos “the moment”, the fleeting opening of opportunity itself that allows us to do the best thing possible.  There’s this Hermetic notion—it’s hard to find the note I was referencing for it, but I’m pretty sure it’s in Copenhaver’s Hermetica or Litwa’s Hermetica II—that we rely on kairos in order to fully carry out the process of rebirth in the Hermetic mystical sense, and that would be determined by the processes of Providence, Necessity, and Fate along with the very will of God.

Along these lines, I wanted to come up with a new cycle of prayers for myself, one specifically for the decans.  Some might know these as faces, the 36 10° segments of the ecliptic, three to a sign of the Zodiac.  The decans are old, as in ancient Egyptian old, and play a part in the astrological prognosticatory and magical literature of the Egyptians, Arabs, Brahmins, and Hermeticists the world over.  We see them referenced in magical-medical texts going back to the classical period, and they also appear in such texts as the Picatrix as well as Cornelius Agrippa (book II, chapter 37).  Though they come up time and time again, they also take so many wildly different forms between traditions and texts, which is fascinating on its own merits.  We even see Hermēs Trismegistus himself talk about the decans and their importance in the Sixth Stobaean Fragment.  In that part of the Hermetic cannon, Hermēs explains to Tat that the decans belong to a celestial sphere between the eighth sphere of the fixed stars and the higher sphere of the All, being a backdrop to the very stars themselves, and thus higher than the constellations and signs of the Zodiac.  These decans exert “the greatest energy” on us and the world, and they drive “all general events on the earth: overthrows of kings, uprisings in cities, famines, plagues, tsunamis, and earthquakes”.  In other Hermetic texts, like the Sacred Book of Hermēs to Asclepius, the decans also rule over specific parts of the body and the injuries and illnesses that afflict them (which is a very Egyptian concept indeed that we see in purer forms of Egyptian religion and spiritual practice).

You can probably guess where I’m going with this: more prayers and a ritual practice dedicated to the decans.  This would consist of two parts:

• An invocation of the powers of the decan itself, according to its specific form and name and virtues, to be done when the Sun enters that decan.
• One prayer per each day the Sun is in a given decan, a set of ten prayers to be recited over a ten day decanal “week”.  Since the Sun spends about one day per degree, this means that each degree of a decan can be considered a separate day, and each day with its own prayer.

After some thinking, I was able to come up with a relatively straightforward set of prayers for the decans themselves at the moment (or the first sunrise following) the Sun’s ingress into them, but it’s the latter part I’m still struggling with.  I have ideas about what to base them on—the ten Hermetic virtues from the Corpus Hermeticum, the Pythagorean symbolism of the first ten numbers, and so forth—but coming up with those prayers is a slow process, indeed.

In the meantime, I’ve been working on a bit of a programming project, something to plan ahead and help me figure out what such a prayer practice would look like scheduled out.  This is basically what I was doing with my Grammatēmerologion project, coding up a variety of astronomical functions to calculate the various positions and attributes of celestial bodies for any given moment, and courtesy of SUBLUNAR.SPACE (whose online customizable almanac is an invaluable and deeply treasured tool for any magician nowadays), I was tipped off to a much easier and faster way to develop such astronomical programs: the Swiss Ephemeris codebase, of which I found a Python extension for even more flexibility.

And that’s when the problems started.  (Beyond the usual mishaps that come along with any nontrivial programming project.)

See, as it turns out, there are more days in a year than there are degrees in a circle—which means that while the Sun moves roughly one degree per day, it actually moves slightly less than one degree per day.  This is why we have 365 days (or 366 days, in leap years) in a year.  To the ancient Egyptians, they considered the civil solar year to only have 12 months of 30 days each, each month consisting of three decans, with a leftover set of five days at the end of the year, considered to be the birthdays of the gods Osiris, Horus, Set, Isis, and Nephthys.  These intercalary (or epagomenal) days were considered a spiritually dangerous and liminal time, but once those days were over, the calendar was brought back into sync with its proper cycle.  However, what I wanted to do is to come up with a 10-day cycle linked to the degrees of the Sun, which means I would have to deal with these epagomenal days throughout the year instead of bundled up all at the end.  My logic was simple:

• Start counting decan day assignments (decan day-numbers) starting from the first sunrise after the March equinox (which is when the Sun enters 0° Aries as well as the first decan).
• Judge the degree of the ecliptical position of the Sun based on sunrise of any given day.
• Take the whole degree of the Sun (e.g. if 9.459°, then 9), divide by 10, take the remainder, and that’s your day in the cycle.  Thus, if o°, then this is our first day; if 1°, the second day; if 2°, the third day;…if 9°, the tenth day.  Thus, when we hit the next o° day, we start the cycle over.
• If the whole degree of the Sun is the same as the previous day (e.g. 7.998° for today and 7.014° for yesterday), then this is an epagomenal day, and we say either no prayer at all or an eleventh special prayer not otherwise used except for epagomenal days.

A relatively simple method, all told.  Or so I thought.  When I actually ran the program, I noticed that there were not five epagomenal days (e.g. 1-2-3-4-5-X-6-7-8-9-10, where X is the epagomenal day) in the final count, but seven, which was…weird.  This would mean that there were 367 days, which would be wrong, except that there were 365 outputs.  It turns out that there were two skipped days (e.g. 1-2-3-4-5-6-7-9-10, but no 8), one in early December and one in mid-February.  On top of that, although I expected the epagomenal days to be spaced out more-or-less equally throughout the year, they were all between early April and mid-September.  After looking into this, and making sure my code was correct (it was), what’s going on is this:

• I made the mistake of assuming that the Sun moves at a constant speed each and every day of the year.  It doesn’t, for a variety of astronomical factors.
• The Sun spends more time in the northern celestial hemisphere (about 185 days) than in the southern celestial hemisphere (about 180 days).
• The Sun moves slower in winter around perihelion than in the summer around aphelion.
• From winter through summer, the sunrise gets earlier and earlier, pushing the judgment-time of each day earlier and earlier, while in summer through winter, the reverse happens.

Talk about vexation: I had here what I thought was a perfectly reasonable method—and to a large extent, it is—yet which results in the cycle just skipping days, which I intensely dislike, since it breaks the cycle.  Without doubling up prayers on the skipped days, which I’d really rather like to avoid, it means that I couldn’t use this otherwise simple method to figure out a decanal 10-prayer schedule that would be in sync with the Sun.

After thinking about it some, I considered five different ways to associate the days to the degrees of the decans:

1. The “Egyptian” method.  This is the most old-school and traditional, and mimics the behavior of the actual ancient Egyptian calendar: starting from the New Year, assign an unbroken cycle of days from day one to day ten 36 times.  This gradually becomes more and more unsynced as time goes on, but we throw in five or six epagomenal days at the very end to catch up all at once before the next New Year.  Simple, traditional, clean, but it’s really the worst of the bunch with the accumulating degree differences that get resolved all at once at the end of the year instead of periodically throughout the year.
2. The “plan-ahead” method. Like the Egyptian”method, this is a pretty artificial way to allocate the days, but elegant in its own way, and spreads out the epagomenal days across the year more-or-less regularly.  We know that, at least for the foreseeable future, we’re going to deal with either normal years of 365 days or leap years of 366 days.  For normal years, we need to have five epagomenal days, so we insert an epagomenal day after the 8th, 15th, 22nd, 29th, and 36th decans (or, in other words, every seventh decan not including the first).  For leap years, we need six epagomenal days, which we insert after the 6th, 12th, 18th, 24th, 30th, and 36th decan (i.e. every sixth decan).  Note that we judge a year to be a normal year or a leap year based on the Gregorian calendar year prior to a given March equinox; thus, for this method, we start assigning days from the March 2020 equinxo using the normal method because the prior calendar year, 2019, was not a leap year; we use the leap year method starting from the March 2021 equinox because the prior calendar year, 2020, was a leap year.
3. The “true degree” method.  This is the method mentioned before: starting with the New Year at the March equinox, when the true degree of the Sun is exactly 0° and using sunrise at one’s location as the reference time, take the degree of the Sun and compare it to the degree at the previous day’s reference time.  If the degree is in the next whole number (e.g. 23.005° and 22.025°), the day proceeds to the next whole number; if the degree is in the same whole number (e.g. 23.985° and 23.005°), then it’s an epagomenal days.  The problem, as stated earlier, is that due to the varying speed of the Sun as the Earth travels between perihelion and aphelion (which also has the effect of the Sun spending more time in the northern celestial hemisphere than in the southern celestial hemisphere), we end up with more epagomenal days than expected around aphelion, and with days that are outright skipped around perihelion.  While the exact match of day to degree is appealing, it’s the skipped days that breaks cycles and which ruins the whole prayer system I was trying to devise.
4. The “average degree” method.  This is a variation on the true degree method, only instead of using the Sun’s true position at the reference time on each day, we take a theoretical position of the Sun based on its average daily motion of 360.0°/365.2421897 days = 0.98564735989°/day.  Starting with the New Year at the March equinox, when both the true degree and average degree of the Sun is exactly 0°, using sunrise at one’s location as the reference time, take the theoretical average degree of the Sun (advancing it by the Sun’s average daily motion day by day at the reference time) and compare it to the degree at the previous day’s reference time, with the same epagomenal rule as before.  The benefit to this method is that it gets us the expected number of epagomenal days which are evenly distributed throughout the year without skipping any other days; the downside is that, as we get closer to the September equinox, the theoretical average position of the Sun drifts further away from the true position by as much as 3.780°, putting us three or four days out of sync with the true position.
5. The “rebalanced true degree” method.  This is an extension of the true degree method above.  We start with the assignments of days to degrees as before, extra epagomenal days and skipped days and all, but we “rebalance” the days by removing some epagomenal days and reinserting them where we were earlier skipping days.  For every skipped day, we alternate between choosing the first and last of the epagomenal days.  So, if we have seven epagomenal days on year days 24, 59, 83, 105, 127, 151, and 182, and we have two skipped days on days 274 and 333, then we first remove the first epagomenal day from day 24 and reinsert it on day 274, and then the last epagomenal day from day 181 (was 182 before we removed the other one) and insert it on day 333.

So, five different methods of assigning days a decan day-number, one of which (the Egyptian method) being the most regular and artificial with the worst drift, one of which (the true degree method) being the most accurate and realistic yet which skips days entirely, and three other methods (plan-ahead, average degree, rebalanced true degree) that vary in terms of computational complexity and accuracy.  We know that the true degree method is the most accurate, so we can plot the various other methods against it to visually see how bad the drift is between it and the other methods.  In the following graphs, the true degree method is given in red, with the other method being compared to it in blue.  Epagomenal days are marked as having a decan day-count number of -1, hence the severe dips at times.  Where the blue and red lines are more in sync, the method is better; where the lines depart, the method gets worse.  The true degree method gives an epagomenal day in decans 3, 6, 8, 11, 13, 15, and 18, and if you look close enough, you can see the skip in the days towards the end of decans 27 and 33.

Just visually looking at these methods, we can see that all four methods start off the same for a little more than the first two decans, but after that, most of them begin to diverge.  The Egyptian method is worse in how often and by how much it diverges, with that nasty flatline of epagomenal days at the end, and the plan-ahead method doesn’t fare much better, either; note also how both of these methods end with epagomenal days for at least the final day of the year.  The average degree method doesn’t look too bad, though it does get worse around the September-October area of the year before it gets better again, eventually getting back in sync for the final three decans of the year.  By far the most pleasing and in-sync graph we see is with the rebalanced true degree method, which does vary a little bit but by no means as bad or as irregularly as the other methods; we have about five decans where they’re in sync, 22 where they’re one day off, and nine when they’re off by two days.

But, besides just looking at them with my eyeballs, how should I best compare the accuracy of all these methods?  What I settled on was a ratio between the day’s decan day-number according to a particular method and the true degree expected for the Sun for that day:

1. If a given day is an epagomenal day, throw out the value entirely, and don’t factor it into calculations.
2. For a given day reckoned at the reference time (sunrise on the March equinox for a given location), find the Sun’s true ecliptic position.
3. Take the whole degree of the Sun (e.g. if 9.227°, 9).
4. Divide the number from the previous step by 10 and take the remainder.
5. Add one to the previous step.
6. Divide a given day’s decan day-number by the previous step.

The shortcut to this method would basically be to divide the method’s decan day-number for a given day against the true degree method’s decan day-number, but I wanted to be sure I was getting the Sun’s true position here for mathematical rigor.  This ratio indicates the general percentage difference we expect; if the ratio is 1, then the given method’s decan day-number is what we’d expect; if more than 1, it’s ahead of what we expect; if less than 1, behind what we expect.

Doing some simple math on these ratios for these given methods gets us the following statistics (omitting the epagomenal days entirely), judged against the year from the March 2020 equinox through the March 2021 equinox (considered a normal year).  I calculated these results based on a prototype decanal calendar starting on March 20, 2020 at 11:12 UTC (the first sunrise after the spring equinox for my town’s given longitude) for 365 days.

In the 2020/2021 year, we can see that it’s the rebalanced true degree method that has the lowest standard deviation and variance, with the mean closest to 1.  This means that the rebalanced true degree method gets us the closest decan day-numbers to what the Sun’s actual position is on the whole, being at worst three days ahead (compared to the potential of being five, six, or eight days ahead with the other non-true degree methods).

For another look, we can also consider the leap year (according to our rule above) for the March 2021 equinox through the March 2022 equinox.  I calculated these results based on a prototype decanal calendar starting on March 20, 2021 at 11:13 UTC for 366 days.

We get even better results during leap years, it’d seem, at least based on this example alone; we’re only a max of two days ahead of the Sun’s true position, and we have even less variance and deviation than before.

If I were to go with any system of assigning a 10-day repeating cycle of prayers to the days to keep more-or-less in sync with the position of the Sun as it goes through the decans, I’d go with the rebalanced true degree method.  Still, even if it’s the most in sync, it’s not truly in sync, as there really isn’t such a system possible without skipping days due to the inconvenient misalignment of physical phenomena with discrete human systems of calendrics.  As SUBLUNAR.SPACE commiserated with me about on Facebook, as he found out when he was coding his own almanac program, the decans “do not like to be pushed into human patterns”, and that we really have to choose degrees or days, because we can’t have both.  In his almanac, he settled with marking things by the actual ingress, which was the common practice in the decan calendars of Ptolemaic times.  On top of that, as far as calculation goes, it’s among the more complicated, requiring manual rebalancing after figuring out the true degree day equivalences first for the whole year until the next March equinox; easy enough to do by a computer program, but tedious or outright difficult to do by hand.

For now, I’m going to content myself with marking the Sun’s ingress into the decans, and leave it at that.  For one, though I’d like to engage in a 10-day cycle of prayers aligned with the decans, and even though I have some sort of system in place to explore that, I still don’t have those damn ten (or eleven) prayers written up for them.  But, at least knowing what the schedule looks like is a start.