Antigram-pairs, their differences, the solar calendar and the circle of the zodiac
As shown above, the 26 trigrams can be seen as 13 antigram pairs that have difference-values from 1 to 13. These differences are unique. There is only one pair of antigrams that differ by 11, or 56 or 143. This fact allows us to organize all n-grams in a linear order, based on their differences.
Two practical applications of this principle follow. There are 729 hexagrams of Trigrammaton, and these include the Zero trigram, and 364 pairs of antigrams. These are attributed to the 365 days of the Solar year, beginning with the Zero on March 20, the Equinox. Thus we can track the path of the Sun throughout the year by using hexagrams.
By removing the 4 equidistant pairs of antigrams, (which are all multiples of 91), there are 360 pairs of antigrams left, which are attributed to the 360 degrees of the Zodiac. This attribution is more generalized, and can be used for various astrological purposes. One could find an antigram pair that matches the degree of their rising sign, for example.
The four pairs that are removed are 91/182, 273/546, 455/637, and 364/728. These are also the pairs that result from the nuclear hexagram process decribed earlier. These eight numbers are the roots of all the hexagrams, and they stand here in pairs as the markers of the four seasons of the year. A similar idea was used by the Chinese in attributing 12 I Ching hexagrams to the months of the year, based on seasonal changes.
In the calendar chart that follows, the lower number of each antigram pair is in the left column. In some cases this is also the value of the difference between them, which appears to the right of the pair, followed by the corresponding day of the Solar Year, and then the degree of the Zodiac Circle, (Note that the Zodiac degrees are not meant to correlate with the solar position; these are two different systems on the same chart). The chart concludes with the Lunar Mansions of Vedic astrology.
Now for some technical data about the antigram sequence…
The initial set of 13 antigram pairs sets the pattern for this sequence by the order of their difference-values. After them, each set of 27 pairs contains the upper trigrams in the same order. Thus the Houses of 27 & 54 are the first set, since their upper trigrams normally differ by 1. These are followed by the Houses of 135 & 189, whose upper trigrams normally differ by 2. The sequence ends with the Houses of 351 and 702, whose upper trigrams differ by 13.
While the upper pair of trigrams in each set of 27 numbers remains the same, the lower pair of trigrams runs through all 13 pairs of antigrams twice. In each case, the set begins with trigrams 13 & 26 in lower position, followed by trigrams 12 & 24, 14 & 25, etc, in reverse order until the center of the sequence, where both hexagrams contain the Zero trigram below.
Then the trigrams switch columns and continue to increase again, until at the end of the sequence of 27 pairs, the lower trigrams are again 13 & 26, but on opposite sides of the fence. This pulsation wave continues ad infinitum through the sequence of hexagrams. This regularity is why only one pair of antigrams can differ by any given number.
The entire series of antigrams that differ by the whole numbers from zero to infinity (in positive and negative directions) is based upon the 13 fundamental antigrams, and the Zero whence they emanate. In order that the sequence increases arithmetically by a value of 1 unit for each subsequent pair, the following 13 pairs are fundamental, the lower number being written first:
01 – 02
05 – 07
03 – 06
04 – 08
17 – 22
15 – 21
16 – 23
11 – 19
09 – 18
10 – 20
14 – 25
12 – 24
13 – 26
Looking at the trigrams based on structure, the sequence of 13 pairs has a nice symmetry to it. Trigrams with no Tao are Planets; with one Tao they’re Zodiacal; and with two Tao they are Elemental. Zero is neutral. From this perspective, the fundamental thirteen pairs have the following flow: 0 E / Z E / Z P / Z P / Z E / Z P / Z P
Since the most basic of all antigrams is the pair of 1 and 2, which represent Yang and Yin, the lower number of an antigram pair is the Yang number, and the higher number is the Yin number. In reference to calendar days, these would be the daylight and darkness portions, respectively.
As the antigram sequence progresses, the Yang numbers always have as their lower trigram the following numbers:
This is nothing more than the numbers given in the chart above, read from bottom to top on the right, and then top to bottom on the left. The sequence of lower trigrams for the corresponding Yin numbers of the antigram pairs just reverses the first pattern:
So what’s happening here is that you have some upper-trigram pair such as 5 and 7 above. Below them on one side you have a sequence from 26 to 0 and then out to 13, while on the other side you have the sequence from 13 to 0 and then out to 26. It’s like a helical twist, with the pair that has the Zero trigram below acting as the pivot for the whole sequence; this puts it at the center of the 27 days.
Each day outward from that center has trigrams below that differ by 1, and then 2, and then 3, etc., until the extremes are reached, and the beginning and end of any 27-day cycle has the 13 and 26 trigrams below. The way the numbers tick upward by 1 unit each time requires that the lower trigrams of any antigram pair will follow the sequence above.
The Hexagrams and the Circle of the Zodiac
The Zodiac circle is divided into 360 degrees, with each of the 12 signs occupying 30 degrees. The hexagrams occur as 364 pairs of opposites, and these have been attributed to the sequence of days in a solar year. A further refinement that is more generalized is to allocate one hexagram pair to each degree of the circle. This, however, will leave four pairs of hexagrams out of the loop. There are three obvious choices of ‘extra’ hexagrams: the first four pairs, the last four pairs, or four equidistant pairs. The last choice is most in keeping with the equanimity of the hexagrams.
The four pairs of opposites in question are: 91-182, 455-637, 273-546, 364-728. These same pairs were shown above as the ideal markers of the four cardinal points of the Zodiac. They represent the exact moment of the Solstices and Equinoxes. These pairs are also the ultimate results of employing the nuclear hexagrams, explained above.
In the process of nucleation, the 729 hexagrams reduce to 81 possible nuclear hexagrams, and these in turn reduce to 9 possible second-generation nuclei. These 9 nuclei are the Zero and all eight of the multiples of 91, represented by the hexagrams above. In this sense, they are the ultimate core of the hexagram tables, and are thus suited to stand outside the other 360 pairs of opposites.
The allocation of the hexagram pairs to the degrees begins with the Zero degree of Aries and ends at the 29th degree of Pisces, in the ordinal sequence of the opposite pairs. This approximates to the attribution of the Solar Year given above, but refines it to employ the Zodiac degrees, rather than one Earth day, (which is slightly less than one degree), for the points of the circle.
The first 90 degrees follow the normal sequence. The first, (zero), degree of Aries is the hexagram pair that differs by 1, the second degree is the pair that differs by 2, and so on. At the 91st degree, the Summer Solstice, one pair is skipped, namely, 91-182. Thus the 91st degree of the Zodiac is the pair that differs by 92, while the 92nd degree is the pair that differs by 93, and so on. The same skipping is done at all four cardinal points, finishing with the antigram pair of 364-728 at the Vernal Equinox, making the pair 363-726 the final degree of Pisces. The specific degrees are enumerated in the chart of the ordinal sequence of opposites, reproduced above.
The total value of all 728 hexagrams is 265,356. By removing the eight multiples of 91, this total is reduced to 262,080, which equals 360 x 728. Since there are 360 pairs of opposites and 360 degrees in the circle, each degree has an average value of 728. This distribution balances the entire circle and also relates it to the 26 trigram, the pure Yin, because the 728 hexagram is simply the doubling of the 26 trigram. The 728 hexagram therefore symbolizes the goddess Nuit, the circumference of the celestial sphere.
One striking result of allocating the hexagrams in this pattern is that the critical pair of 333-666 resides in the last degree of Aquarius. Because of precession, the Equinox point of the Earth’s orbit moves retrograde through the ecliptic, and when it leaves the first degree of Pisces and enters the last degree of Aquarius, then the Age of Aquarius is said to have begun. This Age is consonant with the Aeon of Horus, first adumbrated in The Book of the Law. The harbinger of this new Aeon becomes the sigil of the Beast 666, polarized by 333. The reversals of these hexagrams are 62 and 31, which sum to 93, the number of the “magickal current” of the Aeon of Horus.
The Trigrams and the Nakshatras
There are two different kinds of Lunar months; the synodic month of 29.5 days, the period from one New Moon to the next; and the sidereal month of 27.3 days, which is essentially the Moon’s rotation around the Earth, i.e., the time it takes to return to a fixed star on the celestial sphere.
Vedic Astrology uses the sidereal month as the basis for the 27 “mansions” of the Moon. These mansions are equal divisions of the zodiac, so they each occupy 13 and 1/3 degrees each. The Jyotish astrological system is sidereal, meaning it uses the fixed stars, not the tropical zodiac. This is why they use the sidereal Lunar month for their “mansions”. In their system, the Equinox is at its actual point, near the beginning of Pisces, (not 0 degrees Aries). Nevertheless, their zodiac begins at Aries.
This system has a perfect correlation with TQ, due to the use of the Three Gunas, Rajas-Tamas-Sattvas, to correlate with the 27 mansions. The Gunas are the three fundamental principles of the Universe, similar to the alchemical principles of sulfur, salt, and mercury. They are not “elements” so much as they are modes of operation, related in a certain sense to the “creation-preservation-destruction” cycle of Hindu cosmology.
The Gunas are attributed to the mansions in sets of three, which is identical to the distribution of lines in the trigrams. Rajas is essentially Dynamism, so it represents Yang. Tamas is essentially Inertia, so it represents Yin. This leaves Sattvas, which is the Tao that mediates the other two. Correlating with the trigrams, these are built from the top down, with the top line being primary, the middle line secondary, and the bottom line tertiary.
Along with having three Gunas attributed to each mansion, there is also a planetary ruler. These follow each other in a strict cycle of the seven sacred planets and Ketu and Rahu, (the nodes of the Moon). While the top two lines of the trigram determine the linear order of the mansion, the bottom two lines determine which planet is ruling. Thus each planet rules all three trigrams that reduce to the same digit, meaning each planet rules one of the cells of the “qabalah of nine chambers”.
The 27 Lunar Constellations (Nakshatras) of Vedic Astrology/Jyotish
In this chart, Rajas is abbreviated R, Tamas is T, and Sattvas is S.
Rajas is Yang, (or 1); Tamas is Yin, (or 2); and Sattvas is Tao, (or 0)
It would seem that some ancient Indian astrologers were trying to create a symmetrical classification scheme for the Lunar mansions, and since they required 9 ‘rulers’ to do this, they used the seven visible planets, plus the Lunar nodes, (Ketu and Rahu). These two nodes are the points where the orbital plane of the Moon intersects the orbital plane of the Earth. It is near these points that eclipses take place.
This method requires dividing the zodiac into 27 parts of 13 1/3 degrees each. In the TQ, the antigram-pairs are used to delineate the degrees of the Zodiac. These begin with one set of 13 pairs that have the Zero trigram above, followed by thirteen sets of 27 pairs having upper trigrams that successively differ by every number from 1 to 13.
Since the Vedic Lunar mansions each span 13 1/3 degrees, then the first mansion, Ashwini, (the beginning of Aries), corresponds to the first set of 13 antigram-pairs. After Ashwini, each pair of successive mansions correlates with a full 27-degree cycle.
Within each 27-degree cycle, there are 27 pairs of antigrams; 13 pairs where the lower trigrams differ from 13 down to 1; then a central day where the lower trigrams are Zero, then another set of 13 pairs where the lower trigrams differ from 1 up to 13.
So in effect, each 27-degree cycle can be split into halves, and each half basically corresponds to one of the Hindu mansions. What little bit of discrepancy there is can be absorbed by the central degree of the cycle in question. This makes the TQ Thelemic Calendar correspond with the 27 Nakshatras almost exactly; at least within enough tolerance to make what little discrepancy there is become almost negligible.
The chart of the antigram sequence shows that each Lunar mansion begins with a degree where new sets of antigram pairs begin, or where the center degree of the cycle is located, (within the range of 1 degree, since TQ doesn’t deal with partial degrees).
So for example, the 4th mansion is Rohini, and begins with exactly 10 degrees Taurus. In the TQ calendar, 10 degrees Taurus is the antigram pair 161 and 202, which is the first pair of the cycle that has the 5 and 7 trigrams in upper position. Mansion 6 is Punarvasu, and begins at 20 degrees Gemini; this degree is assigned to the pair 81 and 162, which is the middle degree of the cycle whose upper trigrams are 3 and 6.
In the chart of the antigrams given earlier, the name of the Vedic Lunar mansion is at the right, located on the degree that it begins with in the Zodiac. In every case, the 27 Nakshatras begin within one degree of the beginning or middle of a set of antigram-pairs. Since most of these transitions are partial degrees, there is some leeway in assigning them to one degree or another.
The point is that the antigram-pairs and the mansions are practically identical; in all cases, the Lunar mansion begins either on the initial or central degree of a sequence, or on the previous degree. In all respects, the integration of the Hindu mansions and the TQ Zodiacal degrees works almost perfectly, and it is safe to say that these hexagram pairs delineate basically the same subdivisions of the Zodiac as the Vedic astrology system.