Dice and the Minor Arcana: Outlining the challenge

In past posts, I have discussed possible ways of pairing the 21 Tarot trumps with the 21 possible rolls of two dice, concluding in the end that the "Air Hexactys" system was the best. In this system, dice rolls are ranked first by total (e.g., any roll that totals 7 outranks any roll that totals 6); and then, among rolls with the same total, by the higher of the two numbers rolled (e.g., among rolls totaling 7, 1-6 outranks 2-5, which in turn outranks 3-4). The dice rolls, thus ranked from lowest to highest, were then paired with the trumps from 1 to 21. This system seems to "work" and to make symbolic sense, as briefly discussed in my post "The root trumps of the Air Hexactys."

Now, the main reason for assuming a link between dice rolls and Tarot cards is the fact -- unlikely to be a coincidence -- that there are 21 possible rolls of two dice (2d6) and 56 possible rolls of three (3d6), corresponding precisely to the 21 trumps (the practice of numbering the Fool and counting it as a trump is a relatively recent development) and the 56 suit cards (Minor Arcana). The next step, therefore, is to try to find a dice-to-cards mapping that works for the Minor Arcana as well as the Air Hexactys works for the trumps.

Here are some of the challenges facing anyone who would attempt such a mapping:

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Four suits: The trumps have a simple linear structure, being numbered from 1 to 21; all that is required is to establish a linear ranking of the dice rolls, which is not difficult. The Major Arcana, on the other hand, are organized in four suits. There is no obvious way to divide up the 56 possible rolls of 3d6 into four equal categories.

⁂

Rank within each suit: In most traditional games played with Tarot cards, the long/black suits (swords and clubs) are ranked, beginning with the lowest, A 2 3 4 5 6 7 8 9 10 J C Q K -- the same ranking used in most modern card games, except that aces are low. However, the round/red suits (cups and coins) rank the pips (but not the court cards) in reverse order: 10 9 8 7 6 5 4 3 2 A J C Q K. Although there are a few exceptions (most notably the French jeu de tarot), I think we can take this idiosyncratic ranking system as a very old and probably original feature of the Tarot pack.

⁂

Relative rank of suits: Due to the quirk just mentioned, it seems that if we want to rank all 56 Minor Arcana linearly, we have to group them by suit first and then by rank within each suit. It wouldn't make sense, for example, to begin with the four aces, then the four deuces, and so on, because long aces are low but round aces are high. I suppose we could begin with the long aces and round tens, then the long deuces and round nines, etc., but this seems very unnatural.

The problem with ranking by suit first is that Tarot games do not rank the suits. Players must always either follow suit, play a trump, or discard a card. In no case can cards from two different suits be played in the same trick, so the question of whether, say, the King of Swords outranks the King of Clubs never arises.

Modern "esoteric" or divinatory tarot is not a trick-taking game, so "rank" as such is not an issue. Nevertheless, it has become customary to think of coins/pentacles as the "lowest" suit, followed by swords, then cups, and finally clubs/wands as the highest. Waite's chapters on the Minor Arcana in The Pictorial Key to the Tarot, for example, begin with the King of Wands as the highest card and proceed down the ranks to the Ace of Wands; then the Cups, from King to Ace; then the Swords, and finally the Pentacles. (Waite makes tens high and aces low for all suits, contrary to the Continental tradition of which he was perhaps ignorant; Tarot has never been a game in England.)

Historically, the four suits do have a natural rank, since they can be traced back to Chinese suits representing different denominations of money. Coins, which have preserved their character as single coins, are the lowest denomination. Next come Clubs, which were originally strings of 100 or 1000 coins. (Ancient Chinese coins had a hole in the center and were strung together.) Cups derive from the Chinese character for 10,000; and Swords, from the character for ten -- meaning, in context, ten myriads, or 100,000. This ancient ranking has been preserved in many modern Anglo-French-suited games, where Spades (Swords; cf. Italian spade) are the highest-ranking suit, followed by Hearts (Cups), Clubs, and finally Diamonds (Coins). (Bridge, in which Diamonds outrank Clubs, is an exception.)

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Relation to trumps, especially root trumps: Because each of the trumps has been associated with a roll of 2d6, each roll of 3d6 will be associated with between one and three of the trumps. For example, the roll 1-4-6 would be associated with 1-4 (Lover), 1-6 (Hanged Man), and 4-6 (Sun) -- or, alternatively, with the three root trumps associated with 1, 4,  and 6 -- namely, Magician, Death, and World. For rolls where the same number occurs three times, only one of the trumps will be linked; for example, 4-4-4 should be related to Death. (The Four of Swords would seem a natural choice for this roll.)

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Sparse symbolism: Compared to the trumps, the Minor Arcana contain relatively little imagery. Prior to Waite's innovative "scenic pips," a card like the Seven of Swords portrayed nothing more nor less than seven swords (stylized as arcs in the Marseille tradition). Such cards have no immediately obvious meaning, and the traditional meanings ascribed to them vary widely. This makes it hard to judge whether or not a particular dice-card pairing is a "good" one. This can be viewed either as a bug (because it offers little guidance) or as a feature (because it allows the cards to accommodate a wide variety of systems).

The difference between proof and understanding

The mathematical proofs laid out in my previous post (which I am sure very few of you have bothered to read) left me both satisfied and disappointed. Having had only the patchiest of mathematical educations (basic algebra and statistics, plus such rudiments of set theory and symbolic logic as linguists require; no trigonometry or calculus), I took a certain satisfaction in having been able to do it at all -- but it was disappointing to realize that I didn't seem much closer to understanding the patterns than I had been before. Why are they always palindromic, for instance? Saying that their palindromicity can be expressed algebraically as n(n + 1) ÷ 2 ≡ (2k - (n + 1))(2k - n) ÷ 2 (mod k), and that that equation turns out to be true for all natural number values of n and k, just doesn't count as an answer to that question. I can follow each step of the algebra, but in the end I do not feel enlightened; I do not think, "Oh, now I get it!" It is possible to prove something without really understanding it.

Then I read Kevin McCall's much better proofs of the same postulates. What a difference! Where I had hammered out my proofs by algebraic brute force, McCall had understood. -- and left me thinking, in T. H. Huxley's much-quoted words, "How extremely stupid not to have thought of that!"

⁂

The heart of McCall's proof is the observation that, in modular arithmetic with modulus k, adding k - n is equivalent to subtracting n. You can easily see this in the most familiar everyday use of modular arithmetic, which is our 12-hour clock, with modulus 12. If you want to get from 11:00 to 7:00, for example, you can either subtract 4 hours or add 12 - 4 = 8 hours.

The series of triangular numbers is generated by starting with 0, then adding 1, then adding 2, then 3, and so on through the succession of natural numbers. Due to the fact that k - n ≡ -n (mod k), one you've added numbers up to a certain point, you start doing the modular equivalent of subtracting those same numbers in reverse order, creating a palindrome. For example, if the modulus is 7:

• 0
• +1
• +2
• +3
• +4 ≡ -3 (mod 7)
• +5 ≡ -2 (mod 7)
• +6 ≡ -1 (mod 7) 
• etc.

Obviously, this will create a repeating palindromic pattern with a period of 7.

Why is the period twice as long for even moduli? Consider the case when the modulus is 6.

• 0
• +1
• +2
• +3
• +4 ≡ -2 (mod 6)
• +5 ≡ -1 (mod 6)
• +6 ≡ 0 (mod 6)
• +7 ≡ +1 (mod 6)
• +8 ≡ +2 (mod 6)
• +9 ≡ -3 (mod 6)
• +10 ≡ -2 (mod 6)
• +11 ≡ -1 (mod 6)
• etc.

Because the modulus k is even, k ÷ 2 ≡ -k ÷ 2 (mod k). In this case 3 ≡ -3 (mod 6). That means that, when we go through the first cycle of 6 integers, we add 1, add 2, add/subtract 3, subtract 2, and subtract 1. We can think of this as adding or subtracting 3, but whichever it is, we only perform the operation once in a single cycle, so at the end of the cycle we still have something congruent to 3 (mod 6); we have not yet returned to the 0 with which we began. However, if we go through two cycles, we can think of ourselves as adding 3 the first time around and then subtracting it the second time, bringing us back to our starting point. Thus, the period is twice as long for even moduli, and the palindrome is centered on k ÷ 2, a number which is special because it alone is congruent to its own negation.

⁂

Now I understand -- and just about anyone else can understand, too, without any need to pore over complicated algebraic operations. Congratulations, Kevin McCall; you've really solved this, whereas I was just crunching numbers.

Proving triangular number congruence patterns

For the patterns proven in this post, see here and here.

1. The formula for triangular numbers is T_n = n(n + 1) ÷ 2

This is common knowledge, but I include it for the benefit of any readers who may be even less mathematically inclined than myself.

The n^th triangular number (written T_n) is the sum of the natural numbers from 0 to n. Such numbers are called "triangular" because that number of points can be arranged in a triangular configuration as shown below.

Deriving the formula for triangular numbers is fairly straightforward. Take a given number, n, and write out 1 + 2 + ... + n. The sum of those numbers will be T_n. Now, below that, write the same sum in the opposite direction, n + n-1 + ... + 1. The example below shows what this looks like for n = 5.

Each of the two rows adds up to T_n, so the total of the two rows is 2T_n. If we look at the vertical columns, each of them adds up to n + 1. The first column is 1 + n; the second is 2 + (n - 1); the third is 3 + (n - 2); and so on. There are n columns in all, so 2T_n = n(n + 1). The formula for T_n, therefore, is n(n + 1) ÷ 2.

(Sorry if using an obelus instead of fractional notation makes these equations a little hard to read. It allows me to type them inline instead of inserting an image file for each equation.)

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2. For any modulus k, the series of triangular numbers reduced modulo k repeats, with a period no greater than 2k.

This part was discovered by Kevin McCall, though I've reformulated it somewhat.

It is postulated that, for any modulus k, T_2k ≡ 0 (mod k). This means that T_2k is evenly divisible by k; in other words, that T_2k ÷ k is an integer.

Plugging 2k into our triangular number formula, we get T_2k = 2k(2k + 1) ÷ 2. Dividing by k gives us T_2k ÷ k = 2k(2k + 1) ÷ 2k = 2k + 1. Since k is itself an integer, 2k + 1 is an integer as well. Therefore, T_2k ≡ 0 (mod k).

Now consider the next triangular number in the series, T_{2k + 1}. For any given triangular number,  T_n , T_{n + 1} = T_n  + n + 1. (For example, for n = 10, the 10th triangular number is 55, and 55 + 10 + 1 = 66, which is the 11th triangular number.) Since 2k + 1 ≡ 1 (mod k), this amounts (in the modular arithmetic we are using) to adding 1, and T_{2k + 1} will be congruent to 1 (mod k).

Now the whole series of triangular numbers begins with 0. Then we add 1, then 2, then 3, and so on to infinity to generate the whole series.

• T₀ = 0
• T₁ = T₀ + 1 = 1
• T₂ = T₁ + 2 = 3
• T₃ = T₂ + 3 = 6
• T₄ = T₃ + 4 = 10
• etc.

It should be clear that, when we reach T_2k , the whole process begins again.

• T_2k ≡ 0 (mod k)
• T_{2k + 1} = T_2k + 2k + 1 ≡ 1 (mod k)
• T_{2k + 2} = T_{2k + 1} + 2k + 2 ≡ 1 + 2 ≡ 3 (mod k)
• T_{2k + 3} = T_{2k + 2} + 2k + 3 ≡ 3 + 3 ≡ 6 (mod k)
• T_{2k + 4} = T_{2k + 3} + 2k + 4 ≡ 6 + 4 ≡ 10 (mod k)
• etc.

We can see that any time T_n ≡ 0 and T_{n + 1} ≡ 1 (mod k), the series will repeat. We know that this always happens where n = 2k. Therefore, the period of the repetition must be either 2k or else a number by which 2k is evenly divisible.

⁂

3. The period of repetition is always either k (if k is odd) or 2k (if k is even).

If k is and odd number, the series of triangular numbers reduced modulo k will begin to repeat at k rather than 2k because T_k ≡ 0 (mod k), meaning that T_k ÷ k is an integer. T_k ÷ k = k(k + 1) ÷ 2k = (k + 1) ÷ 2. Since k is odd, k + 1 is even, so (k + 1) ÷ 2 is an integer. T_{k + 1} will then be equal to T_k + k + 1 ≡ 1 (mod k), and so on.

At this point my direct reliance on Kevin McCall's proof stops. Everything below is my own. (Kevin has proved it all, too, but I have not yet read his proof and will not do so until I have first proved it all myself.)

How do we know that the period is never smaller than k? Might it sometimes be k ÷ 2, for instance? No. While it sometimes happens that T_j ≡ 0 (mod k), where j < k, it is never also true that T_{j + 1} ≡ 1 (mod k), which is what is necessary for the series to repeat. Recall that T_{n + 1} = T_n  + n + 1. Therefore, where T_j ≡ 0 (mod k) and j < k, it follows that T_{j + 1} ≡ j + 1 (mod k). The only way j + 1 can be congruent to 1 (mod k) is if j ≡ 0 (mod k), which we have specified that it is not. (With the exception of 0, no number less than k can be congruent to 0 modulo k.)

Therefore, the period of repetition is always precisely k (if k is odd) or 2k (if k is even).

⁂

4. The repeating series is always palindromic.

It is postulated that for any modulus k, the series for triangular numbers from T₀ to T_{2k - 1} reduced modulo k (which series goes on to repeat itself forever, as proven above) is palindromic. This means that

• T₀ ≡ T_{2k - 1} (mod k)
• T₁ ≡ T_{2k - 2} (mod k)
• T₂ ≡ T_{2k - 3} (mod k)
• etc.

Stating this generally, we can say that T_n ≡ T_{2k - (n +1)} (mod k). Replacing T_n with the formula for triangular numbers, we arrive at the following equation: n(n + 1) ÷ 2 ≡ (2k - (n + 1))(2k - n) ÷ 2 (mod k).

Since multiplying both sides by the same number preserves congruence, we can simplify this to: n(n + 1) ≡ (2k - (n + 1))(2k - n) (mod k).

Doing the math, we find that this is equivalent to: n(n + 1) ≡ n(n + 1) + k(4k - 2(2n + 1)) (mod k).

Subtracting n(n + 1) from both sides, we get: k(4k - 2(2n + 1)) ≡ 0 (mod k). Trivially, any multiple of k is congruent to 0 modulo k, so this is true. Therefore, the series is palindromic.

⁂

5. Where k is even, the number at the center of the palindrome is k ÷ 2

The period of the repeating palindrome is 2k where k is even, so the number at the center of the palindrome is T_k = k(k + 1) ÷ 2. This number should be congruent to k ÷ 2 (mod k).

Multiplying both sides by 2, we get k(k + 1) ≡ k (mod k). Trivially, nk ≡ k ≡ 0 (mod k), for any integer value of n, so this is true.

⁂

Now that everything has been proven that I set out to prove, I will look at Kevin McCall's full proof and, if it is different from my own (as I suspect it will be), post it as well.

Recent Engrish finds

Chinese relies a lot more on context than English does. For example, "[verb] carefully" and "be careful not to [verb]" have the same syntactic form. Context and common sense always make it clear which is intended -- but the ambiguity does lead to some interesting results when translated directly into English.

If you're going to slip, at least take the trouble to do it right!

No such explanation suggests itself for this next one, which is Japanese. All I can say is that Nyanta makes some pretty strange choices.

Not a bed of roses

U.E. echoes A.E.

I recently posted on A.E.'s "language of the gods" -- the attempts of the Irish mystic who wrote under that name to intuit the "true" meanings of the individual letters or sounds of which language is composed.

After laying out his alphabet, A.E. goes on to imagine some of the primitive words our distant ancestors may have created. He gives six or seven examples of such words, the first of which -- obviously suggesting to the mind the Semitic word for "God" -- is El.

I imagine a group of our ancestors lit up from within, . . . feeling those kinships and affinities with the elements which are revealed in the sacred literature of the Aryan, and naming these affinities from an impulse  springing up within. I can imagine the spirit struggling outwards making of element, colour, form or sound a mirror on which, outside itself, it would find symbols of all that was pent within itself, and so gradually becoming self-conscious in the material nature in which it was embodied, but which was still effigy or shadow of a divine original. I can imagine them looking up at the fire in the sky, and calling out "El" if it was the light they adored, or if they rejoiced in the heat and light together they would name it "Hel." . . .

⁂

Approximately three days after posting on A.E.'s alphabet, I happened to be reading Serendipities, a translated collection of essays by Umberto Eco, which I had picked up at a used bookstore probably in March or April (when I was writing about the Wheel of Fortune) on the strength of its appropriately serendipitous cover art, which featured (or so it appeared at first glance; more on this below) a Hieronymus Bosch painting which had recently come to my attention during my meditations on that card.

The second essay in that book, called "Languages in Paradise," deals with the evolution of Dante's thoughts regarding the original language spoken by Adam -- specifically with how they changed between De Vulgari Eloquentia and Paradiso -- and speculates about the possible influence of the Kabbalist Abraham Abulafia.

In De Vulgari Eloquentia, writes Eco,

Dante thought that the first sound emitted by Adam could only have been an exclamation of joy that, at the same time, was an act of homage toward his creator. The first word that Adam uttered must therefore have been the name of God, El (attested in patristic tradition as the first Hebrew name of God).

In Paradiso, though, Dante has Adam say that he had originally called God I (thus in the Italian; a single vowel, not the first-person pronoun) and that he became El only later. Eco's translator William Weaver provides what he calls a "literal translation" of the passage in question (Paradiso XXVI, 133-138).

Before I descended into the pains of Hell, on earth the Highest Good was called I, from whence comes the light of joy that enfolds me. The name then became EL, and this change was proper, because the customs of mortals are like leaves on a branch, one goes and another comes.

Having just read A.E.'s account of the ancients "calling out 'El' if it was the light they adored," I was surprised to find the name El here again associated with "the light of joy that enfolds me." I also thought it odd that I didn't remember Dante's ever using that particular turn of phrase -- and, sure enough, he didn't. Looking back at the original Italian, I saw that Weaver's translation was indeed quite literal -- with the single exception of the reference to "light," which does not appear in the original! Dante wrote simply "onde vien la letizia che mi fascia," translated by the peerless Allen Mandelbaum (the gold standard for Dante translations) as "from which derives the joy that now enfolds me." Weaver's addition of "light" is quite unaccountable. Verse translators who fall short of Mandelbaum's virtuosity might fudge like that to meet the demands of meter, but Weaver's translation is in prose and professes to be strictly literal.

The juxtaposition of El and Hell -- echoing the El and Hel of the A.E. passage -- is, obviously, also an artifact of translation.

Eco goes on to speculate as to where Dante got the name I and suggests Abulafia.

[F]or Abulafia, each letter, each atomic element, already had a meaning of its own, independent of the meaning of the syntagms in which it occurred. Each letter was already a divine name: 'Since, in the letters of the Name, each letter is already a name itself, know that Yod is a name, and YH is a name' . . . Paleographers say that in certain codes [sic; codices?] of the Divine Comedy I is written as Y. Why can this not lead us to suppose that the I of Dante was the YOD of Abulafia, a divine name?

This passage, with its reference to each letter having a meaning of its own, was actually the first parallel to A.E. that I noticed. Only after noticing it did I go back and see the various related coincidences discussed above.

⁂

Now, about that cover art.

In my post on some of the early Wheel of Fortune Tarot cards, I noted that one of the creatures on the wheel in the Tarot de Marseille closely resembles the dog in Bosch's painting The Conjuror. I then wrote "Some critics have even identified the other creature, the one in the conjurer's basket, as a monkey, but this is a mistake. The reappearance of this pair in the central panel of Bosch's St. Anthony Triptych leaves no room for doubt that it is a barn owl" and included a relevant detail from that latter painting.

Note that the only reason I mentioned or posted this painting was because of the owl perched on the pig-man's head, which proves that the creature in the conjuror's basket in The Conjuror is also an owl, not a monkey.

The copy of Serendipities I found features on the cover what is immediately recognizable as the St. Anthony Triptych. Not until I wrote this post did I actually take a good look at it and realize that it is not Bosch at all but a terribly amateurish copy -- of which the most striking difference from the original is the complete absence of the owl.

Serendipities cover (left) with detail (right) showing its owllessness

Apparently this is the São Paulo "version" of the painting rather than the more familiar Lisbon one -- and, while I had called the São Paulo painting "a terribly amateurish copy," apparently many art historians are of the opinion that they were both painted by Bosch. After looking at some higher-quality reproductions of the São Paulo painting, I would like to modify my previous statement and say that it is very obviously a terribly amateurish copy! Be that as it may, it's a strange sort of anti-serendipity that the book caught my eye because of the St. Anthony Triptych, that I was interested in that triptych largely because of the owl, and that the version on the book turns out not to have an owl.

Further evidence that Jesus' conversation with Nicodemus ends with John 3:12

Below is John 1:3-21, traditionally considered together as a single "Jesus and Nicodemus" pericope. I have highlighted all first- and second-person pronouns and all occurrences of the verbs say, tell, speak, and answer.

Doesn't this make it easy to tell at a glance where the reported dialogue ends and the author's commentary begins? The dividing line is between vv. 12 and 13, the same point I had previously identified, based on entirely different textual evidence, in this post.

Triangular number congruence patterns proven

Kevin McCall, whose name you may recognize because his thoughts on dice and the Tarot (qv) have appeared on this blog in the past, has worked out a proof of the congruence patterns in the series of triangular numbers which I postulated here. He has proven that the series of triangular numbers reduced modulo k repeats itself, that the repetition has a period of k (if k is odd) or 2k (if k is even), and that the repeating series is always palindromic.

I have only seen his proof of the first of those statements; I am not going to look at the remainder of his proof until after I have proven it myself independently -- at which point I will post both his proof and mine.

A.E.'s "language of the gods"

Sunset, by A.E.

I first encountered the Irish mystic George William Russell, who wrote under the pseudonym A.E., in the pages of Ulysses, where Stephen Dedalus borrows money from him and then quips, "A.E.I.O.U." I thought that was a pretty good pun but, compared to some of the other work of the greatest punster ever to walk the face of the earth, nothing spectacular.

Only recently (as in this week) have I actually gotten around to reading any of A.E.'s work. I found that in The Candle of Vision he makes the letters of the alphabet an object of mystical contemplation, considers the vowels separately from the consonants, and takes pains to get the letters in what he considers to be the correct order. Given that the Candle was published just two years before Ulysses, and that the two authors knew each other, I can only conclude that the allusion to A.E.'s alphabetical mysticism was deliberate.

So, Joyce, I apologize for having underestimated you here. Despite everything (and there's a lot to forgive!), you were, in your own way, God.

⁂

On pp. 116-118, A.E. explains his project of "brooding upon the significance of separate letters":

I was first led to brood upon the elements of human speech by that whisper of the word "Aeon" out of the darkness, for among many thoughts I had at the time came the thought that speech may originally have been intuitive. I discarded the idea with regard to that word, but the general speculation remained with me, and I recurred to it again and again, and began brooding upon the significance of separate letters, and had related many letters to abstractions or elements . . . . I then began to rearrange the roots of speech in their natural order from throat sounds, through dentals to labials, from A which begins to be recognisable in the throat to M in the utterance of which the lips are closed. An intellectual sequence of ideas became apparent. This encouraged me to try and complete the correspondences arrived at intuitively. I was never able to do this. Several sounds failed, however I brooded upon them, to suggest their intellectual affinities, and I can only detail my partial discoveries . . . .

In the following chapter, he lays out these partial discoveries, which may be summarized thus:

1. A: the self, God; a circle
2. R: motion; red; a vertical line
3. H: heat; orange; a triangle
4. L: fire, light, radiation; a shape like an upside-down Y
5. Y: binding, concentration, condensation, gravitation, the will; yellow; a triskelion
6. W: liquidity, water; green; the lower half of a circle, a smile-like shape
7. G: earth; a square
8. K: mineral, rock, crystal, hardness; a square crossed by a diagonal, so oriented that the diagonal is vertical
9. S: impregnation, inbreathing, insouling, the genesis of the cell; a circle with a horizontal line through the center, like the letter theta
10. Z: multiplication, division, reproduction; a circle with a cross in it, like the astronomical symbol for Earth
11. TH: growth, expansion, swelling
12. SH: scattering, dissolution, decay
13. T: individual action, movement, initiative, ego, extroversion; a symmetrical cross, like a plus sign
14. D: silence, sleep, immobility, abeyance, inwardness; the upper half of a circle, with a horizontal line joining its ends, like the letter D rotated 90 degrees counterclockwise
15. J: --
16. TCH: --
17. V: life in water, all that swims; blue
18. F: what lives in air and flies; blue
19. P: masculine life, paternity; indigo
20. B: feminine life, maternity; indigo
21. N: continuance of being, immortality; violet
22. M: close, limit, measure, end, death; violet

A.E. writes, "In all there are twenty-one consonants which with the vowels make up the divine roots of speech. . . . I despair of any attempt to differentiate from each other the seven states of consciousness represented by the vowels." I am not sure why A alone is discussed together with the consonants. At first I had thought it must, like the Hebrew aleph, represent a glottal stop, but A.E. clearly says that there are 21 consonants, which means A is a vowel.

The selection of sounds, and the order in which he puts them, are highly idiosyncratic. These 22 sounds correspond neither to any alphabet I know of nor to the phoneme inventory of any language with which I am familiar. It appears to be based on the sounds of English, as understood by someone deeply ignorant of phonetics. Many native speakers of English do not realize that there are two different "th" sounds (as in thy and thigh, respectively), that the "s" sound in vision is a distinct sound, or that "ng" is not just a combination of the "n" and "g" sounds.

The placing of R in second place implies that A.E. used a uvular ("guttural") "r" sound, as in French. According to Wikipedia, "A guttural/uvular [ʁ] is found in north-east Leinster. Otherwise, the rhotic consonant of virtually all other Irish accents is the postalveolar approximant, [ɹ]." A.E. was from County Armagh, adjoining north-east Leinster, so it is possible that he spoke this way -- or it could be just another sign of his general confusion regarding where in the mouth different consonants are articulated.

The consonants from 7 to 22 appear in pairs, often but not always representing voiced and voiceless versions of the same sound. Sometimes the voiced sound comes first (G-K, J-TCH, V-F), and sometimes the voiceless (S-Z, T-D, P-B). TH and SH are paired, presumably because A.E. was not aware that each had its own voiced counterpart, and because these two "orphan" fricatives seemed vaguely similar. N and M are paired because they are the only nasal consonants of which he was aware. I surmise that A.E. had no knowledge of the voiced/voiceless distinction but simply put together those consonants that seemed intuitively to be similar in sound.

Despite his use of such terms as "dental" and "labial," and despite his account of how he brooded over these sounds, repeating them to himself again and again, A.E. seems not to have been very clear on what was going on in his mouth when he pronounced them. The sounds are supposed to be organized according to place of articulation, from back to front, but there are many puzzling exceptions. L is an alveolar sound, articulated in the same place as T and D, but it is placed far in the back. When he repeated the L sound to himself, he must have been saying "ull, ull, ull" -- giving the consonant its "dark," velarized pronunciation, rather than the "clear" pronunciation found in "luh, luh, luh." The "dark l" sound is unusual in Hiberno-English but apparently does occur in some Ulster dialects.

The placing of the palatal sound Y in the back, behind the velars, is incorrect but understandable. Since the tongue only comes close to the hard palate without actually touching it, the sound's palatal character is not easy to discover by self-observation. In fact, we can see that A.E. put all approximants in the back, regardless of place of articulation, perhaps because it was not easy to observe how they were pronounced.

The interdental TH and the post-alveolar SH, J, and TCH are also also misplaced, less understandably. (I assume this is tch as in Tchaikovsky, thus spelt to differentiate it from ch as in Bach.) If you pronounce TH, S, and SH in succession, I think it's pretty easy to observe the tongue moving from the front of the mouth toward the back.

But the most obvious exception is N, which is placed with the labial consonants apparently on the strength of its similarity to M.

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These amateurish errors, together with the incompleteness of the mappings, demonstrate the sincerity with which the project was carried out. I have not the slightest doubt that A.E. did just what he said he did: brooded over these sounds for a long time and wrote down only those correspondences which were confirmed by intuition. Sometimes he didn't get anything (as with TCH and J), and other times his results were only tentative. (Although my summary does not show it, many of the correspondences are qualified with "I think," or "it vaguely suggested itself to me.")

On one level, this whole list of correspondences seems to be obvious nonsense. Aside from the linguistic difficulties, what does it even mean to say that R corresponds to motion, the color red, and a vertical line? What would follow from such a statement's being true or false? It seems like a classic example of an assertion that is (as Wolfgang Pauli would put it) "not even false." I am reminded of Valentin Tomberg's statement that the traditional planet-metal mappings of astrology (Sun to gold, Moon to silver, Venus to copper, etc.) have been confirmed for him time and again by experience. What sort of experience, I wondered when I read that, could possibly confirm such a thing? What properties of tin unfolded themselves to his understanding when he reflected on the "fact" that tin corresponds to the planet Jupiter? Or what light did tin shed on the nature of Jupiter?

Some of A.E.'s mappings did ring true, but it's hard to be sure why. For example, when I read about F ("what lives in air and flies"), I thought of English fly, flap, flutter, fowl, and feather; of Chinese fēi and Hebrew 'af, both meaning "fly"; and of T. H. White's geese singing "Free, free: far, far: and fair on wavering wings" -- but of course this is just picking cherries, and most flight-related words in most languages do not feature the "f" sound. Does the abundance of such words in English mean that Anglophones have historically been particularly sensitive to the "true" meaning of their consonants? Do Anglophones think "better" or more clearly about flight because they call it flight? Is that why it was the English-speaking Wright brothers who invented the airplane? (On the other hand, the Montgolfier and Breguet brothers were French, and the great American ornithologist Audubon was born Jean Rabin in French-speaking Saint-Domingue.) Or, more likely, are A.E.'s intuitions unconsciously influenced by the vocabulary of his native language? One feels that the influence of English is also at play in A.E.'s mapping of R to red and Y to yellow. (Both of those mappings were found to be particularly common in Sean Day's survey of 43 colored-letter synesthetes, qv; Day regrettably neglected to record the languages spoken by his subjects.)

Still, despite these very deep misgivings, both about the general meaningfulness of the questions A.E. was asking and about the validity of his specific answers, I found that I reacted to many of his mappings with delight, and with a sense that, somehow and for some reason, his project was after all worth doing. I suppose what I am reacting to, more than to anything specific, is the general attitude of taking things seriously, of refusing to take arbitrariness and meaninglessness as the null hypothesis.

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Incidentally, the bit about the alphabet is only a few pages long; the bulk of The Candle of Vision is a serious and sustained meditation of the phenomena of imagination and clairvoyance and on what they mean. It's definitely something I'll be rereading.

Patterns in the digits of triangular numbers

Here's another way of saying what I said in my post on congruence patterns in the series of triangular numbers.

List the triangular numbers, starting with 0. The final digits of the numbers in the series will exhibit a repeating pattern with a period of 20, and the pattern will be palindromic. The penultimate (tens-place) digits will also exhibit a repeating palindromic pattern, with a period of 200. The repeating palindrome for the hundreds-place digits will have a period of 2000, and so on for any "place" you care to choose.

This is assuming you write the numbers in the familiar decimal system -- but any number base will yield corresponding results. If the triangular numbers are written in base b, the pth-to-last digit will exhibit a repeating palindromic pattern with a period of b^p (if b is odd) or 2b^p (if b is even).

(Oswald Wirth, whose writings first alerted me to the existence of such patterns, was doing what he called "Theosophic reduction" -- i.e., adding up the digits of a number until a single digit is arrived at. "Theosophically reducing" a number in base b corresponds to taking its final digit in base b - 1.)

The patterns are easiest to see in very small bases such as binary and ternary, shown below with color-coding to highlight the repeating palindromes.

I have still not figured out why these patterns exist. Even stating the pattern algebraically is proving somewhat difficult for someone whose mathematical training is as limited as mine.

Hungry Eyes

I've always like the Eric Carmen song "Hungry Eyes" (despite having once had a weird nightmare vision triggered by it, as I shall perhaps relate in another post) -- but the English teacher in me cringes every time I hear the repeated line, "I feel the magic between you and I."

Today it suddenly occurred to me that, since it obviously can't be "between you and I," the line in question must actually be "between U and I" -- i.e., between the 21st and 9th letters of the alphabet -- and the letter located precisely halfway between those two is the 15th letter, O.

The Latin letter O looks identical to, and is descended from, the Phoenician letter ʿayin -- whose name means "eye" (cf. the coincidentally similar archaic English plural eyen).

The Chinese word for "hungry" is 餓 -- which, in the Wade-Giles system of romanization, would be transcribed o.

I'm truly sorry

G. at he Junior Ganymede blog has been boasting about having originated "one of the worst puns ever written," so I think it's time to put him in his place.

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A long time ago in Russia there was a an ancient church famed far and wide for its beautiful art and architecture. One night a group of fanatical iconoclasts broke into the church and vandalized it, shocking all of Russia. While many parts of the church were damaged, the act that generated the greatest public outrage was the wanton destruction of the priceless and irreplaceable 10th century carvings of angels that had one adorned the large stone baptismal font. The font itself was still intact but was now bereft of all decoration.

The iconoclasts were apprehended, and such was the notoriety of the case that it came to the attention of the Tsar himself. He ordered that the perpetrators be temporarily locked up while he took his time thinking up a truly suitable punishment for such an enormous crime against art and religion. Finally, perhaps drawing inspiration from Dante, he decided that the vandals would be chained together, crammed into the very font they had vandalized, and "baptized" with boiling water. Despite protests from some that this punishment would be just as much a sacrilege as the vandalism itself, the Tsar stood firm.

However, it happened that one of the iconoclasts came from a wealthy and influential family, and his parents came before the Tsar and begged that their wayward son be shown mercy. The Tsar was a hard man, but such was their persuasiveness that in the end he agreed to modify his original sentence: All the other perpetrators would be tortured as planned, but their son would instead be exiled to Georgia for life, never to return to Russia on pain of death. The tearful parents thanked him for his kindness, and the Tsar sent the modified sentence to the local magistrate to be executed.

But imagine the parents' horror when, on the day of the punishment, their son was chained up in the baptistry and tortured together with his partners in crime! When they sent to the Tsar for an explanation, they found that he was just as astonished as they were. He was adamant that he had sentenced their son to exile only, but the magistrate was just as adamant that he had received orders to punish all the malefactors in the same way. Obviously the sentence had been changed by some unknown hand on its way to the magistrate. The Tsar ordered an investigation into the matter, and it was discovered that the Tsar's own copy editor was to blame. The copy editor confessed but insisted that it had all been a misunderstanding.

"A misunderstanding?" bellowed the Tsar. "How could you, a mere copy editor, possibly think you had the right to change a punishment decreed by the Tsar himself?"

"Begging Your Imperial Majesty's pardon," said the copy editor, "I naturally thought it was just a mistake when I saw that that one sentence was in Georgia while all the other sentences were in the same sans-seraph font."

Texaco's Fire Chief gasoline

As so often, reading about strange coincidences seems to attract them. As I mentioned in my Copper State post, I have been reading the conspiracy classic "King Kill 33" by James Shelby Downard, which ostensibly argues that the Freemasons killed JFK but is in fact an extremely thoroughgoing catalogue of impressive and not-so-impressive coincidences, only a small fraction of which could be explained by a Masonic conspiracy even in principle. Today, for example, I read this.

[Elm Street in Dallas, Texas,] was also the home of the Blue-Front tavern, a Masonic hangout in the grand tradition of "tavern-Masonry." . . . The Blue-Front was the site of the "broken-man" ritual in which various members of the "Brotherhood of the Broom" swept the floor and tended some fierce javelino pigs. The Blue-Front was once a firehouse and was still sporting the pole in the late '20s. This is extremely germane symbolism. The national offices of the Texaco oil corporation are located on Elm Street, Dallas. Its chief products are "Haviland (javelino) oil" and "Fire Chief" gasoline.

As you can see, this is all pretty bizarre even by conspiracy-theory standards. I mean, suppose you were a Masonic Grand Master responsible for orchestrating a presidential assassination. Obviously, the first order of business would be to make sure the president's motorcade passed through a street that was home to a company whose chief products had names reminiscent of random facts about a Masonic hangout that once existed on the same street! Apparently, among Masons this is what passes for leaving one's calling card at the crime scene. What Downard is doing isn't a conspiracy theory; it's just random coincidence-noticing of the sort to which I am myself addicted.

Anyway, when I read this, I thought, "I know Havoline motor oil, which must be what he means, but Fire Chief? I've been to a few Texaco gas stations but never noticed that name."

While I was reading Downard, my wife was watching TV -- one of those reality shows about very fat people. Just minutes after reading about Fire Chief gasoline, I happened to glance up at the TV and saw that the very fat woman who is the star of the show had just ordered a genuine vintage Texaco Fire Chief gas pump and had it delivered to her house.

Those Masons!

A rationale for Waite's orientation of ROTA

Yes, yet another post (see the others) on this very minor bit of Tarot iconography.

In a past post (qv), I wrote that "of the four possible orientations of the ROTA, the only one I can't think of any good reason for is the one actually used by Lévi and Waite, with T at the top!" Now a possible rationale for it has occurred to me.

Notice that the wheel is apparently turning counterclockwise: the snake on the left is descending, and the cynocephalus on the right is ascending. So the letters ROTA correspond, respectively, to the low point, the descent, the high point, and the ascent. Lévi had already identified the A and O of ROTA as corresponding to alpha and omega -- representing the beginning and the end. It seemed to me that the T and R ought to correspond to some similarly fundamental opposition, but nothing came to mind until today. T and R correspond to the Hebrew words tov (טוֹב, "good") and ra (רַע "bad/evil"). Obviously, the top of the wheel represents good fortune; and the bottom, bad fortune. One starts at the bottom, rises (alpha, the beginning), reaches the top, and then falls (omega, the end).

Copper State

The flag of Arizona, featuring a copper-colored star

This afternoon I happened to see someone on the street wearing a T-shirt that said “California: The Golden State.” The thought this prompted was: “California’s the Golden State, and Nevada’s the Silver State. I wonder if there’s a Bronze State.” (I had seen and heard references to the Golden State countless times before without ever wondering that, but this time the thought just popped into my head.)

Several hours later, I read a few pages from James Shelby Downard's "King Kill 33," where I found the sentence "Arizona is the 'Copper State,'" followed by three paragraphs on the supposed occult significance of that nickname.

Congruence patterns in the series of triangular numbers

The Swiss occultist Oswald Wirth writes of what he calls "Theosophic addition and reduction." I don't know what it's got to do with Theosophy, but I found it mathematically interesting.

By "Theosophic addition," Wirth simply means the operation which yields the series of triangular numbers, where the nth triangular number is the sum of all integers from 0 to n, inclusive; in other words, the nth triangular number is equal to (n² + n) ÷ 2. The first several triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666.

"Theosophic reduction" means adding up all the digits of a number, and then repeating the process as necessary until a one-digit number is arrived at. Mathematically, this amounts to finding the smallest positive integer to which it is congruent modulo 9.

Wirth took the first 21 triangular numbers (beginning with 1) and "Theosophically reduced" them, yielding this series: 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6. As you can see, the same series of 9 numbers (1, 3, 6, 1, 6, 3, 1, 9, 9) repeats itself; and if you keep going beyond 21 (which is where Wirth stopped because he was considering the numerology of the 21 Tarot trumps), it becomes apparent that it keeps repeating itself forever.

Wirth took this pattern as confirmation of the traditional (ultimately Pythagorean) numerological idea that the first 9 natural numbers are the building blocks of all the rest, and that that "the decad is a new monad." However, it seemed pretty obvious to me that there is nothing special about the number 9, and that the 9-based pattern Wirth found was almost certainly an artifact of the use of the decimal system in the "Theosophical reduction" -- such that "reduction" meant finding congruence modulo 9 -- and that other moduli would yield other patterns.

I also saw that Wirth had missed an interesting pattern in his integer series because he had started with 1 rather than 0, and because he was thinking in terms of "reduction" rather than congruence. (Nine is congruent to 0 modulo 9, but you can't arrive at that 0 by adding up digits in the "Theosophic" fashion.) If we start with the 0th triangular number (which is 0), and if we "reduce" multiples of 9 to 0 rather than to 9, the repeating series becomes (0, 1, 3, 6, 1, 6, 3, 1, 0) -- which is a palindrome!

I decided to check other moduli, starting with the easiest, which is 10. Any decimal number is congruent modulo 10 to its final digit, so look back at that list of triangular numbers and look at the final digits. The first thing you will notice is that certain final digits (2, 4, 7, 9) never occur at all. Look a little further, and you will see that there is a repeating pattern. It has a period of 20 (not 10, as we might have expected) -- and, sure enough, it is a palindrome: (0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0).

It is natural to jump from this to the induction that every modulus will yield a palindromic repeating pattern, and this turns out to be true for all the moduli I have looked at, as the table below shows. I have used centered alignment to highlight the palindromic nature of the repeating series.

From this sample, it appears that:

1. Any modulus, m, yields a repeating palindromic series of congruences for the series of triangular numbers.
2. If m is odd, the period of the repeating series is equal to m.
3. If m is even, the period is equal to 2m, and each of the two numbers at the center of the palindrome is equal to m/2.

Now, I'm roughly 100% sure that I'm not the first person to have noticed these patterns, and that someone else has already mathematically proven what I have only induced. So, to those of my readers who have had a proper mathematical education -- no spoilers, please! I want to try to figure this out for myself.

Notes on John 3:9-12

Rembrandt, Head of Christ (Philadelphia Museum of Art)

When I first began writing about John 3, I had planned to divide my comments into three posts -- covering, respectively, Jesus' conversation with Nicodemus about being born again (vv. 1-10), Jesus' words to Nicodemus on other subjects (vv. 11-21), and the bit about John the Baptist (vv. 22-36).

After further rereading and thought, I would now divide up the chapter differently:

1. Jesus' conversation with Nicodemus (vv. 1-12); see this post for my reasons for thinking the conversation ends with that verse
2. Commentary by the author (vv. 13-21)
3. John the Baptist's statement about Jesus (vv. 22-30)
4. Further commentary by the author (vv. 31-36)

However, I've already written a post about vv. 1-10 (qv), so now I need to tie up loose ends with a few notes on vv. 11-12 before moving on to the next major section of the chapter. I include vv. 9-10 as well, as necessary context, and take the opportunity of adding some additional notes on those verses.

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[9] Nicodemus answered and said unto him, How can these things be?

[10] Jesus answered and said unto him, Art thou a master of Israel, and knowest not these things?

In English, "master of Israel" would seem to be a reference to the earlier characterization of Nicodemus as a "rule of the Jews," but in fact the word translated master is διδάσκαλος, "teacher" -- the same word used in John 1:38 to gloss the Hebrew term rabbi. Nicodemus is a rabbi, as were all members of the Sanhedrin.

"These things" refers to Jesus' earlier statements about being born again (or born from above), being born of water and the Spirit, etc. Why does Jesus expect that a rabbi, by virtue of being a rabbi, should be familiar with the concept of being "born again"? Should exposure to that concept have been part of a first-century rabbi's education?

Jesus' presumption that a rabbi should already know about being "born again" has led biblical commentators to look for the concept in the Old Testament. Several passages have been proposed. In 1 Samuel 10:6, Samuel tells Saul, "the Spirit of the Lord will come upon thee, and thou shalt prophesy with [the prophets], and shalt be turned into another man." In Ezekiel 37, the Lord breathes new life into dry bones. In Job 33:25, Elihu says of the repentant sinner that "his flesh shall be fresher than a child's; he shall return to the days of his youth." While these are all suggestive, I don't think we can really say that the concept of being "born again" is explicitly present in the Old Testament.

The Talmud comes closer, stating in several places that a convert to Judaism is like a newborn child. For example, in the Yevamot tractate, which deals with questions regarding levirate marriage, it is repeatedly stated, by four different rabbis, that "a man who has become a proselyte is like a child newly born" (and is therefore permitted to enter into what would otherwise be considered an incestuous marriage, since all pre-conversion family ties have been dissolved). While the Talmud did not yet exist in the time of Christ, many of its doctrines were presumably already current as oral traditions, so Nicodemus might have been expected to be familiar with the idea of conversion as a "new birth." However, it seems pretty clear in context that, whatever Jesus may have meant by being "born again," he certainly didn't mean conversion to Judaism! (Nicodemus was already a Jew, and in any case Jesus seems not to have had a very high opinion of Jewish proselytism; see Matthew 23:15.)

All in all, I don't think Jesus is finding fault with Nicodemus's education or implying that being born again is something he should have read about. Rather, he means that, as a teacher of religion, Nicodemus should have been the sort of person who could intuitively grasp the (new) concept of being born again when it was presented to him, and that his obtuseness in this matter implies that he is not spiritually qualified to be a rabbi.

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[11] Verily, verily, I say unto thee, We speak that we do know, and testify that we have seen; and ye receive not our witness.

When Jesus says, "We speak that we do know," he may be speaking of himself and his disciples, or perhaps he means something like "we prophets," alluding to Israel's history of rejecting prophetic messages. ("O Jerusalem, Jerusalem, thou that killest the prophets, and stonest them which are sent unto thee . . . ." Matt. 23:37.) As always in the King James Version, the words ye and you are plural (see this post for details), so Jesus is not reproaching Nicodemus personally with unbelief but rather speaking of a group to which Nicodemus belongs -- presumably "the Jews," or perhaps more specifically the Pharisees or the Sanhedrin. As far as we can tell, Nicodemus himself does believe in Jesus.

Nevertheless, something in what Jesus says here must be applicable to Nicodemus himself. Perhaps he understood Nicodemus's question, "How can these things be?" as an expression of skepticism rather than a request for explanation.

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[12] If I have told you earthly things, and ye believe not, how shall ye believe, if I tell you of heavenly things?

This curious statement implies at least the following:

1. Jesus has thus far taught "earthly things" to Nicodemus and to those like him. (The word you is plural.)
2. He has not heretofore taught "heavenly things" (except perhaps to Nicodemus just now?).
3. Earthly things are easier to believe, such that those prepared to believe heavenly things can be expected to be a subset of those prepared to believe earthly things.

What distinction does Jesus intend when he speaks of "earthly things" and "heavenly things"? Pretty clearly not the familiar distinction between the secular and the sacred, since Jesus was a spiritual teacher from the beginning and did not impart secular learning. Unfortunately, the Gospel provides very little information about the content of Jesus' public teachings up to this point, so we can only speculate as to what he taught and in what sense it was "earthly." John 2:13-22 provides the only relevant data.

We know that Jesus taught that business transactions ought not to be carried out in the Temple. This could be considered an "earthly" teaching because it has to do with worldly commerce in a physical building -- or, more generally, because it refers to the contingent details of a particular (soon to be superseded) religious cultus and in that sense has little to do with what goes on in Heaven.

We also know that he made the enigmatic statement, "Destroy this temple, and in three days I will raise it up." His audience at the time understood him to be speaking literally, in which case this is again an "earthly" teaching about the fate of a particular physical building. After the Resurrection, the disciples reinterpreted this utterance as having been a reference to that event -- something not so readily classified as "earthly."  Perhaps it could be thus characterized because it describes resurrection only in the simplest physical terms -- kill me, and I will come back to life -- without getting into how post-resurrection life differs from mortal life?

What about what Jesus has just now taught Nicodemus, about being "born again"? Is this also included under the heading of "earthly things"? Both answers to that question seem defensible. In the one case, Jesus would be reacting to Nicodemus's apparent skepticism by saying, "See, even when I teach you earthly things like this, you don't believe, so I won't even bother trying to teach you heavenly things." In the other, he would be saying, "I should have known you wouldn't believe heavenly things like this; after all, you didn't even believe the earthly things I taught before." Being born again could be considered an earthly thing because it takes place during earthly (or anyway pre-Heavenly) life, Heaven being for those who have already been born again. Alternatively, it could be considered a heavenly thing because of its spiritual nature, especially as contrasted with teachings about business and the Temple cultus.

Oswald Wirth's version of ROTA

(For context, see my past posts on the Wheel of Fortune Tarot card.)

I found this diagram near the beginning of Oswald Wirth's Le Tarot, des Imagiers du Moyen Age.

Éliphas Lévi, who was the first to introduce into Tarot iconography a wheel marked with letters ambiguously reading either ROTA or TARO, always put T at the top of the wheel. Wirth's diagram is the first example I have found of someone adopting Lévi's idea but changing the orientation of the letters. (In my post on the orientation of ROTA, I spoke favorably of this orientation, with A at the top, because it corresponds with the orientation of the Tarot suits used by both Lévi and Strieber, but still consider R at the top to be the correct orientation.)

Arranging the 22 Major Arcana, rather than the four Minor suits, around the wheel is a little strange, 22 not being divisible by four, but it is central to Wirth's system. Each card is considered to be related to those horizontally and vertically (not diametrically) opposite it on the wheel, as shown in this diagram, also from Wirth.

I haven't yet read and assimilated Wirth's commentary on these proposed mappings, but at first glance the system seems just slightly off. My initial reaction is that I would rotate the cards clockwise 8.18 degrees, so that the horizontal axis lined up perfectly with 0 and 11.

My proposed modification of Wirth's schema

The resulting mappings seem much more intuitively correct -- although, as I say, I have not yet looked in any detail either at Wirth's system or at my proposed alternative.

The need for a new concept of time

The idea that time is just what it appears to be -- that everything just comes and then goes, like a sparrow flying through a mead-hall -- that the past is lost and gone forever, dreadful sorry, Clementine -- is unacceptable. In such a world, everything dies. "Immortality" does not solve this problem, any more than the continued existence of the modern city of Rome can change the fact that ancient Rome is gone forever.

But the alternative -- that time is an illusion, that nothing changes -- that God and the universe as God sees it are simply an eternally static four-dimensional object -- is also unacceptable. In such a world, nothing really lives, for life is an inherently temporal thing. Stasis is not and never can be compatible with life.

What is temporal is temporary, and what is atemporal is lifeless. But Jesus promised eternal life -- really eternal, and really life -- and taking his message seriously means trying to understand what that means.

This is the problem that has been occupying most of my time recently.

Angry Man, Angry Hog

I've been dreaming about books again.

I remember the blurb on the back cover said, "One of the most horrifying books of the decade -- or is it?" and compared the author to Borges. (In fact, I assume the author's name -- a foreign form of "George B.," containing the string -orges -- is a distortion of Jorge Luis Borges, one of whose books I had just bought the day before the dream). There was some sort of "infinite regress" conceit that was central to the plot. I believe there were a lot of ordinary hogs, and a lot of angry hogs that wanted to kill them, then a third tier of angrier hogs that wanted to kill the angry hogs, and so on ad infinitum. I'm not sure how the angry man fit into the scheme of things, but perhaps he was angry about having paid good money for a novel with such a ridiculous plot!

Essential Mozart and Beethoven pieces -- seeking feedback

My appalling ignorance of classical music is something I have had occasion to mention before.

A few days ago I happened, in the course of my work, to listen to the Lacrimosa from Mozart's Requiem, and the spiritual effect was immediate and unmistakable -- a sudden plunge into a deeper, stiller level of consciousness. (Yes, I know "higher" consciousness is the standard metaphor, but that's not how I experience things.) I then went about the rest of my working day, returned home, and, as is my wont, put on some music while I did my evening chores -- the same sort of music I usually listen to, which is to say definitely not Mozart. At some point the contrast hit me, and I thought, What am I doing? Having just been touched by music of astonishing spiritual depth, and having basically all the music every recorded at my fingertips, here I am listening to the Temptations sing "Papa Was a Rollin' Stone"! Granted, "Papa Was a Rollin' Stone" is a great song, perfectum in genere, but is it really what I need right now?

So, after thinking about such things for a bit, I've decided to make one more effort to become familiar with classical music, starting with the two composers generally acknowledged to be the greatest of the great, and working my way down from there.

After comparing and collating dozens of "top 10" lists, I've come up with the following shortlists of what are apparently the essential works of Mozart and Beethoven. However, I cannot overstate the depths of my ignorance in this field, so if anything seems strange about these lists, or if anything absolutely essential has been omitted, my more musically literate readers are invited to chime in in the comments.

Mozart

1. Clarinet Concerto
2. Piano Concerto No. 20
3. The Magic Flute
4. The Marriage of Figaro
5. Symphony No. 40
6. Symphony No. 41
7. Concerto for Flute, Harp, and Orchestra
8. Requiem
9. Don Giovanni
10. Piano Concerto No. 23
11. Sinfonia Concertante for Violin, Viola and Orchestra

Beethoven

1. Symphony No. 3
2. Symphony No. 9
3. Piano Concerto No. 5
4. String Quartet No. 14
5. Symphony No. 5
6. Piano Sonata No. 23
7. Missa Solemnis
8. Symphony No. 6
9. String Quartet No. 15
10. Symphony No. 7
11. Piano Sonata No. 8