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