July 18, 2019

Diagonalizing the Psalms

Mathematics-inspired manipulations of scripture

A Bible open to Psalm 22 — Mathematics of Biblical proportions.

Andrew_Writer *Flickr* (CC BY 2.0)

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

As I was drifting off to sleep one night, I had one of those brilliant ideas that only comes along when you’re drifting off to sleep: diagonalizing the psalms. Earlier that day I had noticed that Psalm 119 was very long—longer than 119 verses, in fact—and wondered how many psalms from the Book of Psalms have at least as many verses as their number in the psalter. One thing led to another, and my subconscious decided the idea of applying Cantor’s diagonal argument to create a new psalm seemed incredibly compelling.

Cantor’s diagonal argument is one of my very favorite proofs. I still get goosebumps sometimes thinking about it. It is a proof that the infinity of the counting numbers 1, 2, 3, 4,… is smaller than the infinity of the real numbers, or even the real numbers between 0 and 1. The real numbers are unlistable, or uncountable. The argument goes like this: Suppose you have a way to list the real numbers. That is, you have some infinite string of decimal digits for the first number on your list, a different infinite string of decimal digits for the second number on your list, and so on. You claim that every real number appears somewhere on this list, but Cantor says he will create a number that is not on the list. He builds this new number this way: The first digit of the new number is anything other than the first digit of the first number. The second digit of the new number is anything other than the second digit of the second number, and so on. (It really doesn’t matter what other digit you choose in each place, as long as it doesn’t match.) The Nth digit of the new number doesn’t match the Nth digit of the Nth number on your list.

How do we know the number doesn’t appear on the list? Well, it can’t be the first number on the list because it doesn’t have the same first digit. It can’t be the second number on the list because it doesn’t have the same second digit. You get the idea. You could add the new number to the list somewhere, but Cantor will just produce another number not on your list, and another one after that, as if pulling rabbits out of a hat. (The argument doesn't work for the list of whole numbers because they have a finite number of digits and doesn't work for rational numbers because it doesn't produce a rational number.)

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My first idea for diagonalizing the psalms was just to make a diagonal psalm: the first verse of Psalm 1, the second verse of Psalm 2, and the Nth verse of any Psalm N that has at least N verses. With the help of the Book of Common Prayer and this list of the psalms ordered by length, I came up with my first diagonal psalm:

A First Diagonal Psalm

Happy are they who have not walked in the counsel of the wicked, *
nor lingered in the way of sinners,
nor sat in the seats of the scornful!
Why do the kings of the earth rise up in revolt,
and the princes plot together, *
against the LORD and against his Anointed?
But you, O LORD, are a shield about me; *
you are my glory, the one who lifts up my head.
Tremble, then, and do not sin; *
speak to your heart in silence upon your bed.
Braggarts cannot stand in your sight; *
you hate all those who work wickedness.
I grow weary because of my groaning; *
every night I drench my bed
and flood my couch with tears.
Awake, O my God, decree justice; *
let the assembly of the peoples gather round you.
All sheep and oxen, *
even the wild beasts of the field,
The LORD will be a refuge for the oppressed, *
a refuge in time of trouble.
The innocent are broken and humbled before them; *
the helpless fall before their power.
He delivered me from my strong enemies
and from those who hated me; *
for they were too mighty for me.
Praise the LORD, you that fear him; *
stand in awe of him, O offspring of Israel;
all you of Jacob's line, give glory.
I have seen the wicked in their arrogance, *
flourishing like a tree in full leaf.
In your sight all the wicked of the earth are but dross; *
therefore I love your decrees.

It definitely doesn’t have the coherence of the actual psalms, but it hits some of the highlights of the book as a whole: wickedness is bad, animals exist, God protects the faithful. I was a little surprised it only has 14 verses. Only Psalms 1-10, 18, 22, 37, and 119 have enough verses to appear in my diagonalized psalms. There are a lot of psalms that are pretty long, but few of them line up well with their numbers. I was most disappointed with Psalm 78, which nearly makes the list but has only 72 verses. (Would anyone mind terribly if we swapped psalms 72 and 78?)

My first diagonal psalm might be a Cliff’s Notes version of the Psalter, but it’s kind of a shame that a poem born of the observation that Psalm 119 is long is so short. It would be nice if a newly created psalm felt longer. One option would be to insert a pause for every psalm that does not have enough verses. It starts out strong and then peters out quickly.

A Dramatic Diagonal Psalm

Happy are they who have not walked in the counsel of the wicked, *
nor lingered in the way of sinners,
nor sat in the seats of the scornful!
Why do the kings of the earth rise up in revolt,
and the princes plot together, *
against the LORD and against his Anointed?
But you, O LORD, are a shield about me; *
you are my glory, the one who lifts up my head.
Tremble, then, and do not sin; *
speak to your heart in silence upon your bed.
Braggarts cannot stand in your sight; *
you hate all those who work wickedness.
I grow weary because of my groaning; *
every night I drench my bed
and flood my couch with tears.
Awake, O my God, decree justice; *
let the assembly of the peoples gather round you.
All sheep and oxen, *
even the wild beasts of the field,
The LORD will be a refuge for the oppressed, *
a refuge in time of trouble.
The innocent are broken and humbled before them; *
the helpless fall before their power.
(7 pauses)
He delivered me from my strong enemies
and from those who hated me; *
for they were too mighty for me.
(3 pauses)
Praise the LORD, you that fear him; *
stand in awe of him, O offspring of Israel;
all you of Jacob's line, give glory.
(14 pauses)
I have seen the wicked in their arrogance, *
flourishing like a tree in full leaf.
(81 pauses)
In your sight all the wicked of the earth are but dross; *
therefore I love your decrees.
(31 pauses)

If you want to choose a set duration for the pauses, this would be a perfect way to emphasize how many psalms are being omitted in a dramatic reading of the psalm. If using this psalm as the text for a choral composition, the voices could rest during the pauses while any instruments played.

An option that would allow all psalms to contribute would be to throw in some modular arithmetic. Modular arithmetic is also called clock arithmetic because it treats numbers as if they wrapped around in a cycle. Five hours after 10:00 is usually called 3:00, not 15:00, because we reset from 12 to 1. To make a diagonal psalm this way, suppose Psalm N is M verses long. The Nth verse of this diagonal psalm is the N (mod M) verse of Psalm N. The first 10 verses of this psalm are the same as my first diagonal psalm. Then we get to Psalm 11, which has only 8 verses. Eleven is 3 mod 8, so we add the third verse of Psalm 11. Psalm 12 also has 8 verses, so we’d use the fourth verse in our psalm because 12 is 4 mod 8. At 150 verses, this diagonal psalm is still shorter than Psalm 119! (But long enough that you'll need to click "Show/hide" below to expand.)

A Modular Diagonal Psalm

Happy are they who have not walked in the counsel of the wicked, *

nor lingered in the way of sinners,

nor sat in the seats of the scornful!

Why do the kings of the earth rise up in revolt,

and the princes plot together, *

against the LORD and against his Anointed?

But you, O LORD, are a shield about me; *

you are my glory, the one who lifts up my head.

Tremble, then, and do not sin; *

speak to your heart in silence upon your bed.

Braggarts cannot stand in your sight; *

you hate all those who work wickedness.

I grow weary because of my groaning; *

every night I drench my bed

and flood my couch with tears.

Awake, O my God, decree justice; *

let the assembly of the peoples gather round you.

All sheep and oxen, *

even the wild beasts of the field,

The LORD will be a refuge for the oppressed, *

a refuge in time of trouble.

The innocent are broken and humbled before them; *

the helpless fall before their power.

When the foundations are being destroyed, *

what can the righteous do?”

Those who say, "With our tongue will we prevail; *

our lips are our own; who is lord over us?”

How long, O LORD?

will you forget me for ever? *

how long will you hide your face from me?

Oh, that Israel's deliverance would come out of Zion! *

when the LORD restores the fortunes of his people,

Jacob will rejoice and Israel be glad.

LORD, who may dwell in your tabernacle? *

who may abide upon your holy hill?

O LORD, you are my portion and my cup; *

it is you who uphold my lot.

Hear my plea of innocence, O LORD;

give heed to my cry; *

listen to my prayer, which does not come from lying lips.

He delivered me from my strong enemies

and from those who hated me; *

for they were too mighty for me.

In the deep has he set a pavilion for the sun; *

it comes forth like a bridegroom out of his chamber;

it rejoices like a champion to run its course.

Send you help from his holy place *

and strengthen you out of Zion;

For the king puts his trust in the LORD; *

because of the loving-kindness of the Most High, he

will not fall.

Praise the LORD, you that fear him; *

stand in awe of him, O offspring of Israel;

all you of Jacob's line, give glory.

You spread a table before me in the presence of those

who trouble me; *

you have anointed my head with oil,

and my cup is running over.

"Those who have clean hands and a pure heart, *

who have not pledged themselves to falsehood,

nor sworn by what is a fraud.

Lead me in your truth and teach me, *

for you are the God of my salvation;

in you have I trusted all the day long.

Test me, O LORD, and try me; *

examine my heart and my mind.

Therefore I will offer in his dwelling an oblation

with sounds of great gladness; *

I will sing and make music to the LORD.

They have no understanding of the LORD'S doings,

nor of the works of his hands; *

therefore he will break them down and not

build them up.

The voice of the LORD splits the flames of fire;

the voice of the LORD shakes the wilderness; *

the LORD shakes the wilderness of Kadesh.

Sing to the LORD, you servants of his; *

give thanks for the remembrance of his holiness.

I will rejoice and be glad because of your mercy; *

for you have seen my affliction;

you know my distress.

You are my hiding-place;

you preserve me from trouble; *

you surround me with shouts of deliverance.

But the LORD'S will stands fast for ever, *

and the designs of his heart from age to age.

Who among you loves life *

and desires long life to enjoy prosperity?

For they have secretly spread a net for me without a cause; *

without a cause they have dug a pit to take me alive.

See how they are fallen, those who work wickedness! *

they are cast down and shall not be able to rise.

I have seen the wicked in their arrogance, *

flourishing like a tree in full leaf.

For I said, "Do not let them rejoice at my expense, *

those who gloat over me when my foot slips.”

Deliver me from all my transgressions *

and do not make me the taunt of the fool.

He lifted me out of the desolate pit, out of the mire and clay; *

he set my feet upon a high cliff and made my footing sure.

The LORD preserves them and keeps them alive,

so that they may be happy in the land; *

he does not hand them over to the will of their enemies.

While my bones are being broken, *

my enemies mock me to my face;

Give judgment for me, O God,

and defend my cause against an ungodly people; *

deliver me from the deceitful and the wicked.

Our heart never turned back, *

nor did our footsteps stray from your path;

All your garments are fragrant with myrrh, aloes, and cassia, *

and the music of strings from ivory palaces makes you glad.

It is he who makes war to cease in all the world; *

he breaks the bow, and shatters the spear,

and burns the shields with fire.

For God is King of all the earth; *

sing praises with all your skill.

Your praise, like your Name, O God, reaches to

the world's end; *

your right hand is full of justice.

For we see that the wise die also;

like the dull and stupid they perish *

and leave their wealth to those who come after them.

Out of Zion, perfect in its beauty, *

God reveals himself in glory.

Create in me a clean heart, O God, *

and renew a right spirit within me.

"This is the one who did not take God for a refuge, *

but trusted in great wealth

and relied upon wickedness.”

See how greatly they tremble,

such trembling as never was; *

for God has scattered the bones of the enemy;

they are put to shame, because God has rejected them.

Render evil to those who spy on me; *

in your faithfulness, destroy them.

I am shaken by the noise of the enemy *

and by the pressure of the wicked;

You have noted my lamentation;

put my tears into your bottle; *

are they not recorded in your book?

I will call upon the Most High God, *

the God who maintains my cause.

The wicked are perverse from the womb; *

liars go astray from their birth.

For you have become my stronghold, *

a refuge in the day of my trouble.

With God we will do valiant deeds, *

and he shall tread our enemies under foot.

For you, O God, have heard my vows; *

you have granted me the heritage of those

who fear your Name.

For God alone my soul in silence waits; *

truly, my hope is in him.

My soul clings to you; *

your right hand holds me fast.

That they may shoot down the blameless from ambush; *

they shoot without warning and are not afraid.

You visit the earth and water it abundantly;

you make it very plenteous; *

the river of God is full of water.

I will enter your house with burnt-offerings

and will pay you my vows, *

which I promised with my lips

and spoke with my mouth when I was in trouble.

Let the nations be glad and sing for joy, *

for you judge the peoples with equity

and guide all the nations upon earth.

Let tribute be brought out of Egypt; *

let Ethiopia stretch out her hands to God.

As for me, I am afflicted and in pain; *

your help, O God, will lift me up on high.

Let all who seek you rejoice and be glad in you; *

let those who love your salvation say for ever,

"Great is the LORD!”

My lips will sing with joy when I play to you, *

and so will my soul, which you have redeemed.

Long may he live!

and may there be given to him gold from Arabia; *

may prayer be made for him always,

and may they bless him all the day long.

Had I gone on speaking this way, *

I should have betrayed the generation of your children.

There are no signs for us to see;

there is no prophet left; *

there is not one among us who knows how long.

Do not toss your horns so high, *

nor speak with a proud neck.’"

How glorious you are! *

more splendid than the everlasting mountains!

The clouds poured out water;

the skies thundered; *

your arrows flashed to and fro;

That the generations to come might know,

and the children yet unborn; *

that they in their turn might tell it to their children;

O God, the heathen have come into your inheritance;

they have profaned your holy temple; *

they have made Jerusalem a heap of rubble.

You have brought a vine out of Egypt; *

you cast out the nations and planted it.

Sing with joy to God our strength *

and raise a loud shout to the God of Jacob.

"How long will you judge unjustly, *

and show favor to the wicked?

Make their leaders like Oreb and Zeëb, *

and all their commanders like Zebah and Zalmunna,

O LORD of hosts, *

happy are they who put their trust in you!

Show us your mercy, O LORD, *

and grant us your salvation.

Bow down your ear, O LORD, and answer me, *

for I am poor and in misery.

I count Egypt and Babylon among those who know me; *

behold Philistia, Tyre, and Ethiopia:

in Zion were they born.

Will your loving-kindness be declared in the grave? *

your faithfulness in the land of destruction?

It shall stand fast for evermore like the moon, *

the abiding witness in the sky.’"

You sweep us away like a dream; *

we fade away suddenly like the grass.

For he shall give his angels charge over you, *

to keep you in all your ways.

For lo, your enemies, O LORD,

lo, your enemies shall perish, *

and all the workers of iniquity shall be scattered.

Ever since the world began, your throne has been established; *

you are from everlasting.

Rise up, O Judge of the world; *

give the arrogant their just deserts.

For he is our God,

and we are the people of his pasture and the sheep of his hand. *

Oh, that today you would hearken to his voice!

As for all the gods of the nations, they are but idols; *

but it is the LORD who made the heavens.

The LORD is King;

let the earth rejoice; *

let the multitude of the isles be glad.

Let the sea make a noise and all that is in it, *

the lands and those who dwell therein.

Proclaim the greatness of the LORD our God

and worship him upon his holy hill; *

for the LORD our God is the Holy One.

For the LORD is good;

his mercy is everlasting; *

and his faithfulness endures from age to age.

Those who in secret slander their neighbors I will destroy; *

those who have a haughty look and a proud

heart I cannot abide.

Let this be written for a future generation, *

so that a people yet unborn may praise the LORD.

Our days are like the grass; *

we flourish like a flower of the field;

You hide your face, and they are terrified; *

you take away their breath,

and they die and return to their dust.

Saying, "Do not touch my anointed *

and do my prophets no harm.”

He saved them from the hand of those who hated them *

and redeemed them from the hand of the enemy.

Let them give thanks to the LORD for his mercy *

and the wonders he does for his children.

For your loving-kindness is greater than the heavens, *

and your faithfulness reaches to the clouds.

Let this be the recompense from the LORD to my accusers, *

and to those who speak evil against me.

The Lord who is at your right hand

will smite kings in the day of his wrath; *

he will rule over the nations.

Hallelujah!

I will give thanks to the LORD with my whole heart, *

in the assembly of the upright, in the congregation.

Their descendants will be mighty in the land; *

the generation of the upright will be blessed.

Hallelujah!

Give praise, you servants of the LORD; *

praise the Name of the LORD.

Judah became God's sanctuary *

and Israel his dominion.

They have hands, but they cannot feel;

feet, but they cannot walk; *

they make no sound with their throat.

O LORD, I am your servant; *

I am your servant and the child of your handmaid;

you have freed me from my bonds.

Praise the LORD, all you nations; *

laud him, all you peoples.

Let Israel now proclaim, *

"His mercy endures for ever.”

In your sight all the wicked of the earth are but dross; *

therefore I love your decrees.

When I was in trouble, I called to the LORD; *

I called to the LORD, and he answered me.

I lift up my eyes to the hills; *

from where is my help to come?

For there are the thrones of judgment, *

the thrones of the house of David.

So our eyes look to the LORD our God, *

until he show us his mercy.

Then would the waters have overwhelmed us *

and the torrent gone over us;

As for those who turn aside to crooked ways,

the LORD will lead them away with the evildoers; *

but peace be upon Israel.

Those who go out weeping, carrying the seed, *

will come again with joy, shouldering their sheaves.

Unless the LORD builds the house, *

their labor is in vain who build it.

You shall eat the fruit of your labor; *

happiness and prosperity shall be yours.

"Greatly have they oppressed me since my youth," *

let Israel now say;

I wait for the LORD; my soul waits for him; *

in his word is my hope.

But I still my soul and make it quiet,

like a child upon its mother's breast; *

my soul is quieted within me.

There will I make the horn of David flourish; *

I have prepared a lamp for my Anointed.

Upon the beard of Aaron, *

and runs down upon the collar of his robe.

Lift up your hands in the holy place and bless the LORD; *

the LORD who made heaven and earth bless

you out of Zion.

He sent signs and wonders into the midst of you, O Egypt, *

against Pharaoh and all his servants.

Who spread out the earth upon the waters, *

for his mercy endures for ever;

As for our harps, we hung them up *

on the trees in the midst of that land.

For you have glorified your Name *

and your word above all things.

LORD, you have searched me out and known me; *

you know my sitting down and my rising up;

you discern my thoughts from afar.

Let hot burning coals fall upon them; *

let them be cast into the mire, never to rise up again.”

O LORD, I call to you; come to me quickly; *

hear my voice when I cry to you.

I pour out my complaint before him *

and tell him all my trouble.

Revive me, O LORD, for your Name's sake; *

for your righteousness' sake, bring me out of trouble.

Happy are the people of whom this is so! *

happy are the people whose God is the LORD!

Your kingdom is an everlasting kingdom; *

your dominion endures throughout all ages.

Put not your trust in rulers, nor in any child of earth, *

for there is no help in them.

He has not done so to any other nation; *

to them he has not revealed his judgments.

Hallelujah!

Fire and hail, snow and fog, *

tempestuous wind, doing his will;

Let the faithful rejoice in triumph; *

let them be joyful on their beds.

Let everything that has breath *

praise the Lord.

Hallelujah!

Show/hide

Although they were inspired by late-night ruminations on Cantor’s diagonal argument, none of these psalms actually employs a key element of Cantor’s argument, changing some aspect of the nth entry. Even if the purpose of the diagonal psalm were to create a psalm not in the Psalter, we would not require this technique. It’s pretty easy to see by inspection that none of the versions we’ve made match any already-existing psalm. But as this exercise was never one of practicality, I decided to write one last psalm based on this part of Cantor’s argument. There are a lot of ways to alter the nth verse of the nth psalm. I decided to incorporate the N+7 idea from French experimental literary group Oulipo. This psalm is created by replacing the first noun in the nth verse of the nth psalm by the noun 7 after it in a dictionary. Because I do not actually own a physical dictionary and online dictionaries are not easy to browse, I used this online N+7 generator. (Its dictionary doesn’t have “arrogance,” so I browsed an online dictionary to replace that one.)

An Oulipian Cantorian Diagonal Psalm

Happy are they who have not walked in the counterbalance of the wicked, *
nor lingered in the way of sinners,
nor sat in the seats of the scornful!
Why do the kips of the earth rise up in revolt,
and the princes plot together, *
against the LORD and against his Anointed?
But you, O LORD, are a shingle about me; *
you are my glory, the one who lifts up my head.
Tremble, then, and do not sin; *
speak to your heartthrob in silence upon your bed.
Branches cannot stand in your sight; *
you hate all those who work wickedness.
I grow weary because of my groaning; *
every nightlight I drench my bed
and flood my couch with tears.
Awake, O my God, decree kayak; *
let the assembly of the peoples gather round you.
All shelves and oxen, *
even the wild beasts of the field,
The LORD will be a regatta for the oppressed, *
a refuge in time of trouble.
The innocent are broken and humbled before them; *
the helpless fall before their praise.
He delivered me from my strong Englishmen
and from those who hated me; *
for they were too mighty for me.
Praise the LORD, you that fear him; *
stand in babble of him, O offspring of Israel;
all you of Jacob's line, give glory.
I have seen the wicked in their arrowsmith, *
flourishing like a tree in full leaf.
In your signature all the wicked of the earth are but dross; *
therefore I love your decrees.

Because the first noun of the first verse is different from the first noun of Psalm 1, we know this is not Psalm 1. Because the first noun of the second verse is different from the first noun of Psalm 2, we know it is not Psalm 2. And so we have triumphed. My dream of a Cantor-style diagonalized psalm has been made reality!

For another exploration of algorithmic religion, check out the Flash Forward podcast episode "Our Father, who art in algorithm."