Kratzke

06 August 2026

Elementary Combinatorics

Dad was being annoying again. He told me not to write a post on permutations, combinations, Pascal's triangle, and binomial coefficients, and then he said, "Just find some time to have a conversation with Drakeson about it." But in order to have a conversation, I need to know what I'm talking about, and for that to happen, I have to write a post.

Combinatorics is a branch of math concerned with counting and arranging. Most word problems that begin with "how many ways" are combinatorics problems. Did you know that there are about 10⁸⁰atoms in the universe and about 10¹²⁰different games of chess? So for every atom in the universe, there are about 10⁴⁰ games of chess. Welcome to the wild world of combinatorics.

The difference between combinations and permutations is that combinations are situations where order does not matter, and permutations are situations where it does. If you have a collection of things, and you don't arrange them in any particular way, this is a combination. Every bag of marbles is a combination. If there is an arrangement, you have a permutation. So the next time your child makes you a beaded bracelet, thank them for the permutation.

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There are four categories of counting and arranging. They are:

Permutations where repetition is not allowed

Permutations where repetition is allowed

Combinations where repetition is not allowed

Combinations where repetition is allowed

Dad likes to talk about how many different ways there are to arrange people at a dinner table. That's a permutation where repetition is not allowed.

I like to talk about flavors of ice cream scoops in an ice cream cone. That's a permutation where repetition is allowed.

Milli tries to bring too many bottles of hand sanitizers on the plane to WA for summer vacation. She's stopped by TSA and has to make some decisions. That's a combination where repetition is not allowed.

When I was growing up, Debbie baked a dozen each of 5 types of Christmas cookies. She would give me a plate and tell me that I could choose any 4. That's a combination where repetition is allowed.

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Dad's family was: Bill, Ruth, William, Mary, Tommy, Steve, and Pete. We'll pretend they're knights and arrange them at a round table, placing them in seats numbered 1, 2, 3, 4, 5, 6, and 7. How many different ways can the Kratzkes arrange themselves?

There are 7 ways to choose the person who sits in seat 1. Because there are only 6 people left, there are now 6 ways to choose the person who sits in the seat 2. So there are 7*6 different ways to choose Kratzkes for seats 1 and 2. For each of these 42 options, there are 5 choices for seat 3, which means there are 7*6*5 ways to designate Kratzkes into the first 3 seats. Continuing this pattern, there are 7 factorial, notated 7!, and meaning 7*6*5*4*3*2*1 ways to seat this family at the round table.

By the by, mathematicians have decided that 0! = 1, which means there is 1 way to order 0 people at a round table.

But what if the round table is pretty small, and it only seats 4 people? How many ways can we seat 4 members of the family? Well, like before, there are 7 ways to fill seat 1 and 6 ways to fill seat 2. With 4 seats, we have 7*6*5*4 ways to fill them. This is the same as 7!/3!.

Permutations with no repetitions: n! / (n - r)! , where we're choosing r out of n things.

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I love Baskin Robbins. Their logo highlights the number 31 because they have 31 flavors. Let's say we're in the mood for a cone, and we can eat 5 scoops of ice cream. Order absolutely matters; chocolates, malts, caramels, and vanillas are more enhanced with the taste and texture of a waffle cone than fruitier flavors. Furthermore, it's kind of annoying to have two brightly flavored scoops touching; nobody would argue with the sensible decision of placing chocolate between mint and strawberry.

We have the choice of 31 flavors for our first scoop. For each of those choices, we have the choice of 31 flavors our second scoop. So there are 31*31 or 31²different ways to choose our first two scoops. Continuing this line of thinking, there are 31⁵different ways to choose 5 scoops of ice cream.

Multiplying everything together also works when you don't have the same number of things to choose from every time. Imagine that your significant other finds you on your stroll home. His pocket doesn't quite conceal the outline of a newly purchased jewelry box, his palms are sweaty, and trying to act nonchalant, he asks you to dinner. Before you can explain that you just ate no fewer than 5 enormous scoops of triple fudge chocolate ice cream, you find yourself gazing upon a prix fixe menu. You may choose 1 of 18 wines, 1 of 4 salads, 1 of 3 appetizers, 1 of 1 palette cleansers, 1 of 5 entrées, 1 of 4 personal sides, 1 of 3 cheese plates, and 1 of 4 desserts. You want the health insurance, so you're going to say yes. To make him feel like the luckiest guy in the restaurant, you explain that each of you has the choice of 18*4*3*1*5*4*3*4 dinners, scribbling down the beginnings of a tree diagram on his beverage napkin.

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If combinations are like permutations, but order does NOT matter, then there ought to be fewer combinations than permutations.

Milli has acquired 24 different types of hand sanitizers, all in 2 oz bottles. When packing for her trip to WA, she has chosen to bring them all. The transportation security officer, Madge, explains the 3-1-1 liquid rule: bottles must be 3.4 oz or smaller, they must all fit into a 1 quart sized clear plastic bag, and each passenger is allowed exactly 1 bag. Milli realizes she can fit 6 hand sanitizers into a quart sized bag and asks Madge how many ways she can choose 6 from 24.

Charmed, Madge replies, "First we need to pretend that order matters and calculate the permutation. Then we'll make that number smaller by dividing in order to get rid of the duplicates. For our first hand sanitizer, we have 24 choices. For our second, 23. So to select all 6 bottles, we have 24*23*22*21*20*19 choices, which can also be expressed as 24!/(24-6)!."

Milli nods in agreement. Madge continues. "But because we don't care about order, we have to divide by something to knock out the duplicate sets. Do you know how many ways there are to order each set of 6 hand sanitizers?" Milli replies, "Of course. Six factorial." Madge nods approvingly and continues, "So if we have 6! duplicates, we need to divide 24!/(24-6)! by 6!, which yields: (24*23*22*21*20*19)/(6*5*4*3*2*1). And because the answer to "how many ways" is always an integer, these fractions will always simplify elegantly."

Milli thanks Madge, procures 4 clear quart sized bags out of nowhere, packages up 4 bags of 6, and distributes them among her dad, mom, and brother.

Let's return our attention to the Kratzkes at the Restaurant of the Round Tables. They had such a wonderful time counting permutations last week that they have returned. But tonight is different - tonight, the hostess has announced that the table for 4 is available, but the table for 7 is occupied for the evening due to some grisly game involving the beheading of potential suitors. Unperturbed, the Kratzkes notice that there's a brand new carnival & batting cage next door, and to attract future business, it's running a promotion where everybody with the letter "z" in their last name attends free of charge. This means they're now faced with the decision: which 4 of them will have dinner, and which 3 will go to the carnival? Oh, how the tables have turned! What was once a permutation problem has now become a combination problem.

Let's focus on one question - which 4 of them will have dinner? We remember from before that there were 7!/3! ways to fill the 4 person table, but for the next few minutes, we don't care about who goes in which seat. For each selection of 4, there are 4! duplicates, so we must divide 7!/3! by 4!.

Combinations with no repetitions: n! / [r!(n - r)!] , where we're choosing r out of n things.

This situation is called "n choose r."

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Before we get to combinations in which repetitions ARE allowed, let's dwell a moment on the combinations-with-no-repetitions phrase "n choose r."

It's all written out in Pascal's triangle.

These hexagonal Pascal's triangle graphics were taken from Archimedes Lab.

Powers of 2

When we're looking for the number "n choose r," we can find it by going down to row (n + 1) and finding the term (r + 1). In the latest case of the Kratzkes, (7!)/(4!3!), we can go down to the 8th row (1, 7, 21, 35, 35, 21, 7, 1) and find the 5th term (35).

The pattern that emerges when reading Pascal's Triangle from left to right is also called "the binomial coefficient."

A polynomial is an expression made of variables and coefficients using addition, subtraction, multiplication, and powers to non-negative integers. Here's an example of a polynomial: 4a³b - b² + 1. The only reason people get so excited about polynomials is that when polynomials are added together, they yield more polynomials.

A binomial is a polynomial with two terms. For example, (a + b) is a really good one. We can make other ones, like (4c⁵- 6.789), but let's stick with (a + b).

There's a trick to expanding (a + b)ⁿ. The expansion yields a string of variables attached to the coefficients from row (n + 1) of Pascal's Triangle. For example,

(a + b)⁹= a⁹ + 9a⁸b + 36a⁷b² + 84a⁶b³+ 126a⁵b⁴+ 126a⁴b⁵+ 84a³b⁶+ 36a²b⁷+ 9ab⁸+ b⁹.

Those coefficients are in row 10.

These strings of numbers in Pascal's triangle, read from left to right, also represent the binomial distribution. There's a physical board that approximates the binomial distribution, and you've probably seen it. It's called a quincunx or a Galton board, and they use one under the name "Plinko" in The Price is Right. These are peg boards in which a ball has an equal chance of bouncing left or right at each junction.

Let's switch our thinking from the quincunx to the classic binomial prop - the penny. The more pennies we flip, the less likely it is we'll flip all heads or all tails, and the more likely it is that our heads and tails counts will be closer together. If we look at the 12th row of Pascal's triangle, we can see the probability of getting some number of heads and tails if we toss a penny 11 times. In 1 out of 2¹¹times, we'll flip all tails. In 11 out of 2¹¹times, we'll flip exactly 10 heads and 1 tail. In (330 + 462 + 462 + 330) out of 2¹¹times, we'll flip 4, 5, 6, or 7 heads out of 11 chances. That's a little over 77% of the time.

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...“and what is the use of a book,” thought Alice “without pictures or conversations?”

Addition

Squares & Sums

Integers - line segments or 1D triangles

Triangular Numbers - triangular grid in 2D (like bowling pins)

Tetrahedral Numbers - pyramids in 3D where all sides are triangular numbers

Pentatope Numbers - hard to understand; triangles in 4D

The Fibonacci Sequence

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Debbie's cookies were a marvel. I had a particular fondness for the peanut butter blossoms and the pink almond maraschino cherry cookies. The other ones might have been chocolate crinkles, jam thumbprints, and Russian tea cookies; let's make it so. You might recall that Debbie said I'm allowed to choose any 4.

Chocolate Crinkles - C

Jam Thumbprints - J

Maraschino Almonds - M

Peanut Butter Blossoms - P

Russian Tea Cookies - R

There's a reason this chapter is last; combinations in which repetitions are allowed are the trickiest of them all.

We can't take 5⁴ and then divide something out, because the cases can't agree on what that number would be. For example, imagine I chose 4 chocolate crinkles (C C C C). There are no duplicates for that pattern. But what if I chose (J M P R)? There are 4! duplicates for that one.

Nor can we pretend the set is (C C C C J J J J M M M M P P P P R R R R) and choose 4. The permutation would be 20*19*18*17, but just like before, we wouldn't know what to divide it by. (C C C C) would yield no duplicates and (J M P R) would yield 4!.

To figure this one out, we actually have to morph the problem into a different "n choose r" situation. (Now would be a good time for me to thank the website mathisfun, which has been guiding me through writing this entire post.)

Let's put Debbie's cookies onto her Christmas cookie platters. Now they look like this:

Instead of tracking our cookies, we're going to write a code that describes our actions.

0 means we took no cookies and moved onto the next plate.

1 means we took 1 cookie and stood still, stupidly staring at the same plate.

So if I choose 4 chocolate crinkles, that looks like this (1 1 1 1 0 0 0 0). And if I chose a jam thumbprint, a maraschino almond, a peanut butter blossom, and a Russian tea cookie, that would look like this (0 1 0 1 0 1 0 1).

This may seem convoluted, but here's where it pays off: both of these strings are 8 digits long. By tracking our actions instead of our cookies, we can represent each choice in a way that lends itself to factorials and division once again. The string will always be exactly r + (n - 1) long, where r is the number of cookies Debbie is letting us have, and n is the number of types of cookies she has made.

Our question has changed topics from a set of 5 cookies to a string of 8 numbers, but we're still choosing 4. And "8 choose 4" on Pascal's triangle is row 9 term 5, which is 70.

Or we can use our "n choose r" formula to get 8!/(4!4!) = 70.

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choosing r out of n	repetition not allowed	repetition allowed
permutations, order matters	n! / (n - r)!	n^r
combinations, order doesn't matter	n! / [r!(n - r)!]	(r + n - 1)! / [r!(n - r)!]

02 August 2026

After the Decimal

This week, Dad asked me to find an algorithm for turning base 10 decimals into binary. He's so annoying.

To review what binary is, we first need to review base 10. We have a counting system called base 10, where we organize our things into piles of 10's using digits 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9. Once we get too many of those piles, we group 10 of them together and call them "hundreds." And we organize those into bigger piles of 10 hundreds, and those are called "thousands." This goes on for quite some time - enough to take care of our counting needs, anyway. We stop understanding numbers easily by one million, which is why we haven't already started a revolution, despite the fact that one person in our country has a trillion dollars.

10³ - one thousand

10⁶ - one million

10⁹ - one billion

10¹² - one trillion
10¹⁵ - one quadrillion
10¹⁸ - one quintillion
10²¹ - one sextillion

10²⁴ - one septillion
10²⁷ - one octillion
10³⁰ - one nonillion
10³³ - one decillion

But the choice of the number "10" and the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9 aren't based on some sort of universal truth, and in fact, they're a bit random. Binary or base 2 is what we call it when we start organizing our things into piles of 2 instead. We only need two digits when we do that, so we use 0 and 1. We can group those piles of 2 into bigger piles of two 2's, and the unit 2² is like our hundreds unit. The unit 2³ is like our thousands.

Here's a helpful exercise. Let's just count a few numbers in binary and see what they look like.

1 - 1

2 - 10

3 - 11

4 - 100

5 - 101

6 - 110

7 - 111

8 - 1,000

9 - 1,001

10 - 1,010

11 - 1,011

12 - 1,100

13 - 1,101

14 - 1,110

15 - 1,111

Let's explore the number: 111,111.

In base 10, that number means 1 pile of 10⁵ + 1 pile of 10⁴ + 1 pile of 10³ + 1 pile of 10² + 1 pile of 10¹ + 1 pile of 10⁰. In base 2, it means 1 pile of 2⁵ + 1 pile of 2⁴ + 1 pile of 2³ + 1 pile of 2² + 1 pile of 2¹ + 1 pile of 2⁰. (Notice that our ones units are counting ones in both systems, because 10⁰ = 2⁰ = 1.)

Now consider this number: .111

In base 10, that's one tenth and one one-hundredth and one one-thousandth.

Another way to look at this is one 10^-1 + one 10^-2 + one 10^-3.

In base 2, that's one half and one fourth and one eighth, or one 2^-1 + one 2^-2 + one 2^-3.

So back to Dad's question. Let's turn some long ugly number with a decimal into binary.

I'll choose 654.32̅1̅.

(If you were dying to convert an irrational number into binary and found yourself reading this post, I'm sorry to tell you that you'll have to round. Life is full of disappointments, and you may as well get used to it now.)

The first step is to work on the part to the left of the decimal.

We can call that the before-the-decimal algorithm.

Divide 2 into the integer and write the remainder in the ones (2⁰) unit.

654/2 = 327 R 0

That's the whole algorithm, so now we'll repeat it for the next unit to the left.

Divide 2 into the new number (327) and write the remainder into the 2's (2¹) unit.

327/2 = 163 R 1

10.

The next remainder goes into the 4's (2²) unit.

163/2 = 81 R 1

110.

8's (2³) unit.

81/2 = 40 R 1

1,110.

16's (2⁴) unit.

40/2 = 20 R 0

01,110.

32's (2⁵) unit.

20/2 = 10 R 0

001,110.

64's (2⁶) unit.

10/2 = 5 R 0

0,001,110.

128's (2⁷) unit.

5/2 = 2 R 1

10,001,110.

256's (2⁸) unit.

2/2 = 1 R 0

010,001,110.

512's (2⁹) unit.

2/1 = 0 R 1

1,010,001,110.

That takes care of the part before the decimal, but before we go on, we ought to check our work.

512 + 128 + 8 + 4 + 2 = 654

Now we turn our attention to the part after the decimal.

It is: .32̅1̅

Did you know that any repeating decimal can be expressed as a fraction?

Let's do that; I know a trick.

First, name your repeating decimal. I'll call this one x.

x = .32̅1̅

Then multiply it by 10^whatever so you're set up to subtract and erase the repeating part.

100x = 32.12̅1̅

100x - x = 31.8

and

99x = 31.8

which means

x = 318/990 = 106/330 = 53/165

We're all ready to get started!

The after-the-decimal algorithm is quite similar to the before-the-decimal one.

We'll be working from the unit nearest the decimal and then moving to the right.

Multiply the fraction by 2.

If the result is ≥ 1, put a 1 in the halves (2^-1) unit, and if it is not, put in a 0.

Once the fraction becomes greater than 1, subtract 1 to find the new fraction.

We start with the 1/2's or (2^-1) unit with the fraction 53/165.

53*2/165 = 106/165 < 1

1,010,001,110.0

1/4's (2^-2) unit.

106*2/165 = 212/165 ≥ 1

1,010,001,110.01

1/8's (2^-3) unit.

47*2/165 = 94/165 < 1

1,010,001,110.010

1/16's (2^-4) unit.

94*2/165 = 188/165 ≥ 1

1,010,001,110.0101

1/32's (2^-5) unit.

23*2/165 = 46/165 < 1

1,010,001,110.01010

1/64's (2^-6) unit.

46*2/165 = 92/165 < 1

1,010,001,110.010100

1/128's (2^-7) unit.

92*2/165 = 184/165 ≥ 1

1,010,001,110.0101001

1/256's (2^-8) unit.

19*2/165 = 38/165 < 1

1,010,001,110.01010010

1/512's (2^-9) unit.

38*2/165 = 76/165 < 1

1,010,001,110.010100100

1/1024's (2^-10) unit.

76*2/165 = 152/165 < 1

1,010,001,110.0101001000

1/2048's (2^-11) unit.

152*2/165 = 304/165 ≥ 1

1,010,001,110.01010010001

That looks like a pretty good place to stop, so we can truncate and call it a day.

I would have stopped at the 1/512's (2^-9) unit, but I thought I'd wait for the next 1 to lock in that 2,048th.

As Skeletor says, "Until we meet again!"

28 July 2026

2027 Hatter Notes

The Hatter's Diary

28 Jul 2026

I'm aware that I'm a year early.

Now that we've solved the riddle, it's time to reacquaint ourselves with the layout of Stuff & Nonsense VII: Mother Goose Suite, which was planned in 2024. The piece will be scored in 6 movements, written in the key of G major for "goose." I normally quote the themes of each set in the postludes of songs 4 and 7. In this case, each of the Stuff & Nonsense themes from songs 1-6 will be paired with a nursery rhyme.

Mother Goose Suite

Here We Go Round the Mulberry Bush

From Through the Looking-Glass and What Alice Found There, Chapter 4.

Alice and the Tweedles were recalled to have sung it.

This one will be used in each of the 6 movements.

Here we go round the mulberry bush,
The mulberry bush, the mulberry bush,
Here we go round the mulberry bush,
All on a frosty morning.

VII.1 Twinkle, Twinkle

From Alice's Adventures in Wonderland, Chapter 7, sung by The Hatter.

The Dormouse interrupts and sings the last 4 twinkles.

Quotes S&N I. The Mouse's Tale, sung by The Mouse.

Twinkle, twinkle, little bat!
How I wonder what you’re at!
Up above the world you fly
Like a tea-tray in the sky.
Twinkle, twinkle—
Twinkle, twinkle, twinkle, twinkle—

VII.2 The Queen of Hearts

From Alice's Adventures in Wonderland, Chapter 12, sung by The White Rabbit.

Quotes S&N IV. The White Rabbit's Evidence, also sung by The White Rabbit.

The Queen of Hearts, she made some tarts,
All on a summer day:
The Knave of Hearts, he stole those tarts,
And took them quite away!

VII.3 Tweedeldum and Tweedeldee

From Through the Looking-Glass and What Alice Found There, Chapter 4, sung by Alice.

Quotes S&N VI. The White Queen's Riddle, sung by The White Queen & The Red Queen.

Tweedledum and Tweedledee
Agreed to have a battle;
For Tweedledum said Tweedledee
Had spoiled his nice new rattle.
Just then flew down a monstrous crow,
As black as a tar-barrel;
Which frightened both the heroes so,
They quite forgot their quarrel.

VII.4 Humpty Dumpty

From Through the Looking-Glass and What Alice Found There, Chapter 6, sung by Alice.

Quotes S&N V. Humpty Dumpty's Poem, sung by Humpty Dumpty.

Humpty Dumpty sat on a wall:
Humpty Dumpty had a great fall.
All the King’s horses and all the King’s men
Couldn’t put Humpty Dumpty in his place again.

VII.5 The Lion and The Unicorn

From Through the Looking-Glass and What Alice Found There, Chapter 7, sung by Alice.

Quotes S&N III. The Lobster Quadrille, sung by The Mock Turtle.

The Lion and the Unicorn were fighting for the crown:
The Lion beat the Unicorn all round the town.
Some gave them white bread, some gave them brown:
Some gave them plum-cake and drummed them out of town.

VII.6 Hush-a-by Lady

From Through the Looking-Glass and What Alice Found There, Chapter 9.

Sung by The Red Queen, who then requested that Alice sing it as well.

Quotes S&N II. The Duchess' Lullaby, sung by The Duchess & The Cook.

Hush-a-by lady, in Alice’s lap!
Till the feast’s ready, we’ve time for a nap.
When the feast’s over, we’ll go to the ball—
Red Queen, and White Queen, and Alice, and all!

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The Hatter's Diary

28 Jul 2026

I'm aware that I finished the last post less than a minute ago.

However, the first post was all about cold, hard facts, and this one will contain some initial ideas and reference material; where the previous post set up the boundaries of my playground, this one will unleash the monsters we'll be playing with.

One idea I had had previously was to build each movement in such a way that it quoted its predecessors, similar to the way many children's stories and songs are structured. Like "There's a Hole in the Bottom of the Sea."

An example of such a structure might be:

1. Mulberry

Twinkle

Mulberry

Twinkle & Mouse's Tale

2. Mulberry

Queen of Hearts

Twinkle & Mouse's Tale

Mulberry

Queen of Hearts & White Rabbit's Evidence

3. Mulberry

Tweedles

Queen of Hearts & White Rabbit's Evidence

Twinkle & Mouse's Tale

Mulberry

Tweedles & The White Queen's Riddle

4. Mulberry

Humpty

Tweedles & The White Queen's Riddle

Queen of Hearts & White Rabbit's Evidence

Twinkle & Mouse's Tale

Mulberry

Humpty & Humpty

5. Mulberry

Lion Unicorn

Humpty & Humpty

Tweedles & The White Queen's Riddle

Queen of Hearts & White Rabbit's Evidence

Twinkle & Mouse's Tale

Mulberry

Lion Unicorn & Lobster Quadrille

6. Mulberry

Hush-a-by Lady

Lion Unicorn & Lobster Quadrille

Humpty & Humpty

Tweedles & The White Queen's Riddle

Queen of Hearts & White Rabbit's Evidence

Twinkle & Mouse's Tale

Mulberry

Hush-a-by Lady & Duchess' Lullaby

Another idea I had was to make sure to quote Twinkle in minor, because it appears that way in the ending fugal section of Father William. How fortunate that this passage was originally scored in G minor, while this suite happens to be in G as well. I'm never reaching for the stars, and yet I'm constantly being inundated with stardust. Somebody or something must love me tremendously, and to quote Twinkle in major would not only be a clumsy misstep, but a rude rejection of such a beautiful gift. We must mind our manners.

In 2024, I had decided that Mulberry Bush, Twinkle, and Hush-a-by (as sung to Purcell's Lillibullero) were worthy melodic quotes, meaning that their scores had been published before 1871 (Through the Looking-Glass and What Alice Found There), and they appear to have been popular enough that Alice readers would have brought them to mind.

My initial conclusions were that the Queen of Hearts "did not pass the test of time," and the Tweedles "didn't lend themselves well to singing." Humpty only passed the margin by 1 year, and there's no indication that either Humpty or The Lion and The Unicorn ever made it big. Though these four contenders aren't as strong as the first three, it's likely I'll use them.

Library of Potential Nursery Rhymes

Sheet Music, 1879

The Singing Master, 1840

Nursery Rhymes with Old Tunes, 1846

British Library Music Collections English Songs Vol. 7, 1775

National Nursery Rhymes and Nursery Songs, 1872

Juvenile Minstrelsy, 1852

The Beggar's Opera, 1735

The Baby's Opera, 1877

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The Hatter's Diary

28 Jun 2026

I'm aware, again, that I finished the last post less than a minute ago. Sometimes it takes three posts before we're organized and ready for hibernation.

I have scores for every nursery rhyme except the Tweedles. But would you know it? Fortune favors the hatters. English poet John Byrom (1692-1763) is credited with the epigram that named the Tweedles in the first place, and those two just so happen to be composers. From 1725:

Some say, compar'd to Bononcini

That Mynheer Handel's but a Ninny

Others aver, that he to Handel

Is scarcely fit to hold a Candle.

Strange all this Difference should be

'Twixt Tweedle-dum and Tweedle-dee!

The nursery rhyme would be printed 80 years later, in Original Ditties for the Nursery. Further investigation into Giovanni Bononcini (1670-1747) reveals that he lived in London from 1720 to 1732, and for that time, he was basically in a popularity contest with Handel. In fact, there was a political divide; the Tories favored Handel while the Whig party favored Bononcini. One of Bononcini's most famous works was his opera Xerse, written in 1694.

Originally, the opera Xerse was composed by Francesco Cavalli (1602-1676) in 1655. The libretto, written by Nicolò Minato, was loosely based on Book 7 of The Histories by Herodotus (484-425 BCE). Nearly 40 years after Cavalli's opera, Bononcini wrote a new opera by the same name, and Minato's libretto was adapted by Silvio Stampiglia. But then in 1738, after another 40 years, Handel (1685-1759) wrote Serse, with Stampiglia's work adapted by an unknown librettist. All three of these operas begin with the aria, "Ombra Mai Fù," in which Xerxes I, the King of Persia, sings tenderly to a tree for providing him shade.

To recap, the original Tweedles were none other than Bononcini and Handel, and both of them are remembered today in part for their renditions of an opera about Xerxes (518-465 BCE), the King of Persia. Coincidentally, the most famous aria in those operas happens to be a song about a tree.

Chapter 4 of Through the Looking-Glass, begins, "They [the Tweedles] were standing under a tree, each with an arm around the other's neck..." Alice then recalls that she and the Tweedles had been singing Here We Go Round the Mulberry Bush, as the music "seemed to come from the tree under which they were dancing..." The white rattle that begins the battle is spotted under a tree. And when the brothers battle, they hit trees. As the monstrous crow enters the scene, Alice hides among the trees. I do believe we have run into some scores for the Tweedles, and composed by the original Tweedles, no less.

Cavalli: Ombra Mai Fù

Philippe Jaroussky

Cavalli: Ombra Mai Fù

Bononcini: Ombra Mai Fù

Cecilia Bartoli

Bononcini: Ombra Mai Fù

Handel: Ombra Mai Fù

Andreas Scholl

Handel: Ombra Mai Fù

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The Hatter's Diary

30 Jul 2026

Today I've been falling ever more in love with Bononcini, but I don't know why I bother, for I don't intend to keep a shred of the beauty (or even the dignity) woven so carefully into these masterful scores.

Meanwhile, I've started thinking about what kinds of mulberries we'll be bringing to this party.

VII.1 - Mode

Twinkle shall make an appearance in G minor, so we'll probably be needing minor mulberries.

Not to be forgotten is Byrd's glorious Picardy third.

VII.2 - Inversion

The White Rabbits' Evidence is all about the horizontal symmetry of the Dorian mode.

I therefore inverted the Mulberry intervals, which resulted in C minor.

Another way to think of this is reflecting the Mulberries over the horizontal G axis.

VII.3 - Retrograde

We are in the presence of the great The White Queen!

I therefore ordered our Mulberries in retrograde.

Another way to think of this is reflecting the Mulberries over the vertical double barline.

VII.4 - Scale

For Humpty Dumpty, I altered the scale to G whole tone.

The harmonies have morphed into augmented triads.

VII.5 - Meter

"Les Lignets" or "Les Lanciers" from The Lobster Quadrille plans to pay a visit.

Meanwhile, our nursery rhyme here is The Lion and the Unicorn.

They're both in duple time!

VII.6 - Tempo

This one is a lullaby, with the gentle lilt of Purcell's dotted rhythm.

Score:

Bushes of Mulberries

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The Hatter's Diary

00 Jun 2027

Score:

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Musescore Audio:

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The Hatter's Diary

00 Jun 2027

Score:

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Musescore Audio:

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The Hatter's Diary

00 Jul 2027

Score:

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Musescore Audio:

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The Hatter's Diary

00 Jul 2027

Score:

Link

Musescore Audio:

Link

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The Hatter's Diary

00 Jul 2027

Score:

Link

Musescore Audio:

Link

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11 July 2026

Rainbow Follies

With the closing of KLCC at the end of December, Malinda and I finally had the time to return to our own programs. We've done it before:

16 songs from 1916

Performed 02 July 2016

Wine, Chocolate & Medleys

Performed 23 December 2017

A Dozen Duets

Performed with Celeste Coburn 29 June 2019

Then covid hit, and with it, the KLCC choir pretty much collapsed. On a good day, we had a choir of five, and after two deaths, there wasn't much to salvage. The congregation itself wasn't all that much bigger than the choir had been, and we continued to meet this way, like a couple of stray cats in an abandoned stadium, for five more years. During that time, Malinda and I funneled our efforts into designing the KLCC Christmas cantatas. We didn't have many resources, but we did what we could to make them special. The idea of Rainbow Follies had been in the works for ages, but finding the time to learn the music, write the script, and plan the party was another thing. Fast forward to 2026, a decade after our first production, and we hardly missed a Sunday. There were the occasional trips or celebrations that prevented us from meeting up, but little by little, we built the program. It's hard to explain exactly what we did on Sundays, but it was always some combination of:

Learning songs - working on the hard parts

Trying them in different ways with different ideas

Changing the order of the program and testing the flow

Writing or editing the script

Shopping for props, stationary, rainbow plates, etc.

Just hanging out and being friends

The hardest step is always to pick a date. One of our goals was to give the KLCC people a reason to get together again, but mostly, we just wanted everyone to have fun. Pianistically, my focus was to feel more secure under pressure, particularly with stride pattern and jumping. I think I achieved that, or at least progressed, but that hardly mattered because I still freaked out due to my kryptonite - page turns - and fumbled pretty badly twice. Anyway, nobody seemed to mind too much.

Things we'll keep:

The craft table!

4:00 pm - people hate driving at night, and the party started 15 minutes early

Programs with questions and clipboards - we loved all the different thoughts

Audience participation/singalong sections

The valise of props

Stats:

10 cancellations

24 guests including our families

13 songs

55 mins including script

3-4 hour party