or more formally
How to fill in a Babbit Square
I imagine that your first question is, "What on earth is a Babbit Square?" and once I tell you what it is your question will likely be "Why on earth would you do this?" To answer your second question, for fun!!! And theoretical purposes, but more on that later.
A Babbit Square is a tool for determining all the different permutations and combinations of a tone row, which is a line of 12-tone music -- music that uses all 12 notes in Western music. The square is 12x12, and can be read left to right (Prime), right to left (Retrograde), top to bottom (Inversion), and bottom to top (Retrograde-Inversion). Every row, and column has the numbers 0-11 only once. This is useful in theory analysis of 20th century music associated with the likes of Arnold Schoenberg, Anton Webern, and - you guessed it - Milton Babbit.
But why Kendra, why???!!! How is this fun??
Umm, how is it not fun? It's like musical Sudoku.
Actually, whenever I embark on 12-tone music, I like to fill out a Babbit Square and then play around with the different rows. But more on that later.
The purpose of this blog entry is to give you, the gracious reader, step-by-step instructions on how to fill out a Babbit Square. Why? Because I'm going to start posting these squares with some regularity as a fun Saturday morning puzzle!
So let's begin at the beginning. And when we get to end, we'll stop.
We begin with the tone row:
As you can see, every 12 notes (or pitch classes) in Western music is used once in the row.
1. Fill in the top row of your Babbit Square with the note names of your tone row
2. Then you need to assign a number from 0-11 to each note name
0 is always assigned to the first note name in your tone row. In this instance D=0.
Now it gets complicated. Thanks Babbit. or Schoenberg.
The rest of the numbers are assigned based on their chromatic order, as in C, Db, D, etc. So what I do when I'm filling out Babbit Squares, is I write out C, Db etc underneath the square, and assign my numbers from there.
So in this instance, D=0; that means the next note name, Eb=1; E=2 and so on until you get to 11.
Are you lost yet? Good. You're doing perfectly.
3. Then I transfer my numbers into my square with the corresponding note name.
4. Before I start figuring out the numbers in my square, I begin with filling out all the zero's (0's). If done correctly, your zero's will create a diagnol line.
Oh, and just to make sure you're still confused, in Babbit square land, 0=12.
5. Now to the fun math stuff! First we will fill out the first column on the left-hand side of the square. You will calculate the number based on the square diagnol to it. The numbers on this diagnol should add up to 12; i.e. top row + x = 12.
In our example, the first row, second square = 9, 9 + x = 12; so the second row, first square = 3.
To look at it non-algebraicly:
0 = 12
9+3 = 12
and so on.
Here's the finished column:
6. Now let's move on to the rows.
Add the number at the beginning of the row, to the numbers in the original tone row.
Let's look at an easy one. On the ninth row, our first number is 1. We add 1 to the number in the top row (original tone row).
0+1 = 1; 9+1 = 10; 5+1 = 6
Making sense? Totes.
Now let's go to the seventh row, where we'll be adding 6 to our top row.
Easy enough - 9+6 = 15
...Hold the phone. What? I thought there was only supposed to be the numbers 0-11. You are correct young grasshopper.
So how we deal with this is by simply subtracting 12. 15-12 = 3.
You can also look at it like the 24 hour clock. 15h00 = 3; 21h00=9. And so on. This may confuse you, but it really helped me (for some reason).
Now let's look at the second row. As you can see, we are adding 3+9. Whenever the answer equals 12, you write a zero. Well would you look at that, we already did that! (See Step 4).
And now you continue on until you're finished filling out all of your squares.
7. Once you've finished filling in all the numbers, you can fill out the note names.
So in this instance, wherever there's a 0, you fill in 'D'.
Whenever there's a 9, you fill in 'B'
And so on.
As you're filling in your square, this will add as a double check. Every note name should only appear once in each row and each column. If something is in there twice, you'll need to go back and check for mistakes.
Here's my finished square (it's in an ugly handmade square I did a while back... yes, I do these often... for fun...):
So there you have it! This is how to fill out a Babbit Square.
Ready to try one out yourself? Here is this article's example:
Then follow me on social media to get the weekly Saturday Morning Babbit Squares!
Post Script - in regard to how I use these Babbit Squares in my own work, I will be posting that in a future blog! Stay tuned - same Bat Time... well... relatively same Bat Time... some Bat Time...but actually same Bat Channel!
Post-Post Script - May 11 - Read about the project I've been using these particular Babbit Squares here.