FIDDLY DETAILS D
How Many Tunes Are Hidden in the Harmony?
I have two boxes in front of me. In the first box there are four pieces of fruit (a banana, an apple, a lemon, and an orange). In the second box there are four useful objects (a pen, a watch, a notebook, and a camera). My job is simply to choose one item from the first box and then choose one item from the second box. When I consider the possibilities, I see that I could have a banana and any one of the four useful objects, or an apple and any of the four, or a lemon and any of the four, or an orange and any of the four. That’s four choices multiplied by four choices, so there are in fact a total of sixteen potential combinations.
If I had a third box with four other things in it, the number of possible ways of doing it would be four multiplied by four multiplied by four (a total of sixty-four). As you can see, the number of possibilities rises very rapidly as you add more boxes.
The four flautists I described at the beginning of chapter nine have a job to do. They have to deliver a sequence of nine groups of four notes to the listeners:
Group 1: C5, G6, E4, C4
Group 2: C5, G6, E4, C4
Group 3: G5, G6, E4, C4
Group 4: G5, G6, E4, C4
Group 5: A5, F6, C5, A4
Group 6: B5, F6, C5, A4
Group 7: C6, F6, C5, A4
Group 8: A5, F6, C5, A4
Group 9: G5, G6, E5, C4
The people in the audience don’t care which flautist plays any particular note as long as they hear the correct groups of notes in the right order. Let’s imagine that we are going to let the flautists choose their own notes. We’ll let flautist 1 have the first pick. She simply has to choose one note from each of the nine groups, and whichever sequence she chooses will be a tune of some sort.
What she is doing is exactly the same process as choosing one of four objects from nine boxes, so the actual number of possible tunes in our simple example is four multiplied by itself nine times: 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 262,144.
When the sheet music was handed out in the first place, the tune was given to flautist 1, and the boring accompaniment notes were given to the other three musicians. Like this:
Group 1
Flute 1: C5
Flute 2: G6
Flute 3: E4
Flute 4: C4
Group 2
Flute 1: C5
Flute 2: G6
Flute 3: E4
Flute 4: C4
Group 3
Flute 1: G5
Flute 2: G6
Flute 3: E4
Flute 4: C4
Group 4
Flute 1: G5
Flute 2: G6
Flute 3: E4
Flute 4: C4
Group 5
Flute 1: A5
Flute 2: F6
Flute 3: C5
Flute 4: A4
Group 6
Flute 1: B5
Flute 2: F6
Flute 3: C5
Flute 4: A4
Group 7
Flute 1: C6
Flute 2: F6
Flute 3: C5
Flute 4: A4
Group 8
Flute 1: A5
Flute 2: F6
Flute 3: C5
Flute 4: A4
Group 9
Flute 1: G5
Flute 2: G6
Flute 3: E5
Flute 4: C4
It’s important to remember that any sequence of notes can be a tune. In fact flautists 2, 3, and 4 are all playing their own tunes. Dull, repetitive tunes—but tunes nonetheless.
These tunes are so dull that our flautists start to rebel. “Why can’t we swap a note here and there?” says flautist number 2. “For example, for those first four notes, you’ve got me playing G6 four times and my friend flautist 3 is playing E4 four times. It’s really boring for both of us. Why don’t we alternate notes? I could play G6, E4, G6, E4, and flautist 2 could play E4, G6, E4, G6. It doesn’t change what the audience hears but it makes our lives a bit more interesting.”
“Yes, good idea!” says flautist 3. “Let’s alternate our notes all the way through.
| “You play: | E4 | G6 | E4 | G6 | C5 | F6 | C5 | F6 | E5, |
| and I’ll play: | G6 | E4 | G6 | E4 | F6 | C5 | F6 | C5 | G6.” |
“Hang on,” says flautist 4. “That still leaves me bored to death. Why don’t we alternate between all three of us? Then it will be even more interesting.”
It’s not long before the three accompanying flautists realize that their day will get even more exciting if flautist 1 joins in with the note swapping. Eventually the four flautists come to an agreement that they should share all the notes—all the accompaniment notes and the “Baa, Baa, Black Sheep” tune.
After half an hour of negotiation (“In group three I’ll swap you my C4 for your G6”) the flautists have redistributed all the notes and they are all a lot happier because they have chosen not to have any boring repeated notes to play. From the 262,144 possibilities they have eventually chosen this one:
Group 1
Flute 1: G6
Flute 2: C4
Flute 3: E4
Flute 4: C5
Group 2
Flute 1: E4
Flute 2: G6
Flute 3: C5
Flute 4: C4
Group 3
Flute 1: G5
Flute 2: C4
Flute 3: E4
Flute 4: G6
Group 4
Flute 1: E4
Flute 2: G5
Flute 3: G6
Flute 4: C4
Group 5
Flute 1: A5
Flute 2: C5
Flute 3: F6
Flute 4: A4
Group 6
Flute 1: F6
Flute 2: A4
Flute 3: B5
Flute 4: C5
Group 7
Flute 1: A4
Flute 2: F6
Flute 3: C5
Flute 4: C6
Group 8
Flute 1: C5
Flute 2: A5
Flute 3: F6
Flute 4: A4
Group 9
Flute 1: G5
Flute 2: G6
Flute 3: C4
Flute 4: E5
This looks more complicated than my original version but it is still just four flutes playing the same nine groups of notes in the same order—so an audience wouldn’t notice any difference between this and the original. They’d hear the melody line of “Baa, baa, black sheep, have you any wool” and the same simple accompanying harmony.
If I asked you to get a pencil and underline the tune notes in this new distribution scheme, I imagine it would take you two or three minutes of cross-checking with my original version. The amazing thing is that our ears always “underline” the melody notes instantly—with no conscious effort.
Here are the tune notes, underlined and in bold.
Group 1
Flute 1: G6
Flute 2: C4
Flute 3: E4
Flute 4: C5
Group 2
Flute 1: E4
Flute 2: G6
Flute 3: C5
Flute 4: C4
Group 3
Flute 1: G5
Flute 2: C4
Flute 3: E4
Flute 4: G6
Group 4
Flute 1: E4
Flute 2: G5
Flute 3: G6
Flute 4: C4
Group 5
Flute 1: A5
Flute 2: C5
Flute 3: F6
Flute 4: A4
Group 6
Flute 1: F6
Flute 2: A4
Flute 3: B5
Flute 4: C5
Group 7
Flute 1: A4
Flute 2: F6
Flute 3: C5
Flute 4: C6
Group 8
Flute 1: C5
Flute 2: A5
Flute 3: F6
Flute 4: A4
Group 9
Flute 1: G5
Flute 2: G6
Flute 3: C4
Flute 4: E5
Even with the help of the underlining and bold print it takes some effort to see the tune hidden in among the other notes—but you can’t hide a tune from our ears. As I explain in chapter eleven, our hearing system uses the principles of similarity, proximity, good continuation, and common fate to pick out the tune from the accompaniment—in this case spotting that one tune from over a quarter of a million possibilities!