BEHIND THE SONG: Morse

"Morse" is a song cowritten by Luis Nasser and Ricardo Gómez. Ricardo is a mathematician at UNAM in Mexico City and a former member of Sonus Umbra. Here they are playing together in Baltimore in 2006 (Ricardo on guitar, Luis on bass). The two of them have completed work recently on a paper involving the math that inspired this song. More on that in a later post.

This song is based on the Thue-Morse sequence (sometimes called the Prouhet-Thue-Morse sequence). It, like Morse code made of dots and dashes, is a binary sequence. The sequence is partly named for Marston Morse, who lived about a century after Samuel Morse of the code fame.

We can write the Thue-Morse sequence as a series of zeroes and ones. Here's what to do: (1) start with a zero, (2) take the opposite of the sequence you have (zero becomes one, one becomes zero), (3) append that opposite on the end, (4) repeat. "Taking the opposite" is formally called "bitwise negation."

Step by step, we have
0
01 (1 is the opposite of 0, put it on the end)
0110 (10 is the opposite of 01, put it on the end)
01101001 (1001 is the opposite of 0110, put it on the end)
and so on.

We can also get the sequence by looking at numbers in binary. The first eight numbers (counting from zero to seven) in binary are:

0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111

Now apply this rule: if the binary number has an even number of "one" digits, that spot in the sequence gets a zero. If it has an odd number of "one" digits, that spot in the sequence gets a one.

Counting the ones gives 0, 1, 1, 2, 1, 2, 2, 3—or if we translate evens and odds to ones and zeros, this becomes 0, 1, 1, 0, 1, 0, 0, 1 (which is step four in the Thue-Morse sequence shown above).

The sequence can be rendered in 2D as well (as Wikimedia Commons does here):

If you look at the bottom row of the final picture here with white being a zero and red being a one, you get the sequence.

This song deserves more than a single post to discuss the nuts and bolts, but there are two key elements in the song that use the sequence. The first movement of the song starts like this:

Movement 1 of "Morse"

The Thue-Morse sequence inspires the rhythm. We translate "0" to a quarter note and "1" to an eighth, and the measures proceed taking on an additional "step" of the sequence each time.

0 | 01 | 0110 | 01101001 | 0110100110010110

And it was decided to stop the main "riff" there. Aaron's guitar part often plays the complement or opposite of this pattern, most noticeably at the end of the piece.

The second movement of the song uses the sequence in a different way. In it, the "0" represents a eight-bar theme and "1" represents a four-bar theme. This uses the sequence to dictate arrangement and where chord changes occur, not just note rhythms. As Gómez and Nasser say in their paper, "the rhythm of the first movement is now translated into the form of the second movement." When we play this section of the song, though, an additional "0" section is included because the composers felt it sounded best artistically. Like in nature, random elements occasionally make things interesting.

It's helpful to reiterate this, as it's central to the Devil's Staircase philosophy: the math and science inspirations inform compositions; they don't provide set prescriptions. My songs using cellular automata are more technical on this count, but even for those we have felt free to change parts and time signatures to make the songs better in the end.

The piece concludes with a mirror image of the first movement with the roles of guitar and bass reversed.

Going forward

This song uses the Thue-Morse sequence to inspire rhythms and arrangements. But the sequence can also inspire different things musically. One can assign zeros and ones to, say, semitones and whole tones and generate different scales on a piano. This is something Gómez and Nasser explore in their paper in greater depth.