Exciting developments (cover, bandcamp release, interview)

So, first, some pressing issues:

We now have a Bandcamp, and soon (if not already now), the album will be ready for pre-order! 

US/NA orders: thedevilsstaircase.bandcamp.com
Europe: https://roth-handle.bandcamp.com/album/the-devils-staircase

Later today, progrock.com is hosting a six-hour event featuring interviews with dozens of artists and producers. Luis will be on at 2 Central (GMT -6) to talk Devil's Staircase. I will join if possible. This 15 minute slot will be preceded by our drummer Mattias Olsson discussing his Döskalle project and followed by Luis, etc., discussing Sonus Umbra.  FB event page.

And next: We have an album cover!

The art was put together by Henning Lindahl (Airwaves Design, Sweden), and in honor of the new cover, I've changed the blog coloring a bit to match.

The art shows a spiral staircase, and the reddish element is the Devil's Staircase function. The Devil's Staircase is connected to the Cantor set which I discussed a bit here, and you can go here for a Wikipedia animation of how this function is put together.

In brief summary: anywhere there's a space in the Cantor set, the Devil's Staircase is flat. Remember the Cantor set can be visualized by a line segment where we "take out" the middle third over an infinite number of steps. The parts taken out are the flat parts in the Staircase function. There are deep mathematical ideas in play here, but basically what you have is a function that's continuous everywhere and flat almost everywhere, and it takes every value from the minimum to the maximum. For those who've had calculus, it's weird in that the function is not the integral of its derivative (for most functions you've seen in math class, this holds up.)

To come in the future: more about the Morse sequence. Luis and Ricardo Gómez (his coauthor on the paper) are working more on this material, and we have a lot more to say mathematically about what is going on with the Morse song. The album divides Morse into pieces, and a future post will go into details and tie into my older post.

Greetings from AAPT

Hi - I (Tim) am now about to present at the Winter AAPT conference in Orlando. I have a short talk today and a longer poster session tomorrow night.  If you're here through any of my QR codes or blog links, welcome. Check out what I've written and see some of the science behind the songs.

And you can click here to hear one of the final studio tunes we're putting out. Enjoy!

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.

BEHIND THE SONG: Cantor's Dust

"Cantor's Dust" is one of the first songs Luis invented for this project—I performed it with him as part of a Columbia College Chicago colloquium before I was hired to work there. Andy Tillotson (our longtime friend and bandmate in Sonus Umbra and Might Could before that) also performed. Although the version we have recorded as The Devil's Staircase rocks it out more now, originally the arrangement was amenable to playing with a small set of acoustic instruments.

I may be inspired to write a longer introduction to fractals later, but for this post I'll explain enough to get at some of the ideas used in the song.

A fractal, as described by Benoit Mandelbrot (hear a favorite fractal-related song of mine related to him here), is a "shape made of parts similar to the whole in some way." The term self-similar is also used a lot in writings.

Fractals are a frequent source of inspiration in the arts, and people usually think of visual art when picturing fractals. A computer image of the Mandelbrot set mentioned in the song linked above artistically shows a fractal. The finger-like projections on the wave tips in Hokusai's famous print shown below have fractal properties.

A representation of the Mandelbrot set (Wolfram MathWorld) and The Great Wave off Kanagawa by Hokusai

The Cantor set

The fractal inspiring our song is the Cantor ternary set, named after mathematician Georg Cantor. It's easy to picture that set. Proceed in steps. First, take a line segment. Then, remove the middle third. Then, remove the middle third of the two pieces left. Then, keep going ad infinitum. What's left is the Cantor ternary set.

Generating the Cantor ternary set (source: Wikimedia commons)

It may seem like if one keeps doing this, there will be nothing left. Picturing some numbers will help if I am more precise. Suppose the first segment represents the interval from zero to one on a number line (including zero and one). Mathematicians write this set as [0, 1], with square brackets used to remind us that the endpoints are included.

The Cantor ternary set instructions have us take out the set (1/3, 2/3), with round parentheses used to remind us that 1/3 and 2/3 are NOT taken out. These are two examples of points in the set that are never taken out. Then the sets (1/9, 2/9) and (7/9, 8/9) are removed, basically preserving the 1/9, 2/9, 7/9, and 8/9 points forever. Other points that aren't endpoints also survive this process.

A weird property of this set is that if you add up the lengths of the pieces removed after an infinite number of steps, you get the length of the whole segment.  But there are infinitely many points in the set.

One further clarification: the term ternary is used in the description to refer to ternary (base-3) numerals.  Any point in the Cantor ternary set from [0,1] does not require the digit 1 to be expressed as a fraction in base 3.  For example, we'd write the normal base-10 fraction 1/3 in base 3 as 0.1 (that is, 1(1/3)), but we can also write it as 0.0222... because 1/3 = 0(1/3) + 2(1/9) + 2(1/27) + 2(1/81)...

Phew, so what's that got to do with the song?

The theme of removing the middle third happens in at least two ways in the song Cantor's Dust.  The introduction of the bass and guitar in the song uses no thirds as the introduction gets underway.  The bass plays the root and fifth note of the C scale starting around 0:33 of the song, whereas the guitar arpeggiates over some kind of suspended chord featuring the root, fourth, and fifth note around 0:42.  It isn't until later in the song where a flatted third shows up, confirming that the song is in C minor.

The first chord I play on acoustic guitar in the song. Note the lack of third. That would be an E or E-flat (a 1 or 2 on the second note of the tablature).

The song is inspired by the Cantor set structurally as well—the first and last third of the song are similar, and the middle sounds completely different. The arpeggios I wrote above appear in the first and last third, but not the middle.

Georg Cantor was an influential thinker in number theory and set theory—for example, his work helped determine how countable certain sets of numbers are. However, he dealt with trouble getting the professorships he wanted and fought philosophically with other thinkers of his time. He suffered from serious depression and died poor in a sanatorium. A minor key was an informed choice for this piece.

Something else cool and related - the Sierpinski carpet

An iteration of the Sierpinski carpet (Wikimedia commons)

The Sierpinski carpet shown above is a fractal that can be seen in light of the Cantor ternary set. In it, start with a solid square, and take out the central square. Repeat for several steps. If you do this with a 3D object, you get a Menger sponge.

Sierpinski carpets are used as antenna shapes. If you stop (like the photo above) after a finite number of steps, you can construct the thing and it contains many different "mini-lengths" of antennae. That makes an antenna shape like this useful for picking up a variety of wavelengths and frequencies.

This is one way to turn the Cantor set 2-D. Another (actually called "Cantor's Dust") is what you get if you take whole middle-third strips out of both the horizontal and vertical dimensions instead of central squares.