After the performance, anticipating the drive back to Seattle late in the snowy evening, I fired up the iPod using an adapter that broadcasts the music to my truck’s radio via a weak FM signal. And while listening to Ingrid Matthews playing a solo Bach partita in the cocoon of the truck cab, wiper blades beating over the hiss of the wet pavement, a wonderful thing happened. A radio signal from a small-town religious station began to come into range, sometimes becoming strong enough to interfere, sometimes ebbing away. Being the electronic music guy that I am, I often let these moments play themselves out rather than fiddling with the equipment, just in case. The station was playing very mellow a cappella gospel music - nearly barbershop, but squarer harmonically. The keys of these tunes were related to the key of the partita, and their slow-moving harmonic rhythms melded the two pieces, in granulated and randomly varying form, together in a completely unexpected way. As the station got stronger, the wiper blades began to interact with the reception, eventually forming a simple rhytmic switching between the two signals. Again, both the tempi and the keys of the pieces (and of the wiper blades) were strangely synchronized. While not identical, they were close enough to fit as though written as counterpoint. Finally, after about 15 minutes of slowly evolving ambience, the station won out, mid-sentence, as the preacher cut in with a colorful and colloquial commentary. Wow.

I could not have created this collage, yet it was a brilliant musical moment. A a fine contrasting palette cleansing after the ultra-carefully-crafted Toensing. A memorable evening, on both counts.

Mascheroni Construction of the Unit Square

The picture to the right is a simple but beautiful Mascheroni construction which results in the points for the unit square. (You can see lines for 2 sides of this square in the drawing.) Melissa Plagemann and Linda Strandberg sang a live musical rendition of this construction at the Anathem launch in San Francisco. A week ago, back in Seattle, they helped me by making a quick recording of the duet for my archives.

Here is a stream of my rough mix:

I’m not going to give a tutorial on CAs; there is a vast amount of information freely available on the web, up to and including Stephen Wolfram’s immense tome, in its informative, yet pompous, entirety. CAs are easy to understand, to notate, and to implement. Diversity and complexity can be generated by very simple machines, without the need for the complicated syntax and evaluation rules used by other common modeling approaches such as L-systems or Chomsky grammars. Perhaps as important as their ability to generate complex patterns from simple beginnings is their ability to generate interesting, yet repetitive, patterns, since the simple breaking of symmetry is far more fundamental to our human enjoyment of music than complexity.

Piano-roll for Simple Automata

Automata are firmly established as tools for creating visual art and music. Musical applications of CAs appeared as early as 1988, and the popularity of Conway’s Game of Life rapidly made them truly commonplace. For this reason, when I decided to use CAs as a basis for the Anathem pieces, I wanted to appeal to listeners’ existing knowledge when present, and yet tread new aural ground. In Anathem, the avout sometimes use musical CAs to perform calculations on-the-fly as a group, and so the obvious place to begin my search was to consider how a group of people might perform continuously evolving CAs from rule notation, rather than from though-composed musical scores.

At the Anathem launch event in San Francisco, I demonstrated the results of this thought experiment, using members of the audience as individual two-state cells. Since not everyone is comfortable with singing, we used rhythmic clapping (8 claps to a cycle) to represent one state, and silence to represent the other. The CAs that we performed together were all synchronous one-dimensional machines, which made it very easy to propagate information left and right down a single-file line of people. (We also performed asynchronous turing machines as a demonstration of less rhythmically rigorous performance possibilities.)

Before starting each rule by counting off eight beats, I gave simple English explanations of the rules, such as the following for the symmetric and pleasing rule 126: “Start clapping on the next cycle if either or both neighbors are clapping. Stop clapping if you and both of your neighbors are all clapping. Otherwise, continue what you are doing on the next cycle.” The chaotic rule 30 was less simple, but still doable: “If one and only one of your neighbors is clapping and you are not, start clapping on the next cycle. If you are clapping, and the person on your left is also clapping, then stop clapping on the next cycle. Otherwise, continue what you are doing on the next cycle.” Even the notorious complexity-producing rule 110 was simple enough to perform: “If you are not clapping, but if either the neighbor on your right is clapping or both neighbors are clapping, start clapping on the next cycle. If you and both of your neighbors are all clapping, stop clapping on the next cycle. Otherwise, continue what you are doing on the next cycle.”

[Details for would-be experimenters: Before starting, you need to set the initial state by telling some members of the line to begin in an "on" state by clapping; you also need to decide how to handle the special cases at both ends of the row. It is also straightforward to have individuals represent more than one cell by cycling through them in order; in this way, small groups can perform large automata. Finally, the possibilities for state mappings that are more complex than simple on/off gestures are infinite and limited only by your imagination.]

Making music is especially enjoyable for amateurs when done in a group, and this experiment confirmed for me that performing cellular automata is a fun, unthreatening, group activity. As detailed above, each cell was represented by individuals clapping, but if those individuals were told to choose a note and sing it, rather than to clap, what would result is Simple Automata. In this piece, the “master of the automata” calls out a rule number (in Orth, on the recording), and the “cells” each pick a pitch before computing their way through a chordal landscape. The resulting chord is fixed until the next automaton is called, but the voicing of the chord changes as singers enter and exit. It is both an extremely simple approach and an interesting thing to which to listen. If the mental gymastics prove too hard, or if you want more control over tonality, the master can pre-assign notes; this is what we did to make the recording. Both approaches are fun to perform.

The most notable feature of cellular automata is their synchrony: like all parallel computing activities, the cells of an automaton must share some notion of synchronization (which is, in this case, the pulse). For musical activity, unlike computing, synchronization is an advantage rather than an impediment.  Performing together is the whole point, after all! Musical performance, like any process unfolding over time, can be thought of as computation. What is interesting about using CAs in this way is that the computation can be purely abstract. The process doesn’t need to produce any results besides the execution of the rule itself, which helps to explain why many researchers working with automata think of them as a way to model life itself.

Get ‘em while they’re hot! Remember, all profits from this project go to the Long Now Foundation.

The age of enlightenment brought with it the hope that all problems, be they physics or metaphysics, could be reduced to the logical manipulation of symbols. As we learned in The Baroque Cycle, Gottfried Leibniz was obsessed not only with metaphysics, with mining technology, with claiming precedence as the inventor of calculus, and with escorting globe-trotting princesses, but also with building a machine that could, by manipulating symbols, separate truth from falsehood. High hopes, such as his, for the power of symbolic representation thrived over the next 200 years as mathematicians, armed with the scientific method, picked apart the basics of arithmetic, algebra, rhetoric, and geometry, teasing out their underlying structures and essential concepts, and coincidentally spawning entire new disciplines such as group theory and mathematical logic. But in the early 20th century, the search for the everything-solver unraveled. Kurt Gödel started the ball rolling with his incompleteness theorem, and around the same time, David Hilbert posed his infamous Entscheidungsproblem (paraphrased by Roger Penrose as “is there some mechanical procedure for answering all mathematical problems?”) Two mathematicians, Alonzo Church and Alan Turing (who appears in Cryptonomicon), independently delivered the coup de grâce by answering Hilbert’s question in the negative, proving once again that there is no free lunch.

As an unintended but very useful side-effect, these guys effectively invented the modern computer.

As part of his proof, Alan Turing invented a “device” capable of executing a finite series of instructions in order to perform a calculation. This device, described using prose alone, was the specification for a very general model of computation now referred to as a Turing machine. A Turing machine has two parts, a “head” and a “tape.” The head can read and write symbols to spots on the tape, and can move along the tape left and right one spot at a time. The head embodies a table of instructions, or “states,” that it uses to perform its computation. The state of the head, plus the value of the symbol currently under the head, completely determine the next action taken, which is determined using a set of transition rules. The initial state of the head and the tape, plus the transition rules, define the computational process.

Because of their generality, it is possible to implement operational Turing machines in diverse ways. (In fact, producing bizarre Turing machine implementations has been a bit of a longtime hobby for computer science grad students with too much time on their hands.) So it should be no surprise that they can be built using musical materials. As an arbitrary example, imagine a solo singer and a long line of choristers. The solo singer plays the role of the Turing machine head, while the individual choristers represent individual storage locations on the tape. The choristers share a palette of distinguishable and repeatable musical events amongst themselves with which to represent symbols, and these symbols are read and written by the soloist as he or she moved left or right along the chorus line. The soloist, in addition to transmitting and receiving symbol values via song, is also free to express both the state of the head and the act of reading and writing as additional musical material, interwoven with and related to the choral sound being emitted by the “tape.”

At the Anathem launch event in San Francisco, I demonstrated exactly how this scenario might sound by leading a choral performance of a three symbol, two state machine that computed binary addition. On the tape, blank spots were indicated by silence, the digit zero was indicated by singing a repeated stacatto note pattern, and the digit one was indicated by singing a sustained tone. The pitches, dynamics, and vowels for these patterns were chosen at liberty by the singer representing the head, who sequentially taught each tape singer what they should be singing as he moved along the tape executing his algorithm. In this performance, the singer playing the “head” had control over changes in musical texture, since changes could not be instigated by the singers constituting the tape. The “head” had additional control: he could move at whatever speed he chose, and could select pitches and patterns based upon the currently sounding musical texture. The result was musically interesting, but the performing life of a member of the tape was pretty boring and unrewarding, since he or she did nothing but repeat motives passed on by the soloist.

A cardboard tetrahedral Turing machine head

To make life as a member of the tape more interesting, I then demonstrated a variation on this technique which reversed these roles and put the individual singers of the tape in control. Rather than embodying the head as a separate all-powerful singer, the “head” in this variation was a physical token that could be passed from singer to singer along the tape, and which was read and interpreted by the singers themselves. It is easier to show this by example than to explain it; look at the photograph to the left to see a token in the form of a tetrahedron, each face of which is a different color. The colors represent the states that the head can take, and the flow of logic within each state is represented by flowcharts printed on each face of the tetrahedron. Changes of state and movement of the head along the tape are meant to be accomplished by the singers as they change the exposed upper face of the tetrahedron (the “current state”) and pass it to their neighbors, following the instructions printed on the token.

In a token-passing performance of this sort, the musical interest is generated by the movement of the tetrahedron along the line of singers, and the decisions that they make while interacting with it. It is easy to adapt such a machine to two dimensions, which also adds depth. In this case the tetrahedron is not only passed left and right, but also forward and backwards. (Two dimension Turing machines such as this are often called Turmites - this link shows the plan for a tetrahedral head for a four state, two symbol, Turmite.) Although the performance is very sensitive to the algorithm used (members of the tape can still be stuck singing the same thing over and over again) the process of producing the music is slightly more interactive, and this technique definitely shows promise as a basis for collaboratively produced generative music.

Turing’s machine was one of the first in a long line of what those of us who study computers or discrete math call finite automata. In another post, I will examine some of the musical features of another form of automaton: the cellular automaton. This machine, coupled with music-specific algorithms, has already proved a fruitful source of structure for many composers, generative musicians, and instrument builders.

Kidnapping Water at the Kubota Gardens

Byron Au Yong’s Kidnapping Water: Bottled Operas is a shapeshifter of a piece. First, it was guerilla art, performed for audiences of one or two or a dozen people from within bodies of water in King County. Next, it was formally packaged, blocked, and turned into a ritual at Bumbershoot. In its current phase, it is an installation at the Jack Straw New Media Gallery, a collaboration between Byron and Randy Moss.

At its heart, the piece is 64 exquisitely crafted songs for solo singer and accompanying percussionist. There is also a percussion quartet. I had heard neither the quartet nor any of the songs written for the other singers, since we all performed on different days in different locations. So for me, the installation was a great chance to hear performances of the rest of the piece - very beautiful, and very restful to listen to in the gallery space with its watery circle of flickering LEDs.

The performances were recorded in the Jack Straw studio after Bumbershoot. Recording water percussion in a studio is an activity fraught with danger! Mics and fancy electronic recording gear don’t necessary mix well with flailing percussionists, tubs of water, and flying droplets. In the end, they were successful: the combination of water, gongs, metal scraps, wooden bowls, stones, bones, and the human voice form an intimate mix that flickers and ripples across many emotional states.

A picture taken by Kenneth Lu at the Anathem launch.

The Long Now Foundation hosted the launch event for Anathem on Tuesday evening in San Francisco, and as part of that event, I had the good fortune to present some of the Anathem math-music, live. All of the singers enjoyed having the opportunity to act as avout ambassadors to the event, and we appreciated the good will of the audience, which seemed to enjoy our performances despite having to endure some long delays and some pretty drastic problems with the sound system. I was impressed, in particular, with the spirited participation in our experiments at creating Turing machines and cellular automata directly out of the attendees themselves. Thanks to all who participated!

In Pursuit of Mysteries has a very nice write-up of the musical side of the event, complete with tasteful pictures of our stylish bolts (whose wrapping techniques were developed by Domini). The martial arts demonstration was also pretty fun, as shown in Kenneth Lu’s Flickr set, which captures some of the dangerous moves. Oh, and did I mention that Neal Stephenson, Danny Hillis, and Stewart Brand were in dialogue? They said some pretty interesting things as well!

I’ve received several inquiries about availability of the Anathem CD in the wake of this event. The copies that were available for purchase at the performing hall had just arrived from manufacturing on Tuesday, and copies from that batch are now available at the store at the Fort Mason pier. As soon as CD Baby has processed the CDs, they will also be available there and for electronic distribution. Meanwhile, you can also listen to most of the tracks as MP3 files, which are posted at the Anathem book site.

Despite the availability of the MP3 files, I still encourage you to buy the CD, since all of the profits from their sale go to the Long Now and also because the sound quality is better! The CD also contains a long and beautiful additional track for women’s voices, which is similar to the meditative duet that was performed Tuesday night.

Thanks to everyone at the Long Now Foundation who worked so hard to make this event possible.

Geometry is one of several mathematical disciplines that has proven fruitful during these experiments. Abstract musical space already contains common notions that relate directly to geometry: “line,” “point,” and “musical shape” are all commonly used terms for speaking about music. There are also less obvious, but still pertinent, concepts that map onto other important geometric entities, such as musical “events,” which map onto incidence, and various musical equivalence relations, which can map to congruence and similarity. The world of music has turned out to be a perfect fit for many forms of geometry: finite projective geometries, differential geometries, and non-euclidean geometries all relate well. I have experimented with each of these while attempting to fuse music, music history, math, and SF.

The most familiar of all geometries, however, plane old Euclidean geometry, has turned out to be a harder fit. And for reasons that will be obvious to those who have read Anathem, Euclidean geometry was important to me. An interpretation of the Adrakhonic Theorem (which on Earth goes by the moniker of “the Pythagorean Theorem”) seemed like required programming for inclusion on the IOLET CD. I tried a number of approaches to setting Euclid’s own proof to music (it is often called the “bride’s chair proof” for reasons that are lost to history), but none seemed entirely satisfactory. I even began setting of an entire book from his Elements, in order to derive a musical language.

In retrospect, the factor that held me back was the problem of accurately representing length, which is central to Euclidean geometry, in the musical system. Certainly music has many different notions of length or distance, rhythm being one, interval another, but I found that to make these useful when doing musical geometry, more resolution and range was necessary than the listening and/or performing brain could handle. Hearing simple units when expressed as rhythms was not a problem, but more detailed roots and ratios (which are definitely needed to work the Pythagorean Theorem) needed more resolution, and the notion of rhythmic length was not up to the  task. Interval seemed a better choice, but by using interval to show length I bolloxed up the rest of the mapping: the representation of shapes then became difficult. One of my goals for this project was to try for morphisms resulting in music that humans can relate to, both in terms of performance and in terms of listening, and I was not succeeding. I wanted a generalizable construct immediately recognizable as a simple geometric shapes, but I couldn’t seem to make this work.

Enter Lorenzo Mascheroni. Lorenzo Mascheroni was an eighteenth-century mathematician who rediscovered what Georg Mohr seems to have known 125 years earlier but not received credit for: that any Euclidean construction can be executed using a compass alone. (Mascheroni has a name that is not an English homonym - hence my choice of naming conventions to Mohr’s detriment.) Mascheroni constructions can be visually quite complicated, but their layers of circles upon circles often paint a graceful picture, and significantly for me, they also translate into music more pleasingly than do measured line segments. A Euclidean construction can be seen as points of incidence and lengths, rather than lines. (Remember elementary school now: “two points define a line,” etc, etc.)

The compass is an instrument of measurement. A circle is nothing more than the sweeping out of a uniform distance from a center. So to sing a circle, one needs a center and a distance. I chose to represent musical circles as symmetric scales or patterns that revolve around a central pitch, repeating themselves over and over. More importantly, one don’t need no stinking straightedges or lines or triangles in a world of circles and points! As a result, I was able to draw the points needed for the Bride’s Chair Proof by starting with a single point and a single circle. (I’ve embedded the diagram below in this post.)

Mascheroni Circles

Armed with the construction, I then prepared to turn it into music by doing an analysis of the centers, radii, and incidences involved. In this particular construction, there were 22 circles and 22 important points. Some of the points were shared by many circles, some not. Some of the points were meetings between circles, some acted only as centers, and some fulfilled both functions. I created a chart based on this information, and started fitting musical patterns to the elements of the chart. And lo, after a few iterations, I had musical elements that were very pleasing to my own ear! As a final nod to the avout, I then turned these musical elements into a game that might be played by fids learning the Adrakhonic Proof. In this game, the musical circles are provided on the page, along with the points within them that are important. Finding the path through them, however, is left as a cooperative exercise to the performers. (See the score for details.)

To my ears, the performance by Linda Strandberg, Melissa Plagemann, and Rebekah Gilmore on the CD is mesmerizing, and it demonstrates that purely mathematical structures can work well as the basis for music. Click on the arrow above to hear an excerpt from the beginning of this performance. The whole piece is fourteen minutes long, and can be found on IOLET::Music from the World of Anathem.

Because of the nature of the site, there was no power in the dome, and so rather than using our normal electronic setup, Perri and I decided to do a totally acoustic ambient piece, using passersby as our source of sound. Perri has a work-in-progress that involves “boxing the compass,” or reciting the points of the compass in order. Not only is this act full of symbol and meaning, but it is also a beautiful source of sound, since there are a small number of familiar words repeated in many combinations. We decided to use this as the sonic palette. As for the structure of the piece, I have been working with simple automata to generate music for the last several years, and so we decided to use an automaton similar to a Turing machine to generate the piece’s triggers.

We were both pleased with the result - I’ve put the score online so that folks can see how it worked, as well as a small excerpt from the performance which follows. (The sound quality is not great, but you can certainly hear the rain pelting down…)

Click on the arrow to listen to a few moments of this half-hour piece.

It is in this vein that I cannot resist linking to Steven Levy’s great profile of Neal Stephenson at Wired, in order to capture the short blurb that it contains for my music:

“to the untrained ear it sounds like the neo-Gregorian chanting that accompanies ritual baby sacrifice in horror films”

Adding this to Al Billings’ “weird shit” and Cory Doctorow’s “spooky” and I’m starting to think a horror film soundtrack should be my next project. I’ve always loved performing vocal special effects for horror movies and singing on their over-the-top scores, so why not?

Just for the record, there are plenty of rituals and many kinds of sacrifice in Anathem, but none of them involve babies. Not in that way, at least.