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a curriculum for mathmusic

The Park School in Baltimore has a math department that has developed its own excellent mathematics curriculum. (And shares it freely!) At its core, the curriculum is based on fourteen habits of mind, enumerated in the screenshot accompanying this post (which I snarfed from the Quantum Progress blog). What a wonderfully concise, and yet comprehensive, expression of the core of mathematical abstraction and process.

14 habits of mindI can easily imagine an integrated music curriculum built atop these same principles, that would begin with the most basic symbolic alternatives for music (neumes, notes, numbers, and/or shapes), and which would continue through rhythm, melody and harmony, eventually leading to sections on species counterpoint, tunings, Bach-style chorale harmonization, pitch-class set analysis, orchestration, and/or other “advanced” topics. Speaking as a composer and theorist who suffered through traditional topics such as harmony and counterpoint, taught as isolated islands of boring pre-requisites by uninterested university profs, these subjects deserve to be presented as integrated whole! They are as intimately related to each other as are arithmetic, logic, geometry, trig, and algebra.

My real question: could you sneak music into the mainline examples of a math class such as that of the Park School? I believe that in the hands of a math teacher who was also a musician, the answer would be yes. Symbolic thinking could certainly be imaginatively illustrated using simplified notations of pitch or rhythm (the global staff-based notation of western notation versus the localized neighborhoods found in Byzantine scores, for example). Set theory and discrete math could be made very concrete by using sets of pitches, or positions in a rhythmic pattern. Logic could be clearly demonstrated using simple two voice counterpoint. And on, and on.

I’d even go as far as to speculate that concepts like the law of sines in trigonometry might be demonstrated and/or generalized using the concept of musical interval and triads. (Such a demonstration could also be used to introduce the salubrious notion of commensurability, via tuning.) After all, what is a musical interval or a time signature but a ratio? A ratio, which is the entity that forms the backbone of mathematical tools such as rational numbers, the notion of metric, cross-ratio invariance, homogenous coordinates and a zillion other useful gizmos.

Math and music. They really do have a lot in common. Thanks to the teachers at the Park School for reminding me.

new works!

It is a good month for brand-spanking new full-length works in Seattle. It started with the American premiere of Ivan Moody’s lush, beautiful, spiritual, and generally wonderful Seven Hymns for St. Sava, sung by Cappella Romana. After that, Byron au Yong provided wonderfully layered music cum field recordings cum spoken word for Spectrum Dance Theatre’s Farewell. This was very experimental music, and Byron pulled it off with aplomb. (Byron also found the time to curate a companion event as counterpoint, a wonderful  art exhibit on related themes also entitled Farewell. This Columbia City exhibit was the best group show that I’ve been to lately, without a doubt.) One weekend later, Garrett Fisher unleashed his latest raga-based opera/dance piece, this time with a Yeats libretto (Yeats, but still featuring haiku!). Finally, Frank Ferko’s 1999 Stabat Mater will soon be performed by Choral Arts Northwest. The Stabat was originally commissioned by His Majesties Clerkes, one of the groups from my Chicago past, and I look forward to performing in the PacNW premiere.

[postscript] The On The Boards’ presentation of Heiner Goebbels’ Songs of Wars I Have Seen is yet another event in this welcome local upwelling of large-scale new works. A riff on the “history repeats itself and is written by the victors/survivors” themes of its spoken Gertrude Stein text, the piece embeds fragments of early music by Matthew Locke into a sampled and processed electronic background, juxtaposing this interesting and unique texture against a second, more by-the-book, new-music chamber orchestra texture. The orchestra was an inspired amalgam of some of Seattle’s finest baroque players and some of Seattle’s finest new music players. The stage layout was by gender rather than by section, bringing the women to the front (both figuratively and literally), from whence they played and took turns reciting the text with its recollections of wartime living. The episodic piece was very effective, especially in its sparer movements. My only gripe would be size of orchestra. A few too many colors for my ears – the contrast of brass and percussion against the baroque strings was very effective, but the piano and harpsichord, and perhaps the woodwinds, seemed superfluous. That’s a small gripe; the piece was thought-provoking and enjoyable.

quaternion actions: the Amazon comes to Burien

The Burien/Interim Arts Space is an installation space by and for the current DIY/guerrilla generation. As far as I can tell, Kathy Justin and Dane Johnson, the project’s artist-instigators, sidled up to the city of Burien (directly next door to SeaTac airport) and said “hey, if you’re not using that empty city block, do you mind if we do?” They have successfully recruited sculptors to transform the wasteland into an urban sculpture park, with unapologetic emphasis on the grittiness of the site and the temporary nature of the installation, and have then hosted a series of live events in this open space.

“Pieces of Eight,” a B/IAS event that will occur on 15 and 16 August, highlights their DIY spirit: a sound installation that features 8 independent speaker stacks, driven by 10,000 watts of amplification. These formidable resources are being made freely available to local composers and performers – 18 at last count. There will be pre-recorded octophonic pieces played during the day, and on Saturday night a smaller number of artists will perform live.

The quaternion group

The quaternion group

Participating in this event was a foregone conclusion for me, since I love experimental public sound art and music. For the pre-recorded portion of the program, I have remixed Mascheroni Circles for eight channels, adding a low drone and some klang in the form of percussive metallic highlights to the voices of Linda, Melissa, and Rebekah. It sounds great in the studio – I can’t wait to hear it outdoors.

For the live performance, I have selected samples from Perri Lynch’s Amazon field recordings, which I will combine using Ableton Live into an 8 channel ambient mix according to the rules of the Quaternion group. (See the illustration, which shows this group’s multiplication table, which I lifted from the very useful open source software tool called Group Explorer.) The quaternion group is useful in this context since it has an order of 8, and its combination of non-abelian complexity and abelian subgroups make for interesting kaleidoscopic combinations of elements. The group action of these quaternions is to trigger samples; I begin by iterating through Cayley and cycle graphs for the group, and follow with algebraic manipulations that seem appropriate for the setting. Although this sounds as though it might be dry and lifeless, no one will know that there is abstract algebra involved! The aural experience is a slowly shifting juxtaposition of the intense sound of the Amazon rain forest set against the desolate urban performance setting of concrete, asphalt, and rusting metal.

As a side-note: quaternions, used as tools for rotational calculations, and the geometry behind Mascheroni Circles are both featured in Neal Stephenson’s Anathem, which is up for the Hugo award this weekend in Montreal. Good luck, Neal!

[Edit: No joy for Neal, but as suspected, the giant sculptures of rusting metal, bonfires, torn-up parking lot, power generators, hulked trucks and buses, and the overall desolate feel of the site made a great foil for electronic noise and loudly amplified insects. Below is a panoramic shot of the site.]

The Burien Interim Art Space

The Burien Interim Art Space

why I am jonesing for M4L

I’m excited about the ambient electronic constructions that I’m currently working on, which combine projective geometry with beautiful field recordings that my friend and collaborator Perri Lynch captured in the Amazon rain forest six weeks ago.

A finite projective plane with 31 points

The finite projective plane PG(2,5)

A finite projective plane with 31 points and 31 lines provides structure for the virtual space that I am creating for the piece. (A mandala-like visualization of this space that I drew using Inkscape should be visible on the left of this post.) Each “point” occupies the tip of one bump on the black ring, while each “line” is a different color. Each line passes through 6 points, and each point has 6 lines passing though it. (For the math geeks, this drawing is an expanded version of the difference set {1, 5, 11, 24, 25, 27}.) The mapping for the piece associates a distinctive sequence of recordings with every point in the space. The spectra and amplitudes of each of these sequences are diffused onto neighboring points using the incidence structure of the underlying geometry.

Projective spaces are famously dual: any theorem stated in terms of the incidence of lines and points can also be stated with the terms “point” and “line” swapped. To emphasize this dual nature as I map musical events onto the space, I am exploiting the two faces of the Fourier transform, a dual space that very familiar to electronic musicians. The Fourier transform ties time, amplitude, phase, and frequency together into a single tidy bundle, using wonderful math. I use it to cross-synthesize and deconstruct the panoramic wide-spectrum jungle landscapes recorded by Perri; the amplitude envelopes from the same sounds are simultaneously used as control signals.

I am currently rendering the piece using a somewhat laborious workflow that involves Ableton Live, Max/MSP, and Logic. I shuttle semi-processed sounds back and forth between the 3 programs using whatever method works: the filesystem, canned plugins, or even rewire when it cooperates. At the same time, the matrix math used for spatialization and signal transformation is done either manually or using a computer algebra package, and then applied manually to the mix.

OK, so what is this M4L thing and why does it relate to this activity? A couple of weeks ago, Ableton and Cycling ’74 (makers of Max/MSP) announced the imminent arrival of a new product named “Max for Live”. If this product functions as hyped, it will greatly improve quality-of-life for us often neglected experimental electronic music folk. For example, I would theoretically be able to automate much of the workflow for the new piece by using Live to trigger and resample sound sequences, while delegating their transformation and spatialization to my own homemade combination of Live racks, Max/MSP patches, and random plugins. I’m sure that sample cutting, final mix, and mastering would still be done in Logic, but the rest of the process, which is the bulk of the work, would become much easier. Cooler still, the “instruments” that I would build to perform this process could then be used over and over again, most notably as part of live expanded performances of this piece or others like it.

The march goes on. What took me months in 1982, laboriously creating soundfiles with a C compiler and hearing them rendered hours later in glorious 12 bit stereo by a dedicated PDP 11, has become a near-realtime programming activity using tools such as Max, SuperCollider and Chuck. To see high-quality mainstream software such as Ableton come to this party is just fantastic. I can’t wait!

found music for the new year

I enjoyed singing in the premiere of Richard Toensing’s lush new Christmas Kontakion in Portland last night. For the occasion, Cappella Romana morphed into a large double choir, with the numerous solo roles pulled from within the choir, much like a passion. It is a beautiful piece, and Cappella’s just-released recording is worth hearing.

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.

constructing a square with circles

Mascheroni Construction of the Unit Square

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:

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why cellular automata work for music

Several people have suggested that I post about cellular automata and music, since two of the pieces on the IOLET CD, Simple Automata and Sixteen-color Prime-generating Automaton, use one-dimensional cellular automata to provide their underlying structure. The subject has also been in the news with recent blog posts about using two dimensional automata for generative music-making. So consider this post to be partly liner notes for the Simple Automata (which can be streamed in its entirety here), and part speculative ramble by a composer who is also a computer guy. How and why would a composer use cellular automata for generative music-making?

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 view of Simple Automata

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.

IOLET::Music from the World of Anathem now available

The mysterious cogs of the distribution system have whirred and turned, and the CD for the Anathem music project is now available from CD Baby [] and from the Long Now Foundation. The Internet tubes are also being filled as I post this, and so the album should also be available in digital form through other online retailers soon.

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

more musical Turing machines

During the Anathem music project, I tried to imagine many different ways that music might be integrated into the daily lives of the avout, not only in the obvious celebratory liturgical uses, but also as a tool for learning, remembering, and computing. One manifestation of this, geometric games for exercising the learning mind, I’ve already posted about. In this post, I’ll cover another: how the avout might perform calculations using music. A warning and an apology in advance: this post contains a pretty high proportion of propeller-head material.

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

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.

installing bottled water

Kidnapping Water at the Kubota Gardens

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.