Mazes, Traveling Salesmen, and the Aesthetics of Spontaneity: Stephen Goss, Labyrinth

Labyrinth, by British composer Stephen Goss, was commissioned by the Guitar Foundation of America for their 2016 International Concert Artist Competition, where it received some outstanding performances, particularly by competition winner Xavier Jara and runner-up Andrea De Vitis. Dedicated to the memory of Italian novelist and philosopher Umberto Eco, Labyrinth takes inspiration from Eco’s book From the Tree to the Labyrinth. Eco contrasts the images of the dictionary or “tree of knowledge,” which organizes the world into a finite, closed loop of connections that can be exhaustively known, and the encyclopedia or “labyrinth” of knowledge, which allows for practically infinite ways of connecting the dots to find order in the world.

Goss sees this distinction as parallel to the difference between a classical labyrinth (traditionally found on the floor of a cathedral), consisting entirely of “a single path from the mouth to the goal,” and a maze, “which gives the traveler choices for the route, some of which lead to dead ends.” An even better metaphor, he suggests, might be a network of points “in which every point can be connected to any other point.” This suggests the classic computational problem of the traveling salesman, who seeks to find the shortest route that visits each of a certain list of cities.

France, Eure et Loir, Chartres, Notre Dame de Chartres Cathedral listed as World Heritage by UNESCO, Labyrinth of the Cathedral

Labyrinth, Chartres Cathedral, France

Thus, Labyrinth is constructed in a series of 13 short fragments, played without pause. These may be literal quotations, clever reworkings, or outright forgeries of works from various periods of music history. Goss specifies which fragments should be played first and last, but instructs the performer to play the remaining fragments in a different order every time, without omitting any. This instance of mobile form (that is, the piece has no determinate order of sections) presents two related challenges to the performer: one practical and one aesthetic.

The practical problem is one of memory. For many musicians, playing confidently from memory is one of the greatest hurdles they have overcome in their training, and they feel a certain pride in being able to play an entire piece, start to finish, without the score. The problem with a work like Labyrinth is that it must not be played start to finish. Nor, if you take Goss’s instructions literally, can it be played in any of the orders one has previously used. No need to worry about running out of possible orders, though: there are 11!, or approximately 40 million, possible permutations!

Instead, the challenge in playing the fragments in a non-linear fashion is to remember not only which fragments one has already played in that performance but also which orders one has followed in previous performances. When Labyrinth was the set piece for the GFA International Concert Artist Competition, the judges employed someone to mark each contestant’s ordering of the sections to ensure no one played in the same way in different rounds of the competition. Some of the contestants I spoke to solved this problem by dividing the fragments into three or four chunks of fragments which always occurred consecutively, so that they only had to remember which chunks they had already played.

In my opinion, this solution follows the letter but not the spirit of Goss’s instructions. One of the beauties of mobile form is its potential for improvisation, the ability to choose in the moment what you will play next. Therefore, as I practiced this piece, I decided to also practice making up the order on the fly. To be sure, this comes at the cost of a less-than-perfectly-smooth transition now and then, but I think the marvelous freedom it affords is worth the price. I think of it like a maze, where each junction requires a choice: sometimes the choice can be made decisively, but other times there will be a slight hesitation, and that’s okay.

Whether the performer plans the order of sections in advance or decides it spontaneously, they will then have to confront the aesthetic problem of performing Labyrinth: how to decide which sections should follow one another. One approach would be to do so randomly or by some statistical process: one could roll dice to determine the order, play every other (or every third, etc.) segment all the way through, follow reverse alphabetical order of the fictional composers’ last names, and so on. However, such a mechanical approach seems lacking in artistry. For me, some segments will naturally follow each order in a more aesthetically pleasing way, especially if they are tonally related or have pitches in common.

In other words, the performer of Labyrinth is obliged to execute a sort of real-world traveling-salesman problem, determining which fragments lie “closest” together in musical space and constructing a route that connects them all in an efficient—and elegant—manner. This process could very truly be described as playful, even competitive. I remember visiting the Human Maze in Winter Park, Colorado as a child and trying to find my way to the stations with the letters M, A, Z, and E to stamp my passport and get out of the maze. At the age of four, I was only able to complete this quest after several trips up the lookout tower in the middle of the maze and a good deal of help from my father. Meanwhile, older children and adults would dash past me, trying to complete the maze in record time. (Two decades later, I read that the creator of that maze started a successful business franchising his setup to other amusement parks and resorts throughout the country.)

Winter Park Amaze'n Maze

Human maze, Winter Park, Colorado

However, you cannot simply calculate the ideal ordering of fragments in Labyrinth in advance and then play that. More specifically, you can never prove that you have found the ideal order, according to your aesthetic tastes or anyone else’s. Remember, once the number of points gets large enough, the actual traveling-salesman problem can only be solved by a computer large enough to pass for a minor planet. Various algorithms can generate reasonably good paths, but it is extremely difficult to prove that no shorter path exists. In particular, so-called “greedy” algorithms, which operate by simply traveling to the nearest unvisited point to one’s current location, tend to fare poorly in finding an optimal solution, since they run the risk of having the last two or three unvisited points be widely separated from each other. Similarly, it would be easy in Labyrinth to get stuck with only one or two segments left to play which didn’t pair well with each or with the ending.


Visualization of the traveling salesman problem

My personal solution to performing Labyrinth was to envision each fragment as a point in a network, with paths connecting each pair of points, but with some paths weighted more heavily than others according to their efficiency. (In computational theory, this is apparently known as an “ant-colony” algorithm.) That is, I will tend to choose certain transitions more often which I think flow more smoothly. These weights may be stronger for certain pairs of segments than others, but in every case, but I will allow myself to choose even “awkward” transitions a small percentage of the time. For example, I particularly liked following up the “Scherzo” segment with “Contrapunctus,” both because of their common pitch B-flat and because of the musical humor of following a quote from a witty scherzo by Beethoven with one from J. S. Bach’s incredibly serious Art of Fugue. I thought of making a rule that these two segments would always appear next to each other in my performances of Labyrinth, but the few times I allowed myself to break this rule, I always seemed to come up with an especially intriguing order! One of my solutions is recorded at this link.

Chartres Labyrinth Outdoor

Garden labyrinth behind Chartres Cathedral, France

Classical performers in the Western tradition are trained to learn pieces of music linearly and by exact repetition. Everything is scripted and rehearsed, thought out beforehand; non-linear, spontaneous, or improvisatory performance feels strange and uncomfortable. Sometimes I wonder if this dynamic carries over into my life outside of music as well; I often find myself rehearsing what I will say in various social situations beforehand so I can put on a polished show when the time comes. Learning and performing Labyrinth has led me to explore an alternative mode of performing—even a mode of being—which I have found incredibly freeing and stretching. Having a flexible repertoire of responses which I can draw from at will, leaning more on tried-and-true favorites if necessary but mixing in new ventures as well, feels empowering, both in music and in life.


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