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mohr mascheroni – math in the music of Anathem

Many people who have heard pieces from the Anathem music project might think that the music is simply a fiction that accompanies the book, and that the science-related titles are a fanciful nod to the plot. As the composer, I certainly hope that the music stands on its own in this way, but for the geeks among us, I also think that I ought to explain that there is another level to the music. Most of the pieces are direct attempts at mapping mathematical structures used or named in the book into music.

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 theĀ  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

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.)

[audio:circlesexcerpt.mp3|titles=Mascheroni Circles (excerpt)|artists=David Stutz]

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.