Geometry of Music

Composers often speak of fitting chords and melodies together, as though sounds were physical objects with geometric shape – and now a Princeton University musician has shown that advanced geometry actually does offer a tool for understanding musical structure.

In an attempt to answer age-old questions about how basic musical elements work together, Dmitri Tymoczko has journeyed far into the land of topology and non-Euclidean geometry, and has returned with a new – and comparatively simple – way of understanding how music is constructed. His findings have resulted in the first paper on music theory that the journal Science has printed in its 127-year history, and may provide an additional theoretical tool for composers searching for that elusive next chord.

“I’m not trying to tell people what style of music sounds good, or which composers to prefer,” said Tymoczko (pronounced tim-OSS-ko), a composer and music theorist who is an assistant professor of music at Princeton. “What I hope to do is provide a new way to represent the space of musical possibilities. If you like a particular chord, or group of notes, then I can show you how to find other, similar chords and link them together to form attractive melodies. These two principles – using attractive chords, and connecting their notes to form melodies – have been central to Western musical thought for almost a thousand years.” (…)

Tymoczko has released software (for Mac and PC) to create these visualizations yourself, as well as a number of demonstration videos.

The piece used in the demonstration above is the fourth of Chopin’s 24 Preludes (Op. 28). Here is a full recording, as well as a link to a rendition sung by the Swingle Singers which I’ve posted about before.

Expanding the sensory range of music is something I like to think about. A friend of mine once wanted a graphic representation of the circle of fifths as a tattoo, and eventually I came up with the following, though motivation for the tattoo had since passed. The center bit shows the progression of key signatures. It’s definitely a rough draft, but the idea is there:

Previously on CN: Touching Music // V-scratch // Whitney Music Box

Sketches of Frank Gehry

Director Sydney Pollack’s first feature length documentary on the acclaimed architect, Frank O. Gehry. The two men have been friends for many years, and Pollack completed the film over a period of five years, starting in 2000. Frank Gehry loves to sketch; it is the beginning of his architectural process.

Opened the 12th of May, though naturally, nowhere near where I live.

What’s In a Face?


The dark markings in the shark embryo pictured above indicate gene expression in the electro-sensory organs in the animal’s head. University of Florida researchers traced the origin of a shark’s electro-sensory powers to the same type of embryonic cell that gives rise to many head and facial features in humans.

Which means, the design of a human face, or any face for that matter, is rooted in the path of least resistance (or highest efficiency) as manifest by neural crest cells – the same cells that comprise the electricity-sensing organs in sharks that allow them to detect energy generated by prey.

Which means (assuming all vertibrates share a common ancestor), the vertibrate face is a tangible representation of millions of years of routing electric energy.

That said, check out this Wikipedia article on facial expression, including a picture from the 1862 book Mécansime de la Physionomie Humaineby Guillaume Duchenne, who used electric stimulation to determine which muscles were responsible for different expressions.

Organic Data Visualization

Treemaps are a method for visualizing information that completely and recursively subdivide a given area into cells, where each cell’s area corresponds to certain attributes in the data set. In other words, you can look at an otherwise large set of data, and instantly distinguish trends and patterns from the wealth of information in front of you.

Take Marcos Weskamp’s Newsmap, where news headlines are aggregated from Google News, organized by subject, displayed in tightly assembled boxes, and sized according to popularity for a quick and informative look at the world’s current events.

Building on that idea are Michael Balzer and Oliver Deussen from the University of Konstanz, Germany, who use an algorithm based on Voronoi tessellations, a method which allows for organizing data into more complex shapes like shown in the image above, producing a more intuitive and flexible treemap that closely resembles the patterns formed by soap bubbles or living cells.

Check out their PDF article for the nitty-gritty math behind this and a few more images.

Electric Sheep

Electric Sheep realizes the collective dream of sleeping computers from all over the internet. It’s a distributed screen-saver that harnesses idle computers into a render farm with the purpose of animating and evolving artificial life-forms.

You can keep your molting pine needles, short-circuiting lights, and expensive fragile ornaments – my Christmas tree holiday bush is a continuous display of cascading three-dimensional fractals and patterns that reproduce based on popularity, breeding an ever-growing royal family of finely tuned visuals – a tree in the purest form of the word.

Vote your favorites up or down to contribute to the gene pool, and download the latest children, or “sheep”, while the screensaver is running. I recommend leaving it on overnight a couple times to quickly build up your library.

The creater, Scott Draves, AKA “Spot”, is a San Francisco VJ and software artist. He does a lot of other interesting work in this field, some of which is displayed in videos and images on his website.


Adrian Ocneanu, professor of mathematics at Penn State, has designed a stainless steel sculpture depicting a 3-dimensional projection of a 4-dimensional “octacube”. The massive sculpture was fabricated by Penn State’s Engineering Services Shop.

Future Feeder < Physorg

I don’t personally subscribe to the idea of a strictly spatial 4th dimension – but who am I to say either way, much less criticize a brilliant piece of artistic engineering. Maybe once I get my PhD in mathematics I’ll try and cross swords with these guys – for now: