Daily Archives: March 2, 2012

Epicycles & Visual Music

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I spent some time this evening looking again at the motion of particles moving along epicyclic trajectories. The motion of a single particle q in the complex plane, z, is given by the following parametric function of t:

z_{q}(t) = Re^{i\omega_{q} t} - re^{i\Omega_{q} t} \;\;\; with \; \Omega_{q} > \omega_{q}, \;\; R > r

The animation above is generated using 50 particles moving according to the same parametric equation, with angular velocities given as integer multiples of a fundamental value:

\omega_{q} = q \omega_{1}, \;\;\; \Omega_{q} = q \Omega_{1}, \;\;\; q = 1, 2, 3 \cdots

With the velocities distributed in this way, the angular positions attain various kinds of harmonic relations, whence the arrangement of the particles in the complex plane falls into simple symmetric patterns. The animation bears some resemblance to the visual music of James and John Whitney. In his later work John Whitney made use of digital computer programs to create visual music animations (Whitney, 1981), using particle systems related to the one described here.

[1] John Whitney, Digital Harmony: On the Complementarity of Music and Visual Art , New York, McGraw-Hill, Inc., (1981).

As Slow As Possible

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A performance of the slowest and longest musical piece in the world is underway in the church of St. Burchardi in Halberstadt, Germany, pictured above. The piece, ASLSP, was written for piano by John Cage in 1985, then re-written for organ in 1987. Cage recommended that the piece be played “as slow as possible”, hence the title ASLSP, but did not specify its exact duration. In the year 2000, a performance of the ASLSP was started using an automatic organ. The duration of the the performance is set to last 639 years. Why Halberstadt and why 639 years? From the website of the project:

Michael Praetorius, a composer of the late 16th and early 17th centuries, wrote that an organ with the first modern keyboard arrangement had been built in Halberstadt’s cathedral in 1361. This organ was the first one with a claviature of 12 notes and this claviature is used on our keyboard instruments today. So one can say that the cradle of modern music was in Halberstadt. Subtract 1361 from the millennial year 2000, and the result is 639.

The performance is streamed over the web and can be listened to at the link: ASLSP. The project description contains the following lovely thought:

In view of our fast moving age, this piece of music is a way of trying to slow down our hectic lives. The “discovery of slowness” and the planting of a “musical apple tree” can be understood as symbols of confidence in the future.

I learned about this project via Prof. Clark Lunberry, who guest lectured in the seminar last year.

n.b. The images above reside on and are linked from the website of the ASLSP project.