Sounds of Manifolds

Carolyn Gordon has a paper in the Mathematical Intelligencer (1989, vol 11, no. 3, pp. 39 - 47) entitled "When You Can't Hear the Shape of a Manifold." Accompanying this article are a sidebar and floppy vinyl record by Dennis DeTurck. Reproduced here, with Dr. DeTurck's permission, are the tracks from the record in .mp3 format; I've included his comments from the sidebar with each track. Two tech-tips: (1) you might need to turn the volume up (I forgot to amplify them when I was editing them) and (2) if you play more than one of these at once, it sounds really weird.

However, I'm not going to discuss the mathematics on this page.

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The "harmonic sequence" of the circle S1 (e.g., sine tones whose frequencies are integer multiples of the first one heard). Then we build a complex S1 -tone (e.g., beginning with the fundamental tone, more and more additional frequencies are mixed in, based on the circle's eigenvalues). Then the complex tone is repeated, with other characteristics added (attack and decay envelope) to get an "electronic piano" sound. As an example, the Dresden Amen is played on this "circular" piano.

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The analog of 1 for the two-sphere S2: the "harmonic sequence" of S2 (just like 1, but with the sphere's eigenvalues), then we build a complex S2 -tone, repeat it with the piano's attack-decay envelope, and perform the on the "spherical piano."

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The effect of dimension: A simple piano exercise is repeated, first on S1, then on S2, then on S3 (which has the same eigenvalues, but not multiplicities, as the complex projective plane C P2), then on S6 (same as the quaternionic projective plane H P2 ), and then on S12 (same as the Cayley plane Ca P2 ).

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The "Romanza" movement, from Beethoven's Sonatina in G is performed on various "spherical pianos." The first section is played on an S1-piano, the second on S2, the next on S3, the next on S6, and a codetta back on S1.

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A chorus of projective planes: The "Bouree" from Handel's Fourth Flute Sonata is performed with C P2 playing the flute part, Ca P2 doing the bass line, and H P2 providing the realization (middle parts).

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The piano exercise is played on S1 again, then on various flat rectangular tori. The first is the square torus T 1:1 , then on tori whose side ratios are 3:5, 11:13, 21:23, and finally 31:33. (Hear the effect of the ratio being close to but not equal to 1?)

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Torus music: Robert Schumann's Wild Rider is performed on a piano whose strings are T 3:5 .

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More torus music: J. S. Bach's Two-part Invention No. 4 in d minor is performed on a T 11:13 piano.

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Spheres and tori together: The "Presto" movement from Scarlatti's Sonata in C (Longo S.3) is performed, with the upper voice played on T 21:23 and the other parts on Ca P2.

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A "one-minute quiz": Chopin's Valse in D-flat,Opus 64, No. 1 ("Minute Waltz"). The oom-pahs are performed on a standard S1-piano. Can you identify the seven manifolds that play different sections of the tune?

In the Intelligencer, answers were provided on p. 79. However, I didn't look at that page, so if you need to know the answers... for now, you'll have to go to the library.

Tech thanks to Jon Fretheim for the embedded-player code and to Sean Kinlin & UNI Electronic Media for letting me use their sound-editing equipment.