Polyhedron of triads

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Polyhedron of triads
On display at Erlebnisland Mathematik
Type Hands-on, exploration
Topics Music

Polyhedron of triads is a musical sculpture on display at Erlebnisland Mathematik.


The exhibit is a column-like polyhedron consisting of triangular faces, as in a stack of five octahedra. The exhibit is 1.5m tall, on a wooden base which is 50cm tall. The edges and vertices are made of metal, the surfaces are equilateral triangles with an edge length of 35cm, that are made of transparent plastic. If you touch a vertex, a musical tone plays. The name of the note is written onto the adjacent surfaces next to the corresponding vertex. The notes that belong to one surface always form a musical triad. The letters on the surface are small if it is a minor chord and capital if it is a major chord. There is a small sign that reads:

The structure displays the geometry of harmonic intervals in diatonic music. The vertices correspond to the tones, the faces to the triads. Touching a vertex will cause the corresponding tone to sound. You can discover major and minor chords and the circle of fifths.

Activities and user interaction

Interaction is by touching and hearing. There is no supervision for this or any exhibit in particular but there are mediators around in the exhibition who you can ask questions. Sometimes there are tours through the exhibition where visitors can explore the music-and-math exhibits, one of which is the polyhedron of triads. In this tour people can for example learn why triads and the twelve-tone system make sense mathematically.

Children love to play with this exhibit but it is also still interesting when one knows a lot about music theory. As many people make music in their free time this is a good way to connect math with people’s hobbies. The exhibit is very aesthetically pleasing.

Mathematical background

If we represent all minor and major triads by triangles, and connect their edges if they have two notes in common, we get a polyhedron which has the topology of a torus. We would get this polyhedron from the exhibit if we glued together the lowest and the upper row of triangles. In the geometry of the polyhedron we can discover all sorts of concepts from music theory: major chords belong to triangles that point upward, minor chords belong to triangles that point down, edges to the upper right belong to perfect fifths, edges to the upper left belong to minor thirds, horizontal edged to major thirds. Chords which are harmonically close, like parallel keys, are also close on the polyhedron. It is very interesting to analyze how typical progressions of chords used by composers look as paths on the polyhedron.


Conceived and designed by Bernhard Ganter.