Geometric Model by A. Harry Wheeler, Moebius Polyhedron (Polyhedron of Musical Chords): Museum Object and Cultural History

The object in the museum catalogue

This plastic model is in roughly the shape of a torus. All the faces are triangles. Twelve are turquoise and twelve white, with the colors alternating. The surface has thirty-six edges and twelve vertices. This would give an Euler characteristic of vertices – edges + faces = 12 – 36 + 24 = 0, which is appropriate for a surface with one hole. Four of the white triangles are numbered. Face 1 also has a tag that reads: 739. Another tag on this side reads: A. Harry Wheeler. Another mark on this side reads: MP. Face 2 is a congruent white triangle on the lower left side, face 3 is a white triangle on the bottom of the back, and face 4 is a triangle on the bottom of the right side.

Wheeler called the surface a “polyhedron of musical chords,” following the German mathematician August F. Moebius, who described the surface in the second volume of his collected works. Wheeler made two other versions of the model, a paper version of the same size with museum number MA.304723.508 and a larger plastic version in yellow and white with museum number MA.304723.404. Musical notes are not indicated on this larger version of the model.

Wheeler’s model shows relationships between the twelve notes in a chromatic musical scale. In the Germanic system, going up by semitones, these are C, C#, D, D#, E, F, F#, G, G#, A , B, H (no flats are used). On a piano, C#, D#, F#, G#, and B would be black keys and the rest white.

If one raises pitches by a major third (four semitones) and keeps going until the original note returns (one octave higher), there are four cyclic sequences:

Each note of the chromatic scale appears in exactly one of these sequences.

Similarly, if one raises pitches by a minor third (three semitones), there are three cyclic sequences, each one note longer:

Again, each note of the chromatic scale appears in one sequence.

Since three and four are divisors of twelve, the sequences of major and minor thirds all take place within one octave. The third musical interval studied is the perfect fifth, consisting of seven semitones. Since seven and twelve are relatively prime, raising the pitch by a fifth produces one multi-octave cycle:

Materials, dates and provenance

• date made: ca 1940
• maker: Wheeler, Albert Harry
• Object Name: geometric model
• Physical Description: plastic (overall material)
• Physical Description: turquoise (overall color)
• Physical Description: white (overall color)
• Physical Description: cut and glued (overall production method/technique)
• Measurements: overall: 3.2 cm x 8 cm x 8 cm; 1 1/4 in x 3 5/32 in x 3 5/32 in
• place made: United States: Massachusetts, Worcester
• Subject: Mathematics
• ID Number: MA.304723.405
• accession number: 304723
• catalog number: 304723.405
• Credit Line: Gift of Helen M. Wheeler
• Data Source: National Museum of American History

Sources and reuse

Catalogue descriptions and facts adapted from National Museum of American History through the Smithsonian's official Open Access dataset. This record explicitly marks its metadata CC0 1.0; formatting and selection have been changed. The displayed image also has an explicit CC0 designation.