Geometric Model by A. Harry Wheeler, Musical Polyhedron, after Moebius
 Description

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 thirtysix 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:

C E G# C, C# F A C#, D F# B D, D# G H D#

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:

C D# F# A C, C# E G B C#, D F G# H D

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 multioctave cycle:

C G D A E H F# C# G# D# B F C

Moebius and Wheeler sought to label the twelve vertices of the torus with notes of the chromatic scale in such a way that edges and triangles represent interesting musical relationships. Recall that two of the most common musical chords are the major triad (such as C E G) and the minor triad (such as C D# G). Any note of the chromatic scale can be the low note in a major triad or a minor triad, making a total of twentyfour triads, which are to be paired up with the twentyfour triangles of the model. The blue triangles of the model represent major triads and the white triangles represent minor triads.

In a major triad, the low and middle note are a major third apart and the middle and high note are a minor third apart, making the low and high note a perfect fifth apart. In a minor triad, the low and middle note are a minor third apart, and the middle and high note are a major third apart, again making the low and high note a perfect fifth apart. It follows that the thirtysix edges in the model need to be divided into three groups of twelve, one group representing a minor third, one group a major third, and the last group a perfect fifth. Each vertex should be incident to two edges of each type, and opposite edges should be of the same type.

We now discuss how the cycles of major thirds, minor thirds, and fifths discussed above are situated on the torus. For a topologist, one of the most significant features of a torus is that there are simple closed curves that cannot be shrunk to a point without leaving the torus. The four edge cycles representing major triads are of this type; they are commonly called meridians of the torus. (There are three edges in each cycle, but they do not bound a triangle on the torus.) The three cycles of minor thirds go the other way around the torus. The cycle of perfect fifths wraps itself around the torus in one continuous band that appears to form a trefoil knot in threespace.

Suppose the model is cut along the four meridians representing major triads (that is to say, cut into four parts at the corners). It is divided into four shapes, each with a six triangles around the edge in a zigzag pattern (an antiprism). Gluing a triangle onto the top and bottom of a set of triangles would produce an octahedron. Thus the model can be thought of as four octahedra glued together in a ring.

For patterns, see 1979.3002.060. For an undated English translation of the relevant pages from Moebius, see 1979.3002.110. Some patterns for this model are labeled in Wheeler’s hand and dated July 1939.

References:

A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.

August F. Moebius, Gesammelte Werke, vol. 2, ed. F. Klein, Leipzig: S. Hirzel, 1886, pp. 553–554.
 Location

Currently not on view
 date made

ca 1940
 maker

Wheeler, Albert Harry
 place made

United States: Massachusetts, Worcester
 Physical Description

plastic (overall material)

turquoise (overall color)

white (overall color)

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
 ID Number

MA.304723.405
 accession number

304723
 catalog number

304723.405
 Credit Line

Gift of Helen M. Wheeler
 subject

Mathematics
 See more items in

Medicine and Science: Mathematics

Science & Mathematics

Mathematical Association of America Objects
 Data Source

National Museum of American History