Triad polyhedron

In 19th century music theory, the relationships of tones and triads were studied using two-dimensional representations. The following tone relationship table can be found in Arthur von Oettingen’s treatise “Harmoniesystem in dualer Entwicklung” from 1866. Tones standing next to each other are fifth-related and tones standing on top of each other are grand-terz-related. Octave relatedness is neglected in this two-dimensional representation and would be thought of as a third dimension if needed.

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Figure 1: Table according to Arthur von Oettingen

The representation by Hugo Riemann (1914) is geometrically a shear of the above. Here the Großterz axis points to the upper right and the Kleinterz axis to the upper left.

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Figure 2: Illustration according to Hugo Riemann

Triads are made up of three notes, and can therefore be visualized as triangles. Major triads, such as f-a-c, c-e-g, or g-h-d, form “standing” triangles, and minor triads, such as d-f-a, a-c-e, or e-g-h, form “hanging” triangles. In the following figures, major triads are shown in red and minor triads are shown in blue.

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Figure 3: Major and minor triads as triangles

The three notes of a triad have different meanings in relation to it. If one chooses any tone, this tone has the meaning of the fundamental in a major triad, which is also named after this tone. But the same tone also has the meaning of the third tone in another major triad, as well as that of the fifth tone in yet another major triad. Similarly, it plays these three roles in corresponding minor triads. The figure below shows those 6 triads in which the note C occurs in the mentioned meanings. In the corresponding node of the graphic, six triangles meet, the centers of which form the basic cell for a hexagonal honeycomb pattern.

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Figure 4: Rolls of a tone in the different triads

The intervals of the fifth and the major third span a plane that extends infinitely and that is completely parceled out by major and minor triads (see the following figure 5). If one wanted to describe the combinatorics of tonal relationships with such a model, then musically this would mean that the fifth and third relationships would be seen as independent forms of tonal relationship.

Combinatorial freedom can be restricted in several ways. In musical notation, for example, fifths and thirds are not independent. Rather, four fifths correspond to a major third (and two octaves). Tones and triads of the same name are then identified with each other.

A further restriction of the repertoire of tones and triads results when triple major thirds are identified with octaves. This is also called enharmonic identification. In the triadic polyhedron, which is constructed as a tower of five floors, each of the six horizontal triangles, which form the floors or ceilings of the floors, corresponds to such a cycle of three major thirds. The concrete pitches sounding there, however, do not result in a cycle of ascending thirds. An interval described here as a major third sounds concretely as a major sixth. Also, the triadic triangles sound in various inversions, which will not be discussed further here.

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Figure 5: The plane parquetized with triads

With a strict limitation to 12 tones and 12 major and 12 minor triads each, a torus is geometrically created, which is covered by 24 triangles. The triad polyhedron with its 30 triangles is a tower whose top floor is enharmonic and octave-equivalent to the bottom floor. It is not closed to a ring.

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Figure 6: Torus of twelve tones

Apart from the combinatorial overview that can be obtained with the help of these representations, the question arises as to their usefulness for musical analysis. Are there typical triadic sequences in pieces of music to which, from a geometrical point of view, special paths on the triadic polyhedron correspond?

The question here is for particularly “economical” triad sequences, in which only one note in a voice is changed in each step and in which a new triad is nevertheless created. For this purpose, there are always three possibilities from each triad:

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Figure 7: Three possibilities after a triad

They are called changes of fifths (e.g. C major — C minor), changes of thirds (e.g. C major — A minor) and changes of leading tones (e.g. C major — E minor). It is an amazing property of major and minor triads that these triadic connections correspond with the smallest pitch changes (down to octaves).

Among the many possible triadic paths that can be obtained from the succession of fifth changes, third changes, and leading tone changes, those in which only two of these connection types alternate with each other are of particular interest. These correspond to three different directions on the triad polyhedron. All three appear sporadically in the music of the 19th century.

1. change of thirds and leading tone

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Figure 8: Changing thirds and leading tones

This triadic progression reaches all 24 (enharmonically identified) major and minor triads. In the 2nd movement of the 9th Symphony by Ludwig van Beethoven, there is a passage (measures 143–176) in which 19 triads are actually run through.

2. Fifth change and leading tone change

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Figure 9: Changing fifths and leading tones

This triadic progression reaches exactly 6 of the 24 enharmonically identified major and minor triads. These also consist of 6 tones in total, which is why one also speaks of the hexatonic cycle here. The piano piece Consolations 3 by Franz Liszt runs through (with the exception of one chord) the hexatonic D-flat major cycle. Here, the triads represent tonal regions touched upon in this piece: D flat major, F minor, F major, A minor, A major, (D flat minor is skipped), D flat major.

3. Fifth change and third change

This triadic progression reaches exactly 8 of the 24 enharmonically identified major and minor triads. These also consist of a total of 8 tones, which is why one also speaks here of the octatonic cycle. The overture to Rosamunde (Andante bars 1–48) Op. 26 / No. 1 by Franz Schubert runs through such a cycle: C minor, E-flat major, E-flat minor, G-flat major, F-sharp minor, A major, A minor, C major. Again, the 8 triads represent tonal regions touched upon in this piece.

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Figure 10: Changing fifths and thirds

Figure 11 below shows the progression of all three triadic progressions on the triadic polyhedron.

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Figure 11: Triad progressions on the triad polyhedron