The Torus of Thirds is the three-dimensional realization of Planet 4D — the same harmonic model you will meet on Floor 4 of the museum, projected onto a torus so you can walk through it in three dimensions before climbing to its full hyperspherical form.

A torus is a doughnut-shaped surface. It is what you get when you take a flat sheet and join the top edge to the bottom, then join the left edge to the right. Two cyclic dimensions, two independent rotations, one continuous surface with no boundary.

The Tonnetz has exactly the right structure for this wrapping. It has two cyclic dimensions: the cycle of major thirds (which closes after three steps: C → E → G♯ → C) and the cycle of minor thirds (which closes after four steps: C → E♭ → F♯ → A → C). 3 × 4 = 12, which gives back all twelve pitch classes — and the Tonnetz wraps cleanly into a torus carrying every major and minor triad.

This visualization is associated with Guerino Mazzola's work in The Topos of Music (2002), which gave the Tonnetz a topological reading as a torus. The hexagonal-Tonnetz layout used here is inspired by Julian Hook's work. Theoretical influences include Thomas Noll's work on diatonic theory.

The torus in MmvM uses a warm color palette — red, orange, yellow — designed by Piran Design (Pyrene, age 10, daughter of Gilles Baroin). It is one of the most beautiful surfaces in the museum.

The visitor can fly through this scene. Move around the torus, through it, inside it. Each chord radiates spatially from its position on the surface.

Demo mode: Press the demo button and a song begins. The torus rotates so that the current chord of the song is always at the visitor's fixed position. This is the user spot mechanism — the visitor does not chase the music, the music comes to the visitor. The harmony unfolds at your stationary location.

A sky display above shows the current position on the torus from a bird's eye view, like looking down on a map.

Seven display modes sit on top of the model — the same seven modes available on the hypersphere in Scene 4. They live on the side panel as seven buttons (Notes · Chicken · Cohn · JQZ on the top row; Chords · CubeDance · Bigo below). A short info line in the GUI names the active mode and credits its originator. The next section walks through them.

In this room: start by watching the torus turn slowly. Then fly closer and let the chords pass through you. Press demo and let the song find its path around the donut.

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From plane to torus — the topology of the Tonnetz

The classical Tonnetz is a planar lattice generated by two intervals: the major third (4 semitones) and the minor third (3 semitones), or equivalently by the perfect fifth and major third. Because the chromatic universe is finite (Z₁₂), the lattice does not extend infinitely — every position recurs after 4 major-third steps in one direction and 3 minor-third steps in the other.

This periodicity wraps the plane into a torus. The result is a closed surface with two cyclic dimensions, perfectly natural for displaying the 24 major and minor triads of triadic harmony.

The torus topology is implicit in Hugo Riemann's harmonic theory and is made explicit in Guerino Mazzola's The Topos of Music (2002), where the Tonnetz is treated as a quotient of the chromatic plane by a periodic lattice — yielding a 2-torus. Theoretical material from Thomas Noll's work on diatonic theory and well-formed scales also informed the design of this scene. The hexagonal layout used on the torus surface is inspired by Julian Hook's Tonnetz imaging (Science, 2006; Exploring Musical Spaces, Oxford, 2023).

Relation to Planet 4D — chronology of the model

The MmvM Torus of Thirds is not a predecessor of Planet 4D but its three-dimensional realization. The four-dimensional hyperspherical model was developed first, as the object of the PhD thesis (Université de Toulouse, 2011). The torus version came afterwards, as a projection that places the same chord structure on a surface visitors can fly through in three dimensions — a stepping stone, in the museum's vertical organization, toward the full hypersphere on Floor 4.

The two scenes share the same code, the same chord nodes, the same display modes. They differ only in the surface onto which the structure is projected: torus here, hypersphere there.

The torus as harmonic space

On the Torus of Thirds:

• One cyclic dimension cycles through the major-third axis (the augmented triad family)
• The other cycles through the minor-third axis (the diminished seventh family)
• Each point on the surface is a chord
• Voice-leading distance corresponds (approximately) to geodesic distance on the torus

This makes the torus a natural home for parsimonious voice-leading analysis. Chord progressions trace curves on the surface; smooth voice leading corresponds to short paths.

The Piran Design color palette

The warm Red-Orange-Yellow color palette used in this scene was designed by Piran Design — the artistic identity of Pyrene, age 10, daughter of Gilles Baroin. The palette is intentionally distinct from the Newton Red-Green-Blue used elsewhere in the museum, marking the Torus of Thirds as a different kind of harmonic space.

Demo mode — the user spot mechanism

The demo song uses the user spot mechanism: the visitor remains stationary while the torus rotates. The current chord of the song is always at the visitor's position. This mechanism is used identically in:

• Scene 3 (Torus) — current scene
• Scene 4 (Planet 4D)
• Scene 10 (Fourier Phases 4D)
• Scene 12 (Entangled Hyperspheres)

The mechanism mirrors how a listener naturally experiences harmony: the chords come to you, not the other way around.

Sky display

A bird's-eye view of the torus appears on a screen above the scene, showing the current chord position from a top-down perspective. This complements the immersive close-up view at the visitor's position.

Display modes

The seven modes share the same model — the Planet 4D chord structure projected onto the torus — and reveal different layers of harmonic relationship on it. Most of them are lifts from the rich two-dimensional tradition of Tonnetz visualization: Hook, Douthett and Steinbach, Cohn, Jedrzejewski, Bigo. Each is brought up to the torus and the hypersphere in MmvM, preserving the music theory.

A short info line in the GUI names the active mode and credits its originator.

Notes

Planet-4D ideographic system — Gilles Baroin

The base layer. Twelve note-planets sit on the torus surface, each a unique pairing of one of four shapes (augmented-triad family) and one of four colors (diminished-seventh family). This is the ideographic system from the PhD thesis — the visual notation of the C₃ × C₄ structure. With this mode alone, the torus reads as a colored constellation of twelve symbols and nothing else: pure pitch-class space, no chords, no edges.

Chords

Planet-4D Chords — Gilles Baroin · Hamiltonian paths: Giovanni Albini

The 24 chord ideograms — 12 major triads and 12 minor triads — appear at their geometric positions on the torus. Major chords are light, minor chords dark; both carry the same shape-and-color encoding as the notes that form them. Giovanni Albini discovered Hamiltonian paths through this network — chord progressions that visit all 24 triads exactly once before returning home, with each transition changing exactly one voice; one of his Hamiltonian compositions is used as a demo here.

Chicken Wire

Chicken Wire — inspired by Julian Hook's hexagonal Tonnetz

The PLR (Parallel / Leading-tone / Relative) connections drawn as a wireframe skeleton on the torus surface. The hexagonal Tonnetz of Julian Hook — a clean, flat reading of triadic voice-leading — is here wrapped around the donut, so that every chord sits at a node and every parsimonious one-voice move is an edge.

Cube Dance

Cube Dance — Jack Douthett & Peter Steinbach · video work with Moreno Andreatta

The Cube Dance overlay highlights the parsimonious voice-leading graph of Jack Douthett and Peter Steinbach (Parsimonious Graphs, 1998), drawn this time on the torus: the 24 major and minor triads plus the 4 augmented triads, connected by every single-semitone voice change. The augmented triads — corners where all three Newton families meet — render in white, the visual mark of their tri-color sum. Moreno Andreatta — a long-standing friend and foundational collaborator at the start of his evolution in mathemusical work — composed The Gunner's Hamiltonian Dream, a Hamiltonian arrangement of Pink Floyd's The Gunner's Dream; the video animation shared with Scene 5 is the result of that joint work.

Hexatonic Regions

Cohn hexatonic system — Richard Cohn

The four hexatonic regions of Richard Cohn (Maximally Smooth Cycles, 1996) drawn onto the torus. Each region is a six-chord cycle generated by alternating parallel and leading-tone transformations, sweeping through a tightly related family of major and minor triads. On the torus the four hexatonic regions partition the 24 triads into four colored bands, making Cohn's classification visible at a glance.

JQZ

JQZ involutions — Franck Jedrzejewski (2019)

A new mode, mutually exclusive with the PLR / Chicken Wire view. Franck Jedrzejewski (“Non-Contextual JQZ Transformations,” MCM 2019) introduced three pointwise pitch-class inversions — J = I₇, Q = I₁₁, Z = I₄ — each an involution: applying it twice returns the original chord. The three involutions generate a dihedral group D₂₄, isomorphic to the PLR group via the mapping J↔P, Q↔L, Z↔R. Unlike PLR, which acts only on major and minor triads, JQZ is non-contextual and acts on any n-chord. The asymmetry that distinguishes JQZ from PLR is its anti-commutation with transposition: J·Tₙ = T₋ₙ·J. The PLR group commutes with transposition; JQZ does not, and that single algebraic difference gives the mode its distinct musical character.

The visual signature is striking. On entering JQZ mode, the twelve note-planets go fully black — the anti-color sign that each note has become “the note the surrounding chords all avoid.” This is the dual of Scene 5's Cube Dance, where the augmented triads render in white as the sum of all three Newton colors; here, the inverted note is the complement. The P, L, R buttons on the side panel rebind to J, Q, Z, and the chord positions reflect to put each chord's three J/Q/Z partners at its three nearest hexagon-neighbor slots.

Generalized Tonnetze

Louis Bigo · Hypersphere of Tonnetze (Baroin × Bigo)

This is the mode most naturally at home on the torus, because the generalized Tonnetze were originally formulated on the 2D torus by Louis Bigo (Bigo, Giavitto, Spicher 2011; Bigo, Andreatta, Giavitto, Michel, Spicher 2013; Bigo & Andreatta 2016). Six chromatic-step Tonnetze, indexed by the step size n: T₁ (semitone), T₂ₐ and T₂ᵦ (the two whole-tone scales — Messiaen's two modes of limited transposition), T₃ (minor-third / diminished-seventh cycle), T₄ (major-third / augmented-triad cycle), T₅ (perfect-fourth cycle). Each Tₙ draws a 12-edge chain around the torus; the visitor can show or hide each chain independently, plus two extra toggles for the note-planets and connecting lines. Default state on entry: T₃ + T₄ + notes ON — the classical Tonnetz layout, augmented triangle plus diminished-seventh cross.

This mode is the direct interactive descendant of the older Hypersphere of Tonnetze — a joint work with Louis Bigo, visible as a film in the Geode (Planet 4D family member #5) — bringing the same generalized Tonnetz construction into a VR space the visitor can fly through.

Further reading

• Mazzola, G. (2002). The Topos of Music. Birkhäuser.
• Cohn, R. (1996). Maximally Smooth Cycles. Music Analysis, 15(1).
• Hook, J. (2006). Exploring musical space. Science, 313(5783), 49.
• Hook, J. (2023). Exploring Musical Spaces. Oxford University Press.
• Douthett, J., & Steinbach, P. (1998). Parsimonious Graphs. Journal of Music Theory, 42(2), 241–263.
• Jedrzejewski, F. (2019). Non-Contextual JQZ Transformations. MCM 2019, LNCS 11502, pp. 149–160. Springer.
• Bigo, L., Giavitto, J.-L., Spicher, A. (2011). Building topological spaces for musical objects. MCM 2011, LNAI 6726, pp. 13–28.
• Bigo, L., Andreatta, M., Giavitto, J.-L., Michel, O., Spicher, A. (2013). Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes. MCM 2013, LNCS 7937, pp. 38–51.
• Bigo, L., & Andreatta, M. (2016). Topological Structures in Computer-Aided Music Analysis. In Meredith (ed.), Computational Music Analysis. Springer.
• Baroin, G. (2011). PhD thesis, Université de Toulouse.

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