Harmonic Orbit Space: A Song's Tonal Center of Gravity

Intuition-first. For method details, see notes. Content generated via LLM collaboration - see authorship.

What the Animation Shows

The moving dot traces a piece of music through embedding space - a 2D projection of the 12-dimensional chroma vector onto the circle of fifths. Position shows which pitch classes dominate at each moment; motion shows how the harmony changes over time.

Center and radius are geometric properties of this projection. A point near the center means low directional bias in this projection. That can happen because many pitch classes are active, or because opposing pulls cancel each other out. A point near the edge means one region of the circle dominates. This is a statement about spectral focus, not directly about consonance or dissonance.

“Tonal center” vs “tonic”: the aggregate center of the orbit cloud points toward the region of the circle the piece spends the most time near. For Bach’s Prelude in C, that’s the C/G neighborhood - consistent with C major, and suggestive of the tonic. But this is a statistical correlation, not a proof. Pieces that modulate widely can have a center that doesn’t correspond to any single key. The two concepts are related but not identical.

Returning inward often feels like resolution or “coming home.” The ear recognizes the return to familiar spectral territory - but it’s a visual/structural reset, not direct evidence that you’ve arrived at the tonic.

Color Modes

The Color dropdown changes what the dot color encodes:

Time (default) - blue at the beginning of the piece, red at the end. Shows the chronological path through embedding space.

Tonal speed - how fast the orbit point is moving between consecutive frames. Green = slow (sustained harmony), red = fast (rapid harmonic change). Ragtime pieces light up red; sustained chords stay green.

Tonal curvature - how sharply the orbit path is bending. Green = straight-line motion (sequential modulation through related keys), red = sharp turns (abrupt harmonic pivots). High curvature often corresponds to deceptive cadences or chromatic substitutions.

Chroma entropy - how evenly the energy is distributed across all 12 pitch classes. Green = concentrated (one or two pitch classes dominate), red = diffuse (many pitch classes active simultaneously). Dense chords and chromatic passages show high entropy.

Tonal focus radius (debug) - distance from the orbit cloud’s geometric center. This measures how far the current moment is from the piece’s average position in the projection. It is a geometric property of the embedding, not a direct measure of consonance, dissonance, or distance from the tonic. It is also different from distance to the projection origin (0,0).

Building Blocks

Chroma: What the Audio Actually Contains

When you play a C on a piano, the string doesn’t just vibrate at one frequency - it produces a fundamental plus a stack of overtones (harmonics). But your ear collapses all of that into a single perception: “that’s a C.” A low C and a high C sound like the same note, just in different registers. This perceptual equivalence across octaves is the basis of pitch class: C3, C4, and C5 all belong to pitch class C.

There are exactly 12 pitch classes in Western music (C, C#, D, … B), and a chroma vector is a 12-element array that answers the question: how much of each pitch class is present in the audio right now?

To build one, you run the audio through a bank of filters - one per pitch class - that sum energy across all octaves. The result is 12 non-negative numbers. Think of each number as “how loud is this note name right now, regardless of which octave it’s in.”

A C major chord (C, E, G sounding together) produces a chroma vector where the C, E, and G bins are large and the other nine are near zero. A dense chromatic cluster lights up many bins at once. Silence gives you all zeros.

The word “chroma” comes from the Greek for color - the idea is that pitch class is to sound what hue is to light. Two C’s an octave apart are the same chroma, just as two reds at different brightnesses are the same hue.

The Circle of Fifths

The circle of fifths places the 12 pitch classes around a ring so that neighbors are a perfect fifth (7 semitones) apart: C -> G -> D -> A -> E -> B -> F# -> C# -> G# -> D# -> A# -> F -> back to C. Notes that sound harmonically natural together end up close to each other on this ring. C major (C, E, G) and its closely related chords (Am, F, G7) all cluster in the same region.

The Projection

Each pitch class gets a fixed unit vector pointing outward from the center of the circle at its assigned position on the ring. For pitch class i:

The factor of 7 comes from the circle of fifths - incrementing the index by 1 jumps 7 semitones (a perfect fifth), cycling through all 12 pitch classes.

The orbit point for a given moment is the weighted average of these unit vectors, weighted by the chroma values:

where $c_i \geq 0$ is the chroma energy of pitch class $i$.

This is the center of mass of the chroma distribution on the unit circle. If only C is sounding, the point is at the C position on the ring - radius 1, distance from origin. If C, G, and D are equally loud (closer to a suspended C/G sonority than Dsus2), the point is somewhere between them.

The point always lives inside the unit disk: $\|\mathbf{p}\| \leq 1$.

What the Pieces Reveal

A few patterns that emerge across the 17 pieces in the viewer:

Bach’s Prelude in C Major - tight cluster in the C/G/D/A region, consistent with diatonic C major harmony. Modulates briefly into F and Am territory, then returns.

Debussy’s Clair de Lune - much broader spread than Bach. Debussy’s impressionist style wanders through many harmonic regions. The orbit visits the Ab/Eb side of the circle, which corresponds to the mediant-heavy chord vocabulary he favored. More scatter, less predictable path.

Chopin’s Nocturne Op. 9 No. 2 - compact orbit with occasional excursions. The nocturne stays close to its home key area for most of its duration, with chromatic passing moments showing up as isolated points.

Scott Joplin’s pieces (Maple Leaf Rag, The Entertainer) - visibly more dynamic motion than the Romantic pieces, consistent with ragtime’s rapid harmonic rhythm. The cloud shows more “path-like” structure rather than a single tight cluster.

Bohemian Rhapsody - the most dramatic orbit of the set. The operatic section scatters points across the entire circle as the harmony shifts rapidly through distant keys. Compare with Imagine, which stays in a compact region throughout.

Billie Jean - tight, repetitive orbit reflecting the song’s minimal harmonic vocabulary (mostly two chords). The path traces the same loop over and over, consistent with the hypnotic groove.

Notes

Frame rate and smoothing: chroma is extracted at one frame per ~45 ms with no inter-frame smoothing. The orbit point can jump between frames; the scatter plot you see is the full unsmoothed trajectory.

100% fifths projection: the viewer uses a pure circle-of-fifths layout. An alternative approach blends in a major-thirds component (placing pitch classes by major thirds instead of fifths), but the viewer here uses fifths only.

Audio source: all 17 pieces were rendered from MIDI files using the SplendidGrandPiano sample library (piano covers), then processed through the same audio chroma pipeline used for YouTube-sourced audio.

References

• Melody as a Log-Derivative - companion post on why intervals are ratios and melody survives transposition.
• Musical Consonance from Frequency Interactions - how consonance emerges from harmonic spectra and auditory physiology.
• Krumhansl, C.L. (1990). Cognitive Foundations of Musical Pitch. Oxford University Press. - foundational work on tonal hierarchies and the circle of fifths as a perceptual space.
• Chew, E. (2014). Mathematical and Computational Modeling of Tonality. Springer. - formal treatment of pitch-class representations including the Spiral Array, a 3D extension of the circle-of-fifths projection used here.

Authorship

I didn’t write any of this content. The text, visualizations, and interactive viewer are the result of iterative collaboration with Claude (Anthropic). The orbit data was extracted from a custom audio analysis pipeline; the interactive canvas component was implemented by Claude Code.