#pianotheory

Music theory often feels like a secret language. Not because it’s complicated, but because of how it’s taught: abstract names, forbidden steps, correct progressions. It feels like you need a degree before you’re allowed to play. That’s where many people give up. Not because they lack musicality, but because the barrier feels too high.

This project began in 2019 while I was home sick, searching for a way to learn piano without drowning in sheet music I’d long forgotten. The answer wasn’t more rules. It was geometry.

The 12-Point Universe To navigate any landscape, you need a map. The piano has 88 keys, but its core is just 12 unique tones that constantly repeat. Every 12 semitones (one octave), the pattern starts again.

Think of this 12-point system as a clock. C is point 0, C# is point 1, D is point 2, and so on to B at point 11. Then the circle begins again at C. This turns infinite complexity into a finite, cyclical map.

All Cs on the piano belong to point 0 on the circle. All C#s belong to point 1. Every note has its fixed position in the cycle.

The Circle is DNA Playing a chord means simultaneously activating three points on the circle. Connect those three points and you see a triangle with three elements: The corners are the notes themselves (0, 4, 7) The sides show the intervals (4, 3, and 5 semitones) The shape as a whole is the chord’s architecture

Playing on the piano makes that architecture physical. The circle shows which notes belong together. Your hand builds that connection.

Not All Triangles Sound Equal Every chord with n notes creates an n-sided polygon in the circle. Three notes make a triangle, four make a quadrilateral, five make a pentagon. But geometry doesn’t decide whether something is a chord. It determines how it sounds.

Symmetrical shapes (augmented, diminished) float without a clear center. Balanced shapes (major, minor) feel stable and consonant. Irregular shapes (clusters, half-diminished) create tension and dissonance.

Rotation Becomes Transposition Here’s where it gets practical. When you rotate a triangle on the circle, you simply shift your hand left or right on the piano. Because the shape stays identical, your hand’s grip stays identical too.

C major exists at points 0, 4, 7. Rotate the entire triangle two steps right. Each point shifts by 2: point 0 becomes 2, point 4 becomes 6, point 7 becomes 9. This gives you D major (2, 6, 9). The sides remain 4, 3, 5. Only the position changed.

On the piano, your hand was on C-E-G. Now it’s on D-F#-A. You’ve shifted two keys to the right, but the distance between your fingers stays exactly the same. Thumb to middle finger is still four semitones, middle finger to pinky still three.

This means you only need to learn one shape for all 12 major chords. Once your fingers remember the 4-3-5 tension, you can place that shape anywhere on the keyboard.

Your Hand Feels the Geometry This geometry lives in your fingers, not just your head. Place your hand in C major (C-E-G) and feel the tension. Thumb to middle finger forms a wide grip, middle finger to pinky a narrower one. This physical tension translates the triangle’s sides into your hand.

Because the shape stays constant, you only need to learn one grip for all 12 major chords. Your fingers remember the distances. You only choose the starting point.

The Pivot Rule: Walking Through Harmony Instead of memorizing progressions, you navigate space using the pivot rule. Hold one or two notes from your current chord as anchors while moving the remaining voices just one or two semitones.

Music theorists call this parsimonious voice leading. You can visualize it using the Tonnetz, a 2D grid where harmonically close chords appear as neighboring triangles. C major and A minor sit side by side because they share two notes. Moving between them is like flipping a triangle over one of its shared edges.

This minimal motion technique powers much film music. Composers like Hans Zimmer and John Williams create massive emotional shifts that feel organic because individual voices barely move. The geometry does the work.

From Linear to Cyclical Without the circle, the piano is a linear row of keys with names attached. The circle makes harmony cyclical. You walk in loops, choosing different routes and returning where you began.

Built on Classical Foundations This approach rests on centuries-old foundations. Pitch class set theory (using numbers 0 through 11 for tones) is standard in conservatories. The circle of fifths organizes the same 12 tones by fifths instead of semitones. And Euler drew chords as triangles in his Tonnetz back in 1739.

What’s new is the translation into a directly playable piano system. You don’t learn the circle to memorize rules. You learn it to feel proximity and navigate harmony like a landscape.

Next in this series: recognizing chord shapes without memorizing their names.