How Metric Modulation Shifts the Beat Using Math

Let me walk you through two connected rhythmic ideas — and by the end you will be able to calculate an exact new tempo from the math alone. A polyrhythm happens when two layers of music divide the same time span into different equal parts simultaneously. The simplest example is 3-against-2 (written 3:2). Picture one measure of 6 eighth notes. One hand groups them as three pairs — ONE-two ONE-two ONE-two — producing three evenly spaced beats. The other hand groups the same 6 notes as two triplets — ONE-two-three ONE-two-three — producing two evenly spaced beats. Both hands start together and end together, but the middle is in constant tension. Polyrhythm is the foundation for the bigger idea: metric modulation. Metric modulation (systematized by composer Elliott Carter, who called it tempo modulation) is a technique where a note value already present in the music is reinterpreted as the new beat unit. Because that note value exists in both sections, the tempo shift is mathematically exact — no rushing, no pausing, just a change in how you count. The formula is straightforward: new tempo = old tempo × (how many pivot notes fit inside one old beat). Here is a worked example. You are at quarter-note = 120 bpm, so one beat lasts 0.5 seconds. Your music contains triplet quarter notes. Three triplet quarter notes fill exactly two old beats (one dotted half note), so each triplet quarter lasts 2/3 of a beat = 0.5 × (2/3) ≈ 0.333 seconds. How many of these fit inside one old beat? Divide: 0.5 ÷ 0.333 = 3/2 = 1.5 pivot notes per old beat. Apply the formula: 120 × (3/2) = 180 bpm. The music moves from 120 to 180 with no interruption — the shift is baked into the notation. Now try a triplet eighth note pivot at the same starting tempo. Three triplet eighth notes fill one old beat, so each lasts 0.5 ÷ 3 ≈ 0.167 seconds. Three pivot notes fit in one beat. Formula: 120 × 3 = 360 bpm. That is extremely fast — which is exactly why composers choose their pivot note carefully before committing to a modulation. Stuck on the formula? Here is a fallback that always works: (1) write the duration of one old beat in seconds, (2) write the duration of the pivot note in seconds, (3) divide old beat ÷ pivot note to get the multiplier, (4) multiply the old tempo by that multiplier. This step-by-step route gives you the same answer without needing to memorize the ratio shortcut.