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warp-math · the integral, made visible

Two charts wired to 01-the-math. The left chart shows 60/BPM(b) — seconds per beat. The shaded area under that curve from 0 to β is the elapsed-seconds value on the right chart at the same β. That equality is the integral t(β) = ∫₀^β 60/BPM(b) db.

tempo regime constant piecewise-constant linear ramp curved (ease)

constant BPM BPM 120

readout at β = 6.00 beats

beatsToSeconds(β) = — s shaded area under 60/BPM from 0 to β = — s residual: — s

60/BPM(b) — seconds per beat. The shaded area from 0 to β is the elapsed seconds on the right.

the warp map t(β). In piecewise mode, drag the coloured dots to pin a beat to a new second.