Note Quantization

Say a piece begins at one tempo, τ₁ and then changes to τ₂ instantaneously.

We'll model this as two sections, each with it's own equation:

t_1i = b_1iτ₁ + T₁

t_2j = b_2jτ₂ + T₂

Here t_1i means the time-offset of the i-th note in section one. b_1i maps notes in section one to beat numbers. T₁ tells us the time offset of the start of the first full bar in the first section.

And the same for the second section, replacing 1 with 2. In the above, I've also used j instead of i to make more explicit that it ranges over a different set of numbers (although I will not always do that).

Note that T₂ is basically the total length of the first section plus the pause between the sections if any.