Tempic Integrations is a musical study that experiments with the use of multiple simultaneous tempi (the plural of tempo) in a musical composition. While Listening to the composition below:

  • Can you hear the parallel musical tempi as they accelerate and decelerate relative to each other?
  • Can you hear when the parallel musical tempi de-synchronize and re-synchronize?
  • How can we design these tempo changes so the layers synchronize at defined musical moments? (Hint: calculus)
  • How can we use this process musically?


Consider the factors that make a musical instrument expressive. The gold standard is the human voice. Can any other instrument be as expressive as the human voice? Probably not, though some may come close. Other instruments are still useful because they extend our capabilities.

Example of a simple polytempic accelerando

This project explores a particular way of extending our sonic palette. It uses integral calculus to unlock a class of previously inaccessible rhythmic patterns. Specifically it shows how simultaneous musical tempi can continuously accelerate and decelerate relative to each other while coming in and out of phase at defined musical points as shown in the image above.

The project had three phases:

  1. Design of a Mathematical algorithm for computing continuous tempo curves required for the polytempic accelerando shown in the image above.
  2. Implementation of Python routines and development of work flow for generating, auditioning, and editing the patterns in a Digital Audio Workstation.
  3. Composition of musical piece using algorithm implementation.

While listening to the composition, listen for the two melodic patterns:

  • Both patterns play the same melody, one octave apart
  • At the start of the piece both patterns play together at the same tempo, synchronized with each other, and with the kick drum.
  • Both patterns accelerate to 1.5 times the original tempo. However, they accelerate at slightly different rates, so they over the course of the piece the go out of phase with each other
  • At 0:57 the two parts re-synchronize with each other and with the drums.

For a detailed explanation of the background and mathematics (and a little bit of music history) read my blog post about Creating Tempic Integrations.