Sampled Harmonies: Algorithms for Additive Synthesis

Introduction

Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.

In this blog post we're going to look at two special cases of additive synthesis and implement efficient algorithms for them.

First we're going to look into modal synthesis with simple exponentially decaying partials and then advance into harmonic synthesis with freely controllable frequency. Instead of having to evaluate trigonometric functions for each of the partials, both of these techniques will only require a couple of multiplications and additions.

Modal Synthesis

With fixed frequencies and decay rates the modal synthesis model looks like this:

This approach can be used to create the sound of drums or idiophones such as the marimba or bells where there are relatively few partials that are related inharmonically.

Let's look at a single partial:

Using a few trigonometric identities we can manipulate the equation into the following recurrence:

Where f_s is the sampling rate. The constants α₁ and α₂ can be calculated once and the recurrence states that to get the next sample of the partial you simply add up two previously generated samples multiplied by the constants, a huge saving over having to evaluate the sin and exp functions for every sample.

Harmonic Synthesis

The modal synthesis above suffers from fixed frequencies and a general lack of control over the amplitude envelopes. If we restrict our attention to a harmonic series where each of the partial frequencies is an integer multiple of the base frequency we get the following equation:

Where the fundamental frequency f and each of the amplitude envelopes a_k depends on time t.
Lets look at a single instant of time:

Where we have absorbed the a_k and ϕ_k terms into a single complex term b_k. The end result is a simple sum of the powers of the complex exponential which can be evaluated with Horner's method:

where I have thrown in a timbral sharpness parameter s for free. The method requires only K-1 complex multiplications in total.

While one complex multiplication per harmonic isn't bad it still requires four real multiplications to be evaluated. We can do better by further reducing our attention by dropping out the ϕ_k terms:

This series can be evaluated with the Chebyshev recurrence:

And then applying the Clenshaw algorithm to this recurrence:

From this recurrence we can see that we only need one real multiplication per harmonic (assuming the 2 v term is computed only once).

What about the case of sines in the series?

This time we use the Chebyshev recurrence of the second kind:

Giving us the final Clenshaw recurrence:

Again there is only one real multiplication per harmonic, but now we had to evaluate both the sin and cos functions once.

Let's do one more. This time with only odd harmonics: