Using Perlin noise in sound synthesis Artem POPOV Gorno-Altaysk, Russian Federation, [email protected] Abstract This paper is focused on synthesizing single- Perlin noise is a well known algorithm in computer cycle waveforms with Perlin noise and its suc- graphics and one of the first algorithms for gener- cessor, Simplex noise. An overview of both algo- ating procedural textures. It has been very widely rithms is given followed by a description of frac- used in movies, games, demos, and landscape gen- tional Brownian motion and several techniques erators, but despite its popularity it has been sel- for adding variations to noise-based waveforms. dom used for creative purposes in the fields outside Finally, the paper describes an implementation computer graphics. This paper discusses using Per- of a synthesizer plugin using Perlin noise to cre- lin noise and fractional Brownian motion for sound ate musically useful timbres. synthesis applications. Keywords 2 Perlin noise Perlin noise, Simplex noise, fractional Brownian mo- Perlin noise is a gradient noise that is built tion, sound synthesis from a set of pseudo-random gradient vectors of unit length evenly distributed in N-dimensional 1 Introduction space. Noise value in a given point is calculated by computing the dot products of the surround- Perlin noise, first described by Ken Perlin in his ing vectors with corresponding distance vectors ACM SIGGRAPH Computer Graphics article to the given point and interpolating between \An image Synthesizer" [Perlin, 1985] has been them using a smoothing function. traditionally used for many applications in com- Sound is a one-dimensional signal, and for puter graphics. The two-dimensional version of the purpose of sound synthesis Perlin noise of Perlin noise is still widely used to generate tex- higher dimensions is not so interesting. While tures resembling clouds, wood, and marble as it is possible to scan Perlin noise in 2D or 3D well as procedural height maps. space to get a 1-dimensional waveform, it's nec- essary to make sure the waveform can be seam- lessly looped to produce a musically useful tim- bre with zero DC offset. For one-dimensional Perlin noise, the noise value is interpolated between two values, namely the values that would have been the result if the closest linear slopes from the left and from the right had been extrapolated to the point in question [Gustavson, 2005]. Thus, the noise value will always be equal to zero on in- Figure 1: 2D Perlin noise as rendered by Gimp teger boundaries. By sampling the resulting 1- plugin \Solid noise" dimensional noise function, it's possible to gen- erate a waveform that can be looped to produce Despite its popularity, Perlin noise has been a pitched tone (Figure 2). seldom used for creative purposes in the fields outside the world of computer graphics. For 3 Simplex noise music applications, Perlin noise has been occa- sionally used for creating stochastic melodies or Simplex noise is an improvement to the original as a modulation source. Perlin noise algorithm proposed by Ken Perlin Figure 2: Perlin noise (left) and Simplex noise Figure 3: 3 octaves of Perlin noise (left) summed (right) with the gradients used for interpolation to generate a fBm waveform (right) himself [Perlin, 2001]. The advantages of sim- while successively incrementing their frequen- plex noise over Perlin noise include lower com- cies in regular steps by a factor called lacunarity putational complexity, no noticeable directional and decreasing the amplitude of the octaves by artifacts, and a well-defined analytical deriva- a factor called persistence with each step [Vivo tive. and Lowe, 2015]. Simplex noise is created by splitting an N- n dimensional space into simplest shapes called X i i simplices. The value of the noise function is a fBm(x) = p · noise(2 · x) (2) sum of contributions from each corner of the i=0 simplex surrounding a given point [Gustavson, Lacunarity can have any value greater than 1, 2005]. but non-integral lacunarity values will result in In one-dimensional space, simplex noise uses non-zero fBm values on the integer boundaries. intervals of equal length as the simplices. For To keep the waveform seamless in a sound syn- a point in an interval, the contribution of thesizer, lacunarity has be an integer number. each surrounding vertex is determined using the A reasonable choice for lacunarity is 2, since equation: bigger values result in a very quick buildup of the upper harmonics (Eq. 2). (1 − d2)4 · (g · d) (1) Fractional Brownian motion is often called Perlin noise, actually being a fractal sum of sev- Where g is the value of the gradient in a given eral octaves of noise. While typically the same vertex and d is the distance of the point to the noise function is used for every octave, different vertex. noise algorithms can be combined in the same Both Perlin noise and Simplex noise produce fashion to create multifractal or heterogeneous very similar results (Fig. 2) and are basically fBm waveforms [Musgrave, 2002]. interchangeable in a sound synthesizer1. For brevity, Perlin noise or noise will be used to 5 Waveform modifiers refer to both algorithms for the scope of this 5.1 Gradient rotation paper, since Simplex noise is also invented by Ken Perlin. One technique traditionally used to animate Perlin noise is gradient rotation [Perlin and 4 Fractional Brownian motion Neyret, 2001]. When gradient vectors in 2- or more dimensional space are rotated the noise is Fractional Brownian motion (fBm), also called varied while retaining its character and detail. fractal Brownian motion is a technique often This technique has been used for simulating ad- used with Perlin noise to add complexity and vected flow and other effects. A similar tech- detail to the generated textures. nique can be applied to 1-dimensional noise to Fractional Brownian motion is created by introduce subtle changes to the sound. summing several iterations of noise (octaves), Rotating gradients is a computationally ex- 1In some cases Perlin noise adds additional low fre- pensive operation and cannot be used with 1- quency harmonics to the sound which may or may not dimensional noise, since the noise is built from be desirable. linear gradients instead of directional vectors. Figure 4: Gradient offsets applied to fBm (left) Figure 5: Simplex noise (left) with domain modify the waveform (right) while preserving warping (right) the timbre It is still possible to apply this technique to 1-dimensional noise by adding a variable offset value to the gradients and symmetrically wrap- ping it when the maximum allowed gradient value (1) is reached. ( 2 − g; g > 1 g0 = (3) −2 − g; g < −1 In a sound synthesizer, gradient rotation does not change the timbre significantly. It does alter the amplitudes of the upper harmonics slightly (Fig. 4), adding variations that can be used in Figure 6: Andes, a JUCE-based synthesizer us- a polyphonic (poly-oscillator) synthesizer. ing Perlin noise 5.2 Domain warping Since the domain of noise is distorted with Another classic technique for adding variation the noise itself, the symmetry of the waveform to Perlin noise is called domain warping. Warp- will remain generally the same as seen on Fig. 5. ing simply means that the noise domain is dis- torted with another function g(p) before the 6 Implementation noise function is evaluated. The presented ideas have been implemented as Basically, noise(p) is replaced with a basic synthesizer plugin called Andes (Fig- noise(g(p)). While g can be any function, ure 6). The plugin has been developed using the it's often desirable to distort the image of JUCE2 framework and is currently available in noise just a little bit with respect to its regular the form of VST, AU, and standalone program behavior. for Windows, MacOS, and Linux3. Then, it makes sense to have g(p) being just At the time of writing, Andes supports gradi- the identity plus a small arbitrary distortion ent rotation, basic warping, up to 16 octaves of h(p) [Qu´ılez,2002]. In the most basic case the noise, and adjustable persistence, which allows distortion can be the noise itself (Eq. 4). a usable range of unique sounds to be produced. The sound of noise is susceptible to aliasing at f(p) = noise(p + noise(p)) (4) higher frequencies, but oversampling has not For the purpose of sound synthesis it is bet- been implemented so far. ter to expose warping as an adjustable param- The resulting sounds resemble early digital eter. Warping modulation can be implemented synthesizers, but also have a unique character by adding a coefficient that is used to control to them and can be described as \distinctively the warping depth (Eq. 5). digital". 2https://juce.com f(p) = noise(p + noise(p) · w) (5) 3https://artfwo.github.io/andes/ 6.1 Predictable randomness level normalization is one of the biggest issues A synthesizer plugin cannot have completely yet to be resolved. randomly sounding timbres when the synthe- The general idea of using unconventional, i.e. sizer is used in certain contexts such as a multi- graphics algorithms in sound and music presents track DAW project. The amplitude and tim- a lot of challenges, but also opens many different bre of the synthesizer cannot change in un- possibilities in both the technical and aesthetic predictable ways to make sure the track won't aspects. break the mix. Predictable randomness in Andes is achieved 8 Acknowledgments by saving the random seed for generating gra- The author would like to thank Maria Pankova dients in the plugin state (preset). The 32-bit for helping the idea of making a noise-based Mersenne Twister 19937 generator from C++ synthesizer to emerge and for assistance with standard library is used explicitly to make sure maths in the early stages of Andes develop- the random numbers generated from the same ment.