(in Swedish)

# New waveforms for the people!

This is a collection of ideas for slightly unusual waveforms for use in wavetable oscillators. Some of the waveforms introduced here are not bandlimited, but most of them have a steep spectral slope and can be used at least at low fundamental frequencies without disturbing amounts of aliasing.

For an introduction to the concepts of wavetable synthesis, the recommended reading section is a good place to start.

## The waveform library

All the waveforms introduced below and many more have been implemented in C++ together with a simple linearly interpolating wavetable oscillator. If you want to play with the code, download it and include it directly in your projects. No complicated compilation process is required. The code is licenced under GNU GPL v.3. Modifications and improvements are encouraged. Absence of errors or bugs and general usefulness is not guaranteed.

## Good wavetable manners

The wavetables are written to an array expected to be of type double, float, or long double, and of odd length. The first and the last entry of the array will be identical, because the wavetables are designed to be used with linearly interpolating oscillators that need the extra guard point at the end.

Wavetables have been used for sound synthesis for a long time because of the enormous efficiency gain compared to direct additive synthesis. Instead of summing a large number of sinusoids, the waveform is pre-calculated and put in a table which is accessed by the phase variable of an oscillator. Wavetables are best suited for perfectly harmonic sounds, but inharmonic sounds may be created by using two or more detuned oscillators that access different partials.

Timbral morphing is possible by interpolating between two or more wavetables that store timbral variants. Similarly, for an oscillator to sound homogenous across a wide frequency range, a set of wavetables should be defined. For low fundamental frequencies the wavetable should contain many partials, and as the fundamental frequency rises the number of partials should decrease correspondingly.

Since the wavetable will be read at different speeds it makes no sense to define its sampling rate as such, but the sampling theorem still means that the highest partial that can put in the table will need at least two samples to complete one period.

Let $w_n = \sin (k2\pi n/L)$ be a sinusoid stored in the wavetable where $n = 0, 1, \ldots, N-1$ is the sample index, L is the wavetable's length and k is the partial number. Then the highest partial would be at $k = L/2$, but this would not be very useful since the samples $\sin(\pi n) = 0$ would be put into the table. Therefore it is better to limit the highest partial to a fourth of the table size or even lower.

Waveforms may be generated in the time domain or by adding sinusoids in the frequency domain. Bandlimited waveforms are useful when aliasing-free harmonically rich sounds are wanted. Choose a waveshape, find its Fourier series, and add up a limited number of partials to make a stricty bandlimited waveform. In the time domain, the spectral slope can be controlled by putting constraints on the smoothness of the wave shape. This is done by assuring that derivatives of increasing order are continuous; the higher the order of continuous derivatives, the steeper the spectral slope.

In particular, a periodic function with a finite number of discontinuities has a Fourier series with terms proportional to $1/k$ where k is the partial number; a piecewise continuous function has terms of the order $1/k^2$; and in general a function whose N-1 first derivatives are continuous but whose Nth derivative is not continuous has a spectrum that drops off like $1/k^N$.

Functions that are not bandlimited will cause aliasing when used as wavetables for oscillators, but worse still, even bandlimited functions will alias as soon as their highest partial exceeds the Nyquist frequency. The result may be anything from dreadful to inaudible. How much aliasing will be considered acceptable will depend on the application.

There is also quantisation noise to worry about. Use a longer wavetable for cleaner sounds.

## Waveform catalog

Some classical bandlimited analog waveforms have been left out of this overview since they are already well known.

Notice that the functions are defined over different ranges. Sometimes they are defined over the integers 0 to $N-1,$ where N is the table size, but often it is more practical to define them over the continuous interval $[0,1]$ or $[-1,1]$. The function then is sampled at N points and written to the table.

All functions are templates with the definition template <typename REAL> where REAL is expected to be a floating point type.

### Expogliss

void make_expogliss(REAL *wt, int N, uint p=5, REAL r=8.0, bool norm=true)

The idea is to generate a smooth waveform which oscillates at a rising frequency while decaying in amplitude. This waveform is continuously differentiable but not bandlimited and takes two parameters, $p \in \mathbb{N}$ and $r > 1$. A diffuse formant may appear around p times the fundamental when r is set to relatively low values.

Top: two periods of the waveform. Bottom: The derivative of the waveform. Where the arrow points the derivative has a corner which makes the second derivative discontinuous.

The wavetable will contain p periods in total of an exponentially decaying sinusoid with rising frequency, covering the frequency range r. The frequency will start off at its lowest, $2\pi\omega_{0}$, and reach $2\pi(\omega_{0}+2)$ at the end of a full period. Working backwards, we have the frequency range $r=\omega_{1}/\omega_{0}$, or $$\omega_{0}=\frac{2}{r-1},$$ where we assume that $r>1$.

Define the waveform as a function $f(t),\ t\in[0,1]$. First the exact right number of periods, p, will be fit into the interval $t\in[0,1]$ so that the slope of the waveform will have the same sign at the beginning and the end. Finally, multiply the waveform with an envelope that ensures that the slopes at the beginning and the end are identical.

First we generate the sinusoid $$x(t)=\sin\gamma\theta_{t} \qquad (1)$$ with variable frequency $\omega_{t}=\gamma\dot{\theta}_{t},$ and where $\gamma$ is a constant to be determined. Setting $\omega_{t}=\omega_{0}+2t$, the phase at time t is found by integrating $\dot{\theta}=\omega_{0}+2t$ from the initial condition $\theta_{0}=0$, giving $$\theta_{t}=\intop_{0}^{t}\omega_{0}+2\tau \,\mathrm{d}\tau=\omega_{0}t+t^{2}. \quad (2)$$ As t goes from $t=0$ to $t=1$ the sinusoid $x(t)$ should complete p entire periods (of decreasing length). This means that the phase $\gamma\theta_{t}$ in (1) must reach $2\pi p$ when $t=1$. From the integral (2) we see that $\theta_{t}|_{t=1}=\omega_{0}+1$, whence $$\gamma=\frac{2\pi p}{\omega_{0}+1}.$$

In order to have a continuous derivative where the table wraps around at the end points $f(0)=f(1)$, we evaluate the derivative $$x'(t)=\gamma(\omega_{0}+2t)\cos(\gamma(\omega_{0}t+t^{2}))$$ at the end points $x'(0)$ and $x'(1)$. Since the increasing frequency makes $x'(1)$ greater than $x'(0)$, we multiply $x(t)$ with a decaying amplitude envelope $e^{-\lambda t}$ with decay parameter $\lambda > 0$ chosen such that the slopes at the end points match, that is, $f'(0)=f'(1)$. Solving $x'(0)=e^{-\lambda}x'(1)$ for $\lambda$, we find that $$\lambda=\ln\left(\frac{x'(1)}{x'(0)}\right)$$ and the waveform becomes $$f(t)=e^{-\lambda t}\sin(\gamma(\omega_{0}t+t^{2})),\ t\in[0,1].$$ This waveshape is once continuously differentiable but its second derivative is discontinuous where the wavetable wraps around.

### Smooth bump functions

void make_bump(REAL *w, int N)
void make_symbump(REAL *w, int N)
void make_diffbump(REAL *w, int N, bool norm=true)

Bump functions are smooth and have compact support (they are zero outside of some finite interval). The first one generates the positive function $$w_{t} = \exp\left(1-\frac{1}{1-t^{2}}\right),\quad t\in[-1,1]$$ which has continuous derivatives of all orders.

The second one makes a function with the same symmetry as a sinusoid by following the bump function as defined above with a negative copy of itself.

When differentiated, the positive bump function remains smooth but turns into a bipolar function. Its shape is similar to that of a sawtooth, but less sharp.

From left to right: the positive bump function, a symmetric bipolar variant and the derivative of the bump function.

### Formant

void make_formant(REAL *wt, int N, int c, bool norm=true)

With some abuse of terminology, put a "formant" frequency at partial c. (Real formants do not generally move along with the fundamental frequency.) Creates an almost symmetrical spectrum centered around the partial c; surrounding partials decay roughy as $a_{k} \sim 1/|k-c|^{2}$. A few extra partials are added to the high end of the spectrum. $$w_{n}=\sum_{k=1}^{p}a_{k}\sin(k2\pi n/N)$$ $$a_{k}=\frac{1}{|k+1/2-c|\times|k-1/2-c|}.$$

### Twin peaks

void make_twinpeaks(REAL *wt, int N, bool naive=false, bool norm=true)

Named after its strong first and second partials.

Waveforms and spectra of naïve version (red) and improved version (blue).

Take the difference of two sinusoids whose periods do not fit into the wavetable and multiply with a function that makes the waveform continuous (naive version) or that also makes the waveform's slope continuous (improved version).

Define the function $g(t), 0 \leq t \leq 1,$ as the difference $$g(t) = \sin (5\pi t/2) - \sin (7\pi t/2)$$ and, for the naive version, multiply it with the ramp function $r(t) = (1-t)$. This will make the endpoints of the waveform meet, but there will be a corner where the slope changes suddenly.

For the improved version, multiply $g(t)$ by a function $p(t)$ such that the derivative $\frac{d}{dt} (p(t) g(t))$ coincides at the end points $t=0$ and $t=1$. Furthermore, we require that $p(1) = 0$ and $p'(1)=-1$. Taken together, these constraints can be met with a quadratic polynomial $p(t)$. Solving for the coefficients using the constraints, we have $$p(t) = (c-1)t^2 + (1-2c)t + c$$ where $c=2/\pi$, and $$w_{t} = p(t) g(t), \qquad t\in[0,1]$$ is the final waveform.

In contrast to the naive version where the third partial is about 27 dB below the first two partials, in the improved version it is about 36 dB below. Both versions have a small DC offset, but it is much smaller for the improved version.

### Chirp

void make_chirp(REAL *wt, int N, REAL c=5, REAL B=12.5)

Generates a chirp or fast downwards glissando multiplied by a window function. In the middle of the waveform when the window funcion reaches its maximum, the instantaneous frequency (in partial numbers) is c. The frequency sweeps linearly from $2c$ down to zero.

The waveform $$w_t = A(t) \sin (2\pi \varphi_t), \quad t \in [-1,1]$$ is constructed by integrating the phase $$\dot{\varphi} = 2c (1 - \frac{t+1}{2})$$ and multiplying by the window function $$A(t) = \frac{1}{1 + \beta t^2} - \frac{1}{1 + \beta}.$$ The parameter $\beta \geq 1$ sets the steepness of the window function; at higher values the window focuses on a narrower region.

Chirp function and its spectrum, here with $\beta=5$ and $c=7$.

### Diphone

void make_diphone(REAL *wt, int N, uint P=5)

Splice two sinusoid segments of equal length together. The first half consists of one period, the second half contains P periods. Match the slopes at the joint by multiplying the amplitude of the second segment by $1/P$. Hence, $$w_t = \begin {cases} \sin(2\pi t), \quad & t \in [-1,0) \\ \frac{1}{P} \sin(P2\pi t), \quad & t \in [0,1]. \end {cases}$$ Apart from a strong second and $2P$th partial, the spectrum consists of odd partials.

Diphone waveform and spectrum with $P = 9$.

void make_noise(REAL *wt, int N, double seed=1/7.)

Generates a noisy signal by iterating the chaotic quadratic map $$w_{n+1}=2w_{n}^{2}-1$$ from the initial condition (seed) $w_{0} \in (-1,1)$; the default initial condition is $w_{0}=1/7$. By a change of variables this map can be transformed into the logistic map.

With a sufficiently long wavetable and at low oscillator frequencies no periodicity will be audible. Aliasing will be noticed at higher frequencies. This is not the kind of signal one would normally think of putting into a wavetable for oscillators.

### The Volterra function

void make_volterra(REAL *w, int N, bool norm=true)

A quirky function often mentioned in real analysis (perhaps in a slightly different form) is $$w_{t}=t^{2}\sin(\pi/t),\quad w_{0}=0$$ which we define over $t \in [-1,1]$. Dividing by t causes the frequency to approach infinity near zero, but multiplying with $t^2$ tames the function to the extent that it is differentiable, although its derivative is not bounded. Definitely not a bandlimited function, reasonably used it will not cause offensive levels of aliasing.

### Darboux nowhere differentiable

void make_darboux(REAL *wt, int N, bool norm=true)

Yet another real analysis example: a nowhere differentiable function—if you have the patience to add an infinite number of partials. This one is actually bandlimited, but contains some very high partials. (The highest partial covers one period over at least four samples in the table.) $$w_{t}=\sum_{k=1}^{p}\frac{\cos(2\pi k!t)}{k!}$$ For best results, use a long wavetable.

### Sparse spectrum

void make_sparse(REAL *wt, int N, int P=55, bool one_over_k=true)

Sparse spectra have widely spaced partials. This waveform is built from partials at the triangular numbers $T_k = 1,3,6,10,15, ...,$ with decaying amplitude. If the parameter one_over_k is true the amplitudes are $a_k = 1/k$, otherwise the amplitude is inversely proportional to the frequency, that is, $a_k = 1/T_k$. This waveform has a peculiar fractal appearance.

Sparse waveform generated with $a_k=1/k$ and spectrum with both spectral slopes shown. The red dots correspond to $a_k = 1/T_k$.