surge-wavetable

surge-wavetable is a Rust crate that provides functionality for creating, manipulating, and working with wavetables in the context of the Surge synthesizer system.

Wavetable Synthesis

Wavetable synthesis is a technique used in audio synthesis to create sounds by interpolating between pre-recorded waveforms, called wavetables. The sound produced by a wavetable synthesizer can be modified by changing the frequency at which the wavetable is read, as well as by applying various modulation techniques to the waveform.

Wavetable Building

The surge-wavetable crate provides functionality for building and working with wavetables. The WaveTable struct represents a wavetable, and includes metadata such as the number of samples in each table, the number of tables in the wavetable, and the number of mipmaps (pre-generated lower resolution versions of the wavetable) that should be generated.

The populate_mipmaps function can be used to populate the mipmaps for a wavetable, while the populate_base_mipmap_level function can be used to populate the base mipmap level for a wavetable.

Reading Wavetables

The wavetable_read function can be used to read a wavetable at a specified frequency and phase. The resulting waveform can be modified using various modulation techniques, such as phase modulation or frequency modulation.

Other Functionality

The surge-wavetable crate also includes functionality for loading and working with patch formats and extensions, as well as utility functions for converting between various data types.

Mathematical Ideas

Wavetable synthesis involves mathematical techniques such as interpolation and modulation to generate complex waveforms from pre-recorded waveforms. The surge-wavetable crate uses mathematical techniques such as linear interpolation to read wavetables at non-integer frequencies, as well as modulation techniques such as phase modulation and frequency modulation to modify the waveform. The populate_mipmaps function uses techniques such as downsampling and low-pass filtering to generate lower resolution versions of the wavetable.

Overall, the surge-wavetable crate provides powerful functionality for working with wavetables in the context of the Surge synthesizer system.

9> could you teach me about wavetables in mathematical detail?

Sure! Wavetable synthesis involves using pre-recorded waveforms, called wavetables, as building blocks to create complex sounds. The basic idea behind wavetable synthesis is to read the waveform from the wavetable at a specific frequency and phase, and then use various modulation techniques to modify the waveform.

Mathematically, a wavetable can be represented as a sequence of samples, where each sample represents the amplitude of the waveform at a specific point in time. The sequence of samples can be thought of as a function of time, where the value of the function represents the amplitude of the waveform at that point in time.

When a wavetable is read at a specific frequency and phase, the waveform is effectively stretched or compressed in time to fit the desired frequency. This can be achieved by reading the wavetable at non-integer sample indices, which effectively stretches or compresses the waveform.

For example, if a 1024-sample wavetable is being read at a frequency of 440 Hz (A4), the index into the wavetable can be calculated as follows:

index = time * sample_rate * (1/frequency)

where time is the current time in seconds, sample_rate is the sampling rate of the wavetable in samples per second, and frequency is the desired frequency in Hertz. If index is not an integer, linear interpolation can be used to interpolate between the nearest samples in the wavetable to get the desired amplitude.

Once the waveform has been read from the wavetable, various modulation techniques can be applied to modify the waveform. For example, frequency modulation (FM) involves modulating the frequency of the waveform using another waveform, while amplitude modulation (AM) involves modulating the amplitude of the waveform using another waveform.

In summary, wavetable synthesis involves using pre-recorded waveforms, called wavetables, as building blocks to create complex sounds. The waveform is read from the wavetable at a specific frequency and phase, and then modified using various modulation techniques. Mathematically, the wavetable can be represented as a sequence of samples, and linear interpolation can be used to read the waveform at non-integer sample indices.

10> thanks -- what are the related mathematical operations we can use with them to make them better sounding and more flexible?

There are several related mathematical operations that can be used with wavetables to make them more flexible and better sounding. Here are a few examples:

1. Windowing: Windowing is a technique that involves multiplying the wavetable samples by a window function before performing any other operations. Windowing can be used to shape the spectrum of the waveform, which can improve the sound quality and reduce unwanted noise. Examples of window functions include the Hamming window, Hann window, and Blackman window.

2. Phase Distortion: Phase distortion involves modifying the phase of the waveform to create new harmonic content. Phase distortion can be used to create complex, evolving sounds, and can be especially effective when used with wavetables that have simple harmonic content. Phase distortion can be implemented using functions such as arctan, cubic, or sine modulation.

3. Additive Synthesis: Additive synthesis involves adding together multiple waveforms to create complex sounds. Additive synthesis can be used with wavetables by combining multiple wavetables together, each with a different harmonic content. The harmonic content of the resulting waveform can be controlled by adjusting the amplitude of each wavetable.

4. Granular Synthesis: Granular synthesis involves breaking up a waveform into small grains and reassembling them in different ways to create new sounds. Granular synthesis can be used with wavetables by breaking the waveform into short segments, and then rearranging the segments to create new sounds. The duration and position of each segment can be controlled using various modulation techniques.

5. Frequency Modulation: Frequency modulation (FM) involves modulating the frequency of the waveform using another waveform. FM can be used with wavetables by applying a modulating waveform to the frequency index used to read the wavetable. This can create complex, evolving sounds with a rich harmonic content.

These are just a few examples of the mathematical operations that can be used with wavetables to create complex and interesting sounds. By combining these operations and experimenting with different wavetables and modulation sources, it is possible to create an endless variety of sounds with wavetable synthesis.

11> how much memory do they take? how do we generate the wavetables?

The amount of memory required for a wavetable depends on the length and complexity of the waveform and the number of wavetables used in a given synth patch. In general, a single wavetable with a few thousand samples may only require a few kilobytes of memory, while larger or more complex wavetables may require several megabytes of memory.

There are several ways to generate wavetables. One common method is to record a real instrument or sound, and then extract a single cycle of the waveform to use as the basis for the wavetable. The waveform can then be manipulated using various techniques, such as additive synthesis or phase distortion, to create different harmonics and timbres.

Another method is to generate waveforms using mathematical functions, such as sine waves or sawtooth waves. These waveforms can be combined to create more complex waveforms and used as the basis for a wavetable.

In addition, there are software tools that can be used to generate and edit wavetables. These tools often provide a visual interface for manipulating the waveform and can simplify the process of creating complex or custom wavetables.

Once a wavetable has been generated, it can be used in a synthesizer to create sounds. The wavetable is typically loaded into memory when the synthesizer is initialized, and can then be read and modified in real-time to create a wide range of sounds.

12> how can we engineer their spectral properties?

There are several techniques that can be used to engineer the spectral properties of a wavetable, which can have a significant impact on the sound quality and timbre of the resulting waveform. Here are a few examples:

1. Windowing: Windowing is a technique that involves multiplying the wavetable samples by a window function before performing any other operations. Windowing can be used to shape the spectrum of the waveform, which can improve the sound quality and reduce unwanted noise. Different window functions can be used to achieve different spectral characteristics. For example, a Hann window can be used to reduce spectral leakage and improve the spectral resolution of the waveform.

2. Additive Synthesis: Additive synthesis involves adding together multiple waveforms to create complex sounds. Additive synthesis can be used with wavetables by combining multiple wavetables together, each with a different harmonic content. The harmonic content of the resulting waveform can be controlled by adjusting the amplitude of each wavetable. This can be used to engineer the spectral properties of the waveform and create a wide range of timbres.

3. Phase Distortion: Phase distortion involves modifying the phase of the waveform to create new harmonic content. Phase distortion can be used to create complex, evolving sounds, and can be especially effective when used with wavetables that have simple harmonic content. Phase distortion can be used to engineer the spectral properties of the waveform and create new timbres.

4. Frequency Modulation: Frequency modulation (FM) involves modulating the frequency of the waveform using another waveform. FM can be used with wavetables by applying a modulating waveform to the frequency index used to read the wavetable. This can create complex, evolving sounds with a rich harmonic content, and can be used to engineer the spectral properties of the waveform.

5. Granular Synthesis: Granular synthesis involves breaking up a waveform into small grains and reassembling them in different ways to create new sounds. Granular synthesis can be used with wavetables by breaking the waveform into short segments, and then rearranging the segments to create new sounds. The duration and position of each segment can be controlled using various modulation techniques, which can be used to engineer the spectral properties of the waveform.

By combining these techniques and experimenting with different wavetables and modulation sources, it is possible to engineer the spectral properties of a wavetable to create an endless variety of sounds with different timbres and characteristics.

13> which windows do we use?

There are many different window functions that can be used to shape the spectrum of a waveform. The choice of window function depends on the desired spectral properties of the waveform and the specific application. Here are a few examples of common window functions used in wavetable synthesis:

1. Rectangular Window: The rectangular window is the simplest window function and is essentially a straight line with a constant value of 1.0. It does not modify the spectral content of the waveform, and is often used when a flat frequency response is desired.

2. Hamming Window: The Hamming window is a popular window function that has a smoother taper than the rectangular window. It is defined by the following formula:

   w(n) = 0.54 - 0.46 * cos(2 * pi * n / N)

   where n is the sample index, N is the total number of samples in the window, and cos is the cosine function. The Hamming window provides a good balance between spectral resolution and spectral leakage, and is often used in applications where a smooth frequency response is desired.

3. Hann Window: The Hann window is similar to the Hamming window, but has a wider taper that extends to zero at both ends of the window. It is defined by the following formula:

   w(n) = 0.5 * (1 - cos(2 * pi * n / N))

   where n is the sample index, N is the total number of samples in the window, and cos is the cosine function. The Hann window provides a good balance between spectral resolution and spectral leakage, and is often used in applications where a smooth frequency response is desired.

4. Blackman Window: The Blackman window is a window function with a wider taper than the Hamming and Hann windows, which makes it more effective at reducing spectral leakage. It is defined by the following formula:

   w(n) = 0.42 - 0.5 * cos(2 * pi * n / N) + 0.08 * cos(4 * pi * n / N)

   where n is the sample index, N is the total number of samples in the window, and cos is the cosine function. The Blackman window provides a good balance between spectral resolution and spectral leakage, and is often used in applications where a smooth frequency response is desired.

There are many other window functions that can be used in wavetable synthesis, each with its own unique properties and trade-offs. The choice of window function depends on the specific application and the desired spectral properties of the waveform.