Note

Click here to download the full example code

# Additive Synthesis¶

**Author**: Moto Hira

This tutorial is the continuation of Oscillator and ADSR Envelope.

This tutorial shows how to perform additive synthesis and subtractive synthesis using TorchAudio’s DSP functions.

Additive synthesis creates timbre by combining multiple waveform. Subtractive synthesis creates timbre by applying filters.

Warning

This tutorial requires prototype DSP features, which are available in nightly builds.

Please refer to https://pytorch.org/get-started/locally for instructions for installing a nightly build.

```
import torch
import torchaudio
print(torch.__version__)
print(torchaudio.__version__)
```

```
2.0.0.dev20230202
2.0.0.dev20230202
```

## Overview¶

```
try:
from torchaudio.prototype.functional import (
oscillator_bank,
extend_pitch,
adsr_envelope,
)
except ModuleNotFoundError:
print(
"Failed to import prototype DSP features. "
"Please install torchaudio nightly builds. "
"Please refer to https://pytorch.org/get-started/locally "
"for instructions to install a nightly build.")
raise
import matplotlib.pyplot as plt
from IPython.display import Audio
```

## Creating multiple frequency pitches¶

The core of additive synthesis is oscillator. We create a timbre by summing up the multiple waveforms generated by oscillator.

In the oscillator tutorial, we used
`oscillator_bank()`

and
`adsr_envelope()`

to generate
various waveforms.

In this tutorial, we use
`extend_pitch()`

to create
a timbre from base frequency.

First, we define some constants and helper function that we use throughout the tutorial.

```
PI = torch.pi
PI2 = 2 * torch.pi
F0 = 344. # fundamental frequency
DURATION = 1.1 # [seconds]
SAMPLE_RATE = 16_000 # [Hz]
NUM_FRAMES = int(DURATION * SAMPLE_RATE)
```

```
def show(freq, amp, waveform, sample_rate, zoom=None, vol=0.1):
t = torch.arange(waveform.size(0)) / sample_rate
fig, axes = plt.subplots(4, 1, sharex=True)
axes[0].plot(t, freq)
axes[0].set(
title=f"Oscillator bank (bank size: {amp.size(-1)})",
ylabel="Frequency [Hz]",
ylim=[-0.03, None])
axes[1].plot(t, amp)
axes[1].set(
ylabel="Amplitude",
ylim=[-0.03 if torch.all(amp >= 0.0) else None, None])
axes[2].plot(t, waveform)
axes[2].set(ylabel="Waveform")
axes[3].specgram(waveform, Fs=sample_rate)
axes[3].set(
ylabel="Spectrogram",
xlabel="Time [s]",
xlim=[-0.01, t[-1] + 0.01])
for i in range(4):
axes[i].grid(True)
pos = axes[2].get_position()
plt.tight_layout()
if zoom is not None:
ax = fig.add_axes([pos.x0 + 0.02, pos.y0 + 0.03, pos.width / 2.5, pos.height / 2.0])
ax.plot(t, waveform)
ax.set(xlim=zoom, xticks=[], yticks=[])
waveform /= waveform.abs().max()
return Audio(vol * waveform, rate=sample_rate, normalize=False)
```

## Harmonic Overtones¶

Harmonic overtones are frequency components that are an integer multiple of the fundamental frequency.

We look at how to generate the common waveforms that are used in synthesizers. That is,

Sawtooth wave

Square wave

Triangle wave

### Sawtooth wave¶

Sawtooth wave can be expressed as the following. It contains all the integer harmonics, so it is commonly used in subtractive synthesis as well.

The following function takes fundamental frequencies and amplitudes, and adds extend pitch in accordance with the formula above.

Now synthesize a waveform

```
freq0 = torch.full((NUM_FRAMES, 1), F0)
amp0 = torch.ones((NUM_FRAMES, 1))
freq, amp, waveform = sawtooth_wave(freq0, amp0, int(SAMPLE_RATE / F0), SAMPLE_RATE)
show(freq, amp, waveform, SAMPLE_RATE, zoom=(1/F0, 3/F0))
```

```
/usr/local/envs/python3.8/lib/python3.8/site-packages/torchaudio/prototype/functional/_dsp.py:60: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient.
warnings.warn(
```

It is possible to oscillate the base frequency to create a time-varying tone based on sawtooth wave.

```
fm = 10 # rate at which the frequency oscillates [Hz]
f_dev = 0.1 * F0 # the degree of frequency oscillation [Hz]
phase = torch.linspace(0, fm * PI2 * DURATION, NUM_FRAMES)
freq0 = F0 + f_dev * torch.sin(phase).unsqueeze(-1)
freq, amp, waveform = sawtooth_wave(freq0, amp0, int(SAMPLE_RATE / F0), SAMPLE_RATE)
show(freq, amp, waveform, SAMPLE_RATE, zoom=(1/F0, 3/F0))
```

```
/usr/local/envs/python3.8/lib/python3.8/site-packages/torchaudio/prototype/functional/_dsp.py:60: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient.
warnings.warn(
```

### Square wave¶

Square wave contains only odd-integer harmonics.

```
def square_wave(freq0, amp0, num_pitches, sample_rate):
mults = [2. * i + 1. for i in range(num_pitches)]
freq = extend_pitch(freq0, mults)
mults = [4 / (PI * (2. * i + 1.)) for i in range(num_pitches)]
amp = extend_pitch(amp0, mults)
waveform = oscillator_bank(freq, amp, sample_rate=sample_rate)
return freq, amp, waveform
```

```
freq0 = torch.full((NUM_FRAMES, 1), F0)
amp0 = torch.ones((NUM_FRAMES, 1))
freq, amp, waveform = square_wave(freq0, amp0, int(SAMPLE_RATE/F0/2), SAMPLE_RATE)
show(freq, amp, waveform, SAMPLE_RATE, zoom=(1/F0, 3/F0))
```

```
/usr/local/envs/python3.8/lib/python3.8/site-packages/torchaudio/prototype/functional/_dsp.py:60: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient.
warnings.warn(
```

### Triangle wave¶

Triangle wave also only contains odd-integer harmonics.

```
def triangle_wave(freq0, amp0, num_pitches, sample_rate):
mults = [2. * i + 1. for i in range(num_pitches)]
freq = extend_pitch(freq0, mults)
c = 8 / (PI ** 2)
mults = [c * ((-1) ** i) / ((2. * i + 1.) ** 2) for i in range(num_pitches)]
amp = extend_pitch(amp0, mults)
waveform = oscillator_bank(freq, amp, sample_rate=sample_rate)
return freq, amp, waveform
```

```
freq, amp, waveform = triangle_wave(freq0, amp0, int(SAMPLE_RATE / F0 / 2), SAMPLE_RATE)
show(freq, amp, waveform, SAMPLE_RATE, zoom=(1/F0, 3/F0))
```

```
/usr/local/envs/python3.8/lib/python3.8/site-packages/torchaudio/prototype/functional/_dsp.py:60: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient.
warnings.warn(
```

## Inharmonic Paritials¶

Inharmonic partials refer to freqencies that are not integer multiple of fundamental frequency.

They are essential in re-creating realistic sound or making the result of synthesis more interesting.

### Bell sound¶

https://computermusicresource.com/Simple.bell.tutorial.html

```
num_tones = 9
duration = 2.0
num_frames = int(SAMPLE_RATE * duration)
freq0 = torch.full((num_frames, 1), F0)
mults = [0.56, 0.92, 1.19, 1.71, 2, 2.74, 3., 3.76, 4.07]
freq = extend_pitch(freq0, mults)
amp = adsr_envelope(
num_frames=num_frames,
attack=0.002,
decay=0.998,
sustain=0.,
release=0.,
n_decay=2,
)
amp = torch.stack([amp * (0.5 ** i) for i in range(num_tones)], dim=-1)
waveform = oscillator_bank(freq, amp, sample_rate=SAMPLE_RATE)
show(freq, amp, waveform, SAMPLE_RATE, vol=0.4)
```

As a comparison, the following is the harmonic version of the above. Only frequency values are different. The number of overtones and its amplitudes are same.

## References¶

**Total running time of the script:** ( 2 minutes 10.545 seconds)