.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "tutorials/additive_synthesis_tutorial.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_tutorials_additive_synthesis_tutorial.py: Additive Synthesis ================== **Author**: `Moto Hira `__ This tutorial is the continuation of `Oscillator and ADSR Envelope <./oscillator_tutorial.html>`__. 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. .. GENERATED FROM PYTHON SOURCE LINES 24-31 .. code-block:: default import torch import torchaudio print(torch.__version__) print(torchaudio.__version__) .. rst-class:: sphx-glr-script-out .. code-block:: none 2.4.0.dev20240419 2.2.0.dev20240420 .. GENERATED FROM PYTHON SOURCE LINES 32-36 Overview -------- .. GENERATED FROM PYTHON SOURCE LINES 36-52 .. code-block:: default try: from torchaudio.prototype.functional import adsr_envelope, extend_pitch, oscillator_bank 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 .. GENERATED FROM PYTHON SOURCE LINES 53-68 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 <./oscillator_tutorial.html>`__, we used :py:func:`~torchaudio.prototype.functional.oscillator_bank` and :py:func:`~torchaudio.prototype.functional.adsr_envelope` to generate various waveforms. In this tutorial, we use :py:func:`~torchaudio.prototype.functional.extend_pitch` to create a timbre from base frequency. .. GENERATED FROM PYTHON SOURCE LINES 70-72 First, we define some constants and helper function that we use throughout the tutorial. .. GENERATED FROM PYTHON SOURCE LINES 73-84 .. code-block:: default PI = torch.pi PI2 = 2 * torch.pi F0 = 344.0 # fundamental frequency DURATION = 1.1 # [seconds] SAMPLE_RATE = 16_000 # [Hz] NUM_FRAMES = int(DURATION * SAMPLE_RATE) .. GENERATED FROM PYTHON SOURCE LINES 86-115 .. code-block:: default def plot(freq, amp, waveform, sample_rate, zoom=None, vol=0.1): t = (torch.arange(waveform.size(0)) / sample_rate).numpy() fig, axes = plt.subplots(4, 1, sharex=True) axes[0].plot(t, freq.numpy()) axes[0].set(title=f"Oscillator bank (bank size: {amp.size(-1)})", ylabel="Frequency [Hz]", ylim=[-0.03, None]) axes[1].plot(t, amp.numpy()) 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() fig.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) .. GENERATED FROM PYTHON SOURCE LINES 116-129 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 .. GENERATED FROM PYTHON SOURCE LINES 131-147 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. .. math:: \begin{align*} y_t &= \sum_{k=1}^{K} A_k \sin ( 2 \pi f_k t ) \\ \text{where} \\ f_k &= k f_0 \\ A_k &= -\frac{ (-1) ^k }{k \pi} \end{align*} .. GENERATED FROM PYTHON SOURCE LINES 149-152 The following function takes fundamental frequencies and amplitudes, and adds extend pitch in accordance with the formula above. .. GENERATED FROM PYTHON SOURCE LINES 152-163 .. code-block:: default def sawtooth_wave(freq0, amp0, num_pitches, sample_rate): freq = extend_pitch(freq0, num_pitches) mults = [-((-1) ** i) / (PI * i) for i in range(1, 1 + num_pitches)] amp = extend_pitch(amp0, mults) waveform = oscillator_bank(freq, amp, sample_rate=sample_rate) return freq, amp, waveform .. GENERATED FROM PYTHON SOURCE LINES 164-166 Now synthesize a waveform .. GENERATED FROM PYTHON SOURCE LINES 167-173 .. code-block:: default 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) plot(freq, amp, waveform, SAMPLE_RATE, zoom=(1 / F0, 3 / F0)) .. image-sg:: /tutorials/images/sphx_glr_additive_synthesis_tutorial_001.png :alt: Oscillator bank (bank size: 46) :srcset: /tutorials/images/sphx_glr_additive_synthesis_tutorial_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none /pytorch/audio/src/torchaudio/prototype/functional/_dsp.py:63: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient. warnings.warn( .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 174-177 It is possible to oscillate the base frequency to create a time-varying tone based on sawtooth wave. .. GENERATED FROM PYTHON SOURCE LINES 178-188 .. code-block:: default 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) plot(freq, amp, waveform, SAMPLE_RATE, zoom=(1 / F0, 3 / F0)) .. image-sg:: /tutorials/images/sphx_glr_additive_synthesis_tutorial_002.png :alt: Oscillator bank (bank size: 46) :srcset: /tutorials/images/sphx_glr_additive_synthesis_tutorial_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none /pytorch/audio/src/torchaudio/prototype/functional/_dsp.py:63: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient. warnings.warn( .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 189-204 Square wave ~~~~~~~~~~~ `Square wave `_ contains only odd-integer harmonics. .. math:: \begin{align*} y_t &= \sum_{k=0}^{K-1} A_k \sin ( 2 \pi f_k t ) \\ \text{where} \\ f_k &= n f_0 \\ A_k &= \frac{ 4 }{n \pi} \\ n &= 2k + 1 \end{align*} .. GENERATED FROM PYTHON SOURCE LINES 204-217 .. code-block:: default def square_wave(freq0, amp0, num_pitches, sample_rate): mults = [2.0 * i + 1.0 for i in range(num_pitches)] freq = extend_pitch(freq0, mults) mults = [4 / (PI * (2.0 * i + 1.0)) for i in range(num_pitches)] amp = extend_pitch(amp0, mults) waveform = oscillator_bank(freq, amp, sample_rate=sample_rate) return freq, amp, waveform .. GENERATED FROM PYTHON SOURCE LINES 219-225 .. code-block:: default 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) plot(freq, amp, waveform, SAMPLE_RATE, zoom=(1 / F0, 3 / F0)) .. image-sg:: /tutorials/images/sphx_glr_additive_synthesis_tutorial_003.png :alt: Oscillator bank (bank size: 23) :srcset: /tutorials/images/sphx_glr_additive_synthesis_tutorial_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none /pytorch/audio/src/torchaudio/prototype/functional/_dsp.py:63: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient. warnings.warn( .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 226-241 Triangle wave ~~~~~~~~~~~~~ `Triangle wave `_ also only contains odd-integer harmonics. .. math:: \begin{align*} y_t &= \sum_{k=0}^{K-1} A_k \sin ( 2 \pi f_k t ) \\ \text{where} \\ f_k &= n f_0 \\ A_k &= (-1) ^ k \frac{8}{(n\pi) ^ 2} \\ n &= 2k + 1 \end{align*} .. GENERATED FROM PYTHON SOURCE LINES 241-255 .. code-block:: default def triangle_wave(freq0, amp0, num_pitches, sample_rate): mults = [2.0 * i + 1.0 for i in range(num_pitches)] freq = extend_pitch(freq0, mults) c = 8 / (PI**2) mults = [c * ((-1) ** i) / ((2.0 * i + 1.0) ** 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 .. GENERATED FROM PYTHON SOURCE LINES 257-261 .. code-block:: default freq, amp, waveform = triangle_wave(freq0, amp0, int(SAMPLE_RATE / F0 / 2), SAMPLE_RATE) plot(freq, amp, waveform, SAMPLE_RATE, zoom=(1 / F0, 3 / F0)) .. image-sg:: /tutorials/images/sphx_glr_additive_synthesis_tutorial_004.png :alt: Oscillator bank (bank size: 23) :srcset: /tutorials/images/sphx_glr_additive_synthesis_tutorial_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none /pytorch/audio/src/torchaudio/prototype/functional/_dsp.py:63: UserWarning: Some frequencies are above nyquist frequency. Setting the corresponding amplitude to zero. This might cause numerically unstable gradient. warnings.warn( .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 262-271 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. .. GENERATED FROM PYTHON SOURCE LINES 273-278 Bell sound ~~~~~~~~~~ https://computermusicresource.com/Simple.bell.tutorial.html .. GENERATED FROM PYTHON SOURCE LINES 278-301 .. code-block:: default 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, 3.0, 3.76, 4.07] freq = extend_pitch(freq0, mults) amp = adsr_envelope( num_frames=num_frames, attack=0.002, decay=0.998, sustain=0.0, release=0.0, 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) plot(freq, amp, waveform, SAMPLE_RATE, vol=0.4) .. image-sg:: /tutorials/images/sphx_glr_additive_synthesis_tutorial_005.png :alt: Oscillator bank (bank size: 9) :srcset: /tutorials/images/sphx_glr_additive_synthesis_tutorial_005.png :class: sphx-glr-single-img .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 302-306 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. .. GENERATED FROM PYTHON SOURCE LINES 307-313 .. code-block:: default freq = extend_pitch(freq0, num_tones) waveform = oscillator_bank(freq, amp, sample_rate=SAMPLE_RATE) plot(freq, amp, waveform, SAMPLE_RATE) .. image-sg:: /tutorials/images/sphx_glr_additive_synthesis_tutorial_006.png :alt: Oscillator bank (bank size: 9) :srcset: /tutorials/images/sphx_glr_additive_synthesis_tutorial_006.png :class: sphx-glr-single-img .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 314-320 References ---------- - https://en.wikipedia.org/wiki/Additive_synthesis - https://computermusicresource.com/Simple.bell.tutorial.html - https://computermusicresource.com/Definitions/additive.synthesis.html .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 4.799 seconds) .. _sphx_glr_download_tutorials_additive_synthesis_tutorial.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: additive_synthesis_tutorial.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: additive_synthesis_tutorial.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_