music/waves.ipynb

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    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Show plots inline.\n",
    "%matplotlib inline\n",
    "\n",
    "# Number of harmonics to include in subsequent plots.\n",
    "Harmonics = 10\n",
    "# Time in seconds.\n",
    "Time = 2.0\n",
    "# Plot resolution.\n",
    "Steps = 200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "# Array of step values.\n",
    "def time_space(sec=Time, samples=Steps):\n",
    "    '''An array of equally spaced samples over a period of time.'''\n",
    "    # One extra sample to also include the endpoint.\n",
    "    return np.linspace(0, sec, num=samples + 1)\n",
    "\n",
    "def sample_rate(sec=Time, samples=Steps):\n",
    "    '''Sampling rate in samples per second.'''\n",
    "    return float(samples + 1) / float(sec)\n",
    "\n",
    "def sample_spacing(sec=Time, samples=Steps):\n",
    "    '''Time between samples.'''\n",
    "    return float(sec) / float(samples + 1)\n",
    "\n",
    "def frequency_spectrum(data):\n",
    "    '''Perform an FFT on `data`.'''\n",
    "    # Get the signal frequencies.\n",
    "    fft = np.abs(np.fft.fft(data))\n",
    "    \n",
    "    # Get the frequencies for a DFT plot (this doesn't depend on the actual signal) based on the \n",
    "    # spacing of our samples.\n",
    "    freqs = np.fft.fftfreq(len(fft), d=sample_spacing())\n",
    "    \n",
    "    return freqs, fft\n",
    "\n",
    "def plot_signal_and_spectrum(domain, data):\n",
    "    '''Plot the signal and spectrum in a stacked plot.'''\n",
    "    plt.figure(figsize=(10,6))\n",
    "    plt.subplot(211)\n",
    "    plt.axhline(color='#666666', linewidth=0.75)\n",
    "    plt.plot(domain, data)\n",
    "    \n",
    "    num_freqs = Harmonics * 2\n",
    "    freqs, mags = (x[:num_freqs] for x in frequency_spectrum(data))\n",
    "    plt.subplot(212)\n",
    "    plt.xticks(range(num_freqs))\n",
    "    plt.bar(freqs, mags)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basic Waves"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sine Wave\n",
    "\n",
    "Sine waves are easy. 😁 The have a single frequency, and no harmonics."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Sine wave.\n",
    "\n",
    "def sine(domain, freq=1.0):\n",
    "    return np.sin(2.0 * np.pi * freq * domain)\n",
    "\n",
    "wave = sine(time_space())\n",
    "plot_signal_and_spectrum(time_space(), wave)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Saw Wave\n",
    "\n",
    "Saw waves consist of every integer harmonic _n_ above the fundamental frequency, where that harmonic's amplitude is n<sup>-1</sup>."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def saw(domain, freq=1):\n",
    "    harmonics = np.asarray([1.0 / h * sine(domain, freq=freq * h) for h in range(1, Harmonics)])\n",
    "    wave = np.sum(harmonics, axis=0)\n",
    "    return wave\n",
    "\n",
    "wave = saw(time_space())\n",
    "plot_signal_and_spectrum(time_space(), wave)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can invert a saw wave to get what many synths call a \"blade\" wave. Instead of gradually falling and then spiking up, blade waves gradually rise and the sharply fall."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_signal_and_spectrum(time_space(), -1 * saw(time_space()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Square Wave\n",
    "\n",
    "Square waves are made of every other harmonic above the fundamental frequency. Again, the amplitude of each harmonic _n_ is _1 / n_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def square(domain, freq=1):\n",
    "    harmonics = np.asarray([1.0 / h * sine(domain, freq=freq * h) for h in range(1, Harmonics, 2)])\n",
    "    wave = np.sum(harmonics, axis=0)\n",
    "    return wave\n",
    "\n",
    "wave = square(time_space())\n",
    "plot_signal_and_spectrum(time_space(), wave)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Triangle Wave"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def triangle(domain, freq=1):\n",
    "    def label(i):\n",
    "        return 2. * i + 1.\n",
    "    harmonics = np.asarray([((-1) ** h) * (label(h) ** -2) * sine(domain, freq=freq * label(h)) for h in range(Harmonics)])\n",
    "    wave = np.sum(harmonics, axis=0)\n",
    "    return wave\n",
    "\n",
    "wave = triangle(time_space())\n",
    "plot_signal_and_spectrum(time_space(), wave)"
   ]
  }
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