intro-solutions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[back to intro](intro.ipynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.367253Z",
     "iopub.status.busy": "2020-11-26T13:08:21.366615Z",
     "iopub.status.idle": "2020-11-26T13:08:21.646657Z",
     "shell.execute_reply": "2020-11-26T13:08:21.647063Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.657225Z",
     "iopub.status.busy": "2020-11-26T13:08:21.656639Z",
     "iopub.status.idle": "2020-11-26T13:08:21.665053Z",
     "shell.execute_reply": "2020-11-26T13:08:21.665539Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4, 5])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.670059Z",
     "iopub.status.busy": "2020-11-26T13:08:21.669463Z",
     "iopub.status.idle": "2020-11-26T13:08:21.671947Z",
     "shell.execute_reply": "2020-11-26T13:08:21.672450Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0. , 0.1, 0.2, 0.3, 0.4, 0.5]), array([0. , 0.1, 0.2, 0.3, 0.4, 0.5]))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(0, 0.6, 0.1), np.arange(6) * 0.1  # two possibilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.676131Z",
     "iopub.status.busy": "2020-11-26T13:08:21.675543Z",
     "iopub.status.idle": "2020-11-26T13:08:21.678005Z",
     "shell.execute_reply": "2020-11-26T13:08:21.678391Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.5, 0.6, 0.7, 0.8, 0.9, 1. , 1.1]), '<-- wrong result!')"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(0.5, 1.1, 0.1), \"<-- wrong result!\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.682129Z",
     "iopub.status.busy": "2020-11-26T13:08:21.681450Z",
     "iopub.status.idle": "2020-11-26T13:08:21.683883Z",
     "shell.execute_reply": "2020-11-26T13:08:21.684285Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.5, 0.6, 0.7, 0.8, 0.9, 1. ]), \"<-- that's right!\")"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(5, 11) * 0.1, \"<-- that's right!\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.687917Z",
     "iopub.status.busy": "2020-11-26T13:08:21.687360Z",
     "iopub.status.idle": "2020-11-26T13:08:21.689883Z",
     "shell.execute_reply": "2020-11-26T13:08:21.690362Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 2., 3., 4., 5., 6.])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(0, 6, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.694598Z",
     "iopub.status.busy": "2020-11-26T13:08:21.694051Z",
     "iopub.status.idle": "2020-11-26T13:08:21.696437Z",
     "shell.execute_reply": "2020-11-26T13:08:21.696855Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0., 1., 2., 3., 4., 5.]), array([0., 1., 2., 3., 4., 5.]))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(0, 6, 6, endpoint=False), np.linspace(0, 5, 6)  # two possibilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.700897Z",
     "iopub.status.busy": "2020-11-26T13:08:21.700376Z",
     "iopub.status.idle": "2020-11-26T13:08:21.702893Z",
     "shell.execute_reply": "2020-11-26T13:08:21.703378Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0. , 0.1, 0.2, 0.3, 0.4, 0.5]), array([0. , 0.1, 0.2, 0.3, 0.4, 0.5]))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(0, 0.6, 6, endpoint=False), np.linspace(0, 0.5, 6)  # again two possibilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.707715Z",
     "iopub.status.busy": "2020-11-26T13:08:21.707083Z",
     "iopub.status.idle": "2020-11-26T13:08:21.709553Z",
     "shell.execute_reply": "2020-11-26T13:08:21.709978Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.5, 0.6, 0.7, 0.8, 0.9, 1. ]), array([0.5, 0.6, 0.7, 0.8, 0.9, 1. ]))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(0.5, 1.1, 6, endpoint=False), np.linspace(0.5, 1, 6)  # and again ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the number of elements is known and the step size should be obtained automatically $\\Rightarrow$ `np.linspace()`  \n",
    "If the step size is known an if it's an integer and the number of elements should be obtained automatically $\\Rightarrow$ `np.arange()`\n",
    "\n",
    "If the step size is not an integer:\n",
    "\n",
    "* If the step size is a fraction of integers, you can use `np.arange()` with integers and divide the result accordingly.\n",
    "\n",
    "* If that's not feasible, calculate the expected number of elements beforehand and use `np.linspace()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.714178Z",
     "iopub.status.busy": "2020-11-26T13:08:21.713610Z",
     "iopub.status.idle": "2020-11-26T13:08:21.717376Z",
     "shell.execute_reply": "2020-11-26T13:08:21.717860Z"
    }
   },
   "outputs": [],
   "source": [
    "dur, amp, freq, fs = 1, 0.3, 500, 44100\n",
    "t = np.arange(np.ceil(dur * fs)) / fs\n",
    "y = amp * np.sin(2 * np.pi * freq * t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "alternative (but inferior) methods to get $t$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.721871Z",
     "iopub.status.busy": "2020-11-26T13:08:21.721329Z",
     "iopub.status.idle": "2020-11-26T13:08:21.723807Z",
     "shell.execute_reply": "2020-11-26T13:08:21.724274Z"
    }
   },
   "outputs": [],
   "source": [
    "t1 = np.arange(0, dur, 1/fs)  # implicit rounding of dur!\n",
    "t2 = np.arange(0, np.round(dur), 1/fs)  # still problematic: arange with floats\n",
    "# wrong if dur isn't an integer multiple of 1/fs:\n",
    "t3 = np.linspace(0, dur, np.round(dur * fs), endpoint=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Length of `y` must be *exactly* 44100 (using a half-open interval for $t$), not 44101 (which would be longer than 1 second).\n",
    "\n",
    "Plotting: 2 ways to zoom (there are probably more): draw a rectangle, drag with the right mouse button in pan/zoom mode.\n",
    "\n",
    "Clicks? Because of discontinuities (also in the derivatives) $\\Rightarrow$ Fade in/out! See [tools.fade()](tools.py)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:21.727905Z",
     "iopub.status.busy": "2020-11-26T13:08:21.727310Z",
     "iopub.status.idle": "2020-11-26T13:08:22.803546Z",
     "shell.execute_reply": "2020-11-26T13:08:22.803108Z"
    }
   },
   "outputs": [],
   "source": [
    "import sounddevice as sd\n",
    "import tools\n",
    "\n",
    "def myplay(data):\n",
    "    \"\"\"Apply fade in/out and play with 44.1 kHz.\"\"\"\n",
    "    data = tools.fade(data, 2000, 5000)\n",
    "    sd.play(data, 44100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:22.806314Z",
     "iopub.status.busy": "2020-11-26T13:08:22.805689Z",
     "iopub.status.idle": "2020-11-26T13:08:22.836772Z",
     "shell.execute_reply": "2020-11-26T13:08:22.836245Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:22.840208Z",
     "iopub.status.busy": "2020-11-26T13:08:22.839763Z",
     "iopub.status.idle": "2020-11-26T13:08:22.841440Z",
     "shell.execute_reply": "2020-11-26T13:08:22.841868Z"
    }
   },
   "outputs": [],
   "source": [
    "def mysine(frequency, amplitude, duration):\n",
    "    \"\"\"Generate sine tone with the given parameters @ 44.1 kHz.\"\"\"\n",
    "    samplerate = 44100\n",
    "    times = np.arange(np.ceil(duration * samplerate)) / samplerate\n",
    "    return amplitude * np.sin(2 * np.pi * frequency * times)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:22.844689Z",
     "iopub.status.busy": "2020-11-26T13:08:22.844161Z",
     "iopub.status.idle": "2020-11-26T13:08:22.851247Z",
     "shell.execute_reply": "2020-11-26T13:08:22.850280Z"
    }
   },
   "outputs": [],
   "source": [
    "z = mysine(440, 0.4, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:22.855419Z",
     "iopub.status.busy": "2020-11-26T13:08:22.854783Z",
     "iopub.status.idle": "2020-11-26T13:08:23.067815Z",
     "shell.execute_reply": "2020-11-26T13:08:23.067378Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:23.071648Z",
     "iopub.status.busy": "2020-11-26T13:08:23.071253Z",
     "iopub.status.idle": "2020-11-26T13:08:23.765922Z",
     "shell.execute_reply": "2020-11-26T13:08:23.766329Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Using matplotlib backend: MacOSX\n"
     ]
    }
   ],
   "source": [
    "%matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def myplot(data):\n",
    "    \"\"\"Create a simple plot @ 44.1 kHz.\"\"\"\n",
    "    samplerate = 44100\n",
    "    times = np.arange(len(data)) / samplerate\n",
    "    plt.plot(times, data)\n",
    "    plt.xlabel(\"Time / Seconds\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:23.769397Z",
     "iopub.status.busy": "2020-11-26T13:08:23.768987Z",
     "iopub.status.idle": "2020-11-26T13:08:23.956377Z",
     "shell.execute_reply": "2020-11-26T13:08:23.956790Z"
    }
   },
   "outputs": [],
   "source": [
    "myplot(mysine(440, 0.4, 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:23.960687Z",
     "iopub.status.busy": "2020-11-26T13:08:23.960269Z",
     "iopub.status.idle": "2020-11-26T13:08:23.982524Z",
     "shell.execute_reply": "2020-11-26T13:08:23.982918Z"
    }
   },
   "outputs": [],
   "source": [
    "import soundfile as sf\n",
    "\n",
    "dur, amp = 1, 0.3\n",
    "frequencies = 400, 500, 600  # Hz\n",
    "fadetime = 2000  # samples\n",
    "\n",
    "for freq in frequencies:\n",
    "    sig = mysine(freq, amp, dur)\n",
    "    sig = tools.fade(sig, fadetime)\n",
    "    sf.write(\"sine_{}hz.wav\".format(freq), sig, 44100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:23.987127Z",
     "iopub.status.busy": "2020-11-26T13:08:23.986639Z",
     "iopub.status.idle": "2020-11-26T13:08:23.997781Z",
     "shell.execute_reply": "2020-11-26T13:08:23.998178Z"
    }
   },
   "outputs": [],
   "source": [
    "from scipy import signal\n",
    "\n",
    "f0, f1 = 100, 5000  # Hz\n",
    "amp = 0.2\n",
    "dur = 2  # seconds\n",
    "fadetime = 2000  # samples\n",
    "fs = 44100\n",
    "\n",
    "t = np.arange(np.ceil(dur * fs)) / fs\n",
    "\n",
    "for method in 'linear', 'log':\n",
    "    sweep = amp * signal.chirp(t, f0, dur, f1, method)\n",
    "    sweep = tools.fade(sweep, fadetime)\n",
    "    sf.write('sweep_{}.wav'.format(method), sweep, fs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.001635Z",
     "iopub.status.busy": "2020-11-26T13:08:24.001235Z",
     "iopub.status.idle": "2020-11-26T13:08:24.008584Z",
     "shell.execute_reply": "2020-11-26T13:08:24.008162Z"
    }
   },
   "outputs": [],
   "source": [
    "sinetone = mysine(frequency=500, amplitude=0.3, duration=1.5)\n",
    "noise = np.random.normal(scale=0.1, size=len(sinetone))\n",
    "sine_plus_noise = sinetone + noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.011715Z",
     "iopub.status.busy": "2020-11-26T13:08:24.010821Z",
     "iopub.status.idle": "2020-11-26T13:08:24.143437Z",
     "shell.execute_reply": "2020-11-26T13:08:24.143875Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(sine_plus_noise)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.146837Z",
     "iopub.status.busy": "2020-11-26T13:08:24.146366Z",
     "iopub.status.idle": "2020-11-26T13:08:24.152536Z",
     "shell.execute_reply": "2020-11-26T13:08:24.152079Z"
    }
   },
   "outputs": [],
   "source": [
    "myplot(sine_plus_noise)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.155700Z",
     "iopub.status.busy": "2020-11-26T13:08:24.155237Z",
     "iopub.status.idle": "2020-11-26T13:08:24.161051Z",
     "shell.execute_reply": "2020-11-26T13:08:24.160585Z"
    }
   },
   "outputs": [],
   "source": [
    "dur = 2\n",
    "amp = 0.2\n",
    "\n",
    "two_sines = mysine(500, amp, dur) + mysine(507, amp, dur)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.164145Z",
     "iopub.status.busy": "2020-11-26T13:08:24.163740Z",
     "iopub.status.idle": "2020-11-26T13:08:24.296020Z",
     "shell.execute_reply": "2020-11-26T13:08:24.296436Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(two_sines)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.300038Z",
     "iopub.status.busy": "2020-11-26T13:08:24.299402Z",
     "iopub.status.idle": "2020-11-26T13:08:24.306560Z",
     "shell.execute_reply": "2020-11-26T13:08:24.306010Z"
    }
   },
   "outputs": [],
   "source": [
    "myplot(two_sines)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two sine tones with similar frequencies create \"beats\", see <http://en.wikipedia.org/wiki/Beat_(acoustics)>.\n",
    "The sum of these two tones is equivalent to an amplitude modulation with a carrier frequency of $\\frac{f_1+f_2}{2}$ and a modulation frequency of $\\frac{f_1-f_2}{2}$.\n",
    "\n",
    "$$\\cos(2\\pi f_1t)+\\cos(2\\pi f_2t) = 2\\cos\\left(2\\pi\\frac{f_1+f_2}{2}t\\right)\\cos\\left(2\\pi\\frac{f_1-f_2}{2}t\\right)$$\n",
    "\n",
    "We don't really *hear* the modulation frequency itself, we only hear the envelope of the modulation, therefore the *perceived* beat frequency is $f_{\\text{beat}} = f_1-f_2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.310222Z",
     "iopub.status.busy": "2020-11-26T13:08:24.309651Z",
     "iopub.status.idle": "2020-11-26T13:08:24.317385Z",
     "shell.execute_reply": "2020-11-26T13:08:24.316794Z"
    }
   },
   "outputs": [],
   "source": [
    "stereo_sines = np.column_stack([mysine(400, amp, dur), mysine(600, amp, dur)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.320633Z",
     "iopub.status.busy": "2020-11-26T13:08:24.320040Z",
     "iopub.status.idle": "2020-11-26T13:08:24.454081Z",
     "shell.execute_reply": "2020-11-26T13:08:24.454563Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(stereo_sines)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first column should be the left channel!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.459030Z",
     "iopub.status.busy": "2020-11-26T13:08:24.458457Z",
     "iopub.status.idle": "2020-11-26T13:08:24.462828Z",
     "shell.execute_reply": "2020-11-26T13:08:24.462293Z"
    }
   },
   "outputs": [],
   "source": [
    "dur, amp = 1, 0.3\n",
    "freq = 500  # Hz\n",
    "delay = 0.5  # ms\n",
    "fs = 44100\n",
    "\n",
    "t = np.arange(np.ceil(dur * fs)) / fs\n",
    "times = np.column_stack((t, t - delay/1000))\n",
    "sig = amp * np.sin(2 * np.pi * freq * times)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.466734Z",
     "iopub.status.busy": "2020-11-26T13:08:24.466207Z",
     "iopub.status.idle": "2020-11-26T13:08:24.599331Z",
     "shell.execute_reply": "2020-11-26T13:08:24.598858Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(sig)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:24.603967Z",
     "iopub.status.busy": "2020-11-26T13:08:24.603379Z",
     "iopub.status.idle": "2020-11-26T13:08:38.179691Z",
     "shell.execute_reply": "2020-11-26T13:08:38.179251Z"
    }
   },
   "outputs": [],
   "source": [
    "dur, amp = 0.5, 0.3\n",
    "frequencies = 500, 1000, 2000  # Hz\n",
    "delays = 0.6, 0.4, 0.2, 0, -0.2, -0.4, -0.6  # ms\n",
    "fs = 44100\n",
    "\n",
    "t = np.arange(np.ceil(dur * fs)) / fs\n",
    "\n",
    "for f in frequencies:\n",
    "    for delay in delays:\n",
    "        times = np.column_stack((t, t - delay/1000))\n",
    "        sig = amp * np.sin(2 * np.pi * f * times)\n",
    "        myplay(sig)\n",
    "        sd.wait()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is supposed to illustrate [Lord Rayleigh's Duplex Theory](http://en.wikipedia.org/wiki/Interaural_time_difference#Duplex_theory) (at least the part about time differences)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.861153Z",
     "iopub.status.busy": "2020-11-26T13:08:38.860604Z",
     "iopub.status.idle": "2020-11-26T13:08:38.864081Z",
     "shell.execute_reply": "2020-11-26T13:08:38.863669Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.00000000e+00],\n",
       "       [2.26757370e-05],\n",
       "       [4.53514739e-05],\n",
       "       ...,\n",
       "       [1.99993197e+00],\n",
       "       [1.99995465e+00],\n",
       "       [1.99997732e+00]])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dur, amp = 2, 0.3\n",
    "frequencies = np.array([200, 400, 600, 800, 1000])\n",
    "fs = 44100\n",
    "t = np.arange(np.ceil(dur * fs)) / fs\n",
    "t.shape = -1, 1\n",
    "t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.867386Z",
     "iopub.status.busy": "2020-11-26T13:08:38.866823Z",
     "iopub.status.idle": "2020-11-26T13:08:38.869352Z",
     "shell.execute_reply": "2020-11-26T13:08:38.868932Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.3  , 0.15 , 0.1  , 0.075, 0.06 ])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "amplitudes = amp * 1 / np.arange(1, len(frequencies)+1)\n",
    "amplitudes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.872011Z",
     "iopub.status.busy": "2020-11-26T13:08:38.871614Z",
     "iopub.status.idle": "2020-11-26T13:08:38.883250Z",
     "shell.execute_reply": "2020-11-26T13:08:38.883650Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(88200, 5)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "five_sines = amplitudes * np.sin(2 * np.pi * frequencies * t)\n",
    "five_sines.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.886322Z",
     "iopub.status.busy": "2020-11-26T13:08:38.885903Z",
     "iopub.status.idle": "2020-11-26T13:08:38.889360Z",
     "shell.execute_reply": "2020-11-26T13:08:38.889764Z"
    }
   },
   "outputs": [],
   "source": [
    "sum_of_sines = five_sines.sum(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.892232Z",
     "iopub.status.busy": "2020-11-26T13:08:38.891833Z",
     "iopub.status.idle": "2020-11-26T13:08:38.899304Z",
     "shell.execute_reply": "2020-11-26T13:08:38.898887Z"
    }
   },
   "outputs": [],
   "source": [
    "myplot(sum_of_sines)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.902259Z",
     "iopub.status.busy": "2020-11-26T13:08:38.901749Z",
     "iopub.status.idle": "2020-11-26T13:08:38.907458Z",
     "shell.execute_reply": "2020-11-26T13:08:38.907864Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(five_sines[:, [0, 1, 2, 3, 4]].sum(axis=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:38.910658Z",
     "iopub.status.busy": "2020-11-26T13:08:38.910190Z",
     "iopub.status.idle": "2020-11-26T13:08:39.042790Z",
     "shell.execute_reply": "2020-11-26T13:08:39.043212Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(five_sines[:, [0, 1, 2, 3]].sum(axis=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.046158Z",
     "iopub.status.busy": "2020-11-26T13:08:39.045697Z",
     "iopub.status.idle": "2020-11-26T13:08:39.179338Z",
     "shell.execute_reply": "2020-11-26T13:08:39.179750Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(five_sines[:, [0, 1, 2, 4]].sum(axis=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.182506Z",
     "iopub.status.busy": "2020-11-26T13:08:39.182030Z",
     "iopub.status.idle": "2020-11-26T13:08:39.315455Z",
     "shell.execute_reply": "2020-11-26T13:08:39.315005Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(five_sines[:, [0, 1, 3, 4]].sum(axis=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.318904Z",
     "iopub.status.busy": "2020-11-26T13:08:39.318324Z",
     "iopub.status.idle": "2020-11-26T13:08:39.450570Z",
     "shell.execute_reply": "2020-11-26T13:08:39.451241Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(five_sines[:, [0, 2, 3, 4]].sum(axis=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.454538Z",
     "iopub.status.busy": "2020-11-26T13:08:39.454083Z",
     "iopub.status.idle": "2020-11-26T13:08:39.586180Z",
     "shell.execute_reply": "2020-11-26T13:08:39.585762Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(five_sines[:, [1, 2, 3, 4]].sum(axis=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<https://en.wikipedia.org/wiki/Harmonic_series_(music)>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.589805Z",
     "iopub.status.busy": "2020-11-26T13:08:39.589273Z",
     "iopub.status.idle": "2020-11-26T13:08:39.591288Z",
     "shell.execute_reply": "2020-11-26T13:08:39.591680Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 200,  400,  600,  800, 1000, 1200, 1400, 1600, 1800, 2000, 2200,\n",
       "       2400, 2600, 2800, 3000, 3200, 3400, 3600, 3800, 4000])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f0 = 200  # Hz\n",
    "partials = 20\n",
    "\n",
    "frequencies = f0 * np.arange(1, partials + 1)\n",
    "frequencies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.594964Z",
     "iopub.status.busy": "2020-11-26T13:08:39.594484Z",
     "iopub.status.idle": "2020-11-26T13:08:39.596852Z",
     "shell.execute_reply": "2020-11-26T13:08:39.597252Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.3       , 0.15      , 0.1       , 0.075     , 0.06      ,\n",
       "       0.05      , 0.04285714, 0.0375    , 0.03333333, 0.03      ,\n",
       "       0.02727273, 0.025     , 0.02307692, 0.02142857, 0.02      ,\n",
       "       0.01875   , 0.01764706, 0.01666667, 0.01578947, 0.015     ])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "amplitudes = amp * 1 / np.arange(1, len(frequencies)+1)\n",
    "amplitudes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.599946Z",
     "iopub.status.busy": "2020-11-26T13:08:39.599545Z",
     "iopub.status.idle": "2020-11-26T13:08:39.634983Z",
     "shell.execute_reply": "2020-11-26T13:08:39.635372Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(88200, 20)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "many_sines = amplitudes * np.sin(2 * np.pi * frequencies * t)\n",
    "many_sines.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.638132Z",
     "iopub.status.busy": "2020-11-26T13:08:39.637712Z",
     "iopub.status.idle": "2020-11-26T13:08:39.641982Z",
     "shell.execute_reply": "2020-11-26T13:08:39.641451Z"
    }
   },
   "outputs": [],
   "source": [
    "sawtooth = many_sines.sum(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.644788Z",
     "iopub.status.busy": "2020-11-26T13:08:39.644338Z",
     "iopub.status.idle": "2020-11-26T13:08:39.651347Z",
     "shell.execute_reply": "2020-11-26T13:08:39.650906Z"
    }
   },
   "outputs": [],
   "source": [
    "myplot(sawtooth)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.654073Z",
     "iopub.status.busy": "2020-11-26T13:08:39.653665Z",
     "iopub.status.idle": "2020-11-26T13:08:39.786334Z",
     "shell.execute_reply": "2020-11-26T13:08:39.786783Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(sawtooth)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://en.wikipedia.org/wiki/Sawtooth_wave"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.789365Z",
     "iopub.status.busy": "2020-11-26T13:08:39.788974Z",
     "iopub.status.idle": "2020-11-26T13:08:39.792598Z",
     "shell.execute_reply": "2020-11-26T13:08:39.792176Z"
    }
   },
   "outputs": [],
   "source": [
    "square = many_sines[:, ::2].sum(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.795020Z",
     "iopub.status.busy": "2020-11-26T13:08:39.794619Z",
     "iopub.status.idle": "2020-11-26T13:08:39.801575Z",
     "shell.execute_reply": "2020-11-26T13:08:39.801161Z"
    }
   },
   "outputs": [],
   "source": [
    "myplot(square)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-26T13:08:39.804213Z",
     "iopub.status.busy": "2020-11-26T13:08:39.803797Z",
     "iopub.status.idle": "2020-11-26T13:08:39.936201Z",
     "shell.execute_reply": "2020-11-26T13:08:39.935736Z"
    }
   },
   "outputs": [],
   "source": [
    "myplay(square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://en.wikipedia.org/wiki/Square_wave"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
    "  <a rel=\"license\"\n",
    "     href=\"http://creativecommons.org/publicdomain/zero/1.0/\">\n",
    "    <img src=\"http://i.creativecommons.org/p/zero/1.0/88x31.png\" style=\"border-style: none;\" alt=\"CC0\" />\n",
    "  </a>\n",
    "  <br />\n",
    "  To the extent possible under law,\n",
    "  <span rel=\"dct:publisher\" resource=\"[_:publisher]\">the person who associated CC0</span>\n",
    "  with this work has waived all copyright and related or neighboring\n",
    "  rights to this work.\n",
    "</p>"
   ]
  }
 ],
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