Sinusoids

import numpy as np
import matplotlib.pyplot as plt
import IPython.display as ipd

$ f(t) = \sin(t), t \in [0, 2 \pi]$

t = np.linspace(0, 8*np.pi, 100)
plt.plot(t, np.sin(t))
plt.plot(t, np.cos(t))
plt.xlabel("t (in radians)")
plt.legend(["$\sin(t)$", "$\cos(t)$"])

<matplotlib.legend.Legend at 0x7f62304c2910>

$ f(t) = A \cos(2 \pi f t + \phi)$

$ f(t) = A \cos(\frac{2 \pi t}{T} + \phi)$

• A = amplitude
• f = frequency (in hz, or cycles per second)
• T = period = 1/f (the length of each repetition, in seconds)
• t = time
• $\phi$ = the phase

A = 1
f = 3
t = np.linspace(0, 5, 1000)
phi = np.pi/3
y = A*np.cos(2*np.pi*f*t + phi)
plt.plot(t, y)
plt.ylim([-5, 5])

(-5.0, 5.0)

sr = 44100
f1 = 440
f2 = 660
t = np.arange(sr)/sr
y = np.cos(2*np.pi*f1*t) #+ np.cos(2*np.pi*f2*t)
ipd.Audio(y, rate=sr)

# Sawtooth wave
t = np.arange(sr)
y = t % 100
plt.plot(y[0:1000])
ipd.Audio(y, rate=sr)

## Formula for frequency of musical notes
### Equal-tempered scale
note  A, A#, B, C, C#, D, D#, E, F, F#, G, G#, A  A#
p     0  1   2  3  4   5  6   7  8  9   10 11  12 13

$f = 440 * 2^{p/12}$

def get_note_freq(p):
    return 440*2**(p/12)

t = np.arange(int(sr/2))/sr
f = get_note_freq(0)
y = np.array([])
for p in range(12):
    f = get_note_freq(p)
    y = np.concatenate((y, np.cos(2*np.pi*f*t)))
ipd.Audio(y, rate=sr)