import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
matplotlib.RcParams._get = dict.get
3.1 The basic sinusoid#
The most elementary periodic function is the basic sinusoid:
It sounds like a “pure tone” — a smooth, featureless hum with no timbral complexity. As we’ll see later in this chapter, the basic sinusoid is a fundamental building block: all periodic sound, no matter how complex, can be expressed as a sum of basic sinusoids.
The basic sinusoid has three parameters: frequency \(f\), amplitude \(a\), and initial phase \(\phi\).
Fig. 3 The sinusoid \(x(t) = 0.8 \sin(2\pi \cdot 2 \, t)\) with \(a = 0.8\), \(f = 2\) Hz, \(\phi = 0\). The dashed red lines mark the amplitude bounds \(\pm a\), and the green dashed lines and arrow mark the period boundaries at \(1/f\).#
Frequency and angular frequency#
Frequency determines pitch. Listen to pure tones at three different frequencies — each sounds higher in pitch than the last:
Pure tones at 220, 330, and 440 Hz. Higher frequency means more cycles per second and a higher perceived pitch.
Why does the basic sinusoid with parameter \(f\) complete exactly \(f\) cycles per second? We can reason about this from the units, building up from what we know about \(\sin\).
Recall from trigonometry that \(\sin\) repeats itself with period \(2\pi\) \(\frac{\text{radians}}{\text{cycle}}\). In our basic sinusoid, at \(t = 1\) second, the argument to \(\sin\) will have accumulated \(2\pi f\) radians. This gives us the angular frequency:
To convert back to frequency in Hertz, we divide by \(2\pi\) \(\frac{\text{radians}}{\text{cycle}}\):
Definition 5 (Angular frequency)
The angular frequency of a sinusoid with frequency \(f\) \(\frac{\text{cycles}}{\text{second}}\) is \(\omega = 2\pi f\) \(\frac{\text{radians}}{\text{second}}\). Equivalently, \(f = \omega / (2\pi)\).
Angular frequency lets us write the basic sinusoid more compactly as \(x(t) = a\sin(\omega t + \phi)\). You will see both forms throughout this book — familiarize yourself with converting between \(f\) and \(\omega\).
Note
A more formal proof that the basic sinusoid has period \(1/f\).
For the mathematically inclined, we can derive this directly. Recall that \(\sin(x) = \sin(x + 2\pi)\):
Therefore \(x(t) = x(t + 1/f)\), confirming that \(x(t)\) is periodic with period \(1/f\). This holds regardless of the values of \(a\) and \(\phi\).
Amplitude#
Amplitude is a comparatively straightforward property. Recall from trigonometry that \(\sin(x) \in [-1, 1]\), so \(\sin\) has a maximum amplitude deviation of 1. Accordingly, \(a \sin(x) \in [-a, a]\), meaning our basic sinusoid has an amplitude of \(a\).
Amplitude determines loudness. Listen to the same 220 Hz tone at three different amplitudes:
Initial phase and instantaneous phase#
At a high level, phase characterizes our position within a cycle. In the basic sinusoid, phase appears in two forms:
The initial phase \(\phi\) — a constant offset in radians that shifts the waveform’s starting point.
The instantaneous phase \(\theta(t) = 2\pi f t + \phi = \omega t + \phi\) — the total phase of the sinusoid at time \(t\), in radians.
The instantaneous phase at time \(t\) equals the radians elapsed based on angular frequency \(\omega\) \(\frac{\text{radians}}{\text{second}}\) plus the initial offset \(\phi\) \(\text{radians}\). We can rewrite the basic sinusoid as \(x(t) = a \sin(\theta(t))\).
Example 1 (Instantaneous phase)
Consider our working example: \(f = 2\) Hz, \(\phi = \pi/2\). The angular frequency is \(\omega = 4\pi\) \(\frac{\text{radians}}{\text{second}}\), so \(\theta(t) = 4\pi t + \pi/2\). At a few specific times:
\(\theta(0) = \pi/2\) radians (the initial phase)
\(\theta(0.5) = 4\pi \cdot 0.5 + \pi/2 = 5\pi/2\) radians (one full period later)
\(\theta(1) = 4\pi \cdot 1 + \pi/2 = 9\pi/2\) radians (two full periods later)
Notice that \(\theta\) increases by \(2\pi\) radians each period — exactly one full cycle.
Our perception of phase differs from that of frequency and amplitude. Listen to a 220 Hz tone at three different initial phases:
The same frequency (220 Hz) and amplitude at three initial phases. The waveforms are visually shifted in time, but they sound nearly identical.
Aside from slightly different “clicks” at the onset and offset of the waveform (caused by the signal’s value at the very first and last sample), these tones sound essentially the same. This is a general property of human hearing: we are largely insensitive to the absolute phase of a sound. This perceptual insensitivity will become important when we discuss additive synthesis below.
The interactive below puts all three parameters in one place. Drag the sliders and watch the waveform respond.
Drag the sliders: the red curve is \(x(t) = a \sin(2 \pi f t + \phi)\), redrawn live. The gray curve keeps the starting parameters for reference, and the dashed teal lines mark one period \(t_0 = 1/f\). The audio card underneath always plays the current settings.