3.0 Periodicity, period, and frequency

Periodicity is the foundation of musical sound. Sounds that we recognize as having a definite “pitch” — a plucked guitar string, a sustained vowel, a flute tone — are periodic in nature. Their waveforms repeat, more or less, at a regular rate. Consider what happens when you pluck a guitar string: the string oscillates back and forth, and the resulting air pressure variations create a waveform whose shape recurs over and over:

Classical guitar, F3, plucked without vibrato. 154030 by Carlos_Vaquero, License: Attribution NonCommercial 4.0.

Fig. 2 A plucked guitar string (F3). Top: the pluck and decay over roughly 1.7 seconds. Bottom: zoomed in to about 23 milliseconds, where the quasi-periodic repetition of the waveform shape is clearly visible.

The waveform in the zoomed view above is not perfectly repetitive. Instead, it’s what acousticians call quasi-periodic, meaning the shape changes slowly over time as the note decays. But over short time scales, the repetition is strikingly regular.

Periodicity

Here we formalize the notion of periodicity, following notational conventions from [McF23]. A continuous signal \(x(t)\) is periodic with period \(T\) if

\[x(t + T) = x(t) \quad \text{for all } t \in \mathbb{R}.\]

The fundamental period \(t_0\) is the smallest strictly positive \(T\) satisfying this condition.

Definition 3 (Periodicity)

A signal \(x(t) : \mathbb{R} \to \mathbb{R}\) is periodic if there exists a finite \(T > 0\) such that \(x(t + T) = x(t)\) for all \(t \in \mathbb{R}\). The fundamental period \(t_0\) is the smallest such \(T\).

If \(t_0\) is the fundamental period, then all integer multiples of \(t_0\) must also be periods:

\[x(t) = x(t + t_0) = x(t + 2 \cdot t_0) = x(t + 3 \cdot t_0) = \ldots\]

More generally, \(x(t) = x(t + k \cdot t_0)\) for any \(k \in \mathbb{Z}\).

Frequency

One full repetition of a periodic waveform is referred to as a cycle. The fundamental period \(t_0\) tells us how long one cycle takes, in units of \(\frac{\text{seconds}}{\text{cycle}}\). Its reciprocal is frequency — how many cycles fit in one second, in units of \(\frac{\text{cycles}}{\text{second}}\):

\[f_0 = \frac{1}{t_0}.\]

This relationship follows directly from the units: if \(t_0\) has units \(\frac{\text{seconds}}{\text{cycle}}\), then \(1/t_0\) has units \(\frac{\text{cycles}}{\text{second}}\).

Definition 4 (Fundamental frequency)

The fundamental frequency of a signal \(x(t)\) with fundamental period \(t_0\) is \(f_0 = 1 / t_0\).

Frequency is measured in Hertz (Hz), where 1 Hz = 1 \(\frac{\text{cycle}}{\text{second}}\).

A waveform with \(t_0 = 0.5\,\mathrm{s}\) has \(f_0 = 2\,\mathrm{Hz}\) (top). Compressing the same shape into a quarter of a second gives \(t_0 = 0.25\,\mathrm{s}\) and \(f_0 = 4\,\mathrm{Hz}\) (bottom).

Frequency is the property most strongly associated with our perception of musical pitch: higher frequencies sound higher in pitch, lower frequencies sound lower. Accordingly, periods follow the opposite rule: the shorter the period, the higher the perceived pitch. If you’ve studied music, each line on a musical staff corresponds to a specific fundamental frequency. We will have more to say about pitch perception in later chapters.