Pitch and frequency

Pitch is the sensation of a sound being "high" (shrill, thin) or "low" (deep, grave) — it is what lets you tell a flute's note from a drum's thump even when both are equally loud. Pitch is not something you can measure with a ruler directly; instead, it is our perception of a physical quantity of the source's vibration called frequency.

Frequency ($f$) is the number of complete vibrations (oscillations) a source makes in one second. Its SI unit is the hertz (Hz), where $1\ \text{Hz}$ means exactly one complete vibration per second. Every vibration also has a time period ($T$), the time taken for one complete vibration, measured in seconds. The two are reciprocals of each other:

where: $f$ is frequency in hertz (Hz), and $T$ is the time period in seconds (s). A short time period packs more vibrations into each second, so $f$ and $T$ move in opposite directions — halve $T$ and you double $f$. The rule linking pitch and frequency is direct: a higher frequency gives a higher (shriller) pitch, and a lower frequency gives a lower (deeper) pitch. Picture a displacement-time graph of the vibration: a high-frequency wave squeezes many crests and troughs into one second, while a low-frequency wave spreads a few crests over the same second.

Pitch must not be confused with loudness. Loudness depends on the amplitude of vibration (how big the swing is), while pitch depends on frequency (how fast the swing repeats). These two are controlled independently, so a sound can be loud and deep (a booming drum) or soft and shrill (a faint whistle). Musical instruments make this rule concrete. When a musician tightens a guitar string, or a singer strains to hit a "high" note, they are increasing the frequency of vibration of the string or the vocal cords, not making the sound louder. Similarly, the small, tightly stretched strings of a violin vibrate faster (higher frequency) than the long, loosely stretched strings of a double bass, which is why the violin's notes sit high and the double bass's notes sit low, regardless of how hard either instrument is played. The human ear can typically detect frequencies roughly between $20\ \text{Hz}$ and $20{,}000\ \text{Hz}$; sounds outside this range — infrasound below $20\ \text{Hz}$ and ultrasound above $20{,}000\ \text{Hz}$ — exist but are inaudible to us, even though other animals such as dogs and bats can detect some of them.

A tuning fork completes $256$ complete vibrations in exactly $1$ second, and a guitar string, plucked separately, takes $0.005\ \text{s}$ to complete one full vibration.

Step 1. For the tuning fork, the count of vibrations per second is already the frequency: $f = 256\ \text{Hz}$. Its time period follows from $T = 1/f$: $T = 1/256 \approx 0.0039\ \text{s}$, about $3.9$ milliseconds per vibration. Since $256\ \text{Hz}$ is a fairly high frequency, this fork produces a fairly high-pitched note — this is why tuning forks marked "256 Hz" are commonly used, as the frequency lies close to the musical note middle C.

Step 2. For the guitar string, the time period is given directly: $T = 0.005\ \text{s}$. Its frequency follows from $f = 1/T$: $f = 1/0.005 = 200\ \text{Hz}$.

Step 3. Now compare with a second, thicker guitar string whose time period is longer, $T = 0.01\ \text{s}$: $f = 1/0.01 = 100\ \text{Hz}$.

Step 4. Rank the three pitches by frequency: the tuning fork ($256\ \text{Hz}$) is the shrillest, the thin guitar string ($200\ \text{Hz}$) is next, and the thick guitar string ($100\ \text{Hz}$) is the deepest. The thick, slower-vibrating string has the lowest frequency and hence the lowest pitch — exactly why the thick strings on a guitar give the bass notes.

• Confusing pitch with loudness. A "high" sound is not the same as a "loud" sound: pitch (shrill/deep) tracks frequency, loudness (soft/loud) tracks amplitude, and the two vary independently.
• Getting the direction of $f = 1/T$ backwards. Since frequency and time period are reciprocals, a shorter time period means a higher frequency, not a lower one — students often guess the opposite.
• Assuming a bigger source always means higher pitch. Generally the reverse holds: longer or thicker strings and larger air columns vibrate more slowly, giving a lower pitch — the double bass is bigger than the violin but plays lower notes.
• Dropping the unit hertz. A frequency must be stated in hertz (Hz); writing just a bare number of vibrations is incomplete as a physical quantity.

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