What determines pitch and loudness?

The characteristics that define how we experience sound—whether it is high or low, soft or roaring—are rooted directly in the physics of the wave itself. The perceived qualities we call pitch and loudness are not abstract concepts; they correspond precisely to measurable attributes of the sound wave as it travels through a medium like air or water. ^[5]^[6] Understanding these determinations moves us from simply hearing a sound to understanding why that sound has the quality it does.

# Pitch and Frequency

The shrillness or flatness of a musical tone, which we perceive as its pitch, is directly linked to how quickly the source is vibrating. ^[7][^8] The technical measurement for this is the wave's frequency. ^[1]^[5]

Frequency quantifies the rate at which oscillations occur—the number of complete wave cycles that pass a fixed point in one second. ^[1]^[2] The standard unit for this measurement is the Hertz ( $\text{Hz}$ ), where $1 \text{ Hz}$ signifies one cycle per second. ^[1]^[4] A high-pitched sound, like a whistle or a woman's voice, corresponds to a high frequency, meaning the sound waves arrive in rapid succession. ^[2]^[7] Conversely, a low-pitched sound, such as a lion's roar or a bass drum, has a low frequency with slower vibrations. ^[2]^[7]

For instance, the standard musical note 'A' above middle $\text{C}$ is often set precisely at $440 \text{ Hz}$ . ^[1]^[4] This means the source vibrates $440$ times every second. ^[1] Related to frequency is the period, which is the time it takes for a single oscillation to complete. ^[1] The relationship is inverse: $\text{Period} = 1/\text{frequency}$ . ^[1] If a string vibrates at $440 \text{ Hz}$ , its period is $1 / 440$ of a second, or about $2.27$ milliseconds. ^[1]

Human hearing is sensitive to a specific band of frequencies, generally ranging from about $20 \text{ Hz}$ to $20,000 \text{ Hz}$ (or $20 \text{ kHz}$ ). ^[5]^[7] Frequencies below this range are classified as infrasonic, and those above are ultrasonic—both typically inaudible to humans, though some animals can perceive them. ^[7][^8]

Musically, pitches whose frequencies are exact multiples of one another share a fundamental sonic relationship; this interval is called an octave. [^8] If one note vibrates at $100 \text{ Hz}$ , the note an octave higher vibrates at $200 \text{ Hz}$ , and the next octave up is $400 \text{ Hz}$ . [^8] This consistency is why musicians can train themselves to identify the intervals between tones, a skill known as relative pitch. [^8] A small change in frequency results in a slight difference in pitch, which is why musicians can tune instruments by listening for the difference between two close frequencies. ^[4]

It is fascinating to see how this mechanical property translates into physical reality. Consider a simple experiment where a plastic ruler is held on a table, with a portion hanging off the edge. When struck, the ruler vibrates, producing a tone. ^[7] If you hang a long length of the ruler over the edge, it vibrates slowly, yielding a low pitch because the effective length allows for a longer period of oscillation. ^[7][^8] Pulling more of the ruler onto the table shortens the overhanging length, forcing quicker, tighter vibrations, which we hear as a higher pitch. ^[7] The physical constraint—how easily the object can move back and forth—is what sets the fundamental frequency, which our ears interpret as pitch. ^[1]

# Loudness and Amplitude

While pitch deals with the "highness or lowness," loudness (often referred to as volume in casual conversation) deals with the "noisiness or quietness" of a sound. ^[4]^[6]^[7] This auditory sensation is determined by the amplitude of the sound wave. ^[1]^[4]

Amplitude is the measure of the wave's intensity or strength—specifically, the maximum displacement of the air pressure from its normal resting state as the wave propagates. ^[3]^[6][^8] A sound wave with a large amplitude involves greater variations in air pressure and will be perceived as loud; a wave with a small amplitude will sound feeble or quiet. ^[2]^[6]

A critical physical relationship exists between amplitude and the energy carried by the wave. The energy exerted by the sound wave is directly proportional to the square of the amplitude. ^[5][^8] This means that if you manage to double the wave's amplitude, the energy—and thus the perceived loudness—increases by a factor of four ( $2^{2}$ ). ^[5]

The loudness we perceive is influenced by several factors beyond the wave's raw amplitude:

Distance from Source: Sound energy spreads out spherically as it travels. ^[5] This spreading causes the intensity to decrease rapidly the farther one moves from the source, following an inverse square law ( $I \propto 1/r^2$ ). ^[5] Doubling your distance from the source reduces the intensity to one-fourth of its previous level. ^[5]
Vibrating Surface Area: A larger surface area that vibrates can move a greater volume of air, increasing the energy transmitted and resulting in a louder sound, even if the initial plucking force (amplitude) is the same. ^[2] This is why instruments often employ soundboards or resonance boxes. ^[6]
Intervening Material: Obstacles between the source and the listener absorb or reflect sound energy, decreasing the amplitude that reaches the ear. ^[1]

When two identical sound waves combine, their amplitudes add together. If they are perfectly in phase, the resulting wave has double the amplitude, making the sound twice as loud. ^[4] Conversely, if two waves are exactly half a cycle out of phase, they cancel each other out, resulting in silence—the principle behind noise-cancelling technology. ^[4]

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# Measuring Intensity: The Decibel Scale

While amplitude can be measured objectively in terms of displacement (like micrometers) or in terms of sound intensity ( $I$ measured in Watts per square meter, $W / m^{2}$ ), ^[4]^[5] the range of intensities the human ear can handle is enormous—from the threshold of hearing ( $10^{-12} \text{ W/m}^2$ ) to levels that cause damage ( $10^3 \text{ W/m}^2$ ). ^[5] To manage these vast numbers, acousticians use the decibel ( $\text{dB}$ ) scale, which is a measure of the Sound Intensity Level ( $\text{SIL}$ ). ^[4]^[5]

The $\text{dB}$ scale is logarithmic, meaning it relates to the ratio of sound intensity relative to the threshold of hearing ( $I_{o}$ ): ^[5]
$\text{SIL} (\text{dB}) = 10 \log \left(\frac{I}{I_o}\right)$
This logarithmic scale has some key practical implications. A ten-fold increase in physical intensity ( $W / m^{2}$ ) only adds $10 \text{ dB}$ to the level. ^[5] More subtly, a two-fold increase in the measured intensity (the physical power) results in an addition of only $3 \text{ dB}$ to the sound level. ^[5]

Here is a brief comparison of sound intensity levels for context:

Source	Approximate Loudness (SIL)
Normal Breathing	$10 \text{ dB}$
Mellow Whisper (at $5 \text{ m}$ )	$30 \text{ dB}$
Basic Conversation	$60 \text{ dB}$
Factory/Heavy Traffic	$70-80 \text{ dB}$
Jackhammer/Jet Takeoff	$120-130 \text{ dB}$

If one instrument produces $40 \text{ dB}$ and you add a second identical instrument, the total level isn't $80 \text{ dB}$ ( $40 + 40$ ); it's $43 \text{ dB}$ (since doubling the intensity adds $3 \text{ dB}$ ). ^[5] Adding a third instrument raises it slightly more, and adding a fourth brings it to $46 \text{ dB}$ ( $40 + 3 + 3$ ). ^[5] This mathematical property explains why stacking many quiet sources can sometimes still produce a sound that feels only moderately louder, whereas doubling a very loud source results in a more dramatic jump in perceived level relative to the scale.

From an energy perspective, the inverse square law is exceptionally harsh when dealing with sound propagation outdoors. Imagine a performer on a stage. If you stand $10$ feet away, you receive a certain level of sound energy. If you move to $20$ feet away, your distance has doubled, meaning the sound energy spread over the area of your ears is only $1 / 4$ as intense. To an audience member, this difference in physical intensity translates to a $6 \text{ dB}$ drop in perceived loudness ( $3 \text{ dB}$ for the first doubling, and another $3 \text{ dB}$ for the second doubling of distance from the original point). ^[5] This rapid drop highlights why sound engineers must strategically place speakers for large outdoor venues to compensate for distance attenuation.

# Subjectivity and Perception

Although pitch is primarily determined by fundamental frequency and loudness by amplitude/intensity, our ears are not perfect scientific instruments; they are biological structures evolved for survival. ^[5] Therefore, the subjective perception does not always map linearly to the objective laboratory measurement. ^[5]

One major deviation lies in loudness perception. The $\text{dB}$ scale measures intensity level ( $\text{SIL}$ ), but the human perception of loudness is measured in phons. ^[5] For sounds around $1000 \text{ Hz}$ , the phon and $\text{dB}$ readings generally agree. ^[5] However, the ear is significantly more sensitive to frequencies in the middle range ( $1000 \text{ Hz}$ to $5000 \text{ Hz}$ ) than it is to very low or very high frequencies. ^[5] For example, a sound wave measured at $36 \text{ dB}$ in the lab at $4000 \text{ Hz}$ might be perceived by the listener as being as loud as a $45 \text{ dB}$ sound at $1000 \text{ Hz}$ . ^[5] This enhanced sensitivity in the speech range is believed to be due to the acoustic properties of the ear canal itself. ^[5]

This perceptual nuance also affects pitch. While frequency determines pitch, if a sound is extremely loud, high frequencies may be perceived as slightly higher in pitch than they objectively are, and low frequencies slightly lower. ^[5]

Furthermore, the ability to distinguish changes—the Just Noticeable Difference ( $\text{JND}$ )—varies. People are better at telling if two sounds are different in loudness when the sounds are already loud ( $\text{JND}$ is smaller, around $0.5 \text{ dB}$ at $80 \text{ dB}$ ) than when they are quiet ( $\text{JND}$ is larger, around $1.5 \text{ dB}$ at $40 \text{ dB}$ ). ^[5] Similarly, our ability to distinguish small pitch differences degrades at higher frequencies. ^[5]

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# Tone Quality

There is a third critical component of sound that separates two sounds even if their measured pitch ( $\text{Hz}$ ) and loudness ( $\text{dB}$ ) are identical: timbre. ^[1][^8] Timbre, or tone quality, is what allows us to instantly tell the difference between a violin and a trumpet playing the exact same note at the exact same volume. ^[1]^[4]

Timbre arises from the waveform—the shape of the wave when plotted against time. ^[1][^8] Most sounds from musical instruments or the human voice are not pure sine waves; they are complex sounds composed of the fundamental frequency (which sets the perceived pitch) plus a series of other frequencies, called overtones or harmonics, present at different amplitudes. ^[5][^8]

The specific mix and relative strength of these higher frequencies determine the resulting waveform's shape. ^[5] A pure tone from a tuning fork produces a simple sine wave because it contains only one frequency. [^8] A clarinet, however, produces a sound wave with a complex shape governed by the relative levels of its fundamental frequency and its higher harmonics, giving it its unique, characteristic tone color. ^[1][^8] In essence, pitch is the address on the scale, loudness is the volume knob setting, and timbre is the unique fingerprint of the sound source encoded in the wave's complexity. ^[1]