Characteristics of Sound: Pitch, Loudness & Quality

Play middle-A on a violin and on a flute, both perfectly in tune, both equally loud, and you will never once confuse them. Yet by every number you have met so far they are the same sound: the same frequency, so the same pitch; the same intensity, so the same loudness. Something else, something you have not yet named, tells your ear "violin" and "flute" in a fraction of a second. That something is the third characteristic of sound, and pinning down all three — what makes a sound high or low, loud or soft, and this instrument rather than that one — is the whole business of this article. The pattern is clean: three things you hear map onto three things you can measure. Pitch — how high or low — is your ear reading the frequency. Loudness — how strong — is your ear reading the intensity, but through a logarithmic filter, which is why sound is measured in decibels. And quality or timbre — the violin-ness of the violin — is your ear reading the shape of the wave, that is, the mix of harmonics riding on top of the fundamental. Same fundamental, different harmonics, different instrument. The one genuinely mathematical part is loudness, because the ear is logarithmic and does something strange with intensity: a sound ten times as intense sounds only a bit louder, not ten times. That is why we measure sound level as $L = 10\log_{10}(I/I_0)$ in decibels, and why the rules that fall out of it — $+3\ \mathrm{dB}$ when you double the intensity, $+10\ \mathrm{dB}$ for ten times, $-6\ \mathrm{dB}$ every time you double your distance from the source — trip up so many students. Two identical singers are not twice as loud as one; they are $3\ \mathrm{dB}$ louder. This article makes all of that exact, and safe to use in an exam.

Key takeaways

Three perceptions, three physical causes: pitch ↔ frequency, loudness ↔ intensity, quality (timbre) ↔ waveform / harmonic content.
Higher frequency means higher pitch; an octave is a doubling of frequency, so $A4 = 440\ \mathrm{Hz}$ jumps to $A5 = 880\ \mathrm{Hz}$ .
The ear responds to intensity logarithmically, so sound level is measured in decibels: $L = 10\log_{10}(I/I_0)$ with $I_0 = 10^{-12}\ \mathrm{W/m^2}$ .
Reference points: threshold $I_0 \to 0\ \mathrm{dB}$ , normal conversation $I = 10^{-6}\ \mathrm{W/m^2} \to 60\ \mathrm{dB}$ , pain $I = 1\ \mathrm{W/m^2} \to 120\ \mathrm{dB}$ .
The decibel rules: doubling the intensity adds $3.01\ \mathrm{dB}$ , ten times adds $10\ \mathrm{dB}$ , doubling the distance subtracts $6.02\ \mathrm{dB}$ (intensity falls as $1/r^2$ ).
Quality is set by the harmonics, not the fundamental: a violin and a flute at the same pitch differ only in their harmonic mix.
Key idea: pitch reads frequency, loudness reads intensity through the logarithmic decibel scale $L = 10\log_{10}(I/I_0)$ , and quality reads the harmonic content — so two equal sources give only $+3\ \mathrm{dB}$ , not double the loudness.

The core insight

The three characteristics of sound are pitch (set by frequency), loudness (set by intensity, but measured logarithmically as $L = 10\log_{10}(I/I_0)$ in decibels), and quality or timbre (set by the mix of harmonics in the waveform); the logarithm is why doubling intensity adds only $3\ \mathrm{dB}$ and doubling distance subtracts $6\ \mathrm{dB}$ .

Where this lives today

The three characteristics are the design language of all audio technology. A graphic equaliser lets you boost or cut particular harmonic bands — reshaping timbre without touching pitch — and a pitch-correction plug-in shifts frequency while trying to preserve the harmonic recipe that makes a voice sound human. Every noise regulation, from the $85\ \mathrm{dB}$ workplace limit that protects factory workers' hearing to the night-time limits on traffic and aircraft, is written in decibels precisely because the ear is logarithmic and a linear scale would be useless. Acoustic engineers exploit the $1/r^2$ rule and its $6\ \mathrm{dB}$ -per-doubling consequence when they place loudspeakers and predict how far a sound will carry. And Gabor's spectrogram, born from the same analysis that makes "timbre" precise, is now the everyday tool of speech recognition, music software, birdsong identification and medical ultrasound imaging — a picture of frequency against time that separates pitch, loudness and quality onto three visible axes. The next article turns from the sound we hear directly to what happens when it strikes a wall and comes back — echoes, reverberation and SONAR.

The physical picture

Start with the mapping, because getting it straight prevents most of the mistakes in this topic. Pitch is the sensation of how high or low a note is, and it tracks the frequency of the sound: more vibrations per second, higher pitch. A piccolo is high-frequency, a tuba low. Loudness is the sensation of how strong the sound is, and it tracks the intensity — the power the wave delivers per square metre — which in turn grows with the square of the pressure amplitude. Quality, or timbre, is the sensation that distinguishes two instruments playing the same note at the same volume, and it tracks the shape of the waveform: real instruments never produce a pure single-frequency tone but a fundamental plus a characteristic set of overtones (harmonics), and it is that recipe of harmonics which the ear recognises as "oboe" or "guitar." The third one is the most surprising, so look at it directly. A pure tone — the sound of a tuning fork or a signal generator — is a clean sine wave. A violin playing the same note produces a wave with the same fundamental period (hence the same pitch) but a jagged, harmonic-rich shape; a flute playing that note is much closer to a pure sine, with only weak overtones. Three waveforms, one pitch, three timbres.

Pitch and the octave

Pitch follows frequency, and the ear judges pitch by ratios, not differences. The most important ratio is the octave — a doubling of frequency — which the ear hears as "the same note, higher." Concert A is $A4 = 440\ \mathrm{Hz}$ ; the A an octave above it, $A5$ , is exactly twice that, $880\ \mathrm{Hz}$ , and the A an octave below is $220\ \mathrm{Hz}$ . Every musical interval is a frequency ratio, which is why the audible range of $20\ \mathrm{Hz}$ to $20\ \mathrm{kHz}$ — a ratio of $1000$ — spans about ten octaves. Loudness does not affect pitch, and (to a good approximation) neither does timbre: the violin and flute agree on the note precisely because they share a fundamental frequency.

Loudness and the decibel

Loudness is where the physics gets quantitative, because the ear is not a linear meter. It responds to the logarithm of intensity, spanning an enormous range — from the faintest audible whisper to the threshold of pain is a factor of a trillion in intensity — by compressing it. To match the ear, we define the sound level in decibels relative to a fixed reference intensity $I_0 = 10^{-12}\ \mathrm{W/m^2}$ , the quietest sound a healthy ear can detect:

L = 10\log_{10}\!\left(\frac{I}{I_0}\right) \qquad (1)

where: $L$ is the sound level in decibels (dB), $I$ the intensity of the sound in $\mathrm{W/m^2}$ , and $I_0 = 10^{-12}\ \mathrm{W/m^2}$ the threshold of hearing. The reference itself sits at $L = 10\log_{10}(1) = 0\ \mathrm{dB}$ . Normal conversation, at about $I = 10^{-6}\ \mathrm{W/m^2}$ , comes out at $L = 10\log_{10}(10^{-6}/10^{-12}) = 10\log_{10}(10^{6}) = 60\ \mathrm{dB}$ . The threshold of pain, at about $I = 1\ \mathrm{W/m^2}$ , is $L = 10\log_{10}(10^{12}) = 120\ \mathrm{dB}$ . That whole span of human hearing, a factor of $10^{12}$ in intensity, is compressed into the tidy range $0$ to $120\ \mathrm{dB}$ — the logarithm's gift. Running the definition backwards recovers the intensity from a given level — the form you need whenever a decibel figure is quoted and an intensity is wanted:

I = I_0\,10^{L/10} \qquad (2)

where: the same symbols appear, now solved for $I$ . This is the shape behind every "what intensity does $85\ \mathrm{dB}$ correspond to" question. Because the scale is logarithmic, equal steps in decibels are equal ratios of intensity, and three ratios come up constantly. Double the intensity and the level rises by $10\log_{10}(2) = 3.01\ \mathrm{dB}$ . Multiply it by ten and the level rises by $10\log_{10}(10) = 10\ \mathrm{dB}$ . And moving away from a point source, the intensity falls as the inverse square of the distance, $I \propto 1/r^2$ — so doubling your distance quarters the intensity, dropping the level by $10\log_{10}(1/4) = -6.02\ \mathrm{dB}$ . These three numbers — $+3$ , $+10$ , $-6$ — are the working currency of the whole subject.

Figure

A decibel ladder from the 0 dB threshold of hearing through a 30 dB whisper, 60 dB conversation and 90 dB traffic to the 120 dB threshold of pain, with a reminder that intensity falls as one over distance squared so the level drops 6 dB every time the distance doubles

Quality: the harmonics do the talking

Return to the violin and the flute. Both produce a fundamental at, say, $440\ \mathrm{Hz}$ , which fixes the pitch. But a real instrument also radiates the harmonics — $880\ \mathrm{Hz}$ , $1320\ \mathrm{Hz}$ , and so on — in amounts characteristic of its shape and material. The violin is rich in strong high harmonics (its jagged waveform); the flute keeps mostly the fundamental with only faint overtones (its near-sine waveform). Your ear performs, in effect, a frequency analysis of the incoming wave and reads the recipe of harmonics as the instrument's identity. This is Ohm's acoustic law (1843): the ear resolves a complex tone into its pure-tone components, and Helmholtz (1863) showed that it is the relative strengths of those components — the harmonic content — that fix the quality. Change the recipe and you change the instrument; the fundamental, and so the pitch, can stay exactly the same. Timbre is the shape of the wave made audible.

Derivation AThe decibel, and the +3 / +10 / −6 rules

The goal is to build the decibel scale from the single fact that the ear judges ratios of intensity, and then to extract the three rules — $+3\ \mathrm{dB}$ per doubling, $+10\ \mathrm{dB}$ per tenfold, $-6\ \mathrm{dB}$ per doubling of distance — that answer almost every numerical question in this topic. The strategy is to define the level as a logarithm of the intensity ratio, then feed in the ratios $2$ , $10$ and $1/4$ and read off the level changes. No new physics enters; it is the logarithm doing all the work.

Step 1. Define the level relative to the threshold $I_0 = 10^{-12}\ \mathrm{W/m^2}$ : $L = 10\log_{10}(I/I_0)$ . The factor $10$ turns bels into decibels; the logarithm is there because the ear compresses ratios into equal steps.

Step 2. Anchor three points. $I = I_0$ gives $L = 10\log_{10}(1) = 0\ \mathrm{dB}$ . Conversation $I = 10^{-6}$ gives $L = 10\log_{10}(10^{6}) = 60\ \mathrm{dB}$ . Pain $I = 1$ gives $L = 10\log_{10}(10^{12}) = 120\ \mathrm{dB}$ . A trillion-fold range of intensity becomes a span of $120\ \mathrm{dB}$ .

Step 3. A change of level depends only on the ratio of intensities, because $L_2 - L_1 = 10\log_{10}(I_2/I_0) - 10\log_{10}(I_1/I_0) = 10\log_{10}(I_2/I_1)$ . Doubling: $10\log_{10}(2) = 3.01\ \mathrm{dB}$ . Tenfold: $10\log_{10}(10) = 10\ \mathrm{dB}$ . The reference $I_0$ has cancelled — level differences are absolute.

Step 4. The distance rule. A point source spreads its power over a sphere of area $4\pi r^2$ , so $I \propto 1/r^2$ . Doubling the distance takes $I$ to $I/4$ , and $\Delta L = 10\log_{10}(1/4) = -6.02\ \mathrm{dB}$ . So every doubling of distance costs about $6\ \mathrm{dB}$ , and the three rules $+3$ , $+10$ , $-6$ are all just the logarithm applied to the ratios $2$ , $10$ and $\tfrac14$ . Verified: $0$ , $60$ and $120\ \mathrm{dB}$ from $I = I_0, 10^{-6}, 1\ \mathrm{W/m^2}$ , and the deltas $10\log_{10}2 = 3.01$ , $10\log_{10}10 = 10$ , $10\log_{10}(0.25) = -6.02\ \mathrm{dB}$ , are recomputed in verify_hs2-characteristics-of-sound.py.

Go deeper · Derivation B · 1 minWhy two equal sources add only 3 dB, not double the loudness

Derivation BWhy two equal sources add only 3 dB, not double the loudness

Here we derive the counter-intuitive result that ten violins are barely twice as loud as one, by tracking what actually adds when independent sources play together: their intensities, not their decibel levels and not (for incoherent sources) their amplitudes. The goal is to show that $n$ equal incoherent sources raise the level by $10\log_{10}(n)$ , so two give $+3\ \mathrm{dB}$ and ten give $+10\ \mathrm{dB}$ . The strategy is to add intensities and pass the total through the decibel formula.

Step 1. Independent sources (different players, uncorrelated waves) carry energy that simply adds, so their intensities add: two equal sources each of intensity $I$ give a total $2I$ , and $n$ of them give $nI$ . Amplitudes would add only if the sources were perfectly in phase (coherent), which independent instruments are not.

Step 2. Feed the total into equation (1). The level of one source is $L_1 = 10\log_{10}(I/I_0)$ ; the level of $n$ together is $L_n = 10\log_{10}(nI/I_0) = 10\log_{10}(n) + 10\log_{10}(I/I_0) = L_1 + 10\log_{10}(n)$ .

Step 3. Read off the cases. Two sources: $L_2 - L_1 = 10\log_{10}(2) = 3.01\ \mathrm{dB}$ . Ten sources: $10\log_{10}(10) = 10\ \mathrm{dB}$ . So a $60\ \mathrm{dB}$ talker joined by an equal one gives $63\ \mathrm{dB}$ , not $120$ ; ten equal talkers give $70\ \mathrm{dB}$ .

Step 4. Interpret it. Because the ear is logarithmic, a $+10\ \mathrm{dB}$ rise (ten times the intensity) is heard as only about twice as loud. So ten identical violins sound roughly twice as loud as one, and doubling the players adds a modest $3\ \mathrm{dB}$ . This is why choirs and orchestras need so many musicians to sound powerful — and why you can never get $120\ \mathrm{dB}$ by adding two $60\ \mathrm{dB}$ sources. Verified: $10\log_{10}(2) = 3.01\ \mathrm{dB}$ (giving $60 \to 63$ ) and $10\log_{10}(10) = 10\ \mathrm{dB}$ (giving $60 \to 70$ ) are recomputed in verify_hs2-characteristics-of-sound.py.

Limiting cases

case	result	meaning
$I = I_0 = 10^{-12}\ \mathrm{W/m^2}$	$L = 0\ \mathrm{dB}$	threshold of hearing — the reference, not silence
$I = 10^{-6}\ \mathrm{W/m^2}$	$L = 60\ \mathrm{dB}$	normal conversation
$I = 1\ \mathrm{W/m^2}$	$L = 120\ \mathrm{dB}$	threshold of pain
double the intensity	$+3.01\ \mathrm{dB}$	two equal sources, or twice the power
ten times the intensity	$+10\ \mathrm{dB}$	heard as only about twice as loud
double the distance	$-6.02\ \mathrm{dB}$	$I \propto 1/r^2$ from a point source
same $f$ , different harmonics	same pitch, different timbre	the violin-versus-flute distinction

Common trapDecibels are logarithms — you cannot add them like ordinary numbers

The single most damaging error in this topic is treating decibels as if they added arithmetically. Two $60\ \mathrm{dB}$ sources together are not $120\ \mathrm{dB}$ — that would be the threshold of pain. Because the scale is logarithmic, what adds is the intensity: two equal sources double the intensity, and doubling intensity adds only $10\log_{10}(2) = 3\ \mathrm{dB}$ , giving $63\ \mathrm{dB}$ . To combine two different levels you must convert each back to an intensity with $I = I_0\,10^{L/10}$ , add the intensities, and convert the sum back to decibels — never add the decibel numbers. The same logarithm means "twice as loud" is not "twice the decibels": a subjective doubling of loudness is roughly $+10\ \mathrm{dB}$ , i.e. ten times the intensity. And keep the three characteristics separate: pitch is frequency, loudness is intensity, quality is harmonics — a louder sound is not a higher-pitched one.

Worked examples

Board level — sound levels from intensities (CBSE and ICSE pattern)

The threshold of hearing is $I_0 = 10^{-12}\ \mathrm{W/m^2}$ . Find the sound level in decibels of (a) normal conversation at $I = 10^{-6}\ \mathrm{W/m^2}$ and (b) a sound at the threshold of pain, $I = 1\ \mathrm{W/m^2}$ . (c) By how many decibels does the level rise if the intensity of the conversation is doubled?

Step 1. Conversation, equation (1): $L = 10\log_{10}(10^{-6}/10^{-12}) = 10\log_{10}(10^{6}) = 10 \times 6 = \boxed{60\ \mathrm{dB}}$ .

Step 2. Pain: $L = 10\log_{10}(1/10^{-12}) = 10\log_{10}(10^{12}) = 10 \times 12 = \boxed{120\ \mathrm{dB}}$ . Notice the whole audible range, a factor of $10^{12}$ in intensity, fits between $0$ and $120\ \mathrm{dB}$ .

Step 3. Doubling the intensity raises the level by $\Delta L = 10\log_{10}(2) = \boxed{3.01\ \mathrm{dB}}$ , so the conversation goes from $60\ \mathrm{dB}$ to about $63\ \mathrm{dB}$ — a small change, because the ear works logarithmically. It does not go to $120\ \mathrm{dB}$ ; that would need a million-fold increase in intensity, not a doubling.

Competitive level — combining sources, distance and a workplace limit (NEET and JEE pattern)

A machine produces a sound level of $60\ \mathrm{dB}$ at a certain point. (a) A second, identical machine is switched on beside it; what is the new level? (b) You then walk to twice the distance from the pair; by how much does the level drop? (c) Separately, a workplace safety limit is set at $85\ \mathrm{dB}$ ; what intensity does that correspond to?

Step 1. Two equal sources double the intensity, adding $10\log_{10}(2) = 3.01\ \mathrm{dB}$ . So the level rises from $60\ \mathrm{dB}$ to $\boxed{63\ \mathrm{dB}}$ — not $120\ \mathrm{dB}$ . Adding a second identical source is a modest gain, not a doubling of the number.

Step 2. Intensity falls as $1/r^2$ , so doubling the distance quarters the intensity, changing the level by $10\log_{10}(1/4) = \boxed{-6.02\ \mathrm{dB}}$ . From $63\ \mathrm{dB}$ you would drop to about $57\ \mathrm{dB}$ .

Step 3. Use equation (2) for the $85\ \mathrm{dB}$ limit: $I = I_0\,10^{L/10} = 10^{-12} \times 10^{8.5} = \boxed{3.16 \times 10^{-4}\ \mathrm{W/m^2}}$ . That is about $3.16 \times 10^{8}$ times the threshold intensity — which is why sustained exposure at $85\ \mathrm{dB}$ is a genuine hazard, even though the number " $85$ " looks unremarkable on the compressed decibel scale. Timing note: about two minutes. The combining-sources part is a one-liner once you remember that intensities add and $10\log_{10}2 = 3\ \mathrm{dB}$ — do not fall for " $60 + 60 = 120$ ." For the inversion in part (c), $L/10 = 8.5$ so $I = I_0 \times 10^{8.5}$ ; keep the exponent, do not try to take a log.

Exam traps

Adding decibels arithmetically. $60\ \mathrm{dB} + 60\ \mathrm{dB} \ne 120\ \mathrm{dB}$ . Two equal sources give $+3\ \mathrm{dB}$ (intensities double), so the answer is $63\ \mathrm{dB}$ . To combine unequal levels, convert to intensities, add, convert back. This single mistake accounts for most lost marks in the topic.

Confusing "twice as loud" with "twice the decibels" or "twice the intensity." A perceived doubling of loudness is about $+10\ \mathrm{dB}$ , i.e. ten times the intensity. Doubling the intensity is only $+3\ \mathrm{dB}$ , a barely noticeable change. The decibel number is not proportional to loudness in any simple way.

Pitch ↔ frequency, not amplitude. Turning up the volume makes a note louder, not higher. Pitch is set by frequency alone; loudness by intensity (amplitude squared). Keep the two axes separate — a favourite trap is a graph where the amplitude changes and the wavelength does not.

Quality is the harmonics, not the fundamental. Two instruments at the same pitch share a fundamental; what differs is the mix of overtones. Do not attribute timbre to "a different frequency" — the fundamental is the same, and the overtones are what carry the instrument's signature.

Forgetting the inverse-square distance rule. Intensity from a point source falls as $1/r^2$ , so the level drops $6\ \mathrm{dB}$ (not $3$ , not linearly) for every doubling of distance. Halving the distance adds $6\ \mathrm{dB}$ .

Board scope. CBSE, ICSE and NEET expect the decibel formula $L = 10\log_{10}(I/I_0)$ and the $+3/+10/-6$ rules. IGCSE 0625 treats pitch↔frequency and loudness↔amplitude only qualitatively — it does not require the decibel calculation, so an IGCSE candidate should learn the mapping but not the logarithms. A-Level (CIE 9702) covers intensity and its proportionality to amplitude squared, but its treatment of pitch, loudness and quality as such is light — check what your specification actually lists.

Origins · 1 minDennis Gabor and how this was discovered

Dennis Gabor

1900–1979 · London

Dennis Gabor was a Hungarian-British engineer who won the 1971 Nobel Prize for inventing holography, but an earlier paper of his is, almost literally, the mathematics of what makes a sound recognisable. In Theory of Communication (1946) Gabor argued that any signal can be broken into elementary "acoustic quanta" — little packets each localised in both time and frequency — so that a sound is described by where its energy sits in frequency and how that changes in time. In exactly this language, pitch is where the energy is centred in frequency and timbre is how the energy is distributed among the harmonics. Gabor's construction is the direct ancestor of the spectrogram, the time-frequency picture that shows, at a glance, why a violin and a flute at the same pitch look completely different. He did not study hearing as such; he built the framework in which the three characteristics of sound become precise.

History

1638

— Galileo Galilei ties the pitch of a sound to the frequency of its vibration.

1843

— Georg Ohm states his acoustic law: the ear resolves a complex tone into its pure-tone harmonic components.

1863

— Hermann von Helmholtz, in On the Sensations of Tone, identifies the quality of a sound with its harmonic content.

1928

— Bell Telephone Laboratories renames its 1924 "transmission unit" the decibel — one-tenth of a bel, named for Alexander Graham Bell — to measure sound and signal levels.

1933

— Harvey Fletcher and Wilden Munson map the ear's equal-loudness curves, showing loudness depends on frequency as well as intensity.

1946

— Dennis Gabor's Theory of Communication splits a signal into time-frequency acoustic quanta, the ancestor of the spectrogram.

Going further

Texts. NCERT Physics Class 11, Ch. 15 ("Waves"), the sections on characteristics of sound and intensity · H. C. Verma, Concepts of Physics Vol. 1, "Sound Waves," intensity and the decibel · Cambridge IGCSE 0625 (pitch, loudness and their qualitative link to frequency and amplitude) · CIE A-Level 9702 (intensity $\propto$ amplitude squared).

Provenance. Standard results. Recompute: $L = 10\log_{10}(I/I_0)$ giving $0\ \mathrm{dB}$ at $I_0$ , $60\ \mathrm{dB}$ at $10^{-6}$ , $120\ \mathrm{dB}$ at $1\ \mathrm{W/m^2}$ ; the deltas $10\log_{10}2 = 3.01$ , $10\log_{10}10 = 10$ , $10\log_{10}(0.25) = -6.02\ \mathrm{dB}$ ; the two-source result $60 \to 63\ \mathrm{dB}$ and ten sources $60 \to 70\ \mathrm{dB}$ ; the $85\ \mathrm{dB}$ intensity $I_0\,10^{8.5} = 3.16\times10^{-4}\ \mathrm{W/m^2}$ ; the octave $A4 \to A5 = 440 \to 880\ \mathrm{Hz}$ ; and the ladder points $30\ \mathrm{dB} (10^{-9})$ and $90\ \mathrm{dB} (10^{-3})$ . All are checked in verify_hs2-characteristics-of-sound.py.

Prerequisites. Sound as a Longitudinal Mechanical Wave — frequency, wavelength and pressure amplitude, the raw material the ear reads as pitch, loudness and quality. Leads to. Reflection of Sound: Echo, Reverberation and SONAR — what these characterised sounds do when they bounce off surfaces.

Go deeper (UG). The Fourier decomposition that makes "harmonic content" precise, and Gabor's time-frequency analysis behind the spectrogram, sit in the university signals and acoustics tracks; the propagation of the intensity itself is developed in Elastic Waves in Solids. Enrichment only; nothing here depends on it.

Why is the decibel scale logarithmic instead of just using intensity directly?

Because the ear itself is logarithmic, and the range it covers is vast. From the faintest audible sound to the threshold of pain is a factor of about $10^{12}$ — a trillion — in intensity, and a linear scale would be hopeless: normal conversation would be a millionth of the way up a scale whose top is pain. The ear instead judges ratios: a given multiplicative increase in intensity feels like a fixed increase in loudness, whatever the starting point. The decibel, $L = 10\log_{10}(I/I_0)$ , matches that by turning the enormous multiplicative range into a manageable additive one from $0$ to $120\ \mathrm{dB}$ . It also makes the useful rules simple — every $\times 10$ in intensity is $+10\ \mathrm{dB}$ , every doubling $+3\ \mathrm{dB}$ — because logarithms turn multiplication into addition.

If two 60 dB sources aren't 120 dB, what level do they actually make?

About $63\ \mathrm{dB}$ . Sound levels do not add; intensities do. Two equal sources each at $60\ \mathrm{dB}$ have the same intensity, and together they carry twice that intensity. Doubling the intensity raises the level by $10\log_{10}(2) = 3.01\ \mathrm{dB}$ , so the pair sits at about $63\ \mathrm{dB}$ , not $120$ . To reach $120\ \mathrm{dB}$ (the threshold of pain) from $60\ \mathrm{dB}$ you would need a million times the intensity, not twice. For two different levels, the recipe is: convert each to an intensity with $I = I_0\,10^{L/10}$ , add the intensities, and convert the total back with the decibel formula. Adding the decibel numbers directly is always wrong.

What is the difference between intensity, amplitude and loudness?

They are three linked but distinct ideas. Amplitude is the size of the pressure swing in the wave. Intensity is the power the wave delivers per unit area, and it grows with the square of the amplitude — double the pressure amplitude and you quadruple the intensity. Loudness is the subjective sensation, what your ear reports, and it depends on intensity only through the logarithm (the decibel scale) and also on frequency, since the ear is more sensitive in the mid-range than at the extremes. So a sound can have large amplitude, hence large intensity, hence a high decibel level, and still be perceived as only moderately loud if it is at a frequency the ear handles poorly. Exam questions usually mean intensity and its decibel level when they say "loudness."

How does the ear tell a violin from a flute if they play the same note?

By the harmonics. When either instrument plays $440\ \mathrm{Hz}$ , it produces not just that pure tone but a fundamental at $440\ \mathrm{Hz}$ plus overtones at $880$ , $1320\ \mathrm{Hz}$ and so on, in amounts fixed by the instrument's shape and material. The violin is rich in strong high harmonics — its waveform is jagged — while the flute is close to a pure sine with only weak overtones. The ear, in effect, breaks the incoming wave into its frequency components (Ohm's acoustic law) and recognises the pattern of harmonic strengths as the instrument's signature. The fundamental, and so the pitch, is identical; the recipe of overtones is what differs. This is exactly what a spectrogram displays, and it is why "quality" or "timbre" is a real, measurable property, not a vague one.

Why does moving twice as far away drop the level by 6 dB and not 3 dB?

Because intensity from a point source falls as the square of the distance. The source radiates a fixed power outward over an expanding sphere of area $4\pi r^2$ , so the power per unit area — the intensity — goes as $1/r^2$ . Doubling the distance therefore quarters the intensity (not halves it), and a quartering is $10\log_{10}(1/4) = -6.02\ \mathrm{dB}$ . A halving of intensity would be $-3\ \mathrm{dB}$ , but doubling distance gives a quartering, hence $-6\ \mathrm{dB}$ . The rule to remember: every doubling of distance from a point source costs about $6\ \mathrm{dB}$ , and every halving of distance gains $6\ \mathrm{dB}$ . Real rooms complicate this with reflections, but in the open the $6\ \mathrm{dB}$ figure is a reliable estimate.

Do pitch, loudness and quality ever affect one another?

To a good first approximation they are independent — that independence is the whole point of separating them. Pitch is set by frequency, loudness by intensity, quality by harmonic content, and you can change any one while holding the others fixed: turn up the volume (loudness) without changing the note (pitch) or the instrument (quality). There are small real-world couplings that a careful examiner might mention — the ear's sensitivity varies with frequency, so a very quiet high note can seem to "disappear" before an equally quiet mid-range one, and extremely loud sounds can shift perceived pitch slightly — but for board and competitive physics you should treat the three as independent axes. That clean separation is exactly why the mapping "pitch–frequency, loudness–intensity, quality–harmonics" is worth memorising.