8.1 Sound as a spectrum — pitch, timbre, and the frequency axis
The Fourier transform (refresher: Fourier) gives us a second, equivalent way to describe a sound. Instead of pressure as a function of time, p(t), we look at amplitude as a function of frequency, ∣p~(ω)∣. The two pictures are mathematically equivalent — one is the Fourier transform of the other — but psychologically they are very different. What we hear as a sound aligns much more closely with the frequency-domain picture than with the time-domain waveform.
Pitch and pure tones
A sinusoidal pressure variation at frequency f Hz is heard as a pure tone of pitch f. The amplitude controls loudness; the frequency controls pitch.
Slide the frequency and hear the pitch change. Note two things:
The mapping from frequency to perceived pitch is logarithmic. Doubling the frequency raises the pitch by an octave; an interval that sounds like a perfect fifth is a frequency ratio of 3:2 (≈ 700 cents); the equal-tempered semitone is a ratio of 21/12≈1.0595.
The audible range spans roughly 20 Hz to 20 kHz for young humans — three orders of magnitude.
Pure tones are perceptually clean but musically lifeless. Real musical and speech sounds are richer.
Harmonics and timbre
A note from a violin or a flute at the same pitch as our sinusoid above doesn’t sound the same. Why?
Look at the spectrum. A bowed violin string playing A4 doesn’t produce a pure 440 Hz sinusoid — it produces a complex periodic waveform whose Fourier series has a large component at 440 Hz and additional components at 880, 1320, 1760, … Hz, the integer multiples. These higher components are the harmonics, and the relative amplitudes of the harmonics determine the timbre — the quality that distinguishes one instrument from another.
440 Hz
0.80
660 Hz
0.40
0°
presets:
Two sinusoids superposed are the simplest non-trivial periodic waveform. With a 2:1 frequency ratio you get something that sounds reedy. With 3:2 you get a perfect fifth — and the two notes blend into a single richer pitch. With a tiny ratio detuning (say 1.01:1) you get beats — a slow envelope at the difference frequency.
Why integer harmonics matter
Most musical instruments produce sounds with integer-related harmonics. The reason traces back to chapter 3: a string, a tube, a membrane, anything with simple boundary conditions on a 1-D or symmetric geometry, supports modes at integer multiples of a fundamental. The resulting sound is consonant — the harmonics align — which is what makes pitched instruments sound “musical”.
A bell, a cymbal, or a wood block has inharmonic overtones — modes at non-integer ratios of the fundamental. The resulting sound has a pitch that’s harder to identify and a quality that feels more percussive than melodic. The whole geography of acoustic music depends on this distinction.
Speech in the spectrum
Speech is even richer. The vocal folds vibrate at a fundamental frequency (around 120 Hz for adult males, 220 Hz for adult females) producing harmonics up to 5 kHz or more. The vocal tract — a closed-open tube of length about 17 cm — acts as a resonator, amplifying certain frequencies and attenuating others. The resonant peaks are called formants, and their positions encode vowel identity.
Ee (as in “beet”): F1 ≈ 280 Hz, F2 ≈ 2250 Hz.
Ah (as in “father”): F1 ≈ 770 Hz, F2 ≈ 1100 Hz.
Oo (as in “boot”): F1 ≈ 290 Hz, F2 ≈ 870 Hz.
Without the formant pattern the vocal-fold buzz is unintelligible. The vocal tract shapes it — and the shaping is what carries linguistic information. The Hearing book picks up this thread when speech meets the cochlea (chapter 4 there).
Why the frequency picture matters now
We need it because most acoustic systems — rooms, instruments, the ear — act on sounds frequency-by-frequency. The next three lessons make this operational:
8.2: spectrograms turn a time-domain recording into a time-frequency image where speech, music, and noise become legible.
8.3: acoustic filters and the room impulse response, viewed as multipliers in the frequency domain.
8.4: resonance is a Lorentzian peak in the frequency response, with width Δω=ω0/Q.
⏳The history— Helmholtz's resonators and the analysis of tone
Hermann von Helmholtz, in his 1863 Die Lehre von den Tonempfindungen, demonstrated that complex musical tones could be analysed into their Fourier components using a set of precisely tuned acoustic resonators — hollow brass spheres, each with a narrow opening, that amplified a single frequency from the ambient sound field. By holding different resonators to his ear, Helmholtz could identify the individual harmonics present in a sung vowel or a bowed string. The experiments provided the first empirical confirmation that Fourier’s mathematics described the physical reality of sound.
The resonators also let Helmholtz demonstrate that timbre — the quality distinguishing a violin from a flute playing the same note — is determined by the relative amplitudes and phases of the harmonics, not by the fundamental frequency alone. This insight connects the physics of sound (this chapter) to the neuroscience of hearing: the cochlea performs the same Fourier-like decomposition that Helmholtz did with his brass spheres, but continuously and in real time.