Acoustics & Music Theory
why it sounds like that
Concept

Spectral envelope — where it appears

The overall shape of a spectrum, as distinct from the individual partials under it. It is what a resonance imposes and what stays put when the pitch changes, which is why it identifies a vowel and a speaker rather than a note.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The vowel in "hod", sung at 110 Hz. The partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.

The sound a listener knows best

A voice is recognisable across every vowel it says, across two octaves of pitch, down a bad telephone line and in a whisper where there is no pitch at all. Nothing that survives all of that can be a frequency. What survives is a ratio: the resonances of a vocal tract are set by its length, so a shorter tract multiplies every formant by the same factor, and identity is a scale on the spectral envelope rather than a position within it. Between an adult man and a child the whole pattern moves by a fifth, and the vowel does not change at all.

timbre · The voice A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there.

The mark that is not a level

There are six of them, they carry no units, and a performer has to turn one into a number before it means anything. What they instruct is not loudness. On a struck string a harder blow shortens the hammer's contact from 2.26 milliseconds to 0.95, which moves the first null of its own pulse from the third partial to the sixth: the partials between those are not quieter at pianissimo, they are gone. A fortissimo is a different sound, and the page has one word for both things it changes.

timbre · Notation One instrument, five spectra. The first 12 partials of a bowed string at 5 pitches, each passed through the same fixed body response and normalised to its own loudest partial. The resonances stay where they are and the partials slide under them, so the pattern is different at every note: the power-weighted centroid runs from 1.02 to 1.71 across the compass and is not monotonic in pitch. A source with no body at all would give 2.35 at every pitch, which is the single number every roughness figure here uses for a string.

An instrument is not one timbre

Every roughness verdict so far uses one partial list per instrument, and a violin does not have one. Its body's resonances stay where they are while the fundamental moves, so the radiated spectrum is different at every pitch: the power-weighted centroid runs from 1.02 to 1.64 across the compass, and it is not monotonic. Which means the result that a spectrum chooses its own scale gives a different scale at every note on the same instrument — three minima in the dissonance curve at the bottom of a violin's range, two in the middle and four at the top.

timbre · Spectrum How many intervals a duet is smooth at. Every ordered pairing of 5 radiators, with the number of wells in its dissonance curve over an octave from 262 hertz. Rows are the instrument underneath and columns the one above, so the grid is not symmetric about its diagonal and that asymmetry is the result. The count runs from 1 to 7 across the grid, and a cell and its mirror need not agree: a violin under a clarinet has 1 and a clarinet under a violin has 2. The fifth is a well in every one of the 25 pairings and no other interval is.

Which instrument is underneath

Every roughness curve so far compares two tones of the same timbre, which is a duet nobody plays. Give the two notes different instruments and the sum stops being symmetric: the same written interval, on the same two players, is up to five times rougher depending on which of them takes the lower note. From G3 upward the fifth is a well in all twenty-five pairings and no other interval is; below it, three pairings lose even that, and all three have a clarinet on top.

timbre · Spectrum Six ways to put three players on three notes. The same chord — G3, B♭3, D4 — played by clarinet, oboe, voice in all 6 possible assignments, scored by the roughness each produces. Every bar is the same pitches and the same instruments; only who is on which note changes. The worst is 1.42 times the best, which is a factor a score can control and a chord symbol cannot express at all. Each row is labelled from the bottom note upward.

Which player on which note

An interval's roughness depends on which instrument is underneath, so the pair does not commute. Three players over three notes is the smallest thing that asymmetry has anywhere to go: six assignments, all of them the same chord, and across 450 of them the roughest averages half again the smoothest and reaches six times it. It is orchestration in the only form that can be computed here — not which chord, and not which voicing, but who is on which note.

timbre · Spectrum C4: the pulse computed and the pulse assumed. Above, the force the hammer delivers to the string at C4, integrated forward against the felt's nonlinear force and the string's returning corner, drawn against the half-sine of 1.60 milliseconds that every earlier figure assumed. The computed contact lasts 2.21 milliseconds and the corner comes home 4.6 times inside it. Below, the excitation each pulse gives to each partial. They agree at the bottom and part company higher up — worst at partial 6, by 34 decibels — because the assumed pulse has nulls the computed one does not.

The pulse that was assumed

Every figure until now low-passes the string's excitation with the spectrum of a half-sine, which is what a hammer would deliver against a rigid wall. An earlier essay said so and declined to do better. Doing better takes forty lines and refuses the prediction that came with it: the corner's round trips govern the spectrum as expected, and the contact time is governed by something else entirely — the mass ratio discovered one essay earlier.

timbre · Excitation point What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series.

What the climb was a search for

The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose.

tuning · Pitch standard A struck note's two ends are the same for every loss law. The partial levels of a string spectrum struck at 80 decibels on 130.8 hertz, and what is left of it when the fundamental itself falls under the threshold of hearing, for three laws relating a partial's decay rate to its number. The left panel is every one of them: a loss law cannot change the spectrum at the instant of the strike, because no time has passed. The other three are every one of them too: whatever the law, the note ends with nothing above the threshold. So both ends of the slide are shared, and everything that distinguishes an exponent of 0.5 from an exponent of 1 from an exponent of 2 is in the middle.

The middle nobody could have guessed

A struck note has no steady state, only a slide from one spectrum to another — so the question is what the middle carries that the ends do not. The answer is exact rather than statistical: every loss law in the family leaves the strike with the same spectrum and ends in the same silence, so both endpoints carry precisely nothing about which of them it is. The whole difference is 41.3 decibels, and it peaks 0.38 seconds in, seven per cent of the way through the note.

timbre · Envelope

Named alongside it

The objects these essays reach for when they reach for this one.

BrightnessTimbreFormantRoughnessSource-filterCritical bandwidthOrchestrationResonanceSpectrumTransientAttack transientBody resonance

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