Generator

Every point on one curve sounds equally loud

The equal-loudness contours of ISO 226:2003, evaluated from the standard's own parameters. The lowest curve is the threshold of hearing. Because the curves are not parallel — they crowd together in the bass and spread apart in the middle — the same change in decibels is a different change in loudness at every frequency, and a spectrum that was balanced at one level is not balanced at another.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 14 of its 14 placements carry sound, built from the same numbers as the drawing, offering these buttons: 50 Hz and 1 kHz at the same level, 50 Hz and 1 kHz, equally loud, Both 20 dB quieter, Both 40 dB quieter, Both 45 dB quieter, Both at 40 phons — the 50 Hz raised to match, One voice, Ten voices, ten decibels louder.

Called by 7 essays

the blast radius of changing it

The quietest thing audible, and why the volume knob is a tone control

A decibel is a fact about air. A phon is a fact about a listener, and the two do not line up — the map between them bends with frequency, and it bends differently at every level. One consequence is that turning a piece of music down does not turn all of it down equally, and the amount by which it does not is a number.

perception · Loudness

What one tone hides, and in which direction. The masked threshold beside a tone masker: any probe below one of these curves is inaudible while the masker sounds. The frequency axis is in Bark, the scale on which the ear's filters are evenly spaced, so the pattern is a pair of straight lines. The upper slope is much shallower than the lower one and gets shallower still as the masker gets louder — masking spreads upward, not downward.

One sound hides another, and it hides upward

A tone can be made completely inaudible by a second tone that is nowhere near it in frequency, and the region it disappears into is lopsided. Masking spreads up the spectrum and barely down it, and the reach grows with the masker's level — so which line in an arrangement vanishes is a prediction, not a matter of taste.

perception · Masking

The most a 6 cm cone can make of a low note. The maximum sound pressure level at one metre from a circular radiator of effective radius 3.2 cm, moving 1.5 mm at its limit, in a system resonating at 250 Hz. Below resonance the cone is already at that limit and the pressure a piston makes goes as the square of frequency, so the curve falls at twelve decibels an octave: 66 dB at 40 Hz, 78 dB at 80 Hz, 94 dB at 200 Hz. The 40 Hz figure is 28 decibels below the 200 Hz one, and that gap is arithmetic about a radius and a displacement rather than a property of any particular loudspeaker. The dots are the harmonics of a 41.2 Hz note with a one-over-n source spectrum; the loudest of them is the 6th.

The bass a small loudspeaker does not make

A three-inch cone at its excursion limit produces sixty-six decibels at forty hertz, which the ear converts to seventeen phons — barely above nothing. The note is heard anyway, because its harmonics are radiated and its fundamental is supplied by the listener. Computing what arrives turns the residue from a curiosity into a design decision, and finds that the fundamental of a low note is not the loudest part of it on any system a listener is likely to own.

instruments · Missing fundamental

Roughness and loudness of one interval against level. A minor third on C4 evaluated at every level from 20 to 100 dB SPL per note, with both quantities drawn relative to their own value at 60 dB. Roughness is the Plomp–Levelt sum over the partials that are above ISO 226's threshold at that level; loudness is the same partials in sones. Between 50 and 95 dB the interval grows 3.2 × 10⁴ times rougher and 24.9 times louder, so roughness grows like loudness raised to the power 3.2.

The same chord is harsher when it is louder

Every roughness number so far was computed at a level nobody stated. Roughness is the product of two partial amplitudes, so it is quadratic in pressure, while loudness is compressive — which makes a minor third at middle C thirty-two thousand times rougher at fortissimo than at pianissimo and only twenty-five times louder. A chord has no single consonance to report.

intervals · Consonance

Crescendo, and what the impression does. A crescendo of 20 dB over 8 seconds, drawn as three loudnesses in sones. The pale line is what is physically sounding, the middle line is the short-term loudness of the moment and the heavy line is the long-term loudness, which is the passage's loudness as a listener would report it. The gap between the last two is the whole of the effect: at its widest the moment is 1.02 times the running impression, and the impression takes 2 seconds to come down against 99 milliseconds to go up.

Loud is relative, and it comes down slowly

The account of loudness had a model of a moment and the account of closure asked it for a model of a form. The published one exists and its content is a pair of numbers that are not the same: a listener's running impression of how loud the music is rises to meet a step in a fifth of a second and takes seven seconds to come back down. A twenty-decibel crescendo spread over eight seconds therefore buys almost no contrast at all, and the same twenty decibels taken as a step buys a factor of two.

form · Loudness $Articulation is worth 2.6 phons, and nobody counts it. A passage of notes at 80 decibels, 2 to the beat, at 7 tempi and 3 articulations, scored against the same level held continuously. The variable is the fraction of each inter-onset interval that is sounding — 0.95 is a legato, 0.4 a staccato — and the vertical axis is what that costs the passage's running loudness in phons. Nothing here is anybody playing harder or softer. At 40 to the beat the span from legato to staccato is 1.47 phons; at 200 it is 2.61, because a staccato note there lasts 60 milliseconds and no longer reaches its own loudness either.$

A staccato is a dynamic mark

Every loudness figure in this collection is of a sound that has been going on long enough, and no note in music has. Run the running-loudness model on notes with lengths in them and an articulation turns out to command 1.5 phons at a slow tempo and 8.1 at a fast one — more than the 1.8 decibels a whole texture commands, on the same page, written down in the same ink, and counted by nobody.

perception · Loudness

A 20-decibel crescendo is 27 phons on a bass note and 20 on a high one. The same change of level, from 60 to 80 decibels, converted to loudness at each register through ISO 226's equal-loudness contours rather than at one kilohertz. The heavy curve gives each note a string spectrum, so its partials are converted in their own bands and summed; the pale one is the fundamental alone. On the spectrum-aware curve the crescendo is worth 27.3 phons at C1 and 20.3 at C7. On the fundamental alone it is 77 at C1, which is not a finding but an artefact: a 60-decibel tone at 33 hertz sits 1.8 decibels above the threshold of hearing and is very nearly nothing. The honest correction is the smaller one, and it is still a difference of 7.1 phons across the compass for a mark written in the same ink.

A subito piano is four seconds longer in the bass

Every loudness figure with time in it converts level to loudness at one kilohertz, and the equal-loudness contours say that no other frequency works that way. Joining the two sorts the published numbers into those that were about the treble and those that were not. Three move a great deal — a twenty-decibel crescendo is worth 27 phons on a bass note and 20 on a high one, and the seven seconds a subito piano takes becomes eleven and a third. Three do not move at all, and the reason they do not is the same reason in every case.

perception · Loudness

All figures · What can be heard