The same chord is harsher when it is louder

Nine rungs of this ladder have printed roughness numbers. Not one of them said how loud the notes were.

That is not an oversight in the captions. It is the model: the Plomp–Levelt sum this ladder built at its second rung takes two spectra and a pair of frequencies and returns a number, and the spectra are lists of relative amplitudes with the fundamental set to one. Nothing anywhere in the chain has a decibel in it. So every dissonance curve, every voicing ranking and every claim about which interval is rougher than which on this site has been made at an unstated level — and level turns out to be the parameter that changes the answer by four orders of magnitude.

Two laws, and they are different laws #

The mechanism is not subtle and it is worth stating before the arithmetic.

Roughness is the product of two amplitudes. In the Plomp–Levelt formulation, each pair of partials contributes an amount proportional to the product of their two amplitudes, times a function of how far apart they are. Turn the whole sound up by a factor of two and each contribution goes up by a factor of four. Roughness is therefore quadratic in sound pressure: ten decibels of level is ten decibels of roughness, exactly, and there is no fitting involved — it falls out of the multiplication.

Loudness is not. Loudness in sones doubles every ten phons, which is a compressive law: ten decibels of level is a factor of two of loudness, not a factor of ten. That is Stevens’s power law, and it is the reason ten violins are not ten times one violin.

So the two grow at different rates, and the ratio between them climbs steadily. Roughness per unit of loudness is not a constant of the chord. It is a function of how hard the chord is being played — and, because the curve above is not flat, of where the chord is.

Which computation produced the number #

A minor third on middle C, taken from fifty decibels to ninety-five — roughly the span between a pianissimo and a fortissimo at a seat in a hall, though a dynamic marking is not a level and the choice of endpoints is the essay’s rather than a measurement.

Roughness grows by a factor of 3.2 × 10⁴. Loudness grows by a factor of 24.9.

Expressed as one number: roughness grows like loudness raised to the power 3.2. Doubling the perceived loudness of the interval roughly multiplies its roughness by nine.

The same computation on a whole C minor triad gives 3.2 × 10⁴ against 24.3, which is the same exponent — the triad has three pairs rather than one and every pair scales the same way, so nothing about chord size changes it.

Four orders of magnitude sounds like more than it is, and this ladder has twice been saved by printing a margin rather than a rank. Nobody hears a factor of thirty thousand in roughness, any more than a factor of a thousand in intensity is heard as a factor of a thousand in anything; the scale is compressive at both ends of the chain. What the number licenses is a comparison rather than a sensation — it says the two curves cannot be the same curve, which is true whatever the units. At the other end, the smallest difference a listener can report in roughness has never been measured as finely as it has for pitch, and is certainly not fractions of a per cent.

That result does not depend on the threshold of hearing, on the critical band or on anything measured. It is a consequence of the model’s own algebra, and it means that a roughness figure with no level on it is reporting a ratio and not a quantity. Which is fine, and is what the previous nine rungs were doing: comparisons between intervals at one unstated level are unaffected by the level, because the same factor multiplies both sides. It is the absolute numbers that were never absolute.

A partial below threshold is not a quiet partial #

The second mechanism is the one this rung was slated around, and it works differently from expected.

A partial whose sound pressure level falls below the threshold of hearing at its frequency is not a faint contributor. It is absent: it cannot beat against anything, and it drops out of the sum entirely. How many partials are above threshold is a function of level, so at low levels the sum runs over fewer pairs.

The reason is in the shape of the threshold curve, and it is worth stating as a number. ISO 226’s threshold of hearing varies by about ten decibels between 500 hertz and 5 kilohertz — nearly flat across the range most partials live in — and then rises steeply below 200, reaching 37 decibels at 65 hertz and 65 at 27. A 1/n spectrum falls by 18 decibels from its first partial to its eighth. So across most of the compass the eighth partial is the faintest thing in the note and disappears first, and in the bottom two octaves the threshold outruns the spectrum: the fundamental is the partial closest to being inaudible, and the note loses its bottom rather than its top.

The expectation in the slate was that quiet playing removes the upper partials, which are the faint ones in a 1/n spectrum, leaving a smoother interval. The computation refused that, and the reason is in the shape of the curve above.

In the bass the first partial to disappear is the fundamental. The threshold of hearing rises by fifty-five decibels between 500 Hz and 31 Hz, far faster than a 1/n spectrum falls, so the lowest partial of a low note is the one closest to being inaudible.

The bottom E of a double bass at 41 Hz needs the note to be at 52 dB before its fundamental is audible at all, and the bottom A of a piano at 27.5 Hz needs 67, because its fundamental is only two decibels below the level of the whole note and the threshold there is 65. Which is a fact the site has met from another direction: the note that is not there is supplied by the listener from the upper partials, and here it is being supplied because the fundamental is genuinely under the threshold rather than because a loudspeaker failed to radiate it.

And the consequence for roughness is almost nothing, because the roughness of a bass interval was never coming from its fundamentals — the seventh rung established that the pairs doing the beating in the bass are high partials of two different tones. So the level effect removes the partial that was not contributing.

Where the level actually changes the answer #

The two mechanisms therefore point in opposite directions, and separating them takes several more runs of the same generator.

The size of that is worth having. Between twenty-five and thirty decibels a low C’s minor third grows rougher by a factor of eight, where the same five decibels above forty would give a factor of three — so at the bottom of the range and the bottom of the dynamic, roughness grows at nearly twice its own asymptotic rate. Measured against what the same spectrum simply scaled would predict, the interval has sixteen per cent of that roughness at 25 dB, forty per cent at 30, half at 35, and all of it at 40.

That makes three registers and three exponents, and they are monotone: 2.8 at C2, 3.2 at C4, 3.4 at C6. What moves is the loudness rather than the roughness — roughness is quadratic in pressure everywhere once every partial is present — so the trend is a reading of the equal-loudness contours crowding together in the bass, which the two loudness figures earlier in this essay draw.

But growing faster is only the amount. The interesting question is whether level changes the order — whether a chord that is rougher than another at one dynamic can be smoother at another — and there the answer is confined and sharp.

So the ranking of intervals by roughness is level-independent above about C3, and level-dependent below it. That is a narrower claim than the slate expected and it is the one the machinery supports. In the register most music is written in, the previous nine rungs’ comparisons survive the level they never stated. In the bottom octave and a half, at levels below a quiet room, they do not.

The thing that was checked and does nothing #

One further level effect suggested itself and had to be computed before it could be dismissed, because dismissing it by argument would have been guessing.

And one further level effect had to be computed before it could be dismissed, because dismissing it by argument would have been guessing. The masked threshold beside a tone is steeper at low levels than at high — about sixteen decibels per Bark at forty and six at ninety — so a loud tone reaches further up the spectrum than a quiet one, and the obvious inference is that a loud chord masks more of its own partials. It does not, and the reason is arithmetic: masker and maskee rise together. Every partial of both notes gains the same number of decibels, so the only thing level changes is the slope, and the slope change buries nothing new. Computed over a minor third at middle C, exactly one partial is masked at forty decibels and exactly the same one at ninety-five. Masking is a large effect between different sounds at different levels — it is what lets one instrument bury another — and between the partials of one chord at one dynamic it is nearly level-invariant.

The obvious inference is that a loud chord masks more of its own partials than a quiet one, which would push roughness the other way. It does not, and the reason is arithmetic: masker and maskee rise together. Every partial of both notes gains the same number of decibels, so the only thing that changes with level is the slope, and the slope change is not enough to bury anything new. Computed over a minor third at middle C, exactly one partial is masked at forty decibels and exactly the same one at ninety-five.

That is a negative result and it is worth the paragraph. Masking is a real and large effect between different sounds at different levels — it is what lets one instrument bury another, and it is what every perceptual audio coder is built on — but between the partials of one chord played at one dynamic it is nearly level-invariant.

What this does to every figure already on the site #

The honest reading of all this is a qualification rather than a retraction, and it has three parts.

The shapes are safe. Every curve, ranking and search on this site compares intervals at one level, and above C3 that comparison is exactly level-invariant — the founding roughness curve’s wells on the fifth, the fourth and the octave sit where they sit at any level above about forty decibels at that register, because level multiplies the whole curve. What is not defined without a level is the height of the axis, and the axis has never carried a unit. Nothing has to be redrawn: not the register sweep of the third rung, not the scale search of the sixth, and not the stretched-spectrum result of the fifth, whose whole content is a ratio between two curves drawn at the same level.

The absolute numbers were never absolute. A caption reporting that an interval scores 0.372 is reporting a number in units of the model’s own normalisation, and it should be read as a ratio to another number on the same figure and never as a quantity.

And the bass figures carry a level they do not state. Every one of them assumes all eight partials are audible, which at the 65 Hz several of them start from means the note is at least 38 decibels, and an octave lower at least 52. That is true of an orchestral bass section and false of the same note played quietly on its own, and it is the one place where a figure on this site would draw something different if the level were changed.

Whose music, and when #

Dynamic markings are not levels and never have been. Pianissimo and fortissimo are instructions relative to an instrument, a hall and a period, and the actual sound pressure at a listener’s ear from a marked ff in a Haydn symphony and one in a Mahler symphony are not the same number.

What the computation says is nonetheless a claim about practice, and it is testable. The same chord, scored identically, is a more dissonant object at a loud dynamic than at a quiet one, and the difference is far larger than the difference in loudness. That prediction fits a piece of orchestration practice stated everywhere in the European scoring manuals from Berlioz onward and derived in none of them: close spacing is tolerated at quiet dynamics and avoided at loud ones, and a cluster that is atmospheric at pp is aggressive at ff. The eighth rung’s search for where to put the third ranked arrangements at a single implied level; this one says the penalty for the bad arrangements grows faster than the music does.

It also gives a specific reading of a specific effect. A brass fortissimo is not merely a loud brass sound: a brass instrument played hard has a much brighter spectrum than one played softly, so the upper partials that supply most of the roughness are stronger relative to the fundamental as well as louder in absolute terms. Both mechanisms push the same way and only the first of them is computed here, so every number in this essay is a floor. The effect is not available on an organ, whose spectrum is the same at every dynamic because its dynamic is a choice of stop rather than a pressure — which is one reason an organ tutti is loud without being harsh in the way a brass tutti is.

Where the model stops #

The spectrum is fixed and a real instrument’s is not. This is the largest omission in the essay and it runs the same way as the result. Real strings, brass and voices get brighter as they get louder; the site’s TIMBRES are constant lists. A model with a level-dependent spectrum would show roughness growing faster than the power of 3.2 computed here, not slower, so the direction of the conclusion is safe and the magnitude is a lower bound.

Loudness is summed over partials and that is an upper bound. Partials inside one critical band do not add as separate loudnesses, so a proper Zwicker loudness of a string spectrum is lower than the sum used here. What the argument needs from it is only the slope — ten phons per doubling, which every summand shares — and no amount of grouping changes a slope.

Four hundred and forty hertz is not a level either. Every level in this essay is the level of one note in a free field. A real chord is played in a room, where the reverberant field adds a copy of every note at every distance, and the level at a seat is not the level at the instrument.

And roughness in the model is not roughness in an ear. That the sum is quadratic in amplitude is a fact about the arithmetic. Whether perceived roughness grows as the square of pressure is an empirical question, and the published measurements of roughness against modulation depth grow steeply with level but not with a settled exponent. The claim that survives either way is the comparative one: roughness grows faster than loudness, so the ratio climbs.

What the picture cannot show #

It cannot show the threshold spread between listeners. ISO 226 is a median of young otologically normal ears, and the threshold at 63 Hz varies by well over ten decibels between individuals. Every crossing level in the partials figure is therefore a median, and the level at which a listener’s bass note loses its fundamental is personal.

It cannot show the duration. The curves are drawn for a held note, and the seventh rung established that a fluctuation needs several cycles to be a fluctuation. A loud short note has the roughness and not the time.

And it cannot show the ear’s own distortion. At high levels the cochlea generates combination tones that are not in the sound at all, and those products grow faster than the primaries. They are extra partials, they appear only when it is loud, and they would add roughness that no model taking the incoming spectrum as its input can see.

Where the ladder ends #

Ten rungs, and the model has now been bounded in every dimension it has. It varies with register, with what the listener has learned, with the spectrum, with transposition, with how long the note lasts, with how the chord is spaced, with the syntax of the style — and, last, with how loud it is.

The last of those is the one this rung adds, and it is the last one available: register, learning, spectrum, transposition, time, voicing and syntax were the seven before it, and there is no eighth free variable in a function of two spectra, two frequencies and a level.

What remains unbounded is not the model. The questions still open about consonance are questions about tuning — which is about where the notes are, not about what happens when two of them meet — about expectation, which the ninth rung showed is where a rule of resolution actually lives, and about orchestration, which is about instruments rather than intervals. Each is a different anchor on this site with its own ladder already under way, and the half of consonance that is preference rather than perception was measured at the fourth rung and belongs to the listener rather than to the sound. This ladder is finished.