A dissonance has to last

Every roughness figure on this site is drawn as though roughness were a quantity a chord has. It is drawn that way because the model computes a number and a number goes on an axis, and the drawing quietly assumes the chord is being held. Nothing on any of those axes is time, and none of the captions mentions how long the chord lasts — which for a sustained organ chord is exactly right and for a passing note in a fast movement is a description of something that never happens.

Roughness is not a quantity. It is a rate — two nearby frequencies swelling and fading at their difference, fast enough that the swellings stop being countable and become a texture. A rate takes time to happen, and a note in music is often very short. That is the whole of this rung, and the number it produces is one the counterpoint literature has been circling for three hundred years without ever writing down.

What the rate is, and where it comes from #

The roughness model sums a contribution over every pair of partials, one from each tone. Each pair has a difference frequency, and that difference is the rate at which that pair beats.

So the rate a listener is exposed to is not the difference between the fundamentals; it is the difference frequency of whichever pairs are actually contributing, and which pairs those are is decided by the critical bandwidth rather than by the interval. Weighting each pair’s difference by how much roughness it contributes gives one number per interval per register, and it comes from the same computation as the roughness — not from a second model bolted on beside it.

That distinction matters immediately, because the answer is not what the fundamentals suggest. A minor second at the bottom of a cello has fundamentals 3.9 Hz apart, which would be a slow throb. The contribution-weighted mean rate is 33 Hz, nearly ten times that.

But the mean is doing something misleading here, and it is worth opening before any of it is used. The single largest contributor to a bass semitone’s roughness is not a high pair at all: it is the fundamentals themselves, beating at 4 Hz, and the pair of second partials at 8 Hz is next. The 33 Hz mean is those slow, heavily weighted contributions averaged against a long tail of faster ones, and its standard deviation is 33 Hz as well — the spread equals the mean, so there is no rate here to speak of, only a distribution with a slow mode and a fast tail.

The decisive figure is how much of the model’s roughness comes from pairs beating below twenty hertz, which is the boundary this essay’s own continuum puts between a countable throb and a texture:

More than half of what the model calls the roughness of a bass semitone is beating slow enough to be counted. That is not roughness under any definition this essay uses; it is the swelling and fading that the beating rung is about, and a listener hears it as a throb rather than as a texture. The mean rate is a summary over two different perceptual phenomena, and in the bass the wrong one dominates the weighting.

The fifth’s row is the same defect at the other end: at C5 and C6 a quarter to a half of its roughness is the near-coincidence of its third and second partials beating at two to four hertz, which is exactly what a tuner listens for when setting a fifth and is not a texture either.

Read as an interval rather than as a number of hertz, the critical band is more than an octave wide at the bottom of the bass and under three semitones at the top of the keyboard — which is why a bass semitone’s fluctuation is spread over so many rates. Two partials inside one band beat at their difference; the lower the register, the more pairs of partials fall inside a band together, and the slower each of their differences is.

The number, and its shape #

Set a criterion of four completed cycles and ask how long each interval needs.

Two things about that table are worth more than the individual numbers.

The requirement is roughly constant from the bass to the middle of the range. A semitone needs 120 ms at C2, 161 at C3 and 132 at C4. It is not the case that a bass dissonance takes much longer than a middle-register one, which is what would be expected if the fundamentals were doing the work.

And it falls by a factor of three in the top two octaves. At C6 a semitone needs 39 ms and a fifth needs 12, because the partials are further apart in hertz and everything beats faster.

So the boundary sits at about 120 milliseconds through the whole lower and middle range — which is a semiquaver at 120 beats a minute, and a quaver at 240. The section below takes the contamination out of that figure and moves it to about seventy, which is a demisemiquaver; the two numbers bracket what the model can honestly say, and the argument that follows is unchanged by which of them is taken, because it turns on the comparison between registers rather than on the absolute.

The table, restricted to the band that is roughness #

The repair is to take the same weighted mean over only the pairs inside the published roughness band, twenty to three hundred hertz, which is the range the essay already quotes and already relies on.

Two things follow and the first is a repair rather than a demolition.

The requirement really is constant across the range, and much more so than the published table shows. A band-restricted semitone needs 62, 66, 71 and 81 milliseconds from C2 to C5 — a spread of a third across four octaves, against the published table’s factor of two with a hump in the middle. The hump was the low-frequency contamination, largest at C3 where seventy per cent of the weight sits below twenty hertz.

And the number is about seventy milliseconds, not a hundred and twenty. That is a demisemiquaver at 120 rather than a semiquaver, and it moves the conclusion drawn below in the direction of the dissonance being more audible than the essay claims: a passing semiquaver of 125 milliseconds clears seventy comfortably at every register, so it is audibly rough everywhere rather than marginal at C4 and failing in the bass.

Where the two columns agree — the fifth, the tritone, everything at C4 and above — the published figures stand unchanged, because there was nothing below twenty hertz to remove.

The other quantity runs the other way #

The interesting part is that roughness itself has the opposite register dependence.

At C2 a minor third scores 0.372 on the model and at C6 it scores 0.036 — a factor of ten smaller, which is the third rung of this ladder and the reason a chord is voiced wide at the bottom. So the bass is where roughness is large and where it is slow, and the treble is where it is small and fast.

Neither register can produce a strong short dissonance. In the bass there is plenty of roughness and no time to hear it; in the treble there is plenty of time and almost no roughness to hear. The maximum audible dissonance in a short note is somewhere in the middle, around C4 to C5, which is where most counterpoint puts its most exposed lines.

That is not an argument that anybody chose the register for that reason. It is an observation that two independent curves cross in the register the music happens to use, and the site has met the same coincidence one rung earlier in a form where it turned out to be uninformative.

The same boundary, arrived at from rhythm #

The site has met this frontier once already and from the opposite direction.

Speed a three-against-two polyrhythm up and somewhere above about twenty events a second it stops being a rhythm and becomes a fifth. That essay is about a rate crossing a boundary upward; this one is about a fluctuation that is already above the boundary and does not have time to establish itself.

Both are the same fact seen twice: a periodic event has to repeat several times before a listener hears it as periodic, and what it is heard as depends on the rate. Below about twenty a second the repetitions are a rhythm, from twenty to a few hundred they are roughness, and above that they are a pitch — but in every band the requirement is the same, that there be several of them.

What this does to the figures already on the site #

Every dissonance curve here is drawn for a sustained pair. That is the honest way to draw it and it should be read with a note length attached.

That is a real qualification rather than a decorative one. A figure that reports a peak of roughness at eleven semitones is reporting something a listener will hear if the note is held for a third of a second and will not if it is held for a twelfth. The curve is a limit that a long note approaches.

The rule this puts a number on #

Species counterpoint permits a dissonance on a weak beat or as a passing note and forbids it on a strong beat unless it is prepared. The permission is always stated in metrical terms and is never stated in milliseconds, which is odd for a rule whose justification is that the dissonance is not really heard.

At a tempo where the crotchet is 500 ms, a passing quaver is 250 ms and a passing semiquaver is 125. The first clears the four-cycle criterion everywhere; the second clears it in the treble, is marginal at C4 and fails at C2 and C3. So the rule’s metrical form and this essay’s acoustic one agree at ordinary tempos and part company at fast ones — and where they part company, the acoustic version says the dissonance is less audible than the notation suggests, not more.

The interesting prediction is about tempo rather than about note value. The same written passage, played twice as fast, changes which of its dissonances are audible as roughness, and it does so with no change to the notes. A passage of passing semiquavers at a slow tempo contains real dissonances and at a fast one contains a texture.

Whose music, and when #

The counterpoint rule is a claim about a repertoire and the repertoire has to be named: it is the pedagogical tradition running from Fux’s Gradus of 1725 through nineteenth-century conservatory teaching, codifying practice of the sixteenth century. It is not a description of what Palestrina did, it is a teaching abstraction of it, and the treatment of dissonance is the part of the abstraction most often criticised as too strict.

The prediction about tempo applies to any music at all, since it is about a perceptual mechanism. Where it has consequences that could be checked is in performance practice: a fast movement played at two different tempos should have measurably different roughness, and the difference should be largest for its shortest dissonances.

And a caution about the direction of the argument. Nothing here suggests the counterpoint rule was derived from roughness. It was derived from what sounded acceptable, over a long period, by people with no measurement of anything. The claim is only that the rule and the acoustics are consistent, and that the acoustics supply a number where the rule supplies a metrical position.

What is left when the roughness is not #

A dissonance too short to be rough is still an event, and it is worth asking what the ear does with it instead, because the answer is another mechanism this site has already measured.

Two tones that start together and stop together, lasting a tenth of a second, are inside the window in which the auditory system integrates rather than resolves. Within that window the louder of two components raises the threshold for the quieter, order is unreliable, and what arrives is one object with a quality rather than two objects with a relationship.

So the short dissonance is not perceived as a weaker version of the long one. It is perceived as a different kind of thing — a colouring of an event — and the counterpoint rule that lets it pass is licensing something other than a quiet dissonance.

Where the model stops #

Four cycles is an assumption, and it is stated as one. What is published is the rate range roughness occupies — a modulation of roughly 20 to 300 Hz, peaking near 70 — and the general principle that a periodic fluctuation needs several periods before it is periodic. Four is the smallest defensible reading of “several”, it is a parameter of the figure, and every conclusion here is a comparison between registers rather than an absolute.

It is also the one assumption in the essay that scales cleanly. A note length is four over a rate, so doubling the criterion doubles every millisecond figure in every table and moves no comparison at all — two cycles halves them, eight cycles doubles them, and the ratio between any two cells is fixed. That is not true of the twenty-hertz band edge, which is the other stated constant here and which enters through a threshold rather than a multiplier: moving it changes which pairs are counted, so it changes different cells by different amounts, and it is the parameter to be suspicious of.

The contribution-weighted mean is one summary of a distribution, and the section on the band restriction is what happens when the distribution is opened. The spread is widest in the bass, where it equals the mean, and the reason recorded here was that the contributing pairs are high partials of two different tones. That is the wrong way round: they are high partials and the fundamentals, and the fundamentals carry more weight. A single rate is a usable summary only for the intervals and registers where nothing is contributing below twenty hertz, which is the wide intervals and the middle of the range.

Attack and decay are not in this at all. A note is not a rectangle. The first fifty milliseconds carry most of an instrument’s identity, and for a plucked or struck note most of the energy is gone before the steady state a roughness model assumes ever exists. On a harpsichord a 125-millisecond note is nearly all attack.

The spectrum is fixed and a real one is not. The rate is computed from a string’s partial amplitudes held constant, and in a real note the upper partials — which are the ones supplying the rate in the bass — decay fastest. So a bass semitone’s fluctuation slows as the note continues, which the model has no term for and which pushes the requirement the wrong way.

And the two notes have to be simultaneous. A passing dissonance in real counterpoint is one voice moving against a held note, so both notes are sounding for the whole of the short one — which is the case computed here. A dissonance between two notes that merely overlap partially is a different and messier calculation.

What the picture cannot show #

It cannot show the reverberation. In a room the previous note is still sounding, so a dissonance in a hall lasts considerably longer than its written value — which is a ceiling on harmonic rhythm the site has computed elsewhere and is a floor under this essay’s note lengths. The stone rooms in which most counterpoint was written extend every dissonance by seconds.

And it cannot show what is heard instead. A fluctuation too short to be roughness is not nothing. It is an event with a particular quality — the site’s own rhythm ladder finds the same boundary from the other side, where a series of events becomes a pitch — and what a two-cycle fluctuation sounds like is a question about attack transients rather than about roughness.

Where the ladder goes next #

Three rungs have now asked the roughness model questions it could not answer or could answer only conditionally. The last one asks it a question it answers cleanly and decisively: given three fixed pitch classes and four octaves to put them in, which arrangement is smoothest — and the answer is a rule every orchestration manual states and none of them derives.