A third is rougher in the bass

Play a major third at the top of the treble staff and it is a clean, bright, entirely unremarkable interval. Play the same major third two octaves below middle C and it is a growl — thick, indistinct, and unpleasant in a way that has nothing to do with wrong notes. Orchestrators have a rule about it, written down since Berlioz: do not voice thirds low.

The ratio is 5:4 in both cases. Whatever is different is not in the interval — and it is not a tuning question either, since both thirds can be made exactly 5:4 and the growl stays.

What is actually being asked #

The dissonance curve on this site sweeps an interval at a fixed root and finds its minima at the simple ratios. That figure answers the question “which intervals are smooth”, and it answers it well.

This one asks the reverse question. Fix the interval and sweep the root. If consonance were a property of the ratio, every curve here would be flat. None of them is, and the differences between them are large: a minor third at 55 hertz is roughly as rough as a semitone, and the same minor third at 880 hertz is nearly as smooth as a fifth.

So consonance is a property of an interval at a register, and the register is not a minor correction. It is comparable in size to the effect of the interval itself.

The mechanism has a width #

The reason is that the ear does not analyse frequency with unlimited resolution. It behaves like a bank of overlapping filters, and two components that fall inside the same filter are not separated — they are summed, and what the listener gets is a single fluctuating loudness rather than two pitches.

That fluctuation is roughness. The band inside which it happens is the critical band, and its width in hertz grows with frequency.

Growing in hertz sounds as though it should make the bass the safe place, and it does the opposite. Music is conducted in ratios, and a band of fixed absolute width is a much larger interval down low than up high.

Converted into semitones the curve falls steeply, flattens out somewhere in the middle of the keyboard, and stays flat. That is the shape that explains the orchestration rule, and it explains it quantitatively rather than as a matter of taste.

Which computation produced the number #

The band width in hertz comes from a published fit; the conversion into semitones is arithmetic.

Glasberg and Moore’s 1990 equivalent rectangular bandwidth, fitted to notched-noise masking data, is

$$\mathrm{ERB}(f) = 24.7\,(0.00437 f + 1) \ \text{hertz}.$$

A band of that width centred on $f$ runs from $f - \mathrm{ERB}/2$ to $f + \mathrm{ERB}/2$, and the musical size of that span is

$$12 \log_2 \frac{f + \mathrm{ERB}/2}{f - \mathrm{ERB}/2} \ \text{semitones}.$$

At 1000 hertz the band is 133 hertz wide, which is 2.3 semitones — narrower than a minor third, so the two notes of a minor third are resolved. At 100 hertz the band is 35 hertz wide, which is 6.2 semitones — wider than a fourth, so the two notes of a major third are firmly inside one band and are not resolved at all.

The crossing point is the useful number, and it is found rather than guessed. Solving

$$12 \log_2 \frac{f + \mathrm{ERB}(f)/2}{f - \mathrm{ERB}(f)/2} = 4$$

by bisection gives $f \approx 202$ hertz. G♯3 is 208 hertz. So on this model a major third stops fitting inside one critical band at about the G below middle C, which is very close to where the orchestration textbooks put the boundary, and the textbooks got there by ear a century and a half earlier.

Two models, and they disagree #

The honest complication is that there is more than one published critical band, and they do not agree in the region the question is about.

Zwicker and Terhardt’s 1980 formula,

$$\mathrm{CB}(f) = 25 + 75\left(1 + 1.4 (f/1000)^2\right)^{0.69},$$

has a floor of about 100 hertz at low frequencies, where the ERB falls to 25. At 100 hertz that is a factor of three, and in semitones it is the difference between a band six semitones wide and one nineteen semitones wide. Both are fits to listening data; they were fitted to different experiments.

Solving the same equation with the Zwicker–Terhardt band gives a crossover for the major third at about 514 hertz — the C above middle C, more than an octave higher than the ERB answer. That is not a small disagreement, and it is drawn in the figure rather than averaged away, because averaging two models to produce a number neither of them supports would be worse than reporting the spread.

What both models agree on is the shape: the band is wide in musical terms down low, narrow up high, and the transition happens somewhere in the middle of the keyboard. Everything the argument needs survives the disagreement. The exact frequency at which to stop writing low thirds does not.

The same solve, for every interval #

Solving that equation once gives one boundary. Solving it for each interval gives the chart the orchestration rule is a summary of.

Read the first column downward and the manuals’ list falls out in order. The octave clears at the bottom of the bass staff, the fifth just above it, and the third not until the octave below middle C — which is precisely “octave, fifth or tenth at the bottom, and never a third”, written as three frequencies rather than as three pieces of advice. The tenth is a third plus an octave, so its two notes are fifteen semitones apart and clear everywhere.

The bottom row is worth having for scale. A whole tone does not clear one critical band anywhere in the musical range on either model, which is why a cluster of adjacent scale degrees is rough at every register and why the low-third rule has no high-register counterpart for seconds.

Where the beating goes #

There is a second way to see the same fact, and it is worth having both.

Two tones separated by less than a critical band produce audible beating at their difference frequency. Maximum roughness occurs when that difference is roughly a quarter of the band width — around 30 to 40 beats a second for most of the range.

A major third at 110 hertz has its two notes 28 hertz apart. That is squarely in the roughest region, and it is why a low third growls. The same arithmetic makes a comma between two high notes a rattle rather than a throb.

A major third at 880 hertz has its two notes 222 hertz apart, far beyond any beating rate, and the two notes are simply two notes.

The partials complicate this, and it is worth computing what they do rather than guessing, because the obvious guess comes out backwards. A complex tone brings its whole harmonic series along, and the upper partials of a low pair are spread out enough to be resolved even when the fundamentals are not — from which it is tempting to conclude that a low third is less objectionable on a bright instrument. It is not.

Roughness rises monotonically with how much spectrum there is, and it rises by nearly a factor of two between a pure tone and a reed. Every partial a timbre carries is another pair of components that can land inside a band, so a bright low third is rougher than a dull one and not smoother.

What the partials do buy is the other thing, and the two pull in opposite directions. A low third’s fundamentals are inside one band and are not resolved, so whatever tells a listener there are two notes rather than one has to come from higher up — and only a spectrum with upper partials has anything up there to supply it. So a bright low third is more identifiable and rougher, and a dull one is smoother and more of a blur.

That is a better account of the orchestration rule than either half alone. A flute third low down is not objectionable because it is rough; it is objectionable because it is a single muddy object. A bassoon third low down is rough and is at least two notes. Two failure modes, two remedies, and the manuals prescribe the same one for both — open the voicing — because opening it fixes both at once.

One thing, then two #

The band has a further consequence that is easier to hear than to read about: what happens between a unison and a resolved interval is a sequence of three distinct experiences, not a gradual change of one.

Start with two tones at exactly the same frequency. They sum to a single louder tone and there is nothing to notice.

Separate them by a hertz or two and the sum swells and fades at the difference frequency. That is countable, and it is what a tuner works from.

Separate them further — past about twenty hertz — and the swelling is too fast to count and becomes a texture. This is the rough region, and it peaks at roughly a quarter of a critical band.

Separate them past the whole band and the roughness disappears and two pitches appear. Nothing was added at that point; the two tones simply stopped sharing a filter.

Plomp and Levelt established this sequence quantitatively in 1965, by asking listeners to rate pairs of pure tones for consonance at many separations and many centre frequencies. Their result — that the rating depends on separation as a fraction of the critical band, not on the interval — is the basis of every roughness computation on this site, and it is the reason those computations are register-dependent at all.

What it does to a voicing #

The rule generalises from intervals to chords in a way that any arranger will recognise.

A close-position triad has its notes three or four semitones apart. Down at the bottom of the bass clef that is inside a band, and the chord is heard as one rough event. The standard remedy is to open the voicing — put the third up an octave, leaving a bare fifth or a tenth at the bottom — which converts a three-semitone gap into a fifteen-semitone one and takes it well outside the band.

That is why the interval at the bottom of an orchestral chord is nearly always an octave, a fifth or a tenth, and almost never a third. It is also why a piano’s left hand plays tenths rather than thirds in a great deal of nineteenth-century writing, and why a bass line and a melody two octaves apart never clash however dissonant they are on paper.

Where the model stops #

Critical bandwidth is a summary of several distinct measurements. The number is derived variously from masking, from loudness summation, from the threshold at which two tones stop beating, and from phase sensitivity. Those experiments do not give identical answers, and “the critical band” is a convenient name for a family of results rather than a single measured quantity.

Both formulas are extrapolations at the bottom, and one of them diverges there. The masking data that the ERB was fitted to thins out below 100 hertz, and the region where the two models disagree most is the region where both are least constrained. Worse than that: Zwicker’s band has a floor of about 100 hertz, so at 55 hertz it runs from 5 to 105 and the lower edge is nearly at zero — which makes its width in semitones 53, more than four octaves, against the ERB’s 9.9. That is not a disagreement between two estimates; it is one formula being used outside the range in which the conversion to a musical interval means anything. The curves are drawn down to 55 hertz here because that is where the music is; it is not where the evidence is.

Roughness is not the whole of dissonance. It is the sensory part — the part that a deaf-to-context model can compute. Everything cultural, everything to do with expectation, and everything to do with what a chord is doing in a progression sits on top of it and is not addressed here at all. A tritone is not unpleasant because it is rough; it is barely rough at all.

Level matters and is ignored. Critical bands widen at high sound levels, so the same interval is rougher when loud. Every number above assumes a moderate level and does not say which.

The curves are computed from one roughness model. The Plomp–Levelt formulation used for the sweeps has known limitations, particularly for very small frequency separations and for partials of very different amplitudes. It reproduces the classic results and it is a fit, not a derivation.

Whose music, and when #

The practical rule is far older than the measurement.

Eighteenth-century keyboard figured-bass practice spaces the left hand widely and the right hand closely, which is exactly the arrangement critical bandwidth recommends. The pattern is so consistent across Bach, Handel and their contemporaries that it functions as a stylistic marker, and there is no theoretical text of the period that explains why.

Berlioz’s Traité d’instrumentation of 1844 states the rule explicitly for the orchestra: thirds and sixths in the low register are muddy and should be avoided, and the interval to use down there is the octave or the fifth. Rimsky-Korsakov repeats it. Every orchestration manual since has repeated it. The reasoning offered is always empirical — it sounds bad — because the psychoacoustics did not exist until Fletcher’s work at Bell Labs in the 1930s and Zwicker’s in the 1950s.

The rule shows up in instrument design too. The low register of the piano is strung with single and double strings rather than triple, and its hammers are voiced softer, both of which reduce the upper partials that would otherwise interact. A bass guitar’s tone controls are almost always used to remove upper harmonics. In both cases the effect is to make the low register less rough by giving the critical bands less to work with.

The counterexamples are as instructive as the rule. Organ mixtures deliberately sound many partials at once and sound clear rather than rough, because those partials are octaves and fifths and their spacing is wide. Barbershop and gospel quartet writing uses close, low voicings that break the rule and gain a characteristic thickness from doing so. And the low third is a standing effect in heavy rock, where the fifth without a third is the default precisely because thirds down there are unusable, and where deliberately using one is a decision.

There is one more design consequence, and it runs the other way. A 32-foot organ stop puts its fundamental at 16 hertz, which is below hearing, and everything audible about the note is its upper partials — which are widely spaced in hertz and therefore comfortably resolved. So the deepest register on the instrument is not the roughest, because almost nothing of it is being heard where the bands are wide. The roughest region on an organ is the one just above it, where the fundamentals are audible and close together, and organ registrations reflect that.

The general form of the rule is worth extracting, because it applies well beyond low thirds. Two components clash when they fall inside one critical band, and everything an arranger does to avoid a clash is a way of getting them out of one: raise the register, open the voicing, thin the spectrum, or move one of them to an instrument whose partials sit elsewhere. Those look like four unrelated pieces of craft and they are one piece of craft, described four ways.

The ladder from here #

Later rungs on this anchor: the roughness of a chord rather than an interval, and whether it decomposes into pairs. Level dependence, measured. Timbre and roughness — why the same interval on two instruments is not equally rough, and what a spectrum has to look like to be safe low down. The relationship between roughness and the beating a tuner counts, which is the same phenomenon at a hundredth of the rate. And the cultural half of consonance, which is everything this anchor deliberately does not model.

The rule about low thirds is two hundred years old, correct, and was arrived at with no access to the quantity that explains it.