Which harmonics carry the pitch

The missing fundamental is presented as a free result. Delete the lowest partial of a note, the pitch does not move, and the auditory system is said to fit the best harmonic series to what remains. Nothing in that account says what the fitting costs, or how many partials it needs, or what happens when there are not enough.

It costs something, and the cost is computable from machinery this site already has.

The argument #

A pattern match needs its evidence one item at a time. If two harmonics land in the same auditory filter, the filter’s output is their sum, and a sum of two sinusoids close together is a single tone whose amplitude fluctuates — which is what roughness is, and which carries no information about the two frequencies that produced it beyond their difference.

So the question which harmonics can a template be matched against has a definite answer, and it is not “all of them”. The ear’s analysis bandwidth grows with centre frequency while the spacing between harmonics stays fixed at $f_0$, so every note has a height above which its harmonics stop being separable. Below that height they can serve as evidence for a fundamental. Above it they cannot, whatever else they do.

That is the whole of this rung, and three consequences follow from it. The residue is not equally available at every pitch. The partials that carry it are not the loudest ones. And the limit at the top of the residue’s existence region — the reason a very high note has no residue pitch worth speaking of — turns out not to be a resolvability limit at all, which is where this ladder’s next rung comes from.

Which computation produced the number #

A harmonic is counted as resolved here when the spacing to its neighbours, $f_0$, exceeds the ear’s analysis bandwidth at that harmonic’s frequency:

$$f_0 \;\ge\; B(n f_0).$$

$B$ is one of the two published bandwidth models this site already carries. Glasberg and Moore’s equivalent rectangular bandwidth, fitted to notched-noise masking, is $24.7(0.00437 f + 1)$ hertz. Zwicker and Terhardt’s critical band is wider, and below 500 hertz it is wider by a factor of three — which is exactly the register the residue arguments are about, so both are computed and neither is treated as the answer.

The criterion itself is a criterion and not a measurement. A more conservative reading demands that the neighbours fall a further quarter of a bandwidth outside the filter, and on the equivalent-bandwidth model that reading takes exactly two harmonics off the count at seven of the eight pitches it was tried at, from 41 hertz to 3.2 kilohertz — so the “about two” is not an estimate but a near-constant. On the wider Bark model it takes off three at 110 hertz, one in the middle, and nothing at all at the bottom, because there the plain count is already zero and there is nothing left to take. What follows is stated so that the conservative reading changes the numbers and not the argument.

Read as an interval rather than as a number of hertz, the bandwidth behind the criterion is more than an octave wide at the bottom of the bass and under three semitones at the top of the keyboard — so a fifth played very low is analysed as one event and a major third played very high comes apart into two notes. The two published models disagree by a factor of three below 500 hertz, which is exactly the region the argument’s one weak point lives in, and everything in this essay is computed from those two curves and nothing else.

The count saturates, and that is why the rule of thumb works #

The received statement is that the first eight or ten harmonics of a note are resolved. It is usually offered as a measurement of one stimulus. It is not: it is what the criterion gives for very nearly the whole musical range.

The flat part is the part that gets quoted. The interesting part is the left-hand end, where it collapses: a note at 41 hertz has three resolved harmonics on the narrower-band model and none at all on the wider one, and a note at 65 hertz has five.

That is the bottom of a bass guitar and the bottom of a cello. Which means the register in which the missing fundamental is most often demonstrated — the bass line coming through a small speaker — is the register in which the mechanism the demonstration is supposed to illustrate has almost nothing to work with.

The partials that decide the pitch are not the loudest ones #

Resolvability says which harmonics are available. It says nothing about which of the available ones the auditory system actually uses, and the answer to that is a measurement.

Ritsma reported in 1967, and Plomp independently in the same year, that when the low harmonics of a complex are shifted against the high ones the reported pitch follows the third to the fifth. Those harmonics dominate even when they are far from the loudest thing present. The literature calls the region of the series they occupy the dominance region.

This is a published measurement and not a computation, and it is worth saying what the argument survives it being wrong by. If the region were the second to fourth harmonics, or the fourth to sixth, every claim below would stand: the point is that the dominant harmonics are low but not the lowest, that they are comfortably inside the resolved range at ordinary pitches, and that they leave the resolved range at the bottom of the bass. A dominance region anywhere between the second and the seventh harmonic gives all three. A dominance region at the first harmonic would break the argument entirely, and would also make the missing fundamental impossible, so it is not a live alternative.

For an ordinary plucked or bowed spectrum, amplitude falls as one over the harmonic number. The loudest partial present is therefore the first, or the second once the first has been deleted — and neither is in the dominance region. The partials that name the note are the ones a spectrum analyser would call middling.

For an ordinary plucked or bowed spectrum amplitude falls as one over the harmonic number, so a string’s loudest partial is the first, and its third to fifth are already down by ten, twelve and fourteen decibels. A clarinet’s list is stranger: the even partials are nearly absent, so of the three harmonics the dominance region is made of, one is strong, one is missing and one is strong again — and the pitch of a clarinet note is decided by a sample of the region with a hole in it, without being thereby ambiguous. The partials that name the note are the ones a spectrum analyser would call middling. Within thirty cents the third, fourth and fifth harmonics of a 200 hertz note are consistent with two fundamentals: 200, whose template they fill without a gap, and 100, which fits every partial exactly and predicts four slots that nothing fills. That second candidate is the octave error a minority of listeners make on residue stimuli, and the arithmetic supports it.

What a channel leaves, and what the ear can use of it #

Two filters sit in series between a source and a listener. One is in the equipment and the other is in the listener, and the interesting cases are the ones where their outputs do not overlap.

The telephone case is the one that made the missing fundamental famous, and the arithmetic shows why it is the easy case. A 300 hertz cutoff removes harmonics 1 and 2 of a 110 hertz voice and leaves harmonic 3 at 330 upward. The dominance region survives intact. The resolved range survives intact. Nothing that the pitch mechanism is known to use has been touched.

Now do the same to a bass guitar.

The complementary case is worth drawing, because it is the one a channel almost never produces and a synthesiser easily can.

That result is the one this rung was written to find, and it was not the one the rung was slated to find. The intersection is empty. Every harmonic the speaker transmits arrives two or more to a filter; every harmonic the ear could separate has been removed. A pattern match over resolved partials has nothing to match. The pitch is still there.

So the demonstration that opens this whole anchor — a bass line at 41 hertz through a laptop, and nobody notices — is not a demonstration of pattern matching. Something else is doing it, and the next rung but two is about what.

Where the model stops, and it stops in an unexpected direction #

The residue’s existence region has an upper edge. Above a fundamental of roughly 1.4 kilohertz the effect fades, and above about 5 kilohertz there is no residue pitch to be had. The obvious explanation is that resolvability fails up there.

It does not. Resolvability improves with frequency and then flattens; it never fails at the top. A tone at eight kilohertz has a filter entirely to itself. Whatever closes the existence region, it is not that the partials have stopped being separable.

Two published numbers meet here and the meeting is worth being careful about. The dominance region is a measurement. The five-kilohertz phase-locking limit is a measurement, and it is quoted in the literature as a range between four and five rather than as a figure. What the drawing does is put them on the same axis, and what it shows is that the fundamental at which the dominance region leaves the phase-locking range is the same fundamental at which the residue is reported to fade. If the phase-locking limit were four kilohertz instead of five the crossing would move down by a fifth, which is well inside the spread of the existence-region measurements themselves.

That is a coincidence with a plausible mechanism behind it and it is not a demonstration of one. What can be said flatly is the negative: the ceiling on residue pitch is not a resolvability ceiling, because resolvability has no ceiling. Something temporal is involved, and a purely spectral account of the missing fundamental cannot explain why the effect stops.

What the picture cannot show #

Level, which is absent from every figure here. The bandwidth models are fits at moderate levels, and the real filter broadens as the input grows — which is why masking spreads further upward at high levels. A loud complex has fewer resolved harmonics than a quiet one, and none of these drawings knows that.

Resolution is not a switch. The criterion draws a line where the real thing is a gradient: a harmonic whose neighbours sit just outside the band is worse evidence than one whose neighbours are two bands away, and the figures count both as resolved. The count is a summary of a continuum, and the conservative criterion moves it by about two harmonics at every pitch.

The criterion is about one steady tone in quiet, and nothing in music is that — which is the one caveat in this list that can be computed rather than conceded, and the answer is larger than the concession was.

The texture count has no amplitudes in it either. A competitor inside the filter is counted whatever its level, so a fortissimo triad and a triad whose upper voices are barely present give the same table — and in the second case the filter’s output is dominated by the partial that belongs to the note in question, which is a resolved partial in every sense that matters. What the count gives is the worst case at equal levels, and the reason to trust the direction rather than the size is that no arrangement of levels turns seven resolved harmonics back on.

And a resolved partial is not the same as a partial a listener can hear out. Hearing a partial out of a complex is a further operation on the filter’s output, it requires attention, and it succeeds for far fewer harmonics than the bandwidth criterion admits. Resolvability is a necessary condition and nothing more.

The classic laboratory stimulus is a 200 hertz tone with its first two partials deleted, leaving the third to the eighth — everything inside the resolved range at that pitch, with the dominance region entirely present. That is what makes the demonstration clean, and it is exactly why it does not transfer unaltered to the bass register where the effect is actually used.

The count in company #

Run the same criterion over a note that is not alone. A partial of the note in question is unresolved when any partial in the texture — its own neighbour or somebody else’s — falls inside its filter, which is the same rule the solo figures use with the pool of competitors widened to what is actually sounding.

A note in a triad has no resolved run at all and none of its dominance region. The seven separable harmonics the solo figures give become one resolvable partial somewhere in the first twelve, and the third, fourth and fifth — the ones a measured pitch judgement follows — are each inside a filter with somebody else’s partial.

That result needed a check before it could be believed, because the obvious way to get it is by accident. A tempered fifth above A2 puts its second harmonic at 329.6 hertz against this note’s third at 330, four tenths of a hertz apart, and a spacing criterion calls that unresolved when it is nothing of the kind: two partials that close produce one component at essentially the right frequency with a slow beat on it, which is the whole subject of the tuner’s ladder and is not a loss of information. Re-running with any competitor closer than fifteen hertz — this site’s own roughness floor — treated as a beat rather than a lump moves the fifth from 2 to 3 and the third from 0 to 0, and leaves every conclusion standing.

And the octave is the one doubling that costs nothing, for a reason worth stating exactly. Every partial of the note an octave up is already a partial of the note below; the sets are nested rather than interleaved, so doubling at the octave adds no competitor to any filter. Every other interval adds partials between the existing ones. The essay’s own orchestration claim below — that doubling a bass line at the octave supplies evidence for the same series — turns out to be understating it: the octave is the only interval that supplies evidence without destroying any, and that is a property of the arithmetic rather than of the instruments chosen.

What this does to the ladder’s argument is the same thing the bass-guitar result did, arriving from the ordinary direction rather than the extreme one. The residue account is a single-note account. Most music is not single notes, and in a plain triad the pattern match has one partial to work with — and the chord still has a bass note with a pitch.

Whose music, and when #

The register argument has a practical edge, and it is about scoring rather than about listening.

A double bass playing its bottom string in a symphonic texture, in the repertoire from roughly Beethoven onward, is producing a note with three resolved harmonics on the more favourable model. Everything above the third is arriving to a listener as an unresolved band. That is one reason orchestral bass lines are so often doubled at the octave by cellos or bassoons: the doubling supplies a partial in the resolved range that belongs to the same series, and it strengthens the fit without adding a new pitch.

The same arithmetic runs the other way in organ registration, which is the subject of the next rung. Mixture stops sound several upper partials of a note at once, and they were built long before anybody could say why they work. They work because the partials they add are the ones the pitch mechanism uses, and they fuse rather than being heard as separate notes.

And it bounds a claim about tuning. Two notes a fifth apart in the bottom octave of a piano are inside one analysis band, which means the beats between them are not evidence about the interval in the way they are two octaves higher. Piano tuners work upward from the temperament octave in the middle of the instrument for exactly this reason, and the arithmetic here is one half of why.

Where the ladder goes next #

This rung has priced the inference and found that the price is paid in the bass. The residue is cheap in the middle of the keyboard, where eight harmonics are separable and the dominance region sits comfortably among them. It is expensive at the bottom, where three are separable at best. And in the case that opens this whole anchor — a low bass note through a small loudspeaker — the two filters between source and listener leave nothing at all.

Three rungs follow from that. An organ’s resultant stop builds a residue deliberately from two partials, which is the fewest that can imply anything, and it is built at 16 hertz where nothing whatever is resolved. A click train band-passed above the resolved harmonics has a pitch that no template could be matching, and jittering its timing destroys it. And a three-inch loudspeaker turns the whole question into an engineering decision, because the fundamental it cannot radiate is the one the listener supplies.

All three are cases where the pattern account — the one this ladder’s first rung endorsed, against Helmholtz’s distortion account, on the strength of the shifted-residue experiment — has run out of partials to work with. The account was right about the experiment it was chosen by. Counting the filters says how far it reaches.