Two pipes for a note neither makes

A pipe that sounds C two octaves below middle C — the note organ builders call thirty-two-foot C — computes to a little over ten metres of open metal or wood. It costs a fortune, it needs a building with a hole in the roof or a chamber to lie down in, and a great many organs that want the note do not have one.

The trade’s answer since at least the early eighteenth century is a stop that sounds two smaller pipes together and lets the listener supply the note. A sixteen-foot rank gives 32.7 hertz and a ten-and-two-thirds-foot rank gives its fifth at 49 hertz. Those are the second and third harmonics of 16.35 hertz, and 16.35 hertz is thirty-two-foot C. The stop is variously called the resultant, the acoustic bass or the quint bass, and organists have been arguing about whether it is any good for as long as it has existed.

The argument #

The previous rung priced the residue in resolved partials: an inference over a pattern needs the pattern’s members separately, and how many of them arrive separately is a computation over the ear’s analysis bandwidth. The resultant stop takes that computation to its limit in both directions at once. It uses two partials, which is the fewest that can imply a series at all, and it puts them at the very bottom of the audible range, where the analysis bandwidth is at its widest relative to the spacing.

Both of those are the wrong side of every boundary, and the stop nevertheless exists in hundreds of instruments and is defended by people with good ears. What the arithmetic supplies is not a verdict but the shape of the disagreement — where the boundary sits, how far the thirty-two-foot case is from it, and which footages if any are on the other side of it, which turns out to be none that anybody builds.

Which computation produced the pipe lengths #

A flue pipe open at both ends supports every whole multiple of a fundamental set by its acoustic length, and that length is the physical pipe plus an end correction at each opening — the distance the wave carries on past the end before it turns round.

At A = 440 and twenty degrees, thirty-two-foot C is 16.352 hertz, and an open pipe sounding it has an acoustic length of 10.49 metres. Subtracting two end corrections for a pipe of 12 cm radius leaves 10.34 metres of physical pipe, which is 33.9 feet. Sixteen-foot C computes to 16.84 feet and eight-foot C to 8.32.

So the footage is not a length. Every rank computes about five per cent longer than its name at modern pitch, and the naming is a note-name rather than a measurement.

A 32-foot open pipe is ten and a half metres of speaking length and a 16-foot is five — which is the whole reason the trick exists, since the first will not fit in most buildings and costs a fortune, and the second and a 10⅔-foot together will.

The five per cent has an explanation worth following, because it is a measurement of something else. Ask instead what pitch standard makes the naming exact: what frequency of thirty-two-foot C gives a physical pipe of thirty-two feet. The answer is 17.32 hertz, which is A = 466. The sixteen-foot rank gives A = 463 and the eight-foot rank A = 457. At fifteen degrees rather than twenty — a cold church rather than a warm one — the three come out at 462, 459 and 454.

That is the high church pitch of the German and Netherlandish organ-building traditions, and it is where the nomenclature comes from. The footage is a fossil of a pitch standard, and the standard was never agreed. The residual spread between the three ranks is larger than the effect of the temperature and smaller than the difference one builder’s scaling makes to another’s, so the arithmetic is consistent with that reading rather than proof of it.

The discrepancy divides cleanly, and dividing it is the check on the reading. A nominal sixteen feet of pipe is 4.877 metres, and the acoustic length that sounds 32.703 hertz is 5.244 — a gap of 126 cents. Thirty-seven of those cents are the two end corrections, which are a real acoustic effect on a real pipe. The remaining eighty-nine are not an acoustic effect at all: they are the ratio between two pitch standards, and eighty-nine cents above A = 440 is A = 463.

And an end correction of a few centimetres is a negligible fraction of a five-metre pipe, so nothing about the trick turns on it — which is worth saying because at woodwind scale the same correction is the dominant error.

So a builder ordering a sixteen-foot rank is not making an error of five per cent. The rank is named for a note, the note has moved down by most of a semitone since the naming, and the pipe is exactly as long as the note now requires.

The two pipes are not resolved from each other #

Here is what the resultant stop is actually doing, and it is not what it is usually described as doing.

The pipes are 16.35 hertz apart and the ear’s analysis band at either of them is close to thirty. They fall inside one filter. What that filter’s output carries is not two frequencies but one, fluctuating in amplitude 16.35 times a second — a flutter rather than a beat and a flutter rather than a tone.

So the resultant is not a template match over resolved partials. There are no resolved partials. Whatever the listener is getting, the spectral account of the missing fundamental cannot be supplying it, and the mechanism has to be the other one: the envelope of the pair repeats at exactly the rate of the note it implies.

And an autocorrelation of the two frequencies alone returns to one at 61 milliseconds, the period of 16.35 hertz — so the temporal account and the pattern-matching account agree here, which they do whenever the partials are exact multiples.

Where the boundary is, on three readings of two models #

The pair of pipes is a fifth apart, and a fifth stops fitting inside one analysis band somewhere. Where that somewhere is decides whether a resultant stop has resolved partials or not, and the site’s two bandwidth models give three defensible answers.

Reading the criterion as the spacing exceeds the bandwidth at the lower partial, the equivalent-bandwidth model puts the boundary at a fundamental of 31.5 hertz; at the upper partial it puts it at 36.5. Reading it as the interval between them is wider than the band — which is how the critical-band essay states it — puts it at 42.4 hertz, and the Bark model, which is three times as wide down there, puts it at 131.5.

Those last two figures were quoted here as 85 and 263, which is exactly twice each, and the doubling is a bookkeeping slip worth naming because it is the kind that survives review. The first two readings evaluate the bandwidth at the lower partial, which is 2f₀ and is where the lower pipe actually is; the second two evaluated it at f₀, which is where nothing is sounding. Corrected, all four readings measure the band where a pipe is, and the spread between them narrows from a factor of eight to a factor of four.

Which footages are on which side #

The point of a boundary is to say which stops clear it, and running the five footages against all four readings gives a less comfortable answer than the one this section was leading to.

The sixteen-foot resultant clears one reading of four. It is not on the other side of the boundary; it is on the near side of three of them and a hertz past the fourth, which is the loosest of the four. The stop that is unambiguously on the other side is the eight-foot one, at 65 hertz, and even that fails the Bark reading — and an eight-foot resultant is not a stop anybody builds, because at that pitch a real pipe is eight feet long and there is nothing to avoid.

That is the awkward shape of the result. The footages where a resultant saves anything worth saving are exactly the footages where its two pipes are not resolved, on nearly every reading of the boundary; and the footages where the partials are cleanly resolved are the ones where nobody needs the trick. The stop is not on the wrong side of a line by an accident of pitch — it is on the wrong side by construction, because what makes a pipe expensive and what makes two partials unresolvable are the same thing, which is being low.

At 32 and 49 hertz a critical band is more than an octave wide, so the two pipes are not resolved into separate tones by the ear’s own filters at all — which is the condition the whole effect needs, and it is why the trick works at 32 feet and would not work an octave or two higher.

Now place the three sizes of the same trick on those boundaries.

A thirty-two-foot resultant implies 16.35 hertz from pipes at 32.7 and 49. It is below every reading, by a factor of two on the most favourable and a factor of sixteen on the least.

A sixteen-foot resultant — the far commoner stop, an eight-foot rank with a five-and-a-third — implies 32.7 from pipes at 65.4 and 98. It is above the most favourable reading, at 31.5, by a whisker: the separation is 32.7 hertz and the bandwidth at the lower pipe is 31.8. It is below the other two.

An eight-foot resultant would imply 65.4 from pipes at 131 and 196, and is above two of the three boundaries. Nobody builds one, because an eight-foot pipe is not expensive.

That is the shape of three centuries of disagreement, and it is not a matter of taste. The stop that organists describe as convincing is the one whose partials sit on the edge of resolution; the stop they describe as a rattle, a growl or a rumble is the one whose partials are well inside a single filter and whose only remaining cue is a sixteen-hertz fluctuation in a very low sound. Both descriptions are accurate reports of different physical situations that have been given the same name.

What a third rank buys, and it is not loudness #

Larger instruments sometimes draw a third rank with the pair — a six-and-two-fifths, sounding the fifth harmonic of the implied note, tuned pure as tierce ranks are. It adds a frequency the ear could hear out on its own, which sounds like a bad idea, and the candidate arithmetic says what it is for.

Two partials fix a ratio and nothing else. Three partials, one of them at an odd harmonic that is not a multiple of the others, fix a fundamental — and the price is that a listener who does hear the tierce out of the sound hears a major third that is not in the music. That is exactly the complaint made about tierce mixtures, and it is why they are drawn in some registrations and not in others.

The one case where Helmholtz and Schouten agree #

This ladder’s second rung exists to rule out the distortion account of the missing fundamental. The ear is nonlinear, two loud tones generate a real difference tone inside it, and Helmholtz proposed that the residue is simply that tone being heard. The shifted-residue experiment killed it: move every partial up by the same number of hertz and the spacing stays put while the reported pitch moves.

For a resultant stop the two accounts give the same answer, and it is worth saying so plainly rather than letting the earlier rung’s verdict do work it cannot do here.

The two pipes are loud, which is the condition the quadratic difference tone needs. So a resultant stop is one of the few musical objects where a genuine, physically present sixteen-hertz component may be generated in the listener. Whether it is audible as a pitch at sixteen hertz is another matter, since that is at or below the bottom of pitch perception altogether.

What the picture cannot show #

Nothing here is about loudness, and the stop is a loudness effect as much as a pitch effect. A resultant is drawn to strengthen the bottom of a full organ chorus, where it is heard through a wall of other sound, and the figures draw two partials in silence.

The room is left out entirely, and at these frequencies a room is not a small correction. A wavelength at 16 hertz is twenty-one metres, longer than most churches, so the building’s own modes decide what actually reaches a listener from a thirty-two-foot rank far more than the rank does. Two listeners in one nave hear different amounts of it.

The pipes are drawn as pure frequencies and a flue pipe is not one. A real sixteen-foot rank has its own harmonics — its second at 65 hertz, its third at 98 — and the ten-and-two-thirds rank has its own on top of those. Some of them coincide with the implied series and some do not, and a full account of what a resultant sounds like has to include the twenty or so partials actually present.

And the candidate ranking is a template argument, not a mechanism. Counting unoccupied slots is a compact way of saying why 16.35 beats 8.18 as an explanation; nothing in the auditory system counts slots.

The sound buttons on this page are transposed upward, and say by how much. A 2:3 pair at 33 and 49 hertz is exactly what the device a reader is holding cannot produce, which is the argument of the last rung of this anchor arriving two essays early. Every ratio is preserved by an octave shift and the register is not, so the buttons demonstrate the pattern and not the experience of a thirty-two-foot stop, which needs a building.

Whose organs, and when #

The resultant is documented from the early eighteenth century and became common in the nineteenth, when instruments grew and thirty-two-foot ranks stayed expensive. Aristide Cavaillé-Coll used quint ranks for the effect in French instruments; English town-hall organs of the same period carry stops labelled acoustic bass that are exactly this; and twentieth-century electronic organs made the same substitution in circuitry rather than in metal.

The practice around it is specific and it follows from the arithmetic. The quint rank is drawn only with the sixteen-foot rank and never alone, because alone it is a fifth. It is voiced softer than its partner, so that the pair is heard as one sound with an implication rather than as two notes. And it is normally restricted to the pedal division and to the bottom octave, above which the pipes are cheap enough that no substitute is wanted — which is also, and not by coincidence, the register in which the two partials start being resolved from each other and the effect starts sounding like a fifth.

Two related pieces of registration belong to the same computation. Mixture stops sound several upper partials of every note at once; they fuse rather than separating because the partials belong to the note’s own series, and they add exactly the harmonics the dominance region uses. And the quintaton, a stopped rank voiced to have a strong third partial and a weak second, is a deliberately ambiguous spectrum — the same machinery, run on a single pipe.

Where the ladder goes next #

The resultant turns out to be the wrong sort of evidence for the account this ladder endorsed. It has no resolved partials, so a template cannot be being matched to it; its two pipes are inside one auditory filter; and the only cue left is that their sum repeats sixteen times a second. If a listener hears a note there at all, something is reading the timing.

That is a claim about a mechanism, and it can be tested with a stimulus built for the purpose rather than for a church. A click train band-passed above every resolved harmonic has a pitch at its repetition rate and nothing spectral to explain it, and jittering the click times destroys that pitch while leaving the average rate and the long-term spectrum where they were. It is the next rung, and it bounds the pattern-matching account from the side this one only points at.

Sideways, the stop is the cleanest instance on this site of a design that was forced. Nobody chose to build a note out of two pipes. A thirty-two-foot rank is thirty-four feet of pipe, buildings are the size they are, and the substitution is what was left — with a defect that has been argued about ever since and that computes, from two published bandwidth models, to a factor of two.