Vol 03 — Open vs Stopped Pipes: Standing Waves & the Resonator

3.4 Open vs stopped, side by side

The two families differ in exactly four coupled ways — pitch law, harmonic content, length for a given pitch, and timbre — as summarised below.

Table 1 — Open vs stopped, side by side

The overblowing rows connect back to Vol 2’s regimes: an over-pressured open pipe jumps to its second mode (2f₁, the octave), which is why the harmonic flute is built at double length and driven there deliberately; an over-pressured stopped pipe has no second-mode octave to jump to and instead leaps to the twelfth (3f₁), the characteristic “quinting” or squealing failure of an over-blown Gedackt. The stopper’s mobility also gives the stopped family a practical advantage the diagram does not show: tuning is done by sliding the cap, a coarse and forgiving motion, whereas an open metal pipe is tuned by cutting, coning, or a tuning slide at the top — one reason stopped wooden ranks are common in small and home-built organs.

3.5 End correction: why real pipes are cut short

The formulas above are idealisations. An open end is not a perfect pressure node exactly at the geometric plane of the opening, because the slug of air just outside the pipe participates in the oscillation — it has inertia and radiates. The pipe therefore behaves as if it were slightly longer than its physical length. The correction added to the acoustic length at a plain open end is

Δ ≈ 0.6·a   (a = pipe internal radius)   — more precisely 0.61·a for an
                                            unflanged circular open end

(Rayleigh; Levine & Schwinger’s exact result gives 0.6133·a for an unflanged pipe; Fletcher & Rossing, ch. 8; corroborated widely, e.g. Hedberg, “Physics of End Correction in Organ Pipes,” ATOS). At the mouth end there is a second, larger and less tractable correction — a mouth (labial) correction Δ_m — because the mouth aperture, its cut-up, ears, and the local jet flow all load the column with extra effective mass. The full effective length is

L_eff = L_phys + Δ_top + Δ_mouth      f = c / (2·L_eff)   [open]
                                       f = c / (4·L_eff)   [stopped]

so the pipe must be cut physically shorter than the naive c/2f or c/4f to land on pitch. For a moderately scaled pipe the two corrections together can shave on the order of a diameter or more off the physical length; the mouth correction in particular is large enough that voicers and scalers treat it empirically rather than from first principles (Nederveen, Acoustical Aspects of Woodwind Instruments; Pykett, pykett.org.uk, “Scaling and the physics of organ pipes”). Because the correction scales with radius, wide-scale (fluty) pipes are cut proportionally shorter than narrow-scale (stringy) pipes of the same pitch — a detail that belongs to scaling (Vol 4). A stopped pipe carries an end correction only at its single open (mouth) end, not at the capped top, which is part of why its half-length rule is not exactly one-half in practice.

This end correction is also why the organ’s “foot-length” names are nominal. An 8′ open rank is so called because its lowest pipe is about eight feet of speaking length, but the modern equal-tempered bottom C (C₂, 65.41 Hz at A₄ = 440 Hz) actually wants an open speaking length near 343/(2·65.41) = 2.62 m ≈ 8.6 ft before end correction — trimmed back toward the nominal eight feet once the corrections are applied. The name is a pitch label, not a tape-measure reading.

3.6 Worked lengths at c = 343 m/s

The table gives ideal speaking lengths (before end correction) for the common pitches, computing both families from L_open = c/2f and L_stopped = c/4f. The sounding note is referenced to the equal-tempered scale at A₄ = 440 Hz; “8′” pitch places C₂ (65.41 Hz) at the bottom of a 61-note manual.

Table 2 — Worked lengths at c = 343 m/s

Read the 8′ and 16′ rows together and the economy of the stopped pipe is obvious: a stopped rank reaches 16′ pitch in the physical length of an open 8′ pipe (≈ 2.62 m). This is precisely why the ubiquitous pedal and manual Bourdon is a stopped 16′ rank — full-length open 16′ pipes (over 5 m) are costly, heavy, and space-hungry, whereas a stopped Bourdon delivers the same fundamental pitch in half the height, at the price of the rounder, odd-harmonic timbre. The same logic makes the stopped Gedackt 8′ the standard soft foundation of small organs: an 8′ voice barely 1.3 m tall.

3.7 Harmonic content: full series vs odd only

The pitch law is only half the story; the timbre difference is carried by which partials the resonator lets through. Vol 2 showed that the jet–lip nonlinearity offers energy at many harmonics; the resonator then reinforces those partials it has a mode for and starves the rest. An open pipe, with a mode at every integer harmonic, therefore radiates a full, gently decaying series; a stopped pipe, with modes only at the odd harmonics, radiates a spectrum with the even partials deeply suppressed — the “hollow” quality the ear reads as a covered, clarinet-adjacent tone. The schematic spectra below (relative amplitudes, illustrative) make the gap visible.

×

×

×

×

1 2 3 4 5 6 7 8 Schematic/illustrative amplitudes. × marks even harmonics the stopped resonator does not support. Real spectra vary with cut-up, scale, and wind (Vol 6).

Perceptually, a spectrum of odd harmonics with a strong fundamental and weak, widely spaced upper partials sounds hollow, dark, and flute-like at low volume; the missing octave partial (2f₁) removes the “brightening” the ear associates with a full series, and the fundamental dominates. This is the tonal signature of the Gedackt and Bourdon. An open Principal’s full series, by contrast, gives the firm, present, “vocal” tone that carries the organ’s chorus. The degree of upper development in either family is then trimmed by cut-up, scale, and wind pressure — the voicing variables of Vol 6 and the scaling of Vol 4 — but the presence or absence of the even harmonics is fixed here, by the boundary condition alone.

3.8 Half-stopped and chimney pipes (Rohrflöte)

Between the two families sits a hybrid: the half-stopped or chimney pipe, of which the Rohrflöte (“reed-” or “tube-flute,” also Chimney Flute, Koppelflöte) is the archetype. The cap of an otherwise stopped pipe is pierced and a short open tube (chimney) — in a wooden pipe, a hole bored through the stopper — is fitted at the top. Acoustically the chimney is a small open resonator in series with the closed body: it is roughly a quarter-wave tuned to a higher partial, and it selectively re-admits some of the harmonics the plain stopper forbade, most audibly reinforcing certain upper partials and adding a measure of even-harmonic content back into the spectrum.

The effect is a stopped-pipe fundamental (so the octave-down, half-length economy is largely kept) carrying a brighter, more “transparent” upper structure than a pure Gedackt — a bell-like or slightly reedy sparkle prized in solo flute stops. The chimney’s proportions set how much and which partials return: Audsley (The Art of Organ Building, 1905) recommends a chimney diameter of about 1/6 to 1/3 of the body diameter and a length of 1/4 to 1/2 of the body length; wider or longer chimneys admit more even harmonics and a brighter tone (organstops.org, “Chimney Flute”; Fletcher & Rossing, ch. 17). The related Quintadena is a stopped pipe deliberately voiced (narrow scale, low cut-up) to let its third harmonic — the twelfth, a quint — sound audibly against the fundamental, a different route to the same idea of coaxing an odd upper partial out of a stopped body.

3.9 The pitch-length convention and mutation ranks

Organ registration labels every rank by the pitch an open pipe of that length would sound, in feet, referenced to the lowest note of the manual. The unison — the rank that sounds the played pitch — is 8′; an octave above is 4′; two octaves, 2′; three, 1′. Sub-unisons run 16′ (an octave below) and 32′. Because the label tracks pitch, not physical size, it applies uniformly across open and stopped ranks — and this is exactly where the stopped pipe’s factor of two becomes a registration fact rather than a curiosity:

A stopped “8-foot” rank is physically about four feet tall, because a 4-ft stopped column sounds the same pitch (c/4L) as an 8-ft open column (c/2L). The nameplate reads 8′ — the pitch — while the pipe is half that length. Likewise a Bourdon 16′ is built from stopped pipes about 8 ft long.

Ranks pitched to a non-octave harmonic of the unison are mutations, and they exist to synthesise timbre additively — each reinforces a specific harmonic of the 8′ series so that drawing several together builds a composite tone (the organ’s version of additive synthesis, and the principle behind the Cornet and Sesquialtera solo combinations):

Table 3 — Sesquialtera solo combinations)

The fractional feet are literal harmonic ratios: 2⅔ = 8/3 (the 3rd harmonic), 1⅗ = 8/5 (the 5th), 1⅓ = 8/6 (the 6th). A quint rank at 2⅔′ can be open or stopped; a stopped Nasard is only about 0.44 m long (half of the open 0.875 m from the worked table), which is why quiet chorus mutations are frequently stopped or chimneyed — small, economical, and mild. A stopped mutation naturally suppresses its own even harmonics, so a stopped Nasard voiced to reinforce the 3rd harmonic contributes a purer fifth-flavoured colour than an open one that also drags in its 2nd and 4th. The boundary condition of this volume thus reaches all the way up into how a registration is designed.

3.10 A note on pitch stability

Every length formula above holds L fixed and lets c vary with temperature, so the sounding pitch of all these pipes, open and stopped alike, rises about +3 cents per °C (from c’s ≈ 0.6 m/s per °C, i.e. ≈ 0.18 %/°C ≈ 3 cents/°C). Because the shift is common to every flue pipe, a flue chorus stays internally in tune as the room warms; it is the mismatch between flue pipes (which track c) and reed pipes (whose tongue frequency is nearly temperature-independent) that detunes an organ against itself. That interaction is the subject of Vol 7; it is flagged here only to stress that the resonator length set by this volume’s boundary conditions is what the temperature term acts upon.

3.11 Summary

A flue pipe’s resonator is a one-dimensional acoustic cavity whose allowed standing waves are fixed by its two ends. The mouth is always an open end (displacement antinode / pressure node). If the far end is also open, the pipe supports f_n = n·c/2L — the full harmonic series — and sounds the bright, complete Principal/flute tone. If the far end is stopped, the pipe supports f_m = (2m−1)·c/4L — odd harmonics only — sounds an octave lower than the open pipe of equal length (equivalently reaching a given pitch at half the length), and produces the hollow, round, quieter Gedackt/Bourdon tone. Real pipes carry an end correction (≈ 0.61·a per open end plus a larger, empirical mouth correction), so they are cut physically short and their foot-length names are nominal pitch labels. Half-stopped/chimney pipes (Rohrflöte) re-admit selected upper and even partials for a brighter covered tone; the Quintadena coaxes out its own twelfth. The pitch-length convention labels ranks by the pitch of an equivalent open pipe, which is why a stopped 8′ is physically ~4 ft and a Bourdon 16′ ~8 ft; mutation ranks (2⅔′, 1⅗′, 1⅓′ …) pitch to non-octave harmonics to build timbre additively, and stopped mutations naturally deliver purer odd-harmonic colour. The drive that excites all of this is Vol 2; how diameter (scale) sets the degree of harmonic development within each family is Vol 4; the voicer’s control of cut-up, wind, and ears is Vol 6.

Sources

• Fletcher, N. H. & Rossing, T. D., The Physics of Musical Instruments, 2nd ed. (Springer) — ch. 8 (standing waves in air columns, boundary conditions, end correction ≈ 0.6a), ch. 15 (cylindrical stopped columns / clarinet analogy, odd harmonics), ch. 17 (flue-pipe resonators, open vs stopped, chimney/half-stopped pipes, overblowing to octave vs twelfth).
• Levine, H. & Schwinger, J., “On the radiation of sound from an unflanged circular pipe,” Phys. Rev. 73 (1948) — exact open-end correction 0.6133·a; Rayleigh, J. W. S., The Theory of Sound — classical 0.6·a result.
• Nederveen, C. J., Acoustical Aspects of Woodwind Instruments — end and mouth/hole corrections, effective length, half-stopped columns.
• Colin Pykett — pykett.org.uk, “Scaling and the physics of organ pipes,” “The physics of voicing organ flue pipes” — end/mouth correction as empirical quantities, open vs stopped spectra, chimney pipes. Rigorous, freely readable.
• Audsley, G. A., The Art of Organ Building (1905) — stopped, half-stopped, and chimney (Rohrflöte) construction; chimney proportions (⅙–⅓ body diameter, ¼–½ body length); mutation-rank practice. Public domain.
• Hedberg, D., “Physics of End Correction in Organ Pipes” (ATOS) — measured end-correction behaviour in real organ pipes.
• organstops.org and Organ Historical Society, “Pipes and Timbres” — Gedackt/Bourdon/Rohrflöte/Quintadena rank definitions and timbres; mutation and foot-length conventions.

Cross-references: the jet-drive that excites these resonances, harmonic generation at the mouth, and overblowing regimes — Vol 2; pipe anatomy and the flue/reed families — Vol 1; how diameter (scale) sets the degree of harmonic development and loudness within each family — Vol 4; reed resonators (a different tuning relationship) — Vol 5; the voicer’s cut-up, ears, and wind-pressure control of the tone — Vol 6; the temperature dependence of c and the resulting pitch drift and flue-vs-reed detuning — Vol 7.