Organ Pipe Scale Calculator

Notes on Voicing and Wind

The Air Sheet and Flue Area

The air sheet (Luftband) is the thin ribbon of air that emerges from the flue and strikes the upper lip. Its cross-section — the flue area — is the product of mouth width and flueway depth. This area, together with the wind pressure, determines how much energy enters the pipe. Flueway depth is not derived from a formula; it is set empirically during voicing, typically in the range of 0.4–1.4 mm for metal pipes and somewhat wider for wood. A deeper flueway means more power and faster speech. A shallower flueway means less power and slower speech. Timbre — the balance between the fundamental and the upper harmonics — is controlled primarily by the cutup height, not the flueway depth.

Wind Pressure and Velocity

Wind pressure in organ building is traditionally measured in millimetres water column (mmWS). The theoretical maximum velocity of air emerging from the flue is given by Bernoulli’s equation:

where P is the pressure in Pascals (1 mmWS = 9.807 Pa) and ρ is the air density (~1.2 kg/m³ at 20°C). At a typical organ wind pressure of 65 mmWS, this gives approximately 32.6 m/s. In practice, the actual velocity at the flue is lower due to friction and turbulence in the foot and windway.

Typical wind pressures in organ building:

• Positiv, Choir organs: 45–55 mmWS
• Great (Hauptwerk): 60–75 mmWS
• Swell: 70–85 mmWS
• Pedal: 75–95 mmWS
• Romantic / symphonic organs: 90–130 mmWS and higher
• House organs: 40–55 mmWS

The Foot Hole and Flue Area Ratio

Both the foot hole (Fußloch) and the flue act as restrictions in the wind path — two orifices in series. Together they determine the internal foot pressure and, through that, the velocity of the air sheet at the mouth. The foot/flue area ratio shows which of the two is the dominant restriction.

• Ratio < 1.0: the foot hole is smaller than the flue. The foot hole is the primary restriction: it reduces internal pressure, power, and speech speed. Common in string voicing and for stops requiring a gentle, controlled onset.
• Ratio > 1.0: the flue is smaller than the foot hole. The flue carries most of the restriction; foot pressure is close to system pressure. Typical of open-toe voicing.

The two restrictions can be used as independent voicing tools, and different traditions have weighted them differently. In the Germanic tradition, the flueway is kept relatively narrow and the toe left fully open. All regulation happens at the flue; the open toe ensures full system pressure at the foot. This produces a direct, energetic tone with a clear, prompt attack. In the French tradition, the flueway is left comparatively wide and the toe is partially closed. The restricted toe reduces effective pressure at the flue independently of the flueway geometry, which allows softer, more controlled voicing on the same wind pressure and gives the voicer separate leverage over the onset character and the steady-state amplitude.

Neither approach is universally correct. Many voicers work with both levers simultaneously, particularly when nicking is involved or when higher wind pressures make fully open-toe voicing difficult to control.

Toe Constant (c)

D. A. Flentrop proposed a formula for the toe diameter that delivers a uniform amount of wind to the pipe across the compass:

where f is the fractional mouth width (mouth width ÷ circumference) and c is a dimensionless constant the voicer adjusts for wind pressure, acoustics, and tonal intent. Rearranged, the pipe diameter cancels and c reduces to:

This means c can be derived directly from the foot hole diameter and mouth width alone, without knowing the pipe diameter. Values of c typically range from about 0.5 to 2.5. Higher wind pressures generally require smaller values of c to avoid over-winding the pipe; lower pressures allow larger values. Comparing c across the compass of a rank reveals whether the toe sizing follows a consistent voicing logic, and comparing it across organs gives a clear picture of different building traditions.

Volume Flow

The volume flow through the flue is the product of wind velocity and flue area. It represents the rate at which air is consumed by the pipe and is relevant for wind system design: the total flow of all sounding pipes at any moment must not exceed the capacity of the bellows and wind trunks, or the pressure will drop and the pitch will sag. In practice, the volume flow shown here is an upper estimate — actual consumption is moderated by the foot/flue area ratio, mouth geometry, and acoustic feedback from the resonator.

Internal Foot Pressure

With foot hole and flue in series, the pressure between them — the internal foot pressure — controls the actual air sheet velocity. Applying Bernoulli’s equation to the two orifices:

When the foot hole is wide open, P_foot ≈ P₀. When the foot hole is smaller than the flue, P_foot drops accordingly — the pipe behaves as if voiced on lower pressure. All values are steady-state estimates; acoustic feedback from the resonator reduces actual wind consumption in a speaking pipe.

Pipe Lengths

Three length values are shown for each pipe. The theoretical length is the acoustic resonance length — λ/2 for open and conical pipes, λ/4 for stopped. The calculated body length is the expected physical pipe length: open pipes use λ/2 − 0.6D, stopped pipes use λ/4 − 0.3D, and conical pipes use the empirical formula λ/2 − 2D_lower − 0.5D_upper. Comparing the measured body length to the calculated value shows how closely a pipe matches its expected dimensions.

The Ising Intonation Number

Hartmut Ising (1971) proposed a dimensionless number that characterises the voicing state of a flue pipe. It relates wind pressure, flue thickness, cutup height, and the pipe’s fundamental frequency:

where P is the blowing pressure (Pa), D is the flue thickness (m), ρ is the air density (~1.2 kg/m³), H is the cutup height (m), and f is the fundamental frequency (Hz). The number captures the essential trading relations between these four voicing parameters. Pipes of different size and pitch but with the same intonation number will have similar voicing character.

• I ≈ 2: optimal efficiency. Most of the wind energy drives the fundamental. This is the region of smooth, vocale tone.
• I well above 2: more energy feeds into upper partials. Typical for principals with a bright, carrying tone.
• I approaching 3: the pipe approaches the overblowing threshold. String stops with a frein (roller beard) may operate here; without stabilisation the pipe tends to jump to the octave.

The formula makes explicit what voicers know intuitively: raising the cutup (H) has a powerful damping effect (H enters as the cube), while increasing wind pressure or flue thickness pushes the number up. A treble pipe at high frequency naturally has a lower I for the same geometry, which is why treble pipes can tolerate relatively higher cutups without losing prompt speech.

Using the Ising Number in Practice

The number is most useful as a consistency check across a rank. If you know the voicing character of one pipe, its Ising number gives you a reference to compare the rest of the compass against. A markedly higher I in the bass suggests the bass is driven harder than the midrange; a drop in the treble may point to a cutup that is too high or a flue that is too thin.

The formula also makes the relative weight of each parameter explicit. Cutup (H) enters as H^3/2 and has by far the strongest influence on I: a small change in cutup moves the number much more than an equivalent proportional change in pressure or flue thickness. Wind pressure and flue thickness both enter as square roots and have roughly equal leverage.

The Ising number shown in the working table is calculated from the estimated internal foot pressure, not from the system wind pressure, so toe regulation is taken into account directly. Note that the aerodynamically effective flue thickness may be somewhat smaller than the mechanical measure taken with a blade gauge, due to boundary layer effects in the flue channel. All values are for the static (non-speaking) condition and should be treated as approximations.

Notes on Scaling Wooden Pipes

The Normscale Reference

The Töpfer Normalmensur provides a standard reference for comparing pipe scales. It is based on a 2′ Principal C with an inner diameter of 55 mm (155.5 mm at 8′ C), with the diameter halving every 16 semitones (the 17th pipe). Deviations from this reference are expressed in semitones (Halbtöne, HT): negative values indicate a narrower scale, positive values a wider one. The Normalmensur is not an ideal to be followed, but a yardstick for comparison — a common language for evaluating and communicating pipe scales.

Equivalent Diameter: Töpfer vs. Nolte

Since Töpfer, the standard practice has been to equate round metal pipes with rectangular wooden pipes by cross-sectional area. The equivalent diameter is:

Nolte challenges this convention. His experience building wooden pipes for orchestrions, nickelodeons, and organs led him to conclude that the cutoff frequency — the frequency above which a pipe no longer supports harmonics — is determined not by the area but by the diagonal of the rectangular cross-section. A rectangular air column behaves acoustically like the circular air column that circumscribes it. The equivalent diameter by this method is simply:

For a typical square pipe, the Nolte method gives a diameter about 5 semitones wider than the Töpfer method. This matches the well-known practical observation that wooden pipes need to be built 4–6 HT narrower than their metal equivalents to achieve a comparable tone. For string stops requiring very rich harmonic development, Nolte reports reductions of up to 10 HT.

The practical implication: a 4″×5″ wooden Gedeckt listed as equivalent to metal scale 48/49 by area actually sounds more like a metal scale 43 — because its diagonal (162.6 mm) is close to that scale’s diameter (162.2 mm).

What the Diameter Does and Does Not Determine

Janke, drawing on 38 years of voicing experience, observes that diameter has a significantly lesser influence on tone than cutup height and wind pressure. Of the many parameters affecting a flue pipe’s sound, the diameter primarily sets one thing: the upper limit of harmonic development. After voicing changes, the effect of diameter often remains audible only as a subtle colouring.

Cutup Scaling

Both Nolte and Janke emphasize that cutup is proportional to frequency, not to mouth width. This means proportional dividers (Reduktionszirkel) are of limited use except when voicing a normscale rank for uniform tonality. Janke proposes a Normaufschnitt (normal cutup) reference derived from the normscale: assuming a 1/4 mouth and a cutup of 1/4 of the mouth width. In practice, he finds:

• Principals: close to the Normaufschnitt values
• Gedeckts: +6 to +9 HT above Normaufschnitt (referenced to sounding length)
• Strings: −6 to −12 HT below Normaufschnitt
• Low wind pressure shifts all values down 2–3 HT

Mouth Width (Labierung)

Mouth width controls amplitude, not harmonic development. A narrow mouth is quieter and darker (emphasising vowels O/U); a wide mouth is louder and brighter (vowels A/E). Two pipes of the same diameter but different mouth widths can be made to sound identical by adjusting cutup and flue depth — the narrower-mouthed pipe will simply be quieter. Janke recommends a labierung of approximately 1:4.5 for domestic rooms and 1:3.8 for large churches.

The Shape of the Scaling Curve

The progression of the scale across the compass is as important as the absolute value at any one point. A romantic Geigenprinzipal 8′ shows a linear rise toward the treble; a baroque Octave 4′ after Silbermann dips to −6 HT in the midrange before rising again in the treble. These different curves serve different musical functions: the Geigenprinzipal supports homophonic playing with a strong soprano, while the baroque Octave reinforces the midrange for polyphonic texture. An accuracy of ±½ HT is sufficient — the trend and shape of the curve matter more than exact values.

Wood-to-Metal Transitions

Janke recommends making the metal continuation 3 HT wider than the preceding wooden pipes to compensate for the tonal difference arising from the different mouth geometry of wood pipes. This is consistent with Nolte’s diagonal theory: the wooden pipe’s effective scale is narrower than its area-based equivalent suggests, and the transition to round metal at the same area-equivalent scale would produce an audible jump in brightness.