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Organ Pipe Scale Calculator

Normscale deviation analysis — Töpfer standard (155.5 mm at 8′ C)
Rectangular pipe scaling — two methods shown:
Töpfer (area): Equiv. diameter = 2√(W×D/π) — equates round and rectangular pipes by cross-sectional area. This is the traditional method, but Nolte argues it overstates the effective scale of wood pipes by ~5 HT.
Nolte (diagonal): Equiv. diameter = √(W²+D²) — equates by the diagonal (circumscribing circle). Nolte’s thesis is that cutoff frequency and harmonic character are determined by the diagonal, not the area. A rectangular pipe sounds approximately like the round pipe whose diameter equals the wood pipe’s diagonal. (J.M. Nolte, “Scaling Pipes in Wood”, ISO Journal N°36, Dec. 2010)
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Pipe Measurements at Octave C′s (mm)

Parameter C01 C13 C25 C37 C49 C61

Normscale Deviation (semitones)

Voicing & Wind Data

Parameter C01 C13 C25 C37 C49 C61

Measured Body Length Interpolation (mm)

Cut-up Interpolation (mm)

Foot Hole Interpolation (mm)

Notes

Using the figures

Use the calculated figures to compare pipes within a rank, or comparable ranks, entered under the same conditions. They are steady-state estimates, not voicing targets.

Foot/flue area ratio: Below 1.0 means the foot hole is the dominant restriction. Above 1.0 means the flue dominates and foot pressure approaches system pressure.

Ising number: Read it as a consistency check across the compass. Values near 2 are fundamental-dominant, values above 2 carry more upper-partial energy, and values approaching 3 approach overblowing.

Wind velocity, flow, and the working Ising number use estimated internal foot pressure when the foot hole and flue dimensions are available. Otherwise they use system wind pressure.

Technical notes

Formulas, reference values, and limitations for the calculated figures.

Air and wind

Flue area is mouth width times flueway depth. Foot hole area is πd2/4. The ratio Afoot/Aflue determines which restriction controls the pressure.

Pfoot = P0 × Afoot² / (Afoot² + Aflue²)
v = √(2P / ρ)
Q = vAflue

P is internal foot pressure in Pa, ρ is air density (about 1.2 kg/m³), and 1 mmWS = 9.807 Pa. Q is untreated volume flow at the flue. The display does not model friction, turbulence, mouth losses, boundary-layer effects, or acoustic feedback.

Reference wind pressures: Positiv and Choir 45-55 mmWS; Great 60-75; Swell 70-85; Pedal 75-95; Romantic or symphonic 90-130 and higher; house organs 40-55.

Toe constant

Flentrop's relation for toe diameter can be simplified once pipe diameter cancels:

c = πdtoe² / (4 × mouth width)

This allows toe scaling to be compared across the compass without entering pipe diameter. Values typically fall between 0.5 and 2.5.

Pipe lengths

Theoretical length is λ/2 for open and conical pipes and λ/4 for stopped pipes. The speed of sound is derived from the entered reference temperature; a ten-degree change alters length by roughly 0.2 percent.

Calculated body length applies the usual end corrections. Open round pipe uses λ/2 − 5D/3. Conical pipe uses λ/2 − 2Dlower − 0.5Dupper. Wood pipe uses the theoretical length minus twice the depth, whether open or stopped. No calculated length is shown for round stopped pipe because stopper position determines sounding length.

Ising number

I = √(2PD / (ρH³)) / f

P is estimated internal foot pressure in Pa, D is measured flue thickness in metres, H is cutup in metres, and f is fundamental frequency in Hz. Cutup enters the denominator under the square root, so I ∝ H−3/2.

The effective dynamic flue thickness can be smaller than the blade measurement. The displayed number is a static estimate: values near 2 are fundamental-dominant, values above 2 indicate greater upper-partial energy, and values approaching 3 approach overblowing. Use it for comparison across a rank, not as a voicing target.

Wooden-pipe scaling

The Töpfer Normalmensur is the scale reference. It uses 55 mm inner diameter at 2′ C (155.5 mm at 8′ C), with diameter halving every 16 semitones. Deviations are expressed in semitones (HT), with negative values narrower and positive values wider.

Deq,Töpfer = 2√(W × D / π)
Deq,Nolte = √(W² + D²)

The Töpfer result equates cross-sectional area. Nolte equates the diagonal, which is about 5 HT wider than the area method for a square pipe. This corresponds to the usual finding that wooden scales run 4-6 HT narrower than their area-equivalent metal scales. Cutup is proportional to frequency, not mouth width.

The Normal Scale cutup extends the Töpfer Normscale to mouth height. It uses a quarter-circumference mouth and a cutup equal to one quarter of mouth width. Relative to this reference, Janke reports: Principals cutups close to Normal Scale; Gedeckts and Bourdons higher at about +6 to +9 HT; and Strings lower at about −6 to −12 HT. Low pressure shifts all values 2-3 HT lower.

Mouth width sets amplitude. Suggested labierung is about 1:4.5 for domestic rooms and 1:3.8 for large churches. When comparing scales, differences within about 0.5 HT may reflect manufacturing and measurement variation and should not be over-interpreted. The progression across the compass matters more than one absolute value. A metal continuation is normally made 3 HT wider than the preceding wooden pipe.

Sources

Johann Gottlob Töpfer, Lehrbuch der Orgelbaukunst, Weimar, 1855.

Hartmut Ising, “Erforschung und Planung des Orgelklanges”, Walcker Hausmitteilungen, Nr. 42, June 1971, pp. 38-57.

John M. Nolte, “Scaling Pipes in Wood”, ISO Journal, N°36, December 2010.

Reiner Janke, “Das Geheimnis der Mensuren”, Die Hausorgel, Heft 25, 2014; reprinted in ISO Journal, N°47, 2015.

Michael McNeil, The Sound of Pipe Organs, Mead, CO: CC&A, LLC, 2012. ISBN 978-0-9720386-5-2.

Courtesy of Rohlfs Pipe Organ Services LLC