Michael McNeil has designed, constructed, voiced, and researched pipe organs since 1973. Stimulating work as a research engineer in magnetic recording paid the bills. He is working on his Opus 5, which explores how an understanding of the human sensitivity to the changes in sound can be used to increase emotional impact. Opus 5 includes double expression, a controllable wind dynamic, chorus phase shifting, and meantone. Stay tuned.
Read part 2 here
Prologue
I am not a good mathematician, but I give fair warning, this article has some math. I have tried to make it accessible. It is an enduring mystery that math appears to describe how the universe works, and it does so with astounding precision. Richard Feynmann once quipped that “It’s the language God talks.”1 There is no gulf between art and science, and if you are interested in the art of voicing, you will gain a deeper appreciation from the science behind the art.
—M. McNeil
The art of voicing organ pipes is often portrayed as a veiled mystery practiced by a few elite and gifted artists. While the underlying physics has some complexity, with some effort we can understand how it works. As readers of The Diapason know, I have attempted to acquire a better understanding of voicing by comparing the work of some of its historical practitioners.2 In this article I will try to draw back the veil on some of the mystery.
In earlier articles I have described a voicer’s basic adjustments: toe diameters, flueway depths, and cutups (mouth heights). Figure 1 shows these features. I demonstrated how we can relate toe diameters to pipe scaling and how voicers use toes to control power and power balances. I have shown how cutups control timbre and how we might visualize cutups on a Normal Scale. Lastly, I wrote that flueway depths not only control power, they also play an important role in pipe speech.3
Voicers know that when you open a toe hole or a flueway the power will normally increase and the timbre will get brighter, but how much brighter and will the speech overblow to the octave? How much will we have to raise the cutup at a higher power to restore the original timbre? If we are voicing new pipes, where do we start? These are not trivial questions, and it is why we encounter recipes. But recipes do not explain how voicing works.
I found some answers to these questions in the inspired work of Hartmut Ising and his fascinating equation that shows how cutups, flueway depths, and wind pressures affect a pipe’s timbre and promptness of speech.4 I have the good fortune of scaling and voicing a principal chorus in my retirement, and I have used that opportunity to explore how we might use Ising’s equation in the real world of voicing.
Ising did not consider how wind pressure in the pipe foot varies with different diameters of pipe toes, and this poses a major problem, for I have shown that all organbuilders close the pipe toe to some degree.5 I addressed this problem by using the voicing of Gottfried Silbermann as a starting point. His voicing, unlike any other I have seen, is extremely consistent in the treatment of toes and flueways. His cutups are his free voicing variable for timbre. In Figure 2 we see Silbermann’s toe constants for 90 millimeters of pressure at the Freiberg Dom (originally 97 millimeters).
Toe constants show us the relative flow of wind in pipes of any diameter or mouth width. A toe with a constant of 2 has twice the area of a toe with a constant of 1. Silbermann is using the same relative flow of wind in pipes of the same pitch regardless of where they appear in his chorus. The wind flow and power in Figure 2 is lowest at 2′ pitch, and it increases in both the bass and treble. The fascinating drop in the flow at 1⁄8′ pitch is explained in a previous article, and it reflects Gottfried Silbermann’s deep mathematical understanding of voicing.6
In Figure 3 we see Silbermann’s flueway depths. These flueways are extremely consistent at any given pitch, and they are very deep, approaching what we see in Romantic voicing. Silbermann’s chorus blend is superb, and the deeper flueways appear to be a major factor in this blend. For my experiment in voicing a new chorus I rigorously adopted these flueway depths from 4′ to 1⁄8′ pitch, allowing only the toe diameters and cutups to vary. This was a new departure for me, and in the higher pitches the flueways are deeper than what I would have normally used. I filed a taper on the end of a brass rod and scribed lines on it to accurately measure flueway depths. Figure 4 shows this tapered rod inserted through the toe and into the flueway of a mixture pipe.
Pipe diameters and mouth widths have been very well understood for centuries—they affect power and timbre, and one halftone of the Normal Scale represents 0.5 dB in power. Larger toe diameters and deeper flueways allow more flow of wind and yield more power, and they also make the timbre much brighter if cutups are not raised. Timbre is a major goal of voicing, but achieving the timbre you want is not a trivial problem: toes, flueways, and cutups all affect timbre, and they interact in complex ways.
Hartmut Ising thought deeply about this problem and derived an equation to understand how these interactions affect timbre. The equation is elegant, and its output is a dimensionless number that represents timbre. A timbre of 2 represents the condition where the pipe has the fastest speech, and the timbre is in the range of a normal principal. The speech becomes slower as values ascend above 2, the timbre gets brighter and more “instrumental,” and at a timbre of 3 the pipe overblows to the octave. The speech remains fast, and the timbre gets smoother as values descend below 2, where a value of 1 is consistent with a smooth flute. The timbre value is called “I” after the German word “Intonation.” Here is Ising’s equation:7
F is the frequency of the pitch in Hz.
D is the depth of the flueway in meters.
H is the height of the mouth in meters.
P is the wind pressure in Pascals. Organ pressure is measured by the height of a water column, and 1 millimeter of pressure = 9.80665 Pascals.
ρ is the density of air in kg/m3, which is 1.2 kg/m3 under standard conditions at sea level.
This equation is simpler than it looks. I gets larger and the timbre gets brighter with more pressure P (which is what happens when we enlarge a toe or add weight to the bellows), the timbre gets brighter as we deepen the flueway D, and the timbre gets smoother very quickly as we increase the cutup height H. We do not worry about the density of air ρ because it does not change unless we move to a different altitude. The term 1/F compensates as the frequency of the pipe F increases with higher pitches. Cutups may appear to be related to mouth widths, but this is a dangerous illusion—cutups and the timbres they control are related to pitch, pressure in the foot, and flueway depth. Very small changes in cutup have enormous effects on timbre because H is cubed, H3!
Ising’s equation shows us how pressures, flueway depths, and cutups interact. To give an example of why this is important, a voicer may be confronted with a pipe that is close to overblowing on slightly higher pressure. Voicers often blow pipes by mouth to test the sound at different pressures.8 An overblowing condition may result from a cutup that is too low or a languid that is too low. Which adjustment should the voicer make? Raising the cutup too far is not easily reversible. Ising tells us what to adjust, and if the cutup is correct, we adjust the languid height and the upper lip position, sparing us the ordeal of having to lower a cutup.
Building a voicing model
Silbermann used extremely regular scales, toes, and flueways, but his cutups in Figure 5 are highly variable for different stops at a specific pitch. These cutups yield different timbres in the chorus.
For the purposes of my experiment with Ising’s equation, I assumed diameters, mouth widths (MW), flueways, toe constants (Tc), and cutups (MH) for Silbermann that are composite values of his higher pressure organs voiced on about 85–90 millimeters:9
Pitch Dia. MW Flue Tc MH, NSMH
16′ 270.0 242.0 2.50 2.00 55.0, -0 HT
8′ 150.0 135.0 1.40 1.69 30.0, -1 HT
4′ 87.8 78.8 1.20 1.20 17.6, -1 HT
2′ 48.8 43.8 0.90 0.88 10.3, -2 HT
1′ 26.3 23.6 0.80 1.00 7.0, +2 HT
1⁄2′ 15.3 13.7 0.60 1.28 5.0, +6 HT
1⁄4′ 9.9 8.9 0.60 1.50 3.0, +6 HT
1⁄8′ 6.9 6.2 0.40 1.20 2.0, +9 HT
The values above were used as a basis for the voicing model shown in Figure 6. The cutups (MH) are also shown in their Normal Scale values (NSMH).
Constructing the model inputs and outputs
I created the model shown in Figure 6 in Excel. The inputs to the model are in the rows with blue backgrounds. These are all self explanatory. The “NS” or “Normal Scale” diameters, mouth widths, and mouth heights are not part of the voicing calculations, but help to visualize the trends.
The toes are entered as toe constants, and the toe diameters in the row “Toe Diameter” equal the square root of (the toe constant × the pipe inside diameter × 4 × the mouth width fraction). The mouth width fraction is simply the mouth width divided by the pipe circumference, and it is shown in the row “Mouth Fraction.” Wider mouths with larger fractions need more wind—this is explained in more detail in the note.10
The row “Ratio, A toe/A flue” is simply a ratio of the area of the toe (π r2) to the area of the flueway (flueway depth × mouth width). This area ratio has a large impact on speech, and in Part 2 we will address this in detail. The row “Ising Timbre” is Ising’s value I, and it is calculated from the Ising Equation inputs in the bottom rows of the model where the dimensions of the flueways and cutups are converted to meters.
Ising assumes a wide open toe (a toe constant of 4 or more), and he does not compensate for the pressure drop in the foot from a closed toe. This problem has prevented the model’s application to voicing with restricted toes, and the solution I propose here is open to many criticisms of which I am well aware. But it shows us a path toward a more general solution, and it allowed me to test Ising’s equation in the voicing of a new chorus.
Addressing the problem of closed toes
Previous experiments by others have shown that typical pressures in a pipe foot are in the range of about 25 to 35 millimeters, and this is vastly lower than the bellows pressure of 85 millimeters in the model. Closing a pipe toe reduces the bellows pressure to a usable value in the foot. In the row “Assumptions” I have defined a toe constant of 1.00 to represent a pressure of 26 millimeters or 255 Pa, and a toe constant of 4.00 (open toe) to represent the static bellows pressure of 85 millimeters or 834 Pa. In the row labeled “Pa, f(Tc)” I have shown a second-order polynomial equation that calculates foot pressures from the toe constants. For example, this equation uses the toe constant of “1.00” circled in red to calculate the pressure “255.0” also circled in red in the row “Pa, f(Tc)”. The foot pressures in Pascals for the different toe constants of the other pitches are shown in this row.
The foot pressures required compensation in the bass where the calculated Ising values were obviously far too bright. Upon reflection, Silbermann tubed the wind for these pipes to the façade or offset racks, and there are pressure losses with offset tubing that likely resulted in the larger toe constants in the bass. In the row labeled “Pa compensation” I introduced a reduction term to lower the pressure to a value that would produce Ising values similar to the value at 2′ pitch. This compensation affects the values in the row “Pa, f(Tc),” and the resulting lower pressures are seen in the last row of data labeled “Compensated Pa.” These values are copied to the row “Pressure, Pa” and are used in the calculation of the Ising values in the row labeled “Ising Timbre.” The model is now “tuned” to an approximation of Silbermann’s voicing.
The flaws in the pressure model
Visualize a completely closed flueway that vents no wind: the pressure in the foot may rise more slowly with restricted toes, but it will eventually and always rise to the bellows pressure. As the flueway is opened, the pressure will vent and drop in the foot. This flueway interaction is why my calculation of foot pressures from toe constants is crude. The model gets us in the ballpark, but a more generally applicable voicing model would include a calculation of foot pressure from the bellows pressure, toe area, flueway area, and the volume of the pipe foot (a rather difficult problem).11
Applying the model
The model can now be used for pipes with any scale diameters, mouth width ratios, flueway depths, and toe constants to find the cutups that will give the voicer any desired timbre. It works like this: input everything but the cutups, then input different cutup values (“Mouth Height”) until you get the Ising timbre you want. Test this on a few pipes at different pitches, raising the cutups by degrees to confirm that the model gives you the right timbre.
The model was applied to a three-rank Fourniture with pitches spanning 1′ to 1⁄8′. This stop is part of a 16′ principal chorus that I have voiced on the model’s assumption of 85 millimeters pressure. We can see the Fourniture voicing model in Figure 7, and it uses different scales than those of the Silbermann model in Figure 6. The toe constants are lower than those of Silbermann, especially in the higher pitches, for a more restrained power on the very large 0.286 mouth width fractions of these pipes. Flueway depths are exactly those of Silbermann. Cutups were adjusted to Ising values that progressed from 2.05 at 1′ pitch to a very smooth timbre of 1.21 at 1⁄8′ pitch. All pipes in the Fourniture of the same pitch are adjusted to the same voicing values regardless of where they appear in the compass.
Validating the model
I started by voicing three test pipes spanning the range of the mixture pitches: 1′ C-sharp, 1⁄3′ F, and 1⁄8′ C. I raised the cutups by degrees, testing the speech and timbre. The modeled cutup values satisfied my expectations of timbre and fast, stable speech (not easily overblown). We can use a spectrum analyzer to visualize the harmonics that make up Ising’s timbre, and your iPhone can be a real-time spectrum analyzer with an app that costs only $20.12 The spectrum analyzer confirmed that I was on the right path, validating the model.
Figure 8 shows the spectra of the 1′ pipe after raising its cutup to the model value. A spectrum analyzer displays a sound in all of its separate frequencies (its spectra), each of which represents a harmonic. A pure tone has only the fundamental harmonic, but as the timbre gets brighter we see more harmonics. This mixture pipe has mild harmonic brightness with an Ising timbre of 2.05. The numbers of each harmonic are seen in red font above the harmonics. Higher peaks are more powerful. The frequencies or pitch of the higher harmonics in Figure 8 are exact multiples of the 551 Hz fundamental harmonic, i.e., the second harmonic “2” at the octave is 551 Hz × 2, or 1,102 Hz, and so on. Note that the power of the harmonics subsides as the harmonics rise in pitch. Smoother timbres result from fewer harmonics and lower harmonic power. When the Ising timbre approaches 3 and a pipe is ready to overblow to the octave, the second harmonic “2” will rise far above the power of the fundamental, and the higher harmonics will also rise in power with a screeching timbre. The fundamental harmonic will disappear when the pipe overblows. The sound in Figure 8 was clear, but not too bright, and it was fast and stable in its speech.
The pipe speaking at about 1⁄3′ in Figure 9 shows far fewer harmonics than those seen in the 1′ pipe in Figure 8. This is consistent with its lower Ising timbre of about 1.60, a smoother timbre than the 1′ pipe.
Figure 10 shows the 1⁄8′ pipe with a nearly pure fundamental harmonic and a barely perceptible second harmonic. This is a very pure, very smooth timbre, and it is consistent with its low Ising timbre of 1.21. The sound of these three pipes and the images of their spectra validated the modeled expectations of timbre and fast, stable speech.
Constructing a voicing worksheet
With the validation of the model, I used the values in Figure 7 to make the worksheet seen in Figure 11 as a guide for voicing every pipe in the mixture.
The two far left columns in Figure 11 are the numbers and names of the notes in the compass, and these are marked on each pipe (this is a meantone organ with a short bass octave having no C-sharp, D-sharp, F-sharp, or G-sharp). The next column shows the composition of the mixture pitches and their break points in green and yellow highlights. The next two columns show the fifth and unison pitches from 1′ to 1⁄8′. The unison pitches at C are highlighted in blue in both columns for quick reference. The voicing is the same for pipes of the same pitch, whether they occur in unisons or fifths. With this three-rank mixture each note will have one or two unisons, which will align with the Unison column values. Each note will also have one or two fifths, and the toes, cutups, and flueways for these fifth-speaking pipes will be the same for the same pitches in the Unison column. So for a fifth-speaking D above the unison G, you look up the D above G in the Unison column to find the voicing values for the fifth-speaking pipe on D. The Fifth column is just a reminder of the note values of the fifth-speaking pipes.
The blue highlighted cells have pipe lengths to help sort the right pitch octave. Modeled voicing and reference values are seen in the columns to the right: toe diameters, toe constants, cutups, flueway depths, ratio of toe and flueway areas, and lastly, the modeled Ising timbre values. The full set of toe diameters and cutups for each note are interpolations between the modeled values. Further to the right we find pipe diameters for fitting tuning slides.13 Next are calculated body lengths in the column with blue font, needed because the overlengths of unvoiced pipes will adversely affect the voicing. These calculated lengths were derived from a model created by
John Brombaugh.14
The group of columns to the far right contain entries of the measured tuned lengths of all of the voiced pipes, their averages, and their surprisingly small deviations from the theoretical values. I found that the final tuned lengths were minimally affected by the toe constants. Smaller toe constants provide less wind and should result in slightly shorter tuned lengths. Actual tuned lengths correlated with the calculated lengths almost perfectly with a mixture that had 0.7 toe constants. These were smaller than the 0.9 toe constants of the Fourniture, which on average had tuned lengths only 0.5% higher than the model from 1′ to 1⁄4′ pitch. The actual tuned length at 1⁄8′ d51 pitch was 8.2% lower than the calculated length, and the source of this very large error, circled in red in Figure 11, is mine, not Brombaugh’s.15
§
In Part 2 we will illustrate the voicing process of the Fourniture and will see how the timbre of the Fourniture relates to the timbre of its principal chorus. We will use the voicing model to investigate the timbre of the sound of Dirk A. Flentrop and see how it differs from and redefines the sound of Gottfried Silbermann. Ising does not address the sources of faster and slower speech that arise from the resistance to the flow of wind in toes and flueways, and we will illustrate those sources with examples, a sound clip, and a short video. Ising’s work was often cited by John Coltman, who created an impedance model to describe how a pipe produces sound. In Part 2 we will see a short video demonstrating a more intuitive mechanical analogy of impedance and its relevance to Ising’s equation.
To be continued.
Notes and references
All images are found in the collection of the author.
1. Steven Strogatz, Infinite Powers (Houghton, Mifflin, Harcourt, 2019), page vii. For those who are interested, a recent book by Stanilas Dehaene explains how our brains perceive math: The Number Sense, How the Mind Creates Mathematics (Oxford University Press, 2011). This book is easily read by the non-specialist, and it provides good answers to that “enduring mystery.”
2. Michael McNeil, “The Sound of Gottfried Silbermann, Part 1,” The Diapason, volume 113, number 12, whole number 1356 (December 2022), pages 12–17, and “The Sound of Gottfried Silbermann, Part 2,” volume 114, number 1, whole number 1357 (January 2023), pages 13–19. See pages 17–18 in part 2 for the explanation of the drop in the toe constant at 1⁄8′ in Figure 2.
3. McNeil, “The Sound of Gottfried Silbermann, Part 1,” and “The Sound of Gottfried Silbermann, Part 2”; Michael McNeil, “The Sound of D. A. Flentrop,” The Diapason, volume 115 number 9, whole number 1378 (September 2024), pages 14–20.
4. Hartmut Ising, “Erforschung und Planung des Orgelklanges,” Walcker Hausmitteilung no. 42, June 1971, pages 38–57. See also Michael McNeil, “The Sound of Pipe Organs at Altitude,” The Diapason, volume 112, number 12, whole number 1345 (December 2021), pages 19–21. This article provides a background on Hartmut Ising and the derivation by Johan Liljencrants of the form of Ising’s equation used in this article.
5. McNeil, “The Sound of Gottfried Silbermann, Part 1,” and “The Sound of Gottfried Silbermann, Part 2.”
6. Ibid.
7. Ising. McNeil, “The Sound of Pipe Organs at Altitude.”
8. Lead poisoning, from which Gottfried Silbermann may have suffered in old age, can result from the contact of organ pipes with one’s skin, especially at the mouth, as organ pipes contain significant amounts of lead. Put the pipe toe in your closed fist and blow into your fist without touching your mouth to the pipe. I wear nitrile gloves when handling organ pipes. Many voicers use a voicing machine, which is just a small windchest, keyboard, and bellows, and this method has some convenience. Blowing a pipe by mouth allows you to change both the pressure and the attack to test if the pipe is close to overblowing—a very useful diagnostic. A YouTube video of the voicing of a Taylor & Boody organ demonstrates the value of varying pressure to find the desired cutups, where we see the partial closing of a slider to reduce the pressure: youtube.com/watch?v=_NT65GJNBrU.
9. Frank-Harald Greß, Die Klanggestalt der Orgeln Gottfried Silbermanns (VEB Deutscher Verlag für Musik, Leipzig, 1989). Data for Silbermann’s scaling and voicing was derived from pages 71, 100–101, 136–146.
10. In conversation, August 25, 1972, Dirk Flentrop defined a fully open toe (where the foot pressure equals the bellows pressure) as:
This equation has an implied toe constant of 4, which historic builders rarely approached, and only in the highest mixture pitches. Mouth widths affect the width and the area of the flueway, and this in turn affects the amount of wind needed from the toe. Flentrop’s typical mouth width is 0.25 × the pipe circumference, and I normalized Flentrop’s mouth width with the term “4 × mouth width fraction,” which in Flentrop’s case would be 4 × 0.25 = 1. Wider mouths like Silbermann’s 0.286 fraction need more wind, and narrower mouths like Ernest M. Skinner’s 0.20 fraction need less. The resulting equation for toe diameters includes a toe constant to vary the flow of wind, the diameter of the pipe, and the mouth width fraction:
An underlying assumption in this equation is a flueway depth of 1 millimeter.
11. Most classical Germanic voicing is considered “open toe,” meaning that the toe area is usually much larger than the flueway area. In this condition the flueway behaves as Ising intended: opening the flueway will increase the velocity of the vortex, the timbre will get brighter, and the pipe will more easily overblow. But as I came to see, the pressure in the foot will drop when the flueway is opened with restricted toes, and this drop is very noticeable when the area of the flueway is larger than the area of the toe—this is the realm of much classical French voicing, where opening the flueway will make the timbre less bright and less likely to overblow. My polynomial equation for foot pressures and toe constants is admittedly crude, and I also experimented with a parabolic function (in vertex form):
Foot PressurePa = -23.1666(Tc-6)2+834
This is a better fit, but that fit requires a fully open toe constant of 6, which seems excessive.
12. Apple sells “Real Time Analyzer,” a very powerful spectrum analyzer app for the iPhone. The version used here is: apps/apple.com/spectrum-analyzer-rta/id490078884.
13. I now use tuning slides, agreeing with Charles Fisk in his observation that cone tuning, while much more stable than slides, inevitably results in severe damage to the voicing when someone unskilled attempts to tune the pipes. If there is one thing that we have learned from Ising’s equation, it is that cutups are extremely critical, requiring accuracy to tenths of a millimeter in mixture pipes. It only takes one inept tuner to collapse the mouths of the pipework, lower the cutups, and change the position of the upper lips. I have rarely seen a cone-tuned organ that survived more than a few decades before someone eventually abused it. A notable exception is the incredible mixture chorus of the Great division of the E. & G. G. Hook organ at the former Church of the Immaculate Conception in Boston. When I documented this organ in 2000, I found all fifteen ranks of the three Great mixtures with exquisitely cone-tuned pipes and completely intact voicing. Fr. Thomas Carroll, SJ, informed me that this was the result of four generations of care by the Lahaise family, who had solely maintained the organ since it was built in 1863 and after the crude interventions of Hook & Hastings in 1902. See the articles by the author, “1863 E. & G. G. Hook, Opus 322, Church of the Immaculate Conception,” in the issues of The Diapason for volume 108, number 7, whole number 1292 (July 2017), pages 17–19; volume 108, number 8, whole number 1293 (August 2017), pages 18–21; and volume 108, number 9, whole number 1294 (September 2017), pages 20–22.
14. On July 24, 2014, John Brombaugh sent me his Excel model for “expected final body lengths” as part of a great many other files constituting what Brombaugh claimed was “half of what I know.” It is a highly complex model that includes inputs for pipe diameters, temperament, temperature, and relative pitch. This file was created for computing the values needed by his pipemaker, and the expected lengths are just one of its many outputs. Brombaugh’s model is an elegant piece of engineering.
15. The significant errors in the tuned lengths at 1⁄4′ to 1⁄8′ pitches in Figure 11 are not due to Brombaugh’s model, which curiously calculates lengths only to 1⁄4′ pitch. My errors resulted from extrapolating the 1⁄2′ to 1⁄4′ lengths by halving them.
