Editor’s note: The Diapason offers a feature at our digital edition—two sound clips. Any subscriber can access this by logging into our website (thediapason.com), click on Magazine, then this issue, View Digital Edition, scroll to this page, and click on each <soundclip> in the text.
This article was written after the author received reed pipes built to specifications in this article by Aug. Laukhuff GmbH. A victim of the economic retraction during the pandemic of 2020 and 2021, Aug. Laukhuff GmbH closed its doors in June of 2021. Established in 1823, it served the needs of organbuilders worldwide. Its unimaginable loss is catastrophic to the profession, not only in the unique products it provided, but also in the centuries-old expertise of its managers and employees. Making an historic reed with the necessary detail will be a challenge.
The sound of an historic reed chorus may strike us like an emotional thunderclap and leave us with the wish that we might make such reeds today. I had one such experience with the sound of the reed chorus from the 1754 Joseph Riepp organ at Dole. Those reeds were a later addition in 1787 by the French builder François Callinet. You can get a sense of the drama of these reeds from a performance by the late Michel Chapuis <soundclip1>.1 The great Romantic organbuilder Cavaillé-Coll often preserved Callinet reeds in the organs he rebuilt.
The sheer complexity of reed construction is daunting, but we might begin to understand reeds by trying to recreate one we like. Although one of my dear colleagues rejoiced that I did not tackle the subject of reeds in my recent book on the sound of pipe organs, this article takes a deep dive into the subject. An exact copy would be prohibitively expensive, but we can capture the sound using the resources available to us from modern suppliers. For this example we used the resources of Aug. Laukhuff GmbH, Weikersheim, Germany, as they described their capabilities in great detail on their website.
There are hurdles to clear before we can reproduce such a reed, and perhaps the greatest of these is sufficient documentation. The detailed data required to really understand a reed is extremely rare, and the greatest practitioner of data collection was Pierre Chéron. We owe him a great debt for the complete measurements of every pipe in the 1774 Isnard organ at St. Maximin.2
Sufficiently complete measurements of the reeds at Dole are unknown to the author, but a wonderful book gives just enough data to get a feel for the scaling of the entire reed chorus of that organ.3, 9
Although we lack sufficient data for the Dole reeds, it is our great fortune that Chéron’s data for the 1st Trompette from the 1790 Callinet organ at Auxonne has been published in exquisite detail by Laurent Plet.4 The complete data from the Callinet reed at Auxonne and the limited data from Dole correlate very well. Chéron took the Auxonne reed data in 1990 prior to the restoration of the organ by Laurent Plet, and you will find a wealth of data and photos on Plet’s website for this organ.5 The complete table may be seen in Figure 25.
Chéron’s table has lettered data columns that are made clear by the drawing of a reed pipe in Figure 1 and the detail of a shallot in Figure 2. Chéron’s dimensions are in millimeters except thicknesses of the tongues, which are in hundredths of a millimeter. Here are descriptions of Chéron’s lettered columns:
Description of the Chéron data
A is the name of the note (there is no bass C-sharp)
B is the inside top diameter of the resonator
C is the thickness of the resonator at the top
E is the inside bottom diameter of the resonator
F is the thickness of the resonator at the bottom
G is the length of the shallot from the block, or “nut”
H is the height of the block, or “nut”
K is the length of the resonator from the block to the top
L is the total tuned length of the reed (there are no tuning slots)
M is the outside diameter of the shallot
N is the outside depth of the shallot at the block
O is the outside depth of the shallot at the end
P is the inside diameter of the shallot
Q is the wall thickness of the shallot
R is width of the shallot face (the slot opening width is R-2Q)
S is the total shallot length, including the section inserted into the block
T is the width of the tongue
U is the thickness of the tongue under the wedge
V is the thickness of the tongue under the tuning wire
W is the thickness of the tongue at the free end
X is the vibrating length of the tongue
Y is the extension of the tongue beyond the end of the shallot
AA is the diameter of the toe hole in the boot.
This seems like a lot of detail, but as you will see, Chéron recorded exactly what is needed to understand a reed.
Let’s start with the top diameters in the table in Figure 3, which control power. The amplitude of a standing wave in a resonator gets larger with larger resonator diameters, and Callinet’s diameters are very robust. In Figure 3, column 1 we see the raw values of the Callinet diameters.
We will use an Excel spreadsheet to analyze the Callinet reed, and we will start by graphing the raw diameters. The beauty of the Excel spreadsheet is that a simple click of the mouse on the graph finds the equation of the line that best fits those diameters. That exponential equation is shown in the graph in Figure 4.
We can use that equation in Excel to calculate the smoothed data we see in Figure 3, column 2, which now fills in all of the missing gaps! Note that there are many missing pipes in the original data. The diameters in Figure 3, column 2 are placed into an Excel table, Figure 5, which will be the basis for the new design.
The diameter of the bottom tip of the resonator is usually not less than the inside diameter of the shallot, and it is often larger. Reiner Janke relates that Cavaillé-Coll used a ratio of approximately 8 for top to bottom diameters.6 The Callinet data shows ratios changing from 10.9 in the bass to 9.3 in the treble. These bottom diameters are narrower than Cavaillé-Coll would have specified (larger ratios yield smaller bottom resonator diameters), but as we will see, Callinet carefully designed the bottom inside resonator diameters to be about one millimeter wider than the inside diameters of his shallots, making a smooth transition from the bore of the shallot to the bore of the resonator.
Using these Callinet ratios we let Excel calculate the inside bottom diameters of the resonators. Callinet tapers the thickness of his resonators in the bass, making them thicker at the bottom for added strength. These thicknesses are entered in the Excel file, and we have the specifications for the resonators. We also added a note to make the resonators of 75% polished tin, the highest tin alloy Laukhuff offered. Figure 5 shows the first steps in this table for the low octave. We will show the entire table when it is complete.
Resonators will be longer with a lower temperature, lower pitch, higher wind pressure, and the flat-tuned intervals in an unequal temperament. For this exercise we will specify temperature at 20 degrees Celsius, wind pressure at 85 mm (the original Callinet pressure), pitch at A = 440 Hz, and Pietro Aaron’s 1⁄4-comma meantone temperament.15
Lengths obviously relate to the pitch, but they also strongly influence the timbre. Resonators voiced to the maximum lengths will have a strong “Bourdon” fundamental in the tone, and resonators voiced shorter will have less fundamental and stronger harmonic fire. The goal in the total tuned length is a resonator where the “flip” or “Bourdon” point preserves considerable fundamental in the tone while not sacrificing harmonic fire. The Bourdon point can be modified to some degree with the curve of the tongue and the diameter of the foot hole during voicing. An article by Reiner Janke has an excellent discussion of the Bourdon point.6, 10 Those interested in the math of the effect of temperature on pitch can refer to a very useful website (www.sengpielaudio.com/calculator-pitchchange.htm). The calculator on this website indicates that pitch changes a full halftone from 32 to 92 degrees Fahrenheit.
As we can see, resonator length is very important, and it is doubly so if we are designing resonators with no tuning slots—like most classical, “dead-length” French resonators. Tuning slots are often used to tune reeds to changing temperatures, and while this maintains the timbre of the reed, such tuning slots nearly always end up molested thus that the pipes become extremely uneven in voicing. The lack of tuning slots in French pipes has preserved the intent of their voicers; the reeds are tuned “on the wire.”
Although we have been talking about resonator lengths, the important length in a reed is its total tuned length, and this includes the length of the resonator, the block to which the resonator attaches, and the extension of the shallot below the block. This total length is what really determines the pitch and timbre of the pipe. Chéron gives us the total tuned length in column L of his data. The original pitch of the Callinet organ was A = 415 Hz, and Plet displays a table for these pipes with the estimated total tuned lengths they would have had at their original pitch.7 The current lengths of the pipes in the Chéron table reflect a pitch closer to A = 440 Hz, and as this pitch is the goal of our new reed, we will use those lengths with the understanding that there is uncertainty in Callinet’s intended pitch of the Bourdon point.
We can enter the total lengths of the Callinet pipes into Excel and fit a curve. Figure 6 shows a table with the raw total tuned length data in column 1. Figure 7 graphs that data for the actual Callinet tuned lengths and shows a fitted curve with its exponential equation. The very high R2 value indicates a close fit to the data.16 With this equation we can fill in the missing gaps with the fitted data in Figure 6, column 2.
We need to add one more wrinkle to the lengths—temperament. The intended 1⁄4-comma meantone temperament will give us an opportunity to show how to make length corrections (the original Callinet temperament with the missing bass C-sharp probably had major thirds of less purity than the Pietro Aaron meantone). We can approximate that correction by calculating an octave of frequencies for this temperament and finding the differences in the intervals as seen in Figure 8. For this type of meantone, pipes speaking C-sharp, E, F-sharp, G-sharp, and B will need to have resonators with lengths that are approximately 2⁄3 of the whole step in which they reside, not the typical half step of equal temperament. Figure 6, column 4 shows the total tuned meantone lengths of new pipes with adjusted lengths in blue font. The values in Figure 6, column 2 work well for equal temperament.
We can now enter these total tuned length values into our Excel table in Figure 9. The estimated block height “H” values are placeholders that will be modified by the pipemaker to suit the Laukhuff blocks. The estimated shallot, from block, “G” values are likewise placeholders; the entered values are approximations based on the vibrating lengths of the tongues plus the necessary clearances of the tuning wires to the wedges. Once the values for “H” and “G” are known, the pipemaker will subtract them from the total tuned length “L” to give the estimated resonators “K,” i.e., K = L - (H + G). The values of the estimated resonator lengths in the table reflect that equation.
It is important to note that the octave ratios of the lengths are 1.94, where the resonators are shorter than a pure doubling of length at the octave as they descend into the bass. This will be addressed in the comments on voicing.
Shallots are usually made of brass, and they provide the surface and openings on which the tongues beat to make the sounds we hear. A variety of shallot and opening sizes and shapes gives us a vast range of power and timbres.17 Callinet shallots are parallel with parallel slot openings. It would be prohibitively expensive to make custom shallots for every pipe, so we will refer to Laukhuff’s selection of shallots to achieve a close approximation of Callinet’s ratios of resonator top diameters to shallot inside diameters. Figure 10 shows a portion of Laukhuff’s shallot selection for their Type II Clicquot shallots with rounded ends, which closely resemble the Callinet shallots in shape (all dimensions are in millimeters).
Callinet’s ratio of “12” for resonators to shallot inside diameters is a crucial part of his sound. Reiner Janke relates that Cavaillé-Coll’s ratio is approximately 11, and this means that Callinet’s shallots are slightly smaller and drive his wide resonators with slightly less energy.6 In the first three rows of our Excel table under the heading “Shallots” in Figure 11 we see the Laukhuff shallot number, the shallot inside diameter “P,” and ratio we are trying to achieve. The Excel file calculates the ratio for us. We cannot achieve exact ratios with the shallot sizes available to us, so we select shallots to get as close as we can to a ratio of 12, i.e., a range of about 11.5 to 12.5. We enter the Laukhuff wall thicknesses in the next row, and the table calculates the outside diameter “M” as simply the inside diameter plus two times the wall thickness.
The slot opening width of the shallot, “CC” in Figure 12, is extremely important; wider slots yield more power and more harmonic fire. While Chéron did not measure this parameter directly, it is implied in his measurement of the face width “R” minus two times the wall thickness “Q,” a calculation that is accurate for shallots that are cut deeply and nearly fully open. The “face” is the surface of the shallot on which the tongue beats. Tables of reed data most often record the height of the cut of the shallot (Chéron’s data “N” and “O”), but this is a very indirect method of getting at the slot width, especially if the wall thickness is unknown. A more direct method is to simply specify that the shallots are cut until they yield the desired slot opening width, and these are the values you see in the next to last row of Figure 11, slot width “CC.” The last row is a calculation of the ratio of the slot width “CC” to the shallot inside diameter “P,” and while the Callinet data is somewhat noisy, it averages about 0.85, a very widely cut shallot. We adjust our slot widths to achieve that ratio. The pipemaker can now cut the shallots of a given Laukhuff number to the same height and slot widths.
We specified Laukhuff’s French blocks for this reed. French boots extend well beyond the block, or the “nut” as it is often called, to support the resonator, a design that minimizes the collapse of the resonator above the block, a common problem in reeds. Bass resonators have a narrower taper, and to keep the boot from extending too far, a “ring” is inserted above the block on the resonators of those pipes. This ring is deleted when the resonator taper sufficiently widens. Figure 13 shows the two classical forms. We specified half-round teak wedges and tinned bronze tuning wires bent at the top at right angles for ease in tuning. The Callinet originals at Dole have wires bent in this manner.
We specified Laukhuff’s tongues made of rolled, spring elastic brass. Laukhuff gave us the option of voiced or unvoiced reeds, and we specified voiced with French tongue curves. The art of reed voicing encompasses power, the Bourdon point and timbre, and the promptness of the speech. With a dead length resonator, the voicer adjusts the diameter of the toe opening in the boot to control the wind along with the stiffness and curve of the tongue.
Stiffness relates to wind pressure and the amount of curve that can be tolerated. With a large curve a thick tongue may speak slowly or not at all. If too thin, it will speak quickly but the sound will be weak in fundamental. A range of timbres can be coaxed from tongues with different curves. We will start by characterizing the Callinet tongues with their stiffness. The power of this method will be demonstrated, and it appears that to some degree Callinet thought in these terms as well.
The stiffness of a tongue, especially if it is a simple parallel tongue, may be understood with some very simple ideas. Stiffness is inversely proportional to length, meaning that a tongue of half the length will be twice as stiff. You can demonstrate this to yourself by taking a metal ruler, holding it down at the edge of a table, and letting a portion of the ruler extend beyond the edge of the table; pull up the ruler at the free end, let it go, and listen to the sound it makes. Now extend the ruler only half as far off the table and listen to the sound it makes—it will sound an octave higher with twice the stiffness. Tongues obviously come in different lengths, and the longer tongues beat at lower pitches. Tongues also come in different widths, and what is intuitively obvious is also correct—a tongue of half the width will have half the stiffness.
Tongues also come in different thicknesses, but our intuition may fail us here—the stiffness will be the cube of the thickness. For example, if we double the thickness of a tongue, its stiffness will be 23, and that means it will be eight times stiffer (23 = 8). That makes thickness by far the most sensitive parameter.
In Figure 14 we see a portion of Laukhuff’s thicknesses for tongue brass. These dimensions, like Chéron’s, are in hundredths of a millimeter, e.g., brass with a thickness of “25” is 0.25 mm thick.
Figure 15 shows a table compiled from Chéron’s data on tongues. Chéron gives us not only tongue thickness in column 6 and width in column 5, he also gives us the vibrating length of the tongue from the tuning wire to its end in column 4, a parameter that is very rarely seen in reed documentation. With this data we can make a calculation of the stiffness of the tongues. These stiffness values involve relatively complex calculations that are illustrated in Figure 24. You do not need to understand them in detail if the math is daunting, and they are already worked into the Excel spreadsheet available from the author.9
The real point of using stiffness values is that we get a method to adapt the tongues to our shallots and to available tongue thicknesses. The value of this may not at first be obvious, but bear with me. We will take it step by step.
We first take the raw values for vibrating lengths in Figure 15, column 4, graph the values, fit an equation, and calculate the smoothed and missing data in column 2. Figure 16 shows the raw data and fitted equation; the fit with an R2 value of 0.99 is quite reasonable.
The next step is to calculate the stiffness of Callinet’s tongues.13 We use the raw data in Figure 15, columns 4, 5, and 6 to calculate the Callinet tongue stiffnesses shown in column 7. We graph those stiffnesses in Figure 17, and fit it to an equation. This data is quite noisy with a low R2 value.16 The new equation yields the smoothed stiffness data for all of the tongues in Figure 15, column 8.
This has been a long slog through data and calculations, but we are now very close to reaping the rewards. In our Excel table for tongues in Figure 18 we have entered the smoothed Callinet vibrating length data in the row for vibrating length “X” and the smoothed Callinet stiffness data in the row for target Callinet stiffness. With this data we can now design our tongues.
We make trial entries in the rows for “width” and “thickness,” and the Excel spreadsheet calculates the resulting stiffness in the row “stiffness.” The last row, “width of tongue-slot/2,” tells us when the width we have entered is so narrow that the tongue will literally fall into the slot opening of the shallot, i.e., when the value in this row is zero or less. Ideally, we would like to have a generous overhang of the tongue on the slot opening of the shallot, and this would be something like a minimum of 2 mm on each side of the bass shallot openings and at least 1 mm on each side of the treble openings. It does not matter if the tongue is significantly wider than the face of the shallot (the slot opening width plus the two shallot walls), but it cannot be less than the slot opening itself.
Now look at the table in Figure 18. The first data column is low C followed by C-sharp, etc. We enter the thickness for low C at 0.47 mm, then we adjust the tongue width until the relative stiffness matches the target Callinet stiffness. A few iterations gives us a tongue width of 14.3 mm and produces a 2.5 mm overhang of the tongue (last row, “width of tongue – slot/2”). If we had selected Laukhuff’s 0.50 mm tongue thickness, we would need to significantly narrow the tongue to achieve the target stiffness, and while it would not fall into the slot, it would be dangerously narrow. This is all there is to it. Just keep doing this for all 49 tongues!
Callinet apparently thought of tongue design in similar terms, although his noisy stiffness data indicates that he was not at all rigorous about this.16 In the table in Figure 19 we see the first two rows of Chéron’s data for the low C and low D pipes. The columns for tongue thickness, “V” and “W,” show that the tongue for D is thinner than the tongue for C. But note in the column for tongue width, “T,” that the 13.0 mm tongue for D is wider than the 12.5 mm tongue for C. Remember that thickness is our most sensitive parameter (it is a cube function of stiffness), while width is simply proportional to stiffness. The thinner tongue for D would lack the necessary stiffness if it were narrower than the tongue for C, and Callinet has compensated by treating tongue widths as an independent variable; he does not care if the tongue is wider than the face of the shallot, and the math for stiffness explains why we should not care, either.
To summarize, we have reproduced Callinet’s stiffness of the tongues with this method. We have achieved this while accommodating the shallots and tongue thicknesses that were available to us from Laukhuff. Figure 20 shows the tongue thicknesses we selected to make the tongues wide enough to extend over the shallot faces, and Figure 21 shows the final adjustments we made to the tongue widths to achieve continuous stiffness. With careful voicing we should be able to reproduce Callinet’s sound <soundclip2>.14
Boot toe hole diameters
Toe hole diameters may be used to control the pressure in the boot, and there is evidence that Callinet made extensive use of this. In the bass and tenor octaves the toe holes are narrower than the shallot inside diameters (the maximum flow of wind is limited by the shallot diameter and its opening). These toe hole diameters, Figure 22, are much narrower than those found in Chéron’s data for a Trompette of similar scales and wind pressure in the Isnard organ at St. Maximin.8 I have entered the fitted data of Callinet’s toe hole diameters in the last data row of Figures 23a and 23b. Like the stiffness data, the toe hole diameters are quite noisy with an R2 value of 0.49, indicating that Callinet used the toe holes as a voicing variable. There is a poor R2 correlation of 0.20 between the tongue stiffnesses and the toe hole diameters, perhaps indicating that Callinet used this variable to accommodate different tongue curves, thicknesses, and widths.
We can now see our completed table in Figures 23a and 23b (the bass and treble are shown separately to make the table more readable). This table specifies our reproduction of the Callinet 1st Trompette at Auxonne, France! The Excel file shown in this article took considerable time to create, and it is available at no charge from the author.9 It can be used as a template for other reeds.
Power balances are greatly affected by scaling and voicing, and Callinet’s data are instructive. Total tuned lengths will be approximately a ratio of 2 for each descending octave if bass power is strong relative to the treble, and we find octave ratios averaging 1.99 for the three Trompettes in the Raisonnance and Grand Orgue of the Isnard organ at Saint Maximin. The Callinet octave ratios are significantly smaller at 1.94, indicating shorter tuned lengths with descending octaves and reduced power in the bass. Reduced wind pressure and reduced power will result in shorter resonators at the same Bourdon point, i.e., at the same timbre.10 This observation is reinforced by the significantly reduced toe hole diameters in the bass and tenor of Callinet’s reeds. The Bourdon point is also affected by the curve of the tongue—a larger curve makes the sound more powerful, the timbre brighter, and moves the Bourdon point to a longer resonator. Larger curves require more pressure and larger toe holes for prompt speech.
We begin to get a feel for Callinet’s power balances: a restrained power in the bass with still brilliant timbre and fundamental warmth—or put another way, a more ascending treble. A smoother, more sonorous treble can be accentuated with a Bourdon point that is closer to the tuned point in the treble, farther in the bass. For example, Reiner Janke relates that Cavaillé-Coll voiced normal-length reeds close to their Bourdon point just before they transitioned to harmonic-length reeds.11
We might also note that Callinet lets his tongues extend by a small and varying amount beyond the end of his shallots, a technique that subtly affects timbre in much the same manner as adding weights to the tongues. See Chéron’s data in column Y, Figure 25.
Reed voicing is a complex subject with much subjective opinion. The goal in voicing this reed, and especially the curve of its tongues, is the scintillating French voicing of Callinet. I can think of no better source for understanding French reed voicing than the wonderful essay on this subject by Reiner Janke.6 I have translated this essay into English, and it is available by contacting him at firstname.lastname@example.org. He has a gift for finding the important variables in scaling and voicing; see the professional documentary in which he shows how to make and voice perfect copies of a Baroque Silbermann pipe and a Romantic Walcker pipe.12
Janke provides further guidance in the voicing of tongues.11 Tongue brass is made by rolling sheets to the desired thickness. The process of rolling produces work-hardened surfaces, and the direction of rolling produces a grain with more stiffness in the direction of rolling. Tongues will be less stiff if they are cut from sheets where the grain is parallel to the width of the tongue. Baroque voicers also commonly filed their tongues to different thicknesses, and this removed the work-hardened surfaces, making their tongues less stiff. A thinner tongue will yield a more rounded timbre, and a thicker and stiffer tongue will be brighter in timbre. Narrowing the tongue will make it louder (it will be less stiff and can be curved more). These are all issues to keep in mind during the voicing process, and they probably account for most of the variability we see in the Callinet stiffnesses in Figure 17 and the toe holes in Figure 22. For those who want to experiment with different tongue configurations, you will need an assortment of reed brass of different thicknesses and a guillotine to cut tongues.
This analysis of a Callinet Trompette helps us understand Callinet’s intentions. For those wishing to replicate the entire Callinet chorus at Dole, the Excel model described in this article may be useful as a template to help us interpret the more limited data available to us for the Dole organ.3, 9 The author ordered and received a Trompette from Laukhuff based on these specifications as part of a new organ to demonstrate the range of wind dynamics, among other things. This new organ will be fully described in a future issue of this journal with sound clips—stay tuned.
There are many ways to design a reed, and the varieties of reeds and their sounds are virtually limitless. This essay on reed design should not be understood as the only way to tackle this complex subject, but taken rather as a contribution to its discussion. When we hear a sound that commands our attention like the Callinet reeds at Dole, this article reminds us that a great deal of time and effort is required before we can understand those sounds. Callinet reeds are worth our time.
Notes & References
1. Michel Chapuis. Michel Chapuis joue la suite gothique à Dole, a Youtube video by Frederic Munoz, January 25, 2018 <soundclip1>. www.youtube.com/watch?v=oxWHMPS6Lp4.
2. Yves Cabourdin and Pierre Chéron. L’Orgue de Jean-Esprit et Joseph Isnard dans la Basilique de la Madeleine à Saint-Maximin. Nice: ARCAM, 1991, 208 pages, ISBN 2-906700-12-6.
3. Pierre-Marie Gueritéy et al. L’orgue de Dole, Canevas Editeur, Frasne, France, 1995, 127 pages, ISBN 2-88382-058-9.
4. Laurent Plet. Restauration de l’orgue d’Auxonne, accessed January 11, 2019. lplet.org/orgues/auxonne/tuyaux/tgo_tr1c.htm.
5. Laurent Plet. Restauration de l’orgue d’Auxonne, accessed January 11, 2019. lplet.org/orgues/auxonne/auxonne.htm#sommaire.
6. Reiner Janke. See the article Grundzüge Zungenintonation, accessed January 3, 2019. www.orgel-info.de/. An English translation is also available. The Cavaillé-Coll ratios are general approximations and were reported to Janke by Pierre Chéron.
7. Plet, Laurent. Restauration de l’orgue d’Auxonne, accessed January 11, 2019. lplet.org/orgues/auxonne/tuyaux/tgo_tr1.htm.
8. L’Orgue de Jean-Esprit et Joseph Isnard dans la Basilique de la Madeleine à Saint-Maximin, pages 139, 155, 156.
9. Michael McNeil. Designing an Historic Reed, Excel file, Mead, Colorado, 2019. Email the author for a free copy: email@example.com. This file also contains data on the Callinet reeds at Dole.
10. See also my discussion of the Bourdon or “flip” point in “1863 E. & G. G. Hook Opus 322, Church of the Immaculate Conception, Boston, Massachusetts, Part 3,” The Diapason, September 2017, pages 20–22.
11. Personal communications, January and March 2019.
12. Reiner Janke. Organ Voicing, Experiential Knowledge of Reiner Janke. www.youtube.com/watch?v=isEWba9rLRY&feature=youtu.be. This professionally produced video by the Universität Göttingen, April 26, 2018, is a revelation for those wishing to understand the voicing of flue pipes. The sound track is in English.
13. In Figure 24 we see the stiffness calculations of the tongues with proper physical units of Newtons of force and Newtons/meter of stiffness. While the force is admittedly applied uniformly over the surface of the tongue and not at the end as modeled in the equations, the resulting stiffness increases uniformly with pitch in the expected manner. See the website: www.tribology-abc.com/calculators/t14_9.htm.
14. Michel Chapuis. Beauvarlet à Dole avec Michel Chapuis, a Youtube video by Frederic Munoz, December 5, 2017
15. This temperament is described in detail in the author’s article, “Exploring the Sound of Keyboard Tunings,” The Diapason, April 2016, pages 20–21.
16. The term “R2” or “R-squared” describes how closely the data fits to the equation on the graph. If all of the data points fall on the line, the R² is equal to “1” and the fit is perfect. If the R2 is zero, the data is just a scatterplot looking much like a shotgun blast with no correlation at all to the line described by the equation. A click of the mouse in Excel will show the R2 for any graphed data fitted to an equation. The noisy fit of R2 = 0.56 for the Callinet tongue stiffnesses in Figure 17 may very likely indicate that Callinet used stiffness as a variable during voicing.
17. Personal communication of March 2019. Janke points out that the shallot functions as a resonator which strongly affects the timbre of the sound. Extremely wide shallots, typical of North German Baroque practice, produce an “oo” vowel with strong harmonics in the range of 200 Hz; less wide shallots will produce an “ah” vowel with strong harmonics in the range of 400 Hz; French Baroque shallots are narrower and produce an “ee” vowel with strong harmonics in the range
of 600 Hz.