The elusive and sonorous meantone of Dom Bédos

August 30, 2020

Michael McNeil has designed, constructed, and researched pipe organs since 1973. He was also a research engineer in the disk drive industry with twenty-seven patents. He has authored four hardbound books, among them The Sound of Pipe Organs, several e-publications, and many journal articles.

Editor’s note: The Diapason offers here a feature at our digital edition—two sound clips. Any subscriber can access this by logging into our website (www.thediapason.com), click on Current Issue, View Digital Edition, scroll to this page, and click on each <soundclip> in the text.

The Clicquot organ at Houdan

The community of Houdan with a current population of just over 3,000 is located about thirty-five miles due west of Paris, France. In 1739 the organbuilder Louis Alexandre Clicquot completed the organ at Houdan in the church of Saint James. He was the father of François-Henri Clicquot who built the organ at the cathedral of Poitiers, and he was also a member of the family who to this day produces the wonderful champagne of that name. 

The organ at Houdan is preserved in virtually every detail of its original construction. For this we have to thank that it was placed in a small community without the resources to “modernize” it—large and prosperous cities have a habit of “improving” the organs in their care and irretrievably losing their history. The organ was unmolested until it went silent in the 1870s.

Fortune struck in the 1960s when Jean-Albert Villard, the organist titulaire of the F.-H. Clicquot organ at the cathedral of Poitiers, intervened when plans were formulated to modernize the instrument and discard most of its heritage. Villard and the efforts of many other preservationists prevailed, and the work of restoring the organ in its original state was entrusted in 1969 to Robert and Jean-Loup Boisseau. This is the organ we hear today. If you ever wondered why French Baroque music sounds lackluster on modern organs tuned in equal temperament, wonder no longer and buy the download of a new recording of the organ at Houdan by Régis Allard.9 It is a revelation; at Houdan the music sounds as it did to its composers and their intent becomes clear.

The limited resources of the community of Houdan were also probably responsible for the incredible economy in the design of the Clicquot organ. It has twenty-one stops distributed on three manuals (the third manual is treble-only), the pedal has no stops and simply couples to the bottom two octaves of the Grand Orgue, and there are no 16′ stops. But what looks at first like a tonal design lacking in grandeur is in fact extremely versatile and very grand. The French Classical scaling and voicing is musical and exciting without being in the slightest overbearing—the grandeur derives from its meantone temperament.

Meantone was an invention of the Renaissance, and one of its earliest practitioners was Pietro Aaron, who described his scheme for tuning it in 1523. While earlier temperaments going back all the way to Pythagoras favored pure fifths, the new system favored pure thirds at the expense of the fifths. In Aaron’s version there were eight pure thirds, four very impure thirds, reasonable fifths that beat about twice as fast as equally tempered fifths, and a wolf fifth that took up the remaining error. Aaron placed that wolf fifth on G-sharp and E-flat. There is always an error to be accommodated with our twelve-tone octave, and in equal temperament we distribute that error equally with the result that no intervals are in tune without beats. 

The wisdom of François Bédos de Celles

Our equally tempered third is a dissonant monster to which we have simply become accustomed. Bédos was well aware of equal temperament and despised it. His thoughts are worth revisiting: 

Among these schemes, two are the most worthy of note. One is called the old system, whereby the fifths are unequally tempered [meantone]; and the other is the new, which diminishes all the fifths to a lesser degree, but equally [equal temperament]. The mathematicians and the music-theorists disagree here. The latter, judging only by instinct and ear, cannot accept the new temperament, which they find harsh and less harmonious than the old one. Indeed, the fifths are diminished by only one-twelfth of a comma . . ., and all equally, with the result that all major thirds sound blurred, which makes a harsh impression on the ear. According to the old temperament, about eleven fifths are diminished by one-fourth of a comma. This is a greater adjustment than one-twelfth of a comma, but it saves, or keeps perfect, eight major thirds. 

Since altering these fifths by one-fourth of a comma still does not lead to a perfect octave, one fifth is sacrificed by having all the rest added to it, making it quite jangled. However, it lies in a seldom-used key. Organ-builders call this fifth the ‘wolf.’ Despite the prestige of the scientists who devised the new system, it has nevertheless been abandoned, even though it is less imperfect than the other, in theory. Music theorists prefer the old system, alleging that fifths may be altered one-fourth of a comma and even more without becoming disagreeable, whereas imperfect thirds must of necessity offend the ear: thus their old system is not inferior to the new . . . .
Moreover, the composer makes use of the very defects of the scale, finding in it resources for emphasizing the character of his various compositions. Whether the tone be gay, mournful, sublime, majestic, etc., he selects a mode suited to the harmonies most expressive of his idea. The new temperament does not offer this resource. Since all the intervals are equal, they all have the same character, with nothing to offset the harshness of the thirds.1  

The point of meantone is its harmonic purity, tension, and color. The heart of meantone is the pure third, which has a benefit not explicated, but intuited by Bédos. A pure fifth sounds the second and third harmonics of a tone an octave lower, e.g., the interval at middle C to G has an audible subtone that sounds tenor C. Equal temperament fifths come close to purity, and we make use of that subtone to create pedal resultants. But here is a key feature of meantone: its pure thirds sound a subtone two octaves lower, e.g., middle C to E sounds the fourth and fifth harmonics of a subtone that sounds low C. When a French Classic composer uses a pure major third in the tenor, that sound contains a very real and audible subtone at 16′ pitch. This is why the Houdan organ sounds grand without a single 16′ stop. Listen to this soundclip of the “Suite du premier ton,” from Livre d’orgue II, Fugue, by Nicolas Clérambault to get a sense of the 16′ subtones created by meantone’s pure thirds <soundclip1>.9  

If you take a deep dive into researching the temperament of the Houdan organ you may find different opinions. The notes in the PDF booklet accompanying the recording heard in the soundclips simply states it to be “mésotonique,” and American organbuilders who have visited this organ report that it is tuned in 1⁄4-comma meantone with the wolf on G-sharp and E-flat, which is Pietro Aaron’s version. But according to Timothy Tikker at least one source reports that the Houdan organ may be tuned in the temperament devised by the French Classic organbuilder Dom Bédos, a variant of Aaron’s version.2  

Are these two temperaments really different, and are the differences important? At first hearing, the recording of the Houdan organ abounds in pure thirds. Aaron’s version has eight pure thirds, and according to Bédos, his version has seven pure thirds and one “slightly diminished” third on B-flat.3 Tikker states that Bédos’s tuning gained widespread favor in late eighteenth century France with its sonority.2

Beat rates describe sonority

There are many ways to compare the two temperaments. We will use beat rates (beats per second) to determine the relative purity of intervals—and beats are what you actually hear when you play an impure interval.4 A pure interval is consonant and has no beats; an impure interval with many beats has dissonant tension. Beat rates depend on actual frequencies, so we must keep in mind that the beat rates we will see in this article are referenced to the specific pitch A = 440 Hz; if the relative pitch is A = 395 Hz, like the organ at Houdan, the beats will be slightly slower. Beats will double for each ascending octave, so if an interval has two beats in the bass, it will have four beats in the tenor, and so on. We use beat rates for this comparison because we want to compare the relative consonances and dissonances of these temperaments, i.e., we want to understand their sonorities. 

Pietro Aaron’s meantone

Beat rates can be calculated for all the common intervals, and a table of the beat rates for Pietro Aaron’s meantone is seen in Figure 1.5 The second column in Figure 1 lists the notes from A-sharp in the bass octave to F in the middle octave. The third column lists the frequencies of those notes. The next columns show the beat rates for the intervals of the minor third (m3), major third (M3), fourth (4), fifth (5), minor seventh (m7), major seventh (M7), and major ninth (M9). A quick glance at this table will show the eight pure major thirds (0 beats) for the notes C, D, D-sharp, E, F, G, A, A-sharp. The wolf fifth on G-sharp–D-sharp in the tenor has 12.9 howling beats per second; it will have twice as many beats in the middle octave. There are four very impure thirds on C-sharp, F-sharp, G-sharp, B—Bédos called these “wolf” thirds. From this table you can get a feel for the extreme variation from consonant purity to dissonant impurity in meantone intervals. Modern conventional wisdom has held that these dissonances are to be avoided, but as Bédos noted, classical composers consciously used these dissonances to enhance emotional effects. Listen to this soundclip of the “Suite du deuxième ton,” Livre d’orgue, I. Plein jeu, by Nicolas Clérambault <soundclip2>.9 This vibrant color is completely lacking in equal temperament.

Key features of the Bédos temperament

So how does the Bédos temperament differ, and does it have any advantages? This question is not easily answered because Bédos left apparently conflicting instructions: these include tables of various types of commas and some specific instructions for tuning his temperament that do not correlate. There is less conflict when we understand that the tables of commas appear to describe conventional meantone; John Brombaugh has analyzed these commas and the author has used Brombaugh’s frequencies to produce a table of beat rates that are virtually identical to the Aaron meantone.6 Bédos’s comma tables and tuning instructions can be found in the Ferguson translation of Bédos’s monumental work, The Organ-Builder.8 Referring to Volume I, §§1142–1145, Bédos clearly states that for his temperament there are “three fifths [D-sharp to A-sharp, G to D, B to F-sharp] that are diminished more than the others [beat faster].” Bédos also clearly states in §1145 that the major third A-sharp to D “should be slightly diminished and beat slowly.” A major third tuned flatter than pure is very unusual! Something seems amiss here, but the beat rate program will shed some light. 

The construction of the Bédos beat rates

In the first column of Figure 1, the beat rate program shows the sequence of tuning the intervals used in the construction of the table. The analysis of the Bédos temperament uses the same sequence until we get to the A-sharp, and from that point it took a few iterations to get it to the point where the advantages of the Bédos temperament became obvious. Those not wanting to dive into the details may skip to the next section.

The use of a plus sign on a beat rate means that we tuned the new note sharp, and a minus sign means we tuned it flat.7 Starting with the A-sharp, instead of using the fourth F to A-sharp = +2.2 beats we will use +3.0 beats; Bédos mentions that the fifth B-flat to F beats faster, so the A-sharp will be tuned higher to make the major third on A-sharp to D diminished. We next tune the “diminished” major third so: A-sharp to D = -1.0, making the D diminished from pure by -1.0 beat (it is -2.0 beats an octave higher in the tenor).  

The major third D to F-sharp is tuned pure. Next, the fourth A-sharp to D-sharp is tuned +2.4 beats rather than the original +1.5 beats, and Bédos states that this interval will beat faster. It was determined by iteration that this preserves the original purity in the D-sharp major third. 

The rest is easy. All of the remaining major thirds, D-sharp to G, G to B, and E to G-sharp are tuned pure. It is very important to note that we have re-tuned G and B; both have new frequencies. Figure 2 shows the beat rates for the presumed Bédos temperament.

The sonority of the Bédos meantone

If the sound of pure major thirds in Aaron’s meantone is impressive, now try to imagine the sound of pure fifths, pure major thirds, and pure minor thirds. One result of Bédos’s instructions is that the keys of C and E now have completely pure major and minor triads. And the purity in the key of C extends to the interval of the major seventh. An inspection of the table in Figure 2 will show that we also achieved Bédos’s seven pure major thirds and one slightly diminished major third on B-flat with two beats in the tenor octave. 

Bédos did not make clear in his instructions that he significantly changed the beat rates of the fifths C to G and E to B when he diminished the third on B-flat and adjusted for its effects—those fifths are now pure! The pure fifths are part of the source of the confusion in his instructions. Bédos starts with instructions for a normal meantone and then modifies it. Perhaps he thought this would be obvious. The steps for tuning the normal meantone in Figure 1 are included at the end of this article.

A price is paid for this new sonority, which we can see in the worse wolf fifth with 15% more beats. The fifths on D-sharp, G, and B are all now degraded, as noted by Bédos. If this is indeed Bédos’s temperament, it has some very interesting sonorities.  

The pure triads are very unusual; only the Kirnberger I temperament has two major triads of such purity (not to be confused with the more common versions, Kirnberger II and III).

Graphics read better than numbers

Tables of numbers are difficult to read, so we can get a better feel for the relative sonority of the Bédos meantone by using color graphics to represent the relative purity and tension between the twenty-four major and minor triads. In Figure 3 we see the Aaron and Bédos meantones represented by major triads in downward facing triangles and minor triads in upwards facing triangles. The lines between the notes are colored to represent purity (bright green), less purity (yellow-green), tension (yellow), and dissonant impurity (orange and red). The actual beat rates are indicated with the numbers placed next to the colored lines. The sonority of the Bédos meantone is now quite evident in the distribution of green.

Key features are satisfied

While we cannot be certain that Bédos’s temperament is represented in Figure 2, the temperament in that table does indeed have an improved sonority, and it follows Bédos’s instructions. The beat rate table in Figure 2 resulted from an attempt to incorporate the diminished third on B-flat with the least amount of adjustment to the Pietro Aaron meantone, and it also resulted in the faster beating fifths on D-sharp, G, and B. These are noted by Bédos as key features of his temperament. The pure major and minor triads on C and E were not a goal of this exercise; they were the surprising result when the adjustments caused by the diminished third were complete!

Unresolved issues

A survey of the literature will show that Bédos’s instructions have been interpreted in many different ways. Some of this confusion results from the conflicts between Bédos’s specific description of his tuning method and his tables of commas.8 Brombaugh has shown that the tables of commas describe normal meantone, but there is one other issue in Bédos’s specific instructions for tuning. He sets the tuning bearings for the third octave of the Prestant, i.e., 1′ to 1⁄2′ pitch, and he specifies that the interval G to D has five or six beats per second.3 While this beat rate is faster than the normal meantone fifth, it is much slower than the beats obtained for that interval in Figure 2: those beats will be faster in the octave of his tuning bearings.

Other interpretations of the Bédos tuning instructions may yield different results than those in Figure 2, but any new interpretation must also satisfy the constraints Bédos has described: a diminished third on B-flat and the faster beating fifths on D-sharp, G, and B. Many interpretations of the Bédos temperament exist, but only the interpretation in Figure 2 meets all of those constraints.

The Clicquot organ at Houdan

It is reasonably certain that the Houdan organ is tuned in 1⁄4-comma Pietro Aaron meantone. But the value of controversies is that they push us to re-explore our previous assumptions, and the exercise in this article may shed new light on the temperament of François Bédos de Celles. 

I hope you enjoy the remarkable meantone sonority of the Houdan organ as much as I do. Two recent recordings are well worth your money.9, 10

Notes & References

1. François Bédos de Celles, O.S.B., The Organ-Builder, an English translation of the original L’Art du facteur d’orgues, 1766–1778, by Charles Ferguson, The Sunbury Press, 1977, pp. 230–231, §1135.

2. Personal communication, January 2019.

3. The Organ-Builder, p. 233, §§1140, 1142.

4. Michael McNeil, “Exploring the Sound of Keyboard Tunings,” The Diapason, April 2016, pp. 20–21. This article gives a description of the tradeoffs when comparing temperaments with cents or beat rates.

5. Detailed and accurate instructions for tuning the Pietro Aaron temperament that appears in this table may be found in Tuning the Historical Temperaments by Ear, by Owen Jorgensen, Northern Michigan University Press, Marquette, 1977, pp. 173–177. An abbreviated version is appended to this article.

6. Personal communication, April 2019.

7. This program and the instructions for its use are contained in a DVD with the author’s book, The Sound of Pipe Organs, CC&A, 2014, Amazon.com. The program is admittedly difficult to use. 

8. The Organ-Builder, pp. 230–234.

9. Régis Allard, Magnificat 1739, Editions Hortus, 2017. Available as a download from www.editionshortus.com. The tuning of the Houdan organ in this recording is spotless.

10. Michel Chapuis and Emmanuel Mandrin, Marc-Antoine Charpentier: Messe pour le Port-Royal, E 8598, Auvidis, France, 1997. This recording showcases the accompanimental balances of the Houdan organ with solo voices. The radiance of the meantone purity with the voices is remarkable.

Method for tuning the Aaron meantone:5

Inputs to the beat rate program are noted in [  ]

1. Tune middle C. [263.2] This will yield A = 440Hz. (McNeil)

2. Tune C below middle C pure to middle C. [C–C]

3. Tune E below middle C pure to C below middle C. [C–E]

4. Tune G-sharp below middle C pure to E below middle C. [E + G#]

5. Test all C, E, and G-sharp for purity.

6. Tune F below middle C pure to C below middle C and then raise F until it beats equally (1.6 beats) between the two Cs. [C + F + 1.6]

7. Tune A below middle C pure to F below middle C. Test that tenor C to A and tenor E to A beat equally (2.0 beats).   [F + A]

8. Tune C-sharp above middle C pure to A below middle C. Test that C-sharp above middle C beats equally with G-sharp and E below middle C (2.5 beats). [A + C#]

9. Tune the C-sharp below middle C pure to the C-sharp above middle C. Test that both C-sharps beat equally with the E below middle C (2.5 beats).   

10. Tune A-sharp below middle C pure to F below middle C, and then raise the A-sharp until it beats at the same rate as the small minor third A-sharp–C-sharp (2.2 beats). [F + A# + 2.2]

11. Tune the A-sharp an octave lower than middle C pure to the A-sharp below middle C. Again, test that A-sharp–F and A-sharp–C-sharp beats equally (1.1 beats), where A-sharp is an octave lower than middle C.   

12. Tune D below middle C pure to the lower A-sharp. Test that the interval D–A beats 1.4 compared to the interval F–middle C at 1.6 beats. [A# + D]

13. Tune F-sharp below middle C pure to D below middle C. Test that the C-sharps above and below the F-sharp beat equally (1.7 beats). [D + F#]

14. Tune D-sharp below middle C pure to the lower A-sharp and then raise the D-sharp until it beats equally between the lower and upper A-sharp (1.5 beats). [A# + D# + 1.5]

15. Tune G below middle C pure to D-sharp below middle C. [D# + G]

16. Tune B below middle C pure to G below middle C. [G + B]

17. Tune B an octave below middle C pure to B below middle C. Test that D-sharp–A-sharp beats equally with E–B in the octave below middle C (1.5 beats).

Photo: the keydesk of the Houdan organ. Photo credit: William T. Van Pelt

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