Beiaarden.nl | Alles over klokken, beiaarden, campanalogie & André Lehr               

Bert Schoofs

Introduction

End November 2006 Dr. André Lehr sent a small e-mail message to the author. It told that he had succeeded to design a new major third bell, showing a perfect twelfth partial tone. It is named GT0611 (Nov. 2006). The Royal Petit & Fritsen Bell Foundry already had molded a test bell of the design with strike note c2. Later, also test bells with strike notes c3 and c4 were molded. Figure 1a gives a picture. After listening the bell in January 2007 we discussed plans to present a corporate contribution to this congress. Then suddenly, end March 2007, André Lehr passed away. Indeed, a very big loss for the carillon community. Thinking it over, later, I realized that major third bells covered his whole professional career. He started working on it as a young campanologist around 1950, and his last action was – in his opinion – the final solution of the major third bell problem. It is an honor for me to present André Lehr’s bell in this congress.

(a) (b)

Figure 1 (a) Molded test bells with strike notes c2 and c3 of bell GT0611. (b) Artist’s impression of bell GT0803

In 2001 the author was involved in the design of a major third bell for the renewed Carillon Museum at Asten, the Netherlands. Finally cymbal bells were chosen for the carillon. “In the competition” was also a typical prototype major third bell, designed by me. Recently I perfected this bell. It is named GT0803 (March 2008). The twelfth partial is only a little under the ideal frequency. The Royal Eijsbouts Bell Foundry will mold a test bell of the design with strike note g2; it is planned to be ready autumn 2008. Figure 1b gives an artist’s impression of the bell. The characteristics of this new bell will be presented here too.

The two new bells will be compared mutually, with former significant major third bells and with a common minor third bell. The comparisons concern both partial tones and strengths of partial tones. Reverberation times of bells will not be discussed in this paper.

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Comparisons of partial tones and strengths of partial tones will be based on computer simulations. These have been carried out using the bell analysis and design program BELLCAD. BELLCAD is a further development, made by the author, on the dynamic optimization program DYNOPT (Schoofs et al., 1994). In this paper a method is proposed to evaluate the so-called acoustic input power; subsequently this is used as a measure for comparing the strengths of partial tones.

The outline of the paper is as follows:

1. Introduction

2. The challenge of developing a major third bell

3. Description of the involved bells

4. Comparison of bells with respect to partial tones

5. Method for computation of acoustic input power

6. Comparison of bells with respect to partial strengths

7. Conclusions

2. Development of a major third carillon bell: a fascinating challenge

Many people have been searching for major third bells for many years. André Lehr is an outstanding example as one of them. Why those search?
The reason is found in a typical property of the bell as musical instrument. It inherently shows a strong minor third chord in its tone structure. This minor third is more or less natural and inevitable for a bell. It comes about almost of itself. In other words: the minor third bell is in the middle of the design space. That is an important reason why in the past the minor third bell could be developed to such a nice musical instrument as it is now.
However, there is a problem if one likes to play music in a major third key on a carillon build of minor third bells. The internal minor chords of the bells conflict with the played major chords. A carillon consisting of major third bells would solve this problem. As a result, the conflict is shifted towards playing music in a minor key! A solution might be to build a carillon, both with minor and major third bells. Except that it would be a very expensive solution, one should ensure that the timbres of the minor and major bells are close enough to make it musically feasible.
With that last statement we just arrived at the question: what may be regarded as the goal in designing a major third bell? There are several aspects which play a role:

1. A major third bell must be an octave bell with the same main tone structure (hum, fundamental, third, fifth, nominal, twelfth and double octave) as the common minor third carillon bell, except the third must be raised to a major third.

2. The heaviness of the bell profile expressed by the so-called fD-value.1

3. The individual strengths of the main partials.

4. Overall strength of the bell sound.

5. Reverberation times of the main partials.

6. The way sound is radiated from the bell.

7. Timbre of the bell given by the tones, strengths and damping values of both the main

   partials and the less important partials.

8. The bell’s shape, both from aesthetics viewpoint and considering the molding

   technology.

1 The fD-value is a measure for the heaviness of the bell or bell profile, and is computed as f times D, where f is the frequency of the hum note (in Hz) and D is the rim diameter (in m). A higher fD-value indicates a heavier bell. Larger and middle-sized carillon bells show an fD-value of about 200.

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9. Weight of the bell with respect to the cost price.

We are currently satisfied to search for and realize points 1 and 2 in a structured way, as this is often difficult enough. Results for the other points have to be taken more or less as they appear in design solutions of point 1 and 2. Fortunately, often there are more solutions available, which enable choices with respect to design goals in the points 3 through 9. Of course, given a certain bell, results for the points 3, 4 and 7 may be influenced very much by choosing an appropriate clapper system. Considering the strengths of individual partials is a main topic in this paper.

The difficulty in designing a major third bell is caused by the strong coupling between the partials third, nominal, twelfth and double octave. These partials show very similar vibration modes in a vertical cross-section of the bell, as can be seen in Figure 2.

Figure 2. Vibration modes (in the vertical cross-section) of a common minor third bell.

As a consequence, those partials are influenced very similarly by shape modifications of the bell profile. Modifying the profile in order to change a minor bell into a major bell, the third must be raised half a tone, hence, the interval third-nominal is reduced half a tone. The intervals nominal-twelfth and nominal-double octave will also be reduced to some extent. In bell design optimization we refer all partials to the nominal. This implies that, if the third is raised with respect to the nominal, the twelfth and the double octave will be lowered. Knowing that in a good minor third bell the twelfth is almost ideal (a fifth above the nominal), the twelfth will be too low in a major bell, often 1/8 to 1⁄4 of a tone. To avoid such a lowering of the twelfth doubles the difficulties in the design of a major bell: in the first place the interval third-nominal must be decreased and at the same time the interval nominal-twelfth

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must be kept constant. Due to the similar vibration modes of the third, nominal and twelfth these are two tough tasks. André Lehr just tackled this problem in his last design of a major bell, and solved it adequately.

3. Description of the bells involved in this research

Figure 3. MINORB: minor third carillon bell.
GT8505: the first major third carillon bell developed in 1985.

GT8706: major bell used in the carillons of Deinze and Garden Grove. GT9518: major bell used in the carillon of Groningen University. GT0611: new major bell designed by André Lehr end 2006.
GT0803: new major bell designed by the author in 2008.

This three minutes video gives an impression of the first five bells and/or carillons in which they were used.
In this paper, bell designs will be considered as given in Figure 3. Compared to the original designs two modifications have been applied. All bells have been scaled such that the nominal equals exactly a2 (so, the strike note is a1), in order to ease comparisons of the bells. The drawing scale is the same for all bells in Figure3. Furthermore, for all bells finite element

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meshes consisting of 38 elements have been used, also to ease comparisons. These modifications do not alter the properties of the bells with respect to the aimed comparisons. In Figure 3 schematic clappers are drawn within the bell profile. These will be discussed later in the chapter on partial strengths. In the next sections the background of the involved bells will be described.

Bell MINORB.

For many years our research laboratory at Eindhoven University of Technology (TU/e) could borrow a nice, tuned, swinging bell with strike note c2. from the National Carillon Museum at Asten. The bell was used for all kind of measurements, and in many experiments such as the swinging behavior, response to the clapper strike, measurements of material and acoustic damping of the partials.

Remembering André Lehr’s statement that this was a very nice minor third bell, the author changed its shape as little as possible in order to get a tempered tuned minor bell (as is common for carillons).

Bell GT8504

In 1982 André Lehr asked our group at TU/e for help in his attempts to find a major third octave bell. He had already been searching for it for almost thirty years. In 1985 a solution was found, after development and applying the necessary software. Royal Eijsbouts built a traveling carillon using the new bell profile. It was a very heavy profile showing an fD-value of 222. The project is extensively described in three papers in the same edition of Music Perception (Schoofs et al., 1987). Many people played the carillon and a vivid discussion emerged. The story is well-known.

Bell GT8706

Some years later the request from Deinze, Belgium, came to Royal Eijsbouts for a carillon consisting of 37 major bells (later the carillon has been extended to 48 bells). André Lehr developed a new bell profile (André Lehr, 1987) for this carillon, using characteristics of a bell profile from 1955, his experience in bell design and tuning graphs computed by DYNOPT. Compared to GT8504, in this bell the inside contour is stretched and the bulge at the outside is less pronounced. Furthermore, the heaviness of the profile is reduced to fD- value 200, a common value for middle sized carillon bells.

The carillon at Deinze2 was realized in 1988. Two years later, at Garden Grove3, USA, a major third carillon consisting of 52 bells was built, using the same bell profile as in Deinze.

Bell GT9518

The aesthetics of the major third bell with the bulge at the waist was not appreciated very much; it was tried to change the shape in that respect. In 1992 the author designed two profiles, GT9231 and GT9302, both without the bulge, see Figure 4a/b. Bell GT9518 looks like GT9302, but there are some distinct differences, see Figure 4c. However, the profiles GT9231 and GT9302 still had some shortcomings. When in 1995 a major third carillon was planned for the Academy Tower of Groningen University, André Lehr used the mentioned profiles as a starting point for the design of a new profile without bulge. The carillon, consisting of 25 bells, was built in 1996 by Royal Eijsbouts; it is automatically and hand- played.

2 See: www.beiaard.org/steden_deinze.html
3 See: www.gcna.org/data/CAGARDEN.HTM

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(a) (b) (c)
Figure 4. Profiles without bulge; a. GT 9231; b.GT9302; c. GT9518 (solid), GT9302 (dotted)

Bell GT0611

End 2006 André Lehr announced, as he called it, “a breakthrough about the major third bell” (André Lehr, 2006). The why and how he describes in “Klok en Klepel” (André Lehr, 2007). He carried out analyses of a number of historical bells: well-sounding, less well and quite ill- sounding ones. He concluded that the cause of the less nice or even bad strike sound (as some people mention it) must be blamed on the – with major third bells – consequently too low twelfth partial. He solves the problem with a surprising new profile design, showing the twelfth at the ideal place: a fifth interval above the nominal. Also surprising is that the bulge has returned, now even more pronounced compared to the first major bell GT8504.

The author does not have information which method was used by André Lehr to find the new profile. Probably he choose a starting profile (maybe bell GT 8706), applied his experience rules on it, together with and some “hand controlled” optimization techniques using BELLCAD. A favorite method of him was to modify the bell shape, applying computer generated tuning graphs for the inside and the outside contour, and a third one for the central line of the bell wall. Certainly, if it has been this way, the outstanding result he achieved is even more remarkable!

The Royal Petit & Fritsen Bell Foundry at Aarle-Rixtel, The Netherlands, molded a test bell of the profile (December 2006) with strike note c2. The bell has been molded “on tune”. Therefore it has not been tuned on carillon standards afterwards. In the course of 2007 Royal Petit & Fritsen also molded two more test bells with strike notes c3 and c4, respectively. The bells can be listened at the National Carillon Museum at Asten.

Bell GT0803

In 2001 we, André Lehr and the author, were searching for a special type of major bell, to be used for the renovated Carillon Museum at Asten. Finally the typical cymbal bells were applied. One of the bells, the author proposed at the time, served as prototype for bell GT0803. The methods used to find it were merely computer optimizations by means of the program BELLCAD (and of course the needed amount of common sense). The first stage was to start with the profile of a common minor third bell. For this bell four rather global shape characteristics were defined: the width of the bell, the height of the bell, how conical is the bell and how curved is the profile. Each characteristic was varied at five distinct values. All possible variations were combined, giving altogether 54 = 625 different bell shapes. So far nothing was optimized; we just had 625 different bell shapes available. Subsequently, in the second stage, optimization runs were carried out, with each of these bell shapes as a starting shape. Of course, the goal was to find bells with the major third partial in the tone structure. In these optimization runs, the bell profile was allowed to be modified much more locally,

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compared to the global variations mentioned above. The results were 625 bell look-alikes with more or less the tone structure of a major third bell. In the third stage the results were sorted looking at the quality of different properties, as there are: the tone structure, the shape with respect to aesthetics and molding technique and the weight of the bell (scaled to a certain strike note). In the last stage the most promising results have been selected, to be optimized individually to a final solution.

As described above it looks like performing a number of straightforward tasks. Unfortunately that is generally not the case, especially not the final optimization. One has to play with the program facilities to achieve the best design. A bell is an obstinate physical system; it is far from easy to manipulate it!

4. Comparison of the bells with respect to the partial tones and some other characteristics

The partial tones which will be considered in this chapter are the most important partials up to and including the double octave, see Table 1. The partials III – 4 and II – 5, located between the twelfth and the double octave, have been omitted because usually they are very weak. In Table 1 also three higher partials are mentioned. Only the triple octave will be discussed in this chapter too, the others will be discussed in the chapter on strengths of partials.

The partials are indicated by their so-called mode, composed of the group number (Roman figure) and the number of meridian node pairs. Where applicable the partial name is mentioned too (such as used in minor third bells). Deviations from musical tones are given in cents, where 100 cents equals half a tone.

Discussion and conclusions

The bells which are presented here are all so-called designed bells, resulting from computer simulations. They are not physical molded, tuned and measured bells. That is no real shortcoming of this presentation. Nowadays computer simulations of the tones of bells are so accurate that there is no doubt about the validity of the results.

Regarding Table 1 the bells will be discussed each on turn on several aspects. Remember that the bells are scaled to have the nominal exactly at a2, so strike note a1 may be assumed. Up to and including the nominal actually little has to be discussed; all partials are pretty close at the ideal place. In minor and major carillon bells, and often also in swinging bells, only these partials are tuned within a few cents (five or less) from zero deviation. The other partials must get their right place at the design stage of the bells. They are not controlled (and little affected) by the tuning procedure.

Bell MINORB.

The minor third bell acts as a reference for the major third bells. Typical for bell MINORB is that the partials II – 4, III – 2 and III – 3 are close to each other. Usually with minor bells, III - 2 and III – 3 are close together, and II – 4 is somewhat further below. With bell MINORB the twelfth is just 5 cents above the ideal e3. The double octave (ais3 – 32) is rather high. It is very well possible to have minor bells where this partial is only 50 cents, or even less, is too high. In common minor a1-bells a double octave at the ideal place a3 is not feasible. The triple octave may be of some importance only in very large bells. Both in minor and in major bells a triple octave at the ideal place a4 is feasible. The dimensions, the weight and the shape of bell MINORB are good averages of common minor bells.

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Table 1. Tone structure, fD-value and dimensions of the bells in this research.

Bell GT8504.

In this bell it is apparent that the partials III – 2 and III – 3 are low in the tone spectrum. It seems to be a property of the bulge. Bell GT0611 and, to a lesser extend, also bell GT8706 show this partial shift. Furthermore, partial III - 3 is frightening close to the nominal. Typically the twelfth is 40 cents too low and the double octave is almost ideal. The fD-value

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is 222, which is very high for a middle-sized bell. At the time the fD-value was not a design goal. It just ended here in the struggle to find an octave bell with a major third. Together with the large relative height (0.910), the high fD-value resulted in a very heavy a1- bell (739 kg).

Bell GT8706.

In bell GT8706 partial III – 2 is close to the nominal. One has to choose a suitable contact point of the clapper in order to avoid beats. The twelfth is the highest of all “historical” major third bells. With (e3 -26) it is just 1/8 of a tone below the ideal e3. The double octave is only 30 cents above a3. So, twelfth and double octave are promising for a rather well defined strike note. Furthermore, the triple octave is only 16 cent above a4, an advantage if very large bells are aimed.
This is relatively a high bell (0.916). At designing this bell André Lehr returned to the common value fD = 200 m/s. Also the shape has returned somewhat in the direction of bell MINORB. The weight of the bell is quite nice (523 kg).

Bell GT9518.

From the major third bells considered here the shape of bell GT9518 is most alike bell MINORB. The partials III - 2 and III – 3 of GT9518 are still lower than those of bell MINORB, although some distance above the nominal. The twelfth (e3 – 47) is almost 1⁄4 tone too low, which may affect the strike note negatively. The double octave (a3 – 5) is almost perfect. The triple octave is not far from a4, enabling this bell to be used for large bells. Although the fD-value (=199) is quite standard, bell GT9518 is a very light one, even 12% lighter than bell MINORB. As can be seen from Table 1, major third bells tend to have increased weight.

Bell GT0611.

Due to the inside and outside bulge, bell GT0611 shows a very low partial III - 2 (g2 – 38), comparable with bell GT8504. If it should disturb the bell’s sound, a suitable striking point must be chosen (at the nodal ring of III - 2). Alternatively this partial might be tuned to musically an appealing tone height. Partial III – 3 is enough above the nominal to avoid problems.

The twelfth (e3) is ideal. André Lehr completely succeeded to reach the goal that he pursued with this bell design. The double octave (ais3 -28) and also the triple octave are rather high. Bell GT0611 is relatively a high and heavy one. The very pronounced bulge asks for special measures in the molding process.

Bell GT0803.

In bell GT0803 the partials III – 2 (d3 + 2) and III – 3 (dis3 + 18) are higher than with bell MINORB, whereas in the preceding major bells these partials are just lower. Partial II – 4 (c3 + 12) is somewhat lower than in all other bells in Table 1. The partials II – 4, III – 2 and III – 3 are all rather close to musical notes (c3, d3 and dis3 respectively), such that tuning to these notes might be considered (if musically interesting). With e3 – 4 the twelfth is almost ideal, certainly strengthening the strike note. The double octave (a3 + 40) is not too much above a3. This bell type is less suited for very large bells because of the high the triple octave (ais4 – 4). The weight of GT0803 is pretty low (513 kg). The widening at the shoulder and the smaller rim diameter (which is below the greatest diameter at the sound bow), ask for special measures at the molding. Although the bell looks low, the relative height (0.807) is quite close to that of bell MINORB. The shape aesthetics of this bell may be disputable. The artist’s impression in Figure 1 will give an idea; however the true shape should be judged at the molded bell.

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5. Computation method for the acoustic input power of individual bell partials; measure for strength of partials

If a solid plate is vibrating in the air as medium, the acoustic power that is put into the air can be computed from (Chen et al., 1995):

P = σ*ρ*c*S*(v2)average (1)

is the acoustic power (Watt)
is the so-called radiation factor giving the effectiveness of the radiation;

0 < σ < 1; for the bell σ = 1 may be taken. is the density of the air (about 1.2 kg/m3) is the sound velocity in air (about 340 m/s)

where: P σ

ρ c S

Formula (1) will be used to compute the acoustic power that is put into the air by separate partials of the vibrating bell. However, the velocity at the bell surface varies per partial, and from place to place. Therefore we regard the bell surface as divided in a large number of small pieces. We compute the acoustic power of each piece according to formula (1). Finally we sum the contributions of all pieces to the total acoustic input power, and call it Pinput. So, each partial gets its own Pinput –value.

It is emphasized that Pinput is not the acoustic power experienced by the listener at some distance of the bell. Therefore an acoustic analysis has to be carried out, and that is not what we do here. Pinput is just the sound power at the bell surface that is generated by a certain partial.
In this paper Pinput will be regarded as a useful measure to compare the strengths of individual partials, both of a particular bell and between different bells.

Computation of surface velocity of the bell, the contact force and contact time.

Say, we like to evaluate formula (1) for a certain bell partial. Than the first task we have is to find the velocity at the bell surface for the considered partial, after the bell has been struck with the clapper.
In the bell design program a module has been implemented to compute mechanical vibration amplitudes during and after the bell has been struck by a clapper (Schoofs et al., 1994). This module needs a combined computation model of the bell and the clapper. For the bell the same finite element model is used as for computation of partial tones. The clapper is modeled as a physical pendulum. Design parameters of the clapper system comprise the pivot point (P), the length and the mass of the clapper and the diameter of the clapper ball (see Figure 5a). Furthermore, material properties of the clapper ball must be given, such as mass density, Poisson’s Ratio and Young’s modulus.

From the geometry of the bell and clapper system the location of the contact point is derived. The contact force as function of time is computed using the theory of Hertz (1882). This theory considers two elastic bodies, both with double curved surfaces, which are pressed against each other, see Figure 5b.

is the surface of the plate (m2)
(v2)average is the average of the squared velocity of the plate perpendicular to its

surface (m/s)2

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Figure 5. (a) Bell and clapper with contact force F(t), the contact point C, and the contact velocity v(t).

(b) Local indentation between bell and clapper ball.

For the simulation, also the initial velocity must be specified with which the clapper hits the bell. From the moment that the clapper makes contact with the bell, the force between bell and clapper starts to build up. Thereby, the clapper velocity decreases and, at the same time, the bell wall begins to move. The equations of motion describing this process are numerically integrated step by step. As a result we get the contact force, accelerations, velocities and displacements of the whole bell surface (for each partial individually), and the clapper movement as function of time. Computed accelerations of an existing minor third bell (0.78 m, 273 kg) and clapper system were compared with measured accelerations. The results agreed very well (Schoofs et al., 1994). Therefore, the computational method can be used to predict tendencies if design parameters of the clapper system are changed.

In Figure 6 results are given of the computed contact force versus time for some clapper configurations with bell MINORB, see Figure 3.

Figure 6. (a) Contact force versus time for clapper velocities 0.2, 0.4 and 0.8 m/s, and at a clapper mass of 3% of the bell mass.

(b) Contact force versus time for clapper masses 2%, 3% and 4% of the bell mass, and at 0.4 m/s clapper velocity.

The bell-clapper contact starts at time t = 0. From Figure 6a it is seen that the contact time decreases at increasing clapper velocity from about 0.95 ms to about 0.74 ms. Figure 6b shows that the contact time decreases with about the same amount (but in a different way), if the clapper mass is reduced from 4% to 2% of the bell mass. It is well-known from practice that higher clapper velocities strengthen higher partials relatively more than lower ones. At

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the same time, it is known that more heavy clappers strengthen lower partials relatively more than higher ones.
Computation of the strength of partials, as pointed out in Chapter 5, will take place immediately after the bell-clapper contact is over. That is within 1 ms, or less, from the initial bell-clapper contact, as can be seen from Figure 6. During the contact remarkable things happen with the higher partials, about from the twelfth and above. From space and time considerations, this phenomenon will not be treated in this paper. It was already treated in (Schoofs et al., 1998), however, in a limited way. The phenomenon is subject of future research.

Bell-clapper configurations, striking point and clapper velocity at initial contact.

To evaluate the partial strengths it is necessary to make choices for the bell-clapper configurations of the different bells. More particularly, choices have to be made for the shape, mass, length and pivot point of the clapper. In Figure 3 the chosen bell-clapper combinations are depicted. In Table 2 the dimensions of the clapper systems are given.

Table 2. Dimensions of the clapper systems.

First, for all bells a clapper mass of 3% of the bell mass was applied, which is quite a usual value in practice. Second, for bell MINORB also clappers of 2% and 4% were applied (indicated with “2.” and “3.” respectively). Third, for bells GT0611 and GT0803 also another clapper configuration was analyzed (indicated with “2.”).

A sphere is chosen as shape of the clapper ball. The clapper length is the distance from the pivot point to the centre of the clapper ball. The distance of the pivot point above the base of the bell is indicated by z-Pivot.
These dimensions determine at which point the bell is hit by the clapper. The dimensions have been chosen such that the clapper hits the bell (about) at the point of maximum thickness of the sound bow. In literature few data are given about the striking point. André Lehr mentions the thickest point as striking point in his old book (André Lehr, 1976). In his new book (André Lehr, 1996) the author missed such data; probably the striking point is not unambiguous. The author regards the choice of the thickest point as a handhold.

For bells GT0611 and GT0803 also another striking point (higher up in the bells) has been analysed, determined by the shift of the pivot point to 640 mm and 670 mm respectively. The purpose of these shifts will be explained later, in Chapter 6.
All bell-clapper configurations have been analysed for three values of the clapper velocity at the time point of first bell-clapper contact, namely 0.2 m/s, 0.4 m/s and 0.8 m/s. Striking with 0.4 m/s may be regarded as a moderate bell strike; striking with 0.2 m/s is a weak one, whereas 0.8 m/s is a really fierce strike.

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Vibration patterns in vertical direction of a bell.

The strengths of the partials are very much influenced by the vibration patterns in vertical direction of the bell. Those vibration patterns tell, for each partial, the location of the nodal rings and the ventral segments (antinodes). A partial does not sound, or just very weak, if the bell is struck at (or close to) its nodal ring. The partial strength is maximized by striking at an antinode.

As an example, in Figure 2 vertical vibration pattern are given for bell MINORB, up to and including the double octave. From space considerations, the vertical vibration pattern of partial (III – 3) is not shown; it is almost the same as for partial (III – 2).

6. Comparison of the bells with respect to the partial strengths

In this chapter at first, for all bells, the strengths of partials will be considered at 0.4 m/s clapper velocity. This velocity can be regarded as an average value. It was decided to define and analyze modified clapper configurations for the new bells. Thereafter differences will be looked at if weak and fierce clapper strikes are applied (velocities 0.2 m/s and 0.8 m/s, respectively). To facilitate that, the concept of the partial amplification factor will be introduced. Finally, implications of partial strengths with respect to strike note and dynamic range will be discussed.

Partial strengths at 0.4 m/s clapper velocity.

In Figure 7 the horizontal axes indicate the partials with a number according to Table 3 (the same order as in Table 14). The vertical axes give the Partial strengths in Watt.

Table 3. Numbering of the partials.

4 In most cases this order is ascending in tone height, but not strictly. The scaling column, No’s 14 in Table 3 and Figure 7, has only been introduced to force the same scale in the pictures of Figure 7.

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Figure 7. Partial strengths of the bells, measured as acoustic input power [W].

Generally only the strength of the partials hum, fundamental, third, nominal and twelfth are significantly above zero, see Figure 7. The partials numbers 4, 6, 7 and 8 are weak, because they all have a nodal ring in the vicinity of the striking point. Since their relative strengths are very sensitive for the precise choice of the striking point, comparison of these partials does not make much sense. The partials double octave (No. 10) and above are weak altogether, and will be discussed later in an other respect. It should be emphasized that not is meant that the weak partials are unimportant for the bell sound; certainly they are.

In Figure 7 it is clearly visible that for all bells the twelfth is much stronger than the double octave. This confirms that a right tone height of the twelfth is much more important for a well defined strike note, than is a right double octave.

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Table 4. Comparison of partial strength with respect to bell MINORB; legend:
--, -, 0, +, ++ means: considerably weaker, weaker, (about) equal, stronger,

considerably stronger.

In Table 4 the results of Figure 7 are globally compared with respect to the strong partials. Overall, partials of the major bells are stronger than with bell MINORB, especially the fundamental and the third (No’s 2 and 3); only the hum and twelfth of bell GT0803 are weaker. For most major bells the hum, nominal and twelfth (No’s 1, 5 and 9) are somewhat stronger than with bell MINORB. Looking at Table 4 bell GT9518 comes most close to bell MINORB.

Other clapper configurations for bells GT0611 and GT0803.

For bell GT0803 the deviations with bell MINORB can be explained. The sound bow of bell GT0803 is very blunt, see Figure 1. Hence, the striking point (point of maximum thickness) is relatively close to the bell’s rim. It has come out that such a point weakens the hum and strengthens the fundamental. Furthermore, the sound bow of bell GT0803 is the most thick one of all bells (see Table 1), although the fD-value is lowest. This indicates that a clapper mass of 3% of the bell mass might be too low.

Therefore a new clapper configuration has been defined, whereby the pivot point is shifted from 650 to 670 mm, and the clapper mass is increased from 15.4 to 16.8 kg, being 3.3% of the bell mass (see Figure 8 and Table 2: bell GT0803, indication “2.”).
Bell GT0611 is relatively heavy in the upper part of the bell, giving perhaps a 3% clapper mass what is too high. Furthermore, the overall stronger partials indicate that the clapper mass should be decreased. Shifting the striking point upwards will weaken the fundamental and the third compared to the hum. These considerations led to the new clapper configuration for bell GT0611, whereby the pivot point is shifted from 610 to 640 mm, and the clapper mass is decreased from 17.3 to 15.5 kg, being 2.7% of the bell mass (see Figure 8 and Table 2: bell GT0611, indication “2.”).

The new configurations have been analyzed in the same way as before. The resulting partial strengths, given in Figure 8, agree rather well with those of bell MINORB in Figure 7.
So far we ignored partials No’s 4, 6, 7 and 8. Looking at the results for bells GT0611 and GT0803 in Figure 7 and Figure 8, interesting changes can be seen with respect to those partials.

With bell GT0611 in Figure 7 partials 4, 6 and 8 are very weak; partial 7 is rather strong. In Figure 8, partials 4 and 6 have increased, while 7 is decreased, but it still can be heard. Generally this partial is not tuned. However, since the partial is low in the tone spectrum (g2 - 38) it might be desirable to optimize and tune to a neighboring regular tone, so to g2 or fis2. Or, as an alternative, the partial could be tuned to just g2 – 50, in order to avoid beats with regular tones. With the new clapper configuration, partial 8 has vanished completely.

With bell GT0803 in Figure 7, partials 7 and 8 are very weak; partials 4 and 6 are relatively strong. In Figure 8, partials 4 and 6 have become even stronger, while 7 and 8 have vanished completely. Partial 6 (c3 + 12) is about a minor third above the nominal; it must be judged

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whether it should be heard or not. If it is desirable that partials 7 and/or 8 are to be heard, again another striking point should be chosen.

Finally, at least two remarks can be made. First, of course, other clapper configurations can be defined, and might be favorable for the other bells too. Second, just choosing a clapper mass of 3% and striking at the thickest point of the sound bow seems too straightforward, and is often not optimal.

Figure 8. Partial strengths of bells GT0611 and GT0803 with modified clapper configurations (red in top figures).

Partial strengths at 0.8 m/s clapper velocity compared to 0.2 m/s clapper velocity. Introduction of the so-called Partial Amplification Factor: PAF
In the past major third bells have been marked upon to have a smaller dynamical range than minor third bells. If the reason of that, right or not, must be found in the partial strengths, it should be visible if the clapper velocity is increased substantially. That is what we do in this section. We analyzed all bell-clapper configurations at 0.2 m/s and 0.8 m/s clapper velocity. Because surprising results came out, first several configurations for bell MINORB were analyzed. Thereafter analyses for the major third bells will be considered.
Figure 9 gives the partial strengths at clapper velocities 0.2 m/s and 0.8 m/s. Please, note that the pictures have different scale.

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Figure 9. Partial strengths of bell MINORB at clapper velocities 0.2 m/s and 0.8 m/s.

The energy of motion of the clapper, so-called kinetic energy, at the first moment of impact, determines the partials strengths to a large extent. The kinetic energy of the clapper is computed as:

Ekin = 1⁄2 * massclapper * (vclapper)2 (2)

With clapper velocities 0.8 m/s compared to 0.2 m/s, formula (2) gives an increase of the kinetic energy with a factor (0.8/0.2)2 = 16. One might expect that sound powers than are amplified with a factor 16 too.
Looking at Figure 9 that fits with the fundamental (amplification is about a factor 0.76/0.047 = 16.2). We will name such factors as “Partial Amplification Factor”, or shortly, “PAF”. However at 0.8 m/s the hum lags behind (PAF is about 0.8/0.054 = 14.8)5. That is what we might expect: at a fierce stroke the lower partials become relatively weaker. The first surprise is that this is true for the hum only! The PAF of the fundamental is about as expected (PAF = 16). All other partials will prove to amplify beyond expectation, for some partials even to an extreme extent: the second surprise. Look at the twelfth (No. 9). PAF is about 0.36/0.07 ~ 51! Furthermore, from Figure 9 it can be seen that also the nominal (No. 5) has become somewhat extra strong. At 0.2 m/s the nominal is lower than the fundamental, while at 0.8 m/s it is higher. However, if we regard the double octave (No. 10) a dramatic amplification has to be expected: PAF is about 0.05 divided by almost zero. With the high partials No’s 11, 12 and 13 it looks like nothing is going on: the third surprise!

In order to illustrate the amplification factors as calculated above, they have been plotted in Figure 10a. Amplification factors increase more or less gradually to somewhat more than 50 for the twelfth (No. 9). Next, PAF of the double octave is about 320. PAF-values drop for partials above the double octave, even under the nominally expected value of 16. The triple octave (No. 13) proves to have PAF = 3. Nevertheless, at 0.8 m/s clapper velocity, the triple octave is still 3 times stronger than at 0.2 m/s. However this is much less than is expected from the four times higher velocity.

Three extra simulations for bell MINORB.

The author suspects that the phenomenon illustrated in Figure 10a is a kind of resonance between the bell and the clapper. To get an idea about that, three other simulations have been carried out. The bell and the clapper, each apart, are so-called linear elastic mechanical systems. Loosely spoken this means that, if something changes in such a system, other things will change linearly proportional. However, the contact between the clapper and the bell is

5 See Table 5 for an overview of the numeric values of the (rounded) amplification factors of all bells. 17

strongly non-linear: the contact force is by no way linearly proportional to the indentation of bell and clapper. This makes the bell-clapper system, as a whole, a non-linear mechanical system. That non-linearity will cause a shift of the peak in Figure 10a if the bell-clapper configuration is changed. The three extra simulations mentioned above comprise:

1. Small bell (with a2 strike note) and clapper mass 3% of bell mass (same as in original a1 bell size).

2. Original a1-bell, however a relative clapper mass of 2%.

3. Original a1-bell, however a relative clapper mass of 4%.

Indeed, notable differences can be seen in Figures 10b through 10d. The results in Figure 10b are quite close to those of Figure 10a up to and including the double octave (No.10). However, the highest three partials in Figure 10b show also high PAF (about 60, 280 and 50 respectively. Here we are lucky! These results confirm that for middle-sized bells, and aiming comparable bell timbre, the same relative clapper mass can be applied. Possible differences at the very high (and overall weak) partials will not disturb the bell sound.

The situation with 2% relative clapper mass is a little different. PAF of the twelfth is somewhat lower: now about 40 instead of about 50 in Figures 10a and 10b. PAF of the double octave is reduced to about 160 (was 300 and more). The highest partials have factors comparable high with Figure 10b, although the order is different. The different results in

Figure 10. Partial Amplification Factors, PAF, for bell MINORB;
i.e. strengths of partials at 0.8 m/s clapper velocity, divided by

corresponding partial strengths at 0.2 m/s clapper velocity.

Figures 10a and 10c could be expected: the lower clapper mass will shift the amplification to the higher partials no’s 11, 12 and 13. However, for the twelfth and the double octave in Figure 10c also higher factors were expected, but they are just lower.

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Finally, the results for the 4% clapper configuration in Figure 10d are quite as we should expect. The heavy clapper amplifies the twelfth with PAF = 30, instead of more than 50 at the 3% clapper. The double octave “only” has about PAF = 60. Partials 11, 12 and 13 drop dramatically, same as in Figure 10a. It looks like if the relative heavier clappers (4% and also 3%) damp the fast bell vibrations of those high partials. This phenomenon is subject of future research.

Overview of amplification factors for the major third bells and the MINORB cases.

For the major third bells and the four cases of bell MINORB the amplification factors are gathered in Table 5. For the new bells GT0611 and GT0803 the modified clapper configurations have been applied.
From Table 5 the following can be seen. Up to and including the nominal (Partial No. 5) all columns show about the same tendency. They start with PAF = 15 for the hum, and increase more or less gradually to about PAF = 22 for the nominal. Next, PAF’s of the weak partials no’s 6, 7 and 8 are somewhere between 23 and 38; they are less interesting. The next milestone is the twelfth (Partial No 9), with PAF between 24 (GT8504) and 53 (MINORB, a1, 3%). For all other major bells PAF is in the range 41 to 47, so, somewhat lower than for bell MINORB (PAF = 53). Clearly, bell GT8504 is far below par.

Table 5. Partial Amplification Factors, PAF, i.e. strengths of partials at 0.8 m/s clapper velocity divided by corresponding partial strengths at 0.2 m/s clapper velocity.

Next, for the double octave (Partial No 10) PAF’s rise almost instantly to values between 26 (bell GT8504) and 331 (GT0803). Other columns, except one, are between 160 and 318. Again, bell GT8504 is far below par. The author suspects that the low PAF’s for the twelfth and the double octave of this bell are caused by its relatively very high weight (739 kg) and high fD-value (222), see Table 1. No further research has been applied on that.

Bell MINORB combined with 4% clapper mass also shows relative low PAF’s for the twelfth and the double octave (31 and 63, respectively), being another indication that heave weights reduce the amplification. However, in this point of view one might also expect lower PAF’s

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with bell GT0803, since its relative thick sound bow and 3.3% clapper mass. But PAF’s for the twelfth and the double octave are high instead, 46 and 331, respectively.
For the high partials, No’s 11, 12 and 13, of the major bells, PAF’s drop similarly rapid as in the case MINORB with 3% clapper mass. Only bell GT9518 has higher PAF’s (up to 47), probably caused by its relative low weight (399 kg), see Table 1.

In the end another, fourth surprise with respect to PAF’s comes out. If we look at Table 5 all columns, except MINORB with 4% clapper and bell GT8504, are quite similar up to and including the double octave. The exceptions concern two “heavy” situations: 4% clapper mass and the heavy bell GT8504. The mentioned similarity means that, at similar bell-clapper characteristics, the timbres of bells, between weak and fierce strikes, change in the same way. The surprise is that this is true, even for such different bells as there are in Table 5.

The mentioned exceptions present another message. Characteristics change substantially for “heavy” situations. For instance, one should be aware of that, especially for carillon design with its thick-walled treble bells.
Bell GT8504 clearly was too heavy to be applied as a middle-sized bell.

Implications of the partial strengths and partial amplification for the strike note and the dynamical range of major third bells.
It is well-known that the nominal, twelfth and the double octave are the far most crucial partials with concern to the strike note, at least for middle-sized bells (Eggen, 1986). Frequency ratio’s of these partials (of common carillon bells) with respect to the fundamental are 2, 3 and 4 respectively. The human hearing system completes this harmonic series with the virtual strike note at frequency ratio 1, so at the same level as the fundamental (which is a physical overtone of the bell). It is peculiar that the fundamental is quite unimportant for the experience of the strike note. Even if the fundamental is shifted, or if it is absent at all, the strike note will be heard at frequency ratio 1 (provided that at least the nominal and the twelfth are right).

Table 6. Partial strengths (in Watt) of the nominal, twelfth and double octave at 0.2 m/s and 0.8 m/s clapper velocity.

But how does the partial amplification affect the strike note? To give an answer, the following considerations have been made.

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In Table 6 the partial strengths (Watt) of the nominal, twelfth and double octave have been listed, both for 0.2 m/s and 0.8 m/s clapper velocity. It is clear that the nominal in all cases is the strongest partial.
First we look at 0.2 m/s clapper velocity. If we neglect bell GT8505, the nominal is about 5.6 times stronger than the twelfth, as an average. The nominal is even about 200 times stronger than the double octave. At 0.8 m/s the nominal is, averaged, about 2.6 times stronger than the twelfth, and about 20 times stronger than the double octave.

It is apparent that the twelfth has become relatively stronger compared to the nominal. The extra increase is a factor 5.6/2.6, as an average; that is an increase of 3.3 dB. Such an increase is clearly recognizable for human.
The double octave is still rather weak at 0.8 m/s, as can be seen in Table 6 (despite the very high amplification factors, seen in Table 5).
For previous major third bells, smaller dynamical ranges compared to minor bells were marked upon in the past. This meant that major bells could not be played as strong as minor bells, in order to avoid ugly strike sounds.
On the one hand, the author thinks that the smaller dynamical range may be caused by too low a twelfth partial tone, in combination with the substantial amplification of the twelfth at fiercer strikes. Say, the twelfth is 50 cents too low. At a soft strike the twelfth is rather weak; the strike note will be dominated by the nominal, and is OK. At a fierce strike the twelfth will be strengthened substantially with respect to the nominal. Hence, the twelfth may become dominant concerning to the strike note, which in turn will be affected negatively.
The research of Eggen (1986), mentioned above, states that there will be a mismatch of the strike note if the twelfth deviates more than 4% from its ideal value (i.e. frequency ratio 1.5 to the nominal). In such cases the strike note will become a subharmonic of the twelfth, hence, not a subharmonic of the nominal anymore. However, 4% too low a twelfth means about 70 cents too low. The twelfth of previous major bells never was more than about 50 cents too low, at a maximum, so considerably less than 70 cents. The author suspects that the limit of 70 cents might be reduced to even less than 50 cents, caused by the abovementioned strengthening of the twelfth at fierce strikes. As a consequence, the strike note may be negatively affected to an unacceptable extent in such cases.
On the other hand, the explanation above implicates that playing previous major bells at moderate strikes is quite well acceptable. Furthermore, dynamic range reduction of bell GT8706 (carillons Deinze and Garden Grove) will be limited (or absent at all), because with this bell the twelfth is not much low, only 26 cents, see Table 1. Another consequence is that the dynamic range of the new major bells GT0611 and GT0803 may equal that of minor bells, due to the right tone height of their twelfths.

7. Conclusions

We will concentrate mainly on the new major bells GT0611 and GT0803. Beside that, some more general conclusions on this paper are given.

Tonal structure of major third bells.

The reader is referred to the section “Discussion and conclusions” at the end of Chapter 4, which gives detailed considerations about individual bells. A few points are recalled here. The tones heights of the hum, fundamental, third, fifth and the nominal are correct in all bells. The main difference between the previous and the new major bells is, purposely, the tone height of the twelfth. In the previous major third bells the twelfth is 1/8 to 1⁄4 tone too low. In

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the bells GT0611 and GT0803 it is at the (almost) perfect tone height. This will give these new bells an indisputable strike note.
The double octave of previous major bells usually is close to the ideal tone height: one octave above the nominal. With the new major bells the double octave is above the ideal, comparable with common minor bells. However, in Chapter 6 it was pointed out that too high a double octave does not give serious problems, due to its low strength.

Except bell GT0803, the major bells tend to have lower partial tones for the modes III – 2 and III – 3 (in minor bells called: 1st and the 2nd eleventh, respectively). That is caused by the typical bulge in the waist of some major bells. Usually, tones are lower at a more pronounced bulge. Especial partial III – 2 of bell GT0611 needs attention, because it is low in the tone spectrum. On has to decide whether it should be tuned, and if so, to which tone.

Shape and weight of the bells

Bell GT0611 has gotten again the bulge, while bell GT0803 shows a serious widening at the shoulder; the rim of the last bell has moved inwards. Therefore special measures have to be taken at the casting of both bells. Of course, the aesthetics of the bells is disputable. It is the price for an (almost) exact partial tone of the twelfth. The weight of bell GT0611 is rather high: 576 kg for an a1-bell. The author has indications that the weight can be reduced considerably, while keeping the bell’s characteristics intact.

Acoustic input power as measure for strengths of partials.

As a measure for the strength of a partial, the concept of the acoustic input power was introduced. Together with bell-clapper simulations, it provides a valuable tool to visualize the strengths of individual partials. The strengths of partials are computed immediately after the bell-clapper contact is over (usually within 1 ms from the initial contact).
Again, it is emphasized that the acoustic input power is computed directly at the bell’s surface. It is not the sound power the listener will experience at some distance from the bell. However, acoustic input power values are expected to be rather linear proportional to those listener’s experiences.

Partial strength.

From Table 4 it was seen that partial strengths of major bells overall are stronger than for the minor bell, especially the fundamental and the third. Bell GT0803 showed lower strengths for the hum and the double octave. Modified clapper configurations were defined for both new bells, GT0611 and GT0803. The resulting partial strengths agreed rather well with bell MINORB. It was nice to see how even the weak partials (fifth, major tenth, 1st and 2nd eleventh) could be monitored.

This exercise showed that clapper configurations can be designed in a more or less structured way. The importance of that for campanology should be well recognized. Anyway, it came out that a fixed relative clapper mass and fixed striking point often is too straightforward, and not optimal.

Partial Amplification Factor: PAF

Another new concept was introduced: the so-called Partial Amplification Factor (PAF). PAF tells to what level the strength of a partial is increased between a weak and fierce clapper strike. “Weak” and “fierce” were defined, rather ad hoc, by 0.2 m/s, respectively 0.8 m/s clapper velocity at the moment of impact. It came out that, at higher clapper velocity, the increase of partial strengths is (much) more than could be expected from the increased kinetic energy of the clapper.

Four surprises emerged from the simulations at fiercer strikes: 22

1. Only the hum is weakened; the fundamental strengthens conform kinetic clapper energy.

2. Higher partials (up to and including the double octave) strengthen extra, sometimes extremely.

3. Usually, above the double octave the strengthening drops suddenly.

4. Comparable bell-clapper configurations show the same “PAF-behavior”, even for

   quite different bells.

Dynamic range reduction explained.

Partial amplification was used to explain possible dynamic range reduction of major third bells, showing a twelfth which is considerably too low. From that, it is expected that with bell GT8706 (Deinze, Garden Grove) such reduction will be quite limited (or absent at all). Furthermore, due to their right twelfths, it is expected that the dynamic range of the new major bells, GT0611 and GT0803, is comparable with that of qualitative minor third bells.

Test bells.

The author received some comments on the c2-test bell GT0611, molded by Royal Petit & Fritsen Bell Foundry. The bell has been mold on tune; hence, it is not tuned on carillon standards afterwards, which is a pity. Nevertheless, comments are very positive in the sense of: lovely sounding, clear strike note, no ugly strike sound whatsoever, good reverberation time. In other words, the bell is very promising to be used as base for a fine carillon.

At the moment of writing, the test bell GT0803, to be mold by Royal Eijsbouts Bell Foundry, has not been finished. The most pressing question is about its reverberation times; measuring the test bell will give answers. Partial tones are nice, partial strengths too. Furthermore, the “PAF-behavior” (as the author named it) is rather the same as with bell MINORB (see Table 5), and is OK.

Finally

In the e-mail, mentioned on top of this paper, André Lehr questioned whether his new design GT0611 might be the final solution for the major third bell problem. Anyway, the author thinks it a large and deciding step forward. In major bell designs, solutions improved gradually over the past twenty-three years. Improvements will be always possible, albeit only some weight reduction of GT0611. André Lehr also questioned whether it should be possible to remove the reintroduced bulge. Bell GT0803 provides another shape, however, all merits of that design are not known yet.

In his splendid career André Lehr created fine hand played major third carillons in Deinze, Garden Grove and Groningen. Especially, the author was pleasantly surprised by the data of bell GT8706 (Deinze, Garden Grove). As early as 1987, it was already a very good design, especially, since it is not far from a perfect twelfth partial tone.

It is hoped that the new major third bell designs will encourage carillon advisors and bell founders together to realize new fine musical instruments. That would honor André Lehr at most.

Literature

Chen P.T., Ginsberg J.H. [1995]. “Complex power, reciprocity, and radiation modes for submerged bodies”, Journal of the Acoustical Society of America, 98(6), p.3343-3351. J.H. Eggen & A.J.M. Houtsma [1986]. “The pitch perception of bell sounds”, IPO Eindhoven, in IPO Annual Progress Report, 21, p. 15-23.

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André Lehr [1976]. “Leerboek der Campanologie”, Nationaal Beiaardmuseum, Asten. André Lehr [1987]. “The designing of swinging bells and carillon bells in the past and present, Athanasius Kircher Foundation, page 60.
André Lehr [1996]. “Campanologie”, Koninklijke Beiaardschool ‘Jef Denyn’, Mechelen. André Lehr [2006]. “Een doorbraak rond de grote terts-klok”, Berichten uit het Beiaardmuseum, No. 44, Dec. 2006.

André Lehr [2007]. “De rol van de slagtoon in de klank van de grote terts-klok”, Klok en Klepel, No. 98, March 2007.
Schoofs, A., Asperen, F. van, Maas, P., Lehr, A. [1987]. “A Carillon of major-third bells. I. Computation of Bell Profiles Using Structural Optimization”, Music Perception, Vol. 4, No. 3, p. 245-254.

A. Schoofs, D. van Campen [1998]. “Analysis and Optimization of Bell Systems”, in ”Proceedings of the 11th World Carillon Congress”, Mechelen-Leuven, p. 208-227. A.J.G.Schoofs, P.J.M. Roozen-Kroon, D.H. van Campen [1994]. “Optimization of structural and acoustical parameters of bells”, in “Proceedings of the 5th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization”, Sept. 7-9, Panama City, Florida, ISBN 1-56347-097-7, p. 1167-1180.

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