J. Acoust.Soc. Jpn. (E)10, 5 (1989)

Acoustics of Eastern and Western , old and new

Thomas D. Rossing

Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA

Bells have many modes of vibration whose modal shapes are characterized by the numbers of nodal meridians and nodal circles. In Western bells and bells, the is shaped so that the lowest mode frequencies are harmonically related. The subjective strike note is determined by three strong partial tones with frequencies nearly in the ratios 2:3:4. The pitch of a , on the other hand, is determined by the frequency of the fundamental tone. In ancient Chinese bells, the vibrational modes occur in pairs, and two different tones result from striking at the sui and gu strike points. A new type of carillon bell has been developed in which the traditional partial is replaced by a partial, thus changing the timbre of the bell. PACS number: 43. 75. Kk, 43. 40. At

ers in the Low Countries, especially the Hemony 1. INTRODUCTION brothers (Francois and Pieter) and Jacob van Eyck, Bells are one of the oldest and most cherished took the lead in tuning bells, and many of their fine musical instruments in human history. In fact, bells bells are found in today. are probably much older than human history. The carillon also developed in the Low Countries. According to historical legend, bells existed in China Chiming bells by pulling ropes attached to the clap- as early as the twenty- century B.C. A number pers had been practiced for some time before the of bells from the Shang dynasty (1600-1100 B.C.) idea of attaching these ropes to a keyboard or hand- are found in museums throughout the world. In clavier occurred to bell ringers in the 16th century. 1978 a remarkable set of 65 bells from the 5th cen- Many mechanical improvements during the 17th tury B.C. was discovered in the tomb of the Marquis and 18th centuries, including the breached wire Yi in Hubei province (see Fig. 1). Bells probably system and the addition of foot pedals for playing existed in India and other parts of Asia only a little the larger bells, led to development of the modern later than in China. carillon. Today, the term carillon is reserved for an The oldest bells in Japan of the dotaku type date instrument of 23 (two ) or more tuned bells from the Yayoi culture (250 B.C. to 250 A.D.). They played from a clavier (smaller sets are called are more or less conical with an oval cross section, "chimes") . The largest carillon in existence is the and they have a flange which extends from the rim up 74-bell (6 ) carillon at Riverside Church in one side, over the top, and down the other side, as New York with a of more than 18,000 kg shown in Fig. 2(a). The introduction of Buddhism to (20 tons), shown in Fig. 3. Japan in the sixth century led to the sho or temple also date back at least several centuries bell. The oldest such bell cast in Japan is probably B.C., although tuned handbells of the present-day the Ojikicho bell, cast on Kyushu in 698 and now type were developed in England in the eighteenth in Kyoto (see Fig. 2(b)). century. One early use of handbells was to provide Bells developed as Western musical instruments tower bellringers with a convenient means to prac- in the 17th century when bell founders discovered tice . In more recent years, handbell how to tune their partials harmonically. The found- choirs have become popular in schools and churches;

241 J. Acoust. Soc. Jpn. (E) 10, 5 (1989)

Fig. 1 A set of 65 bells from the 5th century B.C. found in Hubei province in China.

Fig. 3 The 74-bell Laura Spellman Rocke- feller carillon at Riverside Church in New York. With a bourdon of more than 18,000 kg, it is the largest carillon in the world.

by the author.7) In this paper, the modes of vibration of Western church bells and carillon bells are discussed in some detail in order to make a comparison with the modes observed in tuned handbells and ancient Chinese Fig. 2 (a) Dotaku dating from the Yayoi and Japanese bells. It will be shown how the applica- culture (250 B.C. to 250 A.D.); (b) Sho tion of modern technology has led to a new type of or , cast on Kyushu in 698. bell with a major-third partial.

2. VIBRATIONAL MODES OF WESTERN some 2,000 choirs are reported in the USA alone. CHURCH BELLS AND CARILLON BELLS Excellent histories of bells appear in recent books by Price,1) Lehr,2-4) Spear,5) and Elphick.6 Several Analysis of the rich sound of a Western church historical papers are reprinted in a collection edited bell or carillon bell reveals many components or

Fig. 4 The first five vibrational modes of a tuned or carillon bell. Dashed lines indicate the nodes. Frequencies relative to the prime and names of the corresponding partials are given below each diagram.9)

242 T. D. ROSSING: ACOUSTICS OF EASTERN AND WESTERN BELLS

partials, each associated with a different mode of vibration of the bell. The various partials in the

sound of a church bell or carillon bell are given such

descriptive names as "hum," "prime," "tierce," "quint ," "nominal," etc. The most prominent par- tials in the sound of a tuned bell, like those of most musical instruments, are of a fundamental. Fig. 5 Motion of a bell for inextensional

The first five modes of a church bell or carillon modes of small m. Modes with m=0 and

bell are shown in Fig. 4. Dashed lines indicate the m=1 require one or more nodal circles

locations of the nodes. The numbers (m, n) at the (n>0).

top denote the numbers of complete nodal meridians "swinging" modes extending over the top of the bell (half the number . As m increases, these modes of nodes observed along a circumference), and the have radial components which become increasingly numbers of nodal circles respectively. Note that larger compared with their tangential ones. there are two modes with m=3 and n=1, one with The m=2 mode is easily excited in a bell-shaped a circular node at the waist and one with a node near wine glass by running a moistened finger around the the soundbow. Thus, we follow the suggestion of rim. Rayleigh12) pointed out that the tangential Tyzzer8' and others and denote the one as (3, 1#) in component of the motion makes this possible. In Fig. 4. The ratio of each modal frequency to that of this case, the diagram in Fig. 5 rotates so that the the prime is given at the bottom of each diagram. point of maximum tangential motion follows the A detailed study of the vibrational modes of a finger around the rim. church bell has compared the normal modes com- The (2, 1#) mode deserves further discussion. Its puted by a finite element method to the first 134 circular node in a D5 church bell was observed to be modes observed in the laboratory.10) Modes such 16 cm above the mouth, as compared with 29 cm as (2, 0), (3, 1), and (4, 1) are classified as "ring in the (3, 1) mode (the tierce or minor third) and 10 driven" since, in the vicinity of the soundbow, they cm in the (3, 1#) mode (the quint or fifth). Thus, have many of the characteristics which the thick insofar as positions of nodal circles are concerned, ring at the soundbow would exhibit if it were able to the mode fits into group II better than group I, but vibrate in its various inextensional radial modes as in a sense it serves both as the (2, 1) and the (2, 1#) an independent system. These modes are referred modes in the mode classification scheme. Note that to as "group I" modes by Lehr.11) They are excited the (2, 0) mode (hum) is the only normal mode in strongly by the clapper, and they radiate most of the the modern church bell without a circular node. strong partials in the bell sound. The second im- Figure 6 is a periodic table showing some of the portant family, designated as "group II", includes modes observed in a D5 church bell. The relative the (2, 1#), (3, 1#), (4, 1#) and higher modes. They modal frequencies and the locations of the nodes are are classified as "shell driven" modes, as are other indicated. To the groups previously suggested has important families having n=2, 3, 4, ... and referred been added a group 0 with a single member, the (2, 0) to as groups III, IV, V, ... They are characterized mode. This classification into groups makes it easier by a nodal circle near the mouth. Like the "ring to compare church bell modes with those of other driven" modes, they are inextensional in the sense types of bells. that a neutral circle in each plane normal to the bell's Vibrational frequencies of groups 0-IX in a church symmetry axis remains unstretched. This means that bell with a D5 strike note are shown in Fig. 7. Also the radial and tangential components of the motion, shown are the relative strengths of several partials in u and v, respectively, are related by u+„qv/„qƒÆ=0, the bell sound. Arrows denote the three partials in where ƒÆ is the polar angle in the plane concerned.12) group I that determine the strike note. Graphical Thus, we may write u=m cos mƒÆ and v= sin mƒÆ. displays of several modes computed by the finite Motion of the bell for m=0, 1, 2, and 3 is illustrated element (PAFEC) method are shown in Fig. 8. in Fig. 5. For m=0, the motion is purely tangential, In addition to the inextensional modes we have and we can describe these modes as "twisting" discussed, there are a number of vibrational modes modes. Modes with m=1 might be described as that involve stretching of the bell metal. In these

243 J. Acoust. Soc. Jpn. (E)10, 5 (1989)

Fig. 6 Periodic table of inextensional modes of vibration in a church bell. Below each drawing are the modal frequencies of a D5 church bell relative to the prime (whichhas es- sentially the same frequency as the strike note in a bell of high quality). At lower left (m, n) gives the number of nodal meridians 2m and nodal circles n.13)

Fig. 7 Vibrational frequencies of groups Fig. 8 Modal shapes in a church bell pre- IX in a D5 church bell. Also shownO- on dicted by finite element calculation for the the right are the relative strengths at impact inextensional modes with m=3: (a) (3,1); of several partials in the bell sound. Arrows (b) (3,1#); (c) (3,2); (d) (3,3); (e) (3,4); (f) denote the three partials in group I that (3,5); (g) (3,6).10) determine the strike note.9)

244 T. D. ROSSING: ACOUSTICS OF EASTERN AND WESTERN BELLS extensional modes, the radial and tangential motions Terhardt and Seewann also found a rather wide are related by v+„qu/„qƒÆ=0 so that u=cos mƒÆ and distribution in the tuning of partials in historical u=m sin mƒÆ. For m=0, in this case, the motion German bells.18) is entirely radial, but with increasing m the tangen- tial motion increases until it takes over from the 4. THE STRIKE NOTE radial motion. The m=0 extensional mode can be When a large church bell or carillon bell is struck described as a "breathing" mode. by its metal clapper, one first hears the sharp sound Another family of ring driven modes, not much of metal on metal. This atonal strike sound includes discussed in literature, are the "ring axial" modes in many inharmonic partials that die out quickly, giv- which the soundbow twists and moves in a longi- ing way to a strike note or that is domi- tudinal direction.10) Although such modes can occur nated by the prominent partials of the bell. Most quite low in the frequency spectrum (the m=2 case observers indentify the metallic strike note as having occurs at 2.76 times the prime in this D5 bell), they a pitch at or near the frequency of the strong second radiate very weakly as do the inextensional m=0 partial (prime or fundamental), but to others its pitch "twisting" and m=1 "swinging" modes and all the is an octave higher. Finally, as the sound of the bell families of extensional modes. ebbs, the slowly-decaying hum tone (an octave The modal frequencies of a flat circular plate are below the prime) lingers on. (For a historical ac- found to follow an empirical relationship fmn=c(m+ count of research on the strike note, see Ref. 7)) 2n)p for large values of (m+2n), a relationship some- The strike note is of great interest to psychoacous- times called Chladni's law.14) Although church bells ticians, because it is a subjective tone created by have a geometry substantially different from flat three strong nearly- partials in the bell plates, it is found that the modal frequencies of a sound. The octave or nominal, the twelfth, and the church bell can be fitted to a modified form of upper octave normally have frequencies nearly in Chladni's law fmn=c(m+bn)p by selecting different the ratios 2:3:4 (See Table 1). The ear assumes values of c, b, and p for large and small m.15) these to be partials of a "missing fundamental," which it hears as the strike note, or perhaps we should 3. TUNING OF WESTERN BELLS say, as the primary strike note. Bell founders usually tune the lowest five modes Figure 9 illustrates the role of each partial in of church bells and carillon bells so that their vibra- determining the pitch of the strike note. The sound tional frequencies are in the ratios 1:2:2.4:3:4. of a Hemony bell was recorded, and by means of a This is done by carefully thinning the inside of the digital filter, each of the first nine partials was raised bell at selected heights while it is mounted on a bell and lowered in frequency up to 10 % while listeners lathe. When this tuning is done, another five or six judged the pitch of the resulting strike note in a partials take on a nearly harmonic relationship, pitch-matching experiment. The results show that thus giving the bell a strong sense of pitch and a partials five and six (the octave and the twelfth) are very musical quality. The various names of im- the most important, followed by partial seven (the portant partials are given in Table 1. Also given are upper octave). The other partials, including partial the relative frequencies in an "ideal bell" (just tun- two (the prime, which coincides closely to the strike ing) and a bell with partials tuned to equal tem- note in frequency), have very little effect on the pitch perament. of the strike note, as indicated on the vertical axis.") Notice that the highest four partials in Table 1 are In very large bells, a secondary strike note may raised by as much as 4 % above those of the "ideal" occur a musical fourth above the primary strike note bell. A similar relationship is seen in other tuned and may even appear louder under some condi- church bells and carillon bells.16) This "stretching" tions.20) This secondary strike note is a subjective of the partial series may very well contribute a tone created by four partials beginning with the desirable quality to the bell sound.17) upper octave. These partials are from the (6, 1), Not all church bells have harmonically tuned (7, 1), (8, 1), and (9, 1) modes of vibrations, whose partials, however. A study of 363 church bells in frequencies are nearly three, four, five, and six times Western Europe revealed that only 17 % have the that of the secondary strike note (See Table 1). In hum, the prime and the nominal tuned in octaves.4) a large bell (800 kg or more), these partials lie below

245 J. Acoust. Soc. Jpn. (E) 10, 5 (1989)

Table 1 Names and relative frequencies of of important partials of a tuned church bell or carillon bell.13)

3,000 Hz, where the residue pitch is quite strong.21) In small bells, the higher partials lie at a very high frequency where the virtual or residue pitch is weak, and so both the primary and secondary strike notes are weak. The pitch is then determined mainly by the hum, the prime, and the nominal, which nor- mally are tuned in octaves. It is sometimes difficult to decide in which octave the pitch of a small bell lies, especially if the frequencies of these three partials are not exactly in 1:2:4 ratio. Difference tones between these partials may be heard. Some authorities question the existence of sub- jective strike notes, and suggest that the strike note is an auditory illusion resulting from the nominal, the strongest partial during the first few seconds.22) Because of this, some English founders denote the pitch of the bell as the pitch of the nominal. There is some feeling that a bell rung in full circle sounds a different pitch from a bell struck somewhat more gently in a carillon. Whether a founder tunes the nominal or the strike note makes little difference, Fig. 9 Effect of nine different partials in however, because the nominal is one of the partials determining the strike note of a bell. d PIP that determines the tuning of the strike note. is the relative change in pitch resulting from a change in relative frequency AP' of 5. HANDBELLS a partial. Note the great importance of the The vibrational frequencies of a C5 handbell are fifth and sixth partials (octave and twelfth) shown in Fig. 10. A comparison with Fig. 7 shows followed by the seventh partial (upper oc- tave); other partials are relatively unim- some interesting similarities and differences. Note portant.19) that each curve in Fig. 10 drawn for n=1, 2, 3, ...

246 T. D. ROSSING: ACOUSTICS OF EASTERN AND WESTERN BELLS shows a minimum in frequency at about m=n+2. strain due to stretching and bending in a cylindrical This is similar to the behavior of a cylinder with a shell with fixed ends decreases with m until it reaches fixed end cap, in which the stretching energy be- a minimum, then increases with m.24) In larger G2 comes substantial for small m.12) Thus the total and G3 handbells, the minima occur at about m= n+3. Vibrational modes of a handbell are arranged in a periodic table in Fig. 11. Once again the numbers at lower left give (m, n), the numbers of complete nodal meridians and nodal circles. Unlike those of a modern church bell, their order in frequency depends upon the size and shape of the handbell. The (2, 0) and (3, 0) modes are always the modes of lowest frequency. The next mode may be the (3, 1), (4, 0) or (4, 1) mode, however, depending upon the size of the handbell; the (2, 1) mode occurs at a considerably higher frequency.25) The modes of vibration calculated in a C5 handbell using finite element methods were found to be in good agree- ment with the modes observed by hologram inter- ferometry.23) Although they are cast from the same bronze material and cover roughly the same range of pitch, the sounds of church bells, carillon bells, and hand- bells have distinctly different timbres. In a handbell, only two modes of vibration are tuned (although Fig. 10 Vibrational frequencies of a C5 there are three harmonic partials in the sound), handbell.23) whereas in a church bell or carillon bell at least five

Fig. 11 Periodic table of vibrational modes in a handbell. Below each drawing are the rela- tive modal frequencies in a Malmark C5 handbell. At lower left (m, n) gives the number of nodal meridians 2m nodal circles n.13)

247 J. Acoust. Soc. Jpn. (E) 10, 5 (1989) modes are tuned harmonically. A church bell or carillon bell is struck by a heavy metal clapper in 6. ANCIENT CHINESE TWO-TONE BELLS order to radiate a sound that can be heard at a great Much interest has developed in the acoustical distance, whereas the gentle sound of a handbell behavior of ancient Chinese bells whose almond- requires a relatively soft clapper. In the so-called English tuning of handbells, followed by most handbell makers in England and the USA, the (3, 0) mode is tuned to three times the frequency of the (2, 0) mode. The fundamental (2, 0) mode radiates a rather strong second harmonic par- tial, however, so that the sound spectrum has promi- nent partials at the first three harmonics.25) Some Dutch founders aim at tuning the (3, 0) mode in handbells to 2.4 times the frequency of the funda- mental, giving their handbell sound a minor-third character somewhat like a church bell. Such bells are usually thicker and heavier than bells with the English-type tuning.26) A handbell, unlike a church bell, appears to sound its fundamental pitch almost from the very onset of sound. There are several reasons for the absence of a separate strike note. First of all, there is no group of harmonic partials to create a strong subjective tone. Secondly, handbells employ a relatively soft non-metallic clapper, so that there is no sound of metal on metal, and the partials develop a little more Fig. 12 Modal frequencies in a Chinese two- slowly after the clapper strikes the bell. tone bell. Note that most modes occur in pairs.28)

Fig. 13 First six modes in: (a) Chinese two-tone bell; (b) church bell; (c) handbell. Dashed lines indicate locations of nodes.28)

248 T. D. ROSSING: ACOUSTICS OF EASTERN AND WESTERN BELLS

Table 2 Intervals between sui and gu tones.28,29)

shaped cross section consists of two segments of a bells of circular cross section. Larger and larger circle joined at the xian or spine. These bells emit bells were cast, often with a scalloped rim and tones with two different pitches when struck at the elaborate inscriptions. Casting of large temple bells sui and gu strike points. Two different families of reached its zenith during the Ming dynasty (1368 N vibrational modes are excited by striking the bell at 1620 A.D.). The largest bell in China was cast during these two points. the reign of the emperor Yongle (1403-1424). Its Holographic studies of similar bells at the Shang- bronze body, 4.5 m high and with a maximum di- hai Museum27) and at Northern Illinois University28) ameter of 3.3 m, is inscribed with 227,000 characters. showed that the vibrational modes tend to occur in Its is estimated to be 52 tonnes. The frequency pairs, one with a node at the spine or xian, and one of the fundamental (2, 0) mode is 22 Hz, but promi- with an antinode at that location. The mode with a nent partials in its sound spectrum at 129, 164, 218 node at the spine generally has the higher frequency. Hz, which are near C3, E3, and A3, give the bell a Modal frequencies in a copy of a bell from the Zhou period (1027-256 B.C.) are shown in Fig. 12. Note the similarities, and also the differences, between the mode families in Fig. 12 and those of a church bell (Fig. 7) and a handbell (Fig. 10). Figure 13 compares the first six modes of vibra- tion of the Chinese bell with the first six modes in a church bell and a small handbell. Note that the (2, 0)a, (3, 0)b, and (4, 0)a modes would be efficiently excited by a blow at the sui strike point (S), but the (2, 0)b, (3, 0)a, and (3, 1)a modes would not. A blow at the gu strike point (G), on the other hand, would excite the (2, 0)b, (3, 0)a, (3, 0)b, and (3, 1)a modes quite strongly. Intervals between the sui and gu fundamental pitches (determined by the (2, 0)a and (2, 0)b mode frequencies) in 88 two-tone bells studied by Ma Chengyuan27) and the 64 Zeng bells29) are compared in Table 2.

7. CHINESE TEMPLE BELLS Fig. 14 Mode frequencies of the Yongle temple bell as a function of the number of After Buddhism came to China, the casting of nodal meridians m and nodal circles n two-tone bells gradually gave way to large temple (data from Ref. 30)).

249 J. Acoust. Soc. Jpn. (E)10, 5 (1989)

rather musical sound.30) The sound pressure has been measured to be 92 dB some 25 m away. The frequencies of ten modes of vibration are shown in Fig. 14.

8. JAPANESE TEMPLE BELLS Acoustical measurements on a Japanese temple bell 30.3 cm in diameter and 42.5 cm high were reported by Obata and Tesima.31' It was found to have modal frequencies of 247, 624, 850, 1,100, 1,380, 1,640 and 2,050 Hz, which are nearly in the ratios 2:5:7:9:11:13:16. The lowest three modes have m=2, 3 and 4; that is, they have 4, 6 and 8 nodal meridians distributed around the circum- ference. The investigators report a subjective strike note corresponding to a frequency of about 260 Hz, which is the difference between the prominent 1,380 Hz and 1,640 Hz tones. However, based on the Fig. 15 Profiles of major-third bells (dashed curves) compared to a minor-third bell.4) experience with church bells, one might be more likely to expect a strike tone around 275 Hz (the "missing fundamental" of 1 ,100, 1,380 and 1,640 Hz) or possibly even 124 Hz (the nearly-common factor in all seven frequencies above).

9. MAJOR-THIRD BELLS A new type of carillon bell has been developed at the Royal Eijsbouts Bellfoundry in The . The new bell replaces the dominating minor-third partial with a major-third partial, thus changing the tonal character of the bell sound from minor to major. This requires an entirely new bell profile. The idea of a bell with a major third partial is not new. For years some carillonneurs have felt that a composition in a major key, especially if it includes chords with many notes, might sound better if played on bells with a major character. Some authorities have even suggested a "neutral" third, lying between a major and minor third. Previous efforts to fabricate such bells have not been successful, however, since changing the bell profile to raise the third partial Fig. 16 Relative frequencies of the first eight invariably changes the other harmonic partials as partials in two major-third bells compared well. to those in a traditional minor-third bell. The new bell design evolved partly from the use Note that the hum, prime, fifth, and octave of a technique for structural optimization using finite partials remain the same (adapted from Ref. 4)). element methods on a digital computer at the Tech- nical University in Eindhoven.32) This technique allows a designer to make changes in the profile of an Based on results of the structural optimization existing structure, and then to compute the resulting procedure, Andre Lehr and his colleagues have changes in the vibrational modes. designed two entirely different bells, both having a

250 T. D. ROSSING: ACOUSTICS OF EASTERN AND WESTERN BELLS

Table 3 Note names and relative frequencies of partials in major- and minor-third bells.4)

Chichester, 1988). 7) T. D. Rossing, Acoustics of Bells (Van Nostrand Reinhold, New York, 1984). 8) F. G. Tyzzer, "Characteristics of bell vibrations," J. Franklin Inst. 210, 55-56 (1930) (Reprinted in Ref. 7)). 9) T. D. Rossing, "Acoustics of bells," Am. Sci. 72, 440-447 (1984). 10) R. Perrin, T. Charnley, and J. de Pont, "Normal modes of the modern English church bell," J. Sound Vib. 90, 29-49 (1983) (Reprinted in Ref. 7)). 11) A. Lehr, "Hedendaages Nederlandse klokkengiet- kunst," Publ. No. 7, Neth. Acoust. Soc., 20-49 (1965) (English translation in Ref. 7)). 12) Lord Rayleigh (J. W. Strutt), The Theory of Sound, Fig. 17 Two new major-third bells (left) Vol. 1 (Macmillan, London, 1894) (Reprinted by compared to conventional (minor-third) Dover, New York, 1945). bells with the same pitches. 13) T. D. Rossing and R. Perrin, "Vibration of bells," Appl. Acoust. 20, 41-70 (1987). 14) T. D. Rossing, "Chladni's law for vibrating plates," major-third partial, as shown in Fig. 15. The first Am. J. Phys. 50, 271-274 (1982). bell, introduced in 1985, has more rapidly decaying 15) R. Perrin, T. Charnley, H. Banu, and T. D. Rossing, "Chladni's law and the modern English church bell partials, whereas the second bell has a longer decay, ," more nearly that of the traditional minor-third bell.4) J. Sound Vib. 102, 11-19 (1985). 16) A. Lehr, "Partial groups in the bell sound," J. Frequencies of the first eight partials in the two Acoust. Soc. Am. 79, 2000-2011 (1986) (See Table major-third bells are compared to those of a minor- V). third bell in Fig. 16. Notice the large differences in 17) F. H. Slaymaker, "Chords from tones having partials 7 and 8, both of which belong to group III stretched partials," J. Acoust. Soc. Am. 47, 1569- (III-2 and III-3, respectively). Note names and 1571 (1970). relative mode frequencies are given in Table 3. 18) E. Terhardt and M. Seewann, "Auditive and objec- tive Bestimmung der Schlagtonhohe von historischen Kirchenglocken," Acustica 54, 129-144 (1984). REFERENCES 19) J. H. Eggen, The Strike Note of Bells (Inst. for Per- 1) P. Price, Bells and Man (Oxford Univ. Press, Oxford, ception Research, Eindhoven, 1986). 1983). 20) J. F. Schouten and J. 't Hart, "De slagtoon van 2) A. Lehr, Van Paardebel tot Speelklok (Europese klokken," Publ. No. 7, Neth. Acoust. Soc., 8-19 Bibliotheck, Zaltbammel, 1981). (1965) (English translation in Ref. 7)). 3) A. Lehr, Klokken en Klokkenspelen in het Oude 21) R. J. Ritsma, "Frequencies dominant in the percep- China (Athanasius Kircher-Stichting, Asten, 1985). tion of pitch," J. Acoust. Soc. Am. 42, 191-198(1967). 4) A. Lehr, The Designingof SwingingBells and Carillon 22) M. J. Milsom, "Tuning of bells," Ringing World, Bells in the Past and Present (Athanasius Kircher September 3, 1982, 733. Foundation, Asten, 1987). 23) T. D. Rossing, R. Perrin, H. J. Sathoff, R. W. 5) N. Spear, Jr., A Treasury of Archaeological Bells Peterson, and A. Lehr, "Vibrational modes of a (Hastings House, New York, 1978). tuned handbell," J. Acoust. Soc. Am. 76, 1263-1267 6) G. Elphick, The Craft of the Bellfounder(Phillimore, (1984).

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24) R. N. Arnold and G. B. Warburton, "Flexural and H. J. Sathoff, "Vibrational modes of Chinese vibrations of the walls of thin cylindrical shells hav- two-tone bells," J. Acoust. Soc. Am. 83, 369-373 ing freely supported ends," Proc. R. Soc. London (1988). Ser. A 197, 238-256 (1949). 29) A. Lehr, "The tuning of the bells of Marquis Yi," 25) T. D. Rossing and 1-1.J. Sathoff, "Modes of vibra- Acustica 67, 144-148 (1988). tion and sound radiation from tuned handbells," J. 30) C. Tong and Z. Darui, "Acoustical properties of the Acoust. Soc. Am. 68, 1600-1607 (1980). Yongle bell," Chin. J. Acoust. 5, 375-381 (1986). 26) T. D. Rossing, "Acoustics of tuned handbells," 31) J. Obata and T. Tesima, "Experimental investiga- Overtones 27 (1), 4-10, 27 (1981). tions on the sound and vibration of a Japanese 27) M. Chengyuan, "Ancient Chinese two-pitch bronze hanging-bell," Jpn. J. Phys. 9, 49-73 (1933-34). bells," Chin. Music 3, 81-86 (1980); 4, 18-19, 31-36 32) B. Schoofs, F. van Asperen, P. Maas, and A. Lehr, (1981). "Computation of bell profiles using structural 28) T. D. Rossing, D. S. Hampton, B. E. Richardson, optimization," Music Percept. 4, 245-254 (1987).

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