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

Size-frequency relation and tonal system in a set of ancient Chinese bells: Piao-shi bianzhong

Junji Takahashi MusicResearch Institute, Osaka College of Music, Meishin-guchi1-4-1, Toyonaka, Osaka, 561 Japan (Received5 April 1989)

Resonancefrequencies and tonalsystem are investigatedon a setof ancientChinese bells named"Piao-shi bianzhong." According to the measurementof 12 bellspreserved in Kyoto,it is foundthat a simplerelation between frequency and sizeholds well. The relationtells frequency of a bellinversely proportional to squareof its linearmeasure. It is reasonableto concludethat the set wascast for a heptatonicscale which is very similarto F# majorin our days. Precedingstudies in historyand archaeologyon Piao- shi bellsare also shortlyreviewed. PACSnumber: 43. 75.Kk

•Ò•à) in Kyoto. In ƒÌ ƒÌ 2 and 3, historical and 1. INTRODUCTION archaeological studies on the set will be reviewed.

In China from 1100 B.C. or older times (Zhou In ƒÌ 6, musical intervals of the set will be examined and it will be discussed for what tonal system the period), bronze bells named "zhong" (•à) of special shape with almond-like cross section and downward bells were cast. mouth had been cast and played in ritual orchestra. 2. SHORT REVIEW ON After the excavation of a set of 64 zhong bells in 1978 PIAO—SHI BIANZHONG from the tomb or Marquis Yi of Zeng (˜ðŒò‰³),

vigorous investigations have been concentrated upon In about 1928, many bronze wares including some

zhong bells from many fields of researches as ar- zhong bells were excavated from tombs of Jincun

chaeology, history, musicology, and, acoustics. ) near Luoyang (—Œ—z). Through unknown (‹à‘º pro-

In general several or more zhong bells were cast and cesses, 2 bells of them were brought to Canada and now preserved at the Royal Ontario Museum. Other played in a set, hence called bian-zhong (•Ò•à), and served to give some standard pitches to a tonal sys- 12 bells were brought to Japan and added to the

tem in those times. Sumitomo Collection and now preserved at SEN-

A zhong bell produces two tones. When struck OKU HAKKO KAN museum in Kyoto. These

at the center of front face, it emits a lower tone, 14 bells are confirmed to have been cast as one set

which is often refered as "suiyin" (詉¹) and will be by their styles and inscribed patterns.

called in this paper as a-tone. When struck at the By epigraphic interpretation and reference to the history of Warring State period (475-221 B.C.), position about halway from the center to a lateral spine called "xian" (‘L), it emits an upper tone, often the established conclusion shows that the bianzhong

refered as "guyin" (ŒÛ‰¹) and called b-tone.1) was casted for a memory of a victory of a feudal

In this paper a simple empirical formula between load Piaoqiang (éŠã³) in the battle of 404 B.C.,

size and resonance frequency in a set of zhong bells therefore named as Piao-shi bianzhong.2)

will be proposed, according to the measurement on 12 bells in Kyoto are numbered as 1 to 12 accord-

a set of 12 bells named "Piao-shi bianzhong" (銎• ing to their sizes from the largest to the smallest.

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

According to its size the large bell in Canada (called

as ROM1 in this paper) is to be placed between

No. 3 and 4 of the Kyoto bells, and the small Cana-

dian bell (called as ROM2) between No. 6 and 7

of Kyoto's. At first such numbering was mere

convenience of the musium, but it is found the num-

bering is same as that intended by the original bell

maker; for numeral letters inscribed in mirror image

on the inside of the top plates of the bells are dis-

covered," which seem to have been for craftsmen's

convenience. On the Nos. 1, 2, 3, 4, 5, 6, 7, and 8

bells, numbers 1, 2, 3, 5, 6, 7, 9, and 10 are inscribed Fig. 1 Names of parts of a zhong bell. respectively. (left) front view, (right) side view.

3. PROPORTIONALITY IN SIZE

Names of the parts of a zhong bell are given in Size of many parts of Piao-shi bianzhong was Fig. 1. The length of xian is called "xianchang" measured by Okamura3) and Dohrenwend,4) the

(‘L’·). The length of a line passing across the center value are listed in Table 1.

of the front face from the bottom to the top is called All the Piao-shi bells are made in a similar shape. "zhengchang" (•Þ’·) . The interval between two tips Okamura indicated that the historically significant

of the xians is called "xianjian" (‘LŠÔ) The xianjian, ratioes of parts are very near to a constant value

called as "tip-to-tip interval" hereafter in this paper, 0.8. That is, the ratio of xianjian to xianchang falls

is used as a representative of a bell size, because it in 0.81•}0.02 for all the 14 bells. The ratio of gujian

can be measured accurately. The interval between to xianjian is in 0.80•}0.02, and that of wuguang to

the two mid-points of the bottom arch is called wuxiu in 0.78•}0.02 (1 exception). Considering the "gujian" (ŒÛŠÔ) . The longest distance on the top errors in measurements caused by a little rust on the plate, which connects the two tops of the xians, is surface, it is reasonable to conclude that the bell- called "wuxiu" (•‘ãù). The shortest distance on the maker intended to shape the bells exactly under a top plate, which acrosses the longest distance at certain quantitative rule. right angle, is called "wuguang" (•‘œA). The above conclusion is also confirmed by the

Table 1 Measures of fundamental parts of Piao-shi bianzhong.

300 J. TAKAHASHI: SIZE-FREQUENCY RELATION IN CHINESE BELLS

classical text. A standard proportion of parts of a target partial wave is extracted by a digital band- zhong bell was written in "Kao-gong-ji' (•l•H‹L), pass-filter and lastly the period of the wave is in the official documents "Zhou-li"(ŽüâX) of Zhou evaluated. dynasty. In a section beginning with "Fu-shi wei A small ascending tendency of frequency is zhong" (é莕ਕà, Family Fu makes zhong bells), observed in several hundred milliseconds after we can read as follows; reached to a stable state, such a tendency is quite 十分其銑,去 二以爲鉦,以 其鉦爲之銑間。 common to pitched percussion instruments. For Divide the xian (chang) into 10 equal parts, precise determination, it is reasonable to define its and remove 2 parts, then let it be zheng (chang). frequency when its vibration amplitude goes infin- Let this zheng (chang) be (equal to) xianjian of itesimal, which is independent of the initial am- the bell. plitude.6) 去 二分以爲之鼓 間。 Dependence of the frequency on temperature is Remove 2 parts, then let it be guijian of the bell. also observed. If the temperature goes 10•Ž higher, 以其鼓間爲之舞脩。 the frequency goes about 5 to 15 cents lower. Of Let this gujian be (equal to) wuxiu of the bell. course frequencies of the bells of the same set must 去二分以爲舞廣。 Remove 2 parts, then let it be wuguang. Table 2 Frequencies and relative pitches of Commonly the above text has been interpreted Piao_shi bianzhong(at 30•Ž) as; the ratioes xianchang: zhengchang: xianjian : gujian: wuxiu: wuguang =10:8:8:6:6:4 However Okamura3) proposed an alternative inter- pretation as; xianchang: zhengchang: xianjian =10:8:8 xianjian: gujian=10:8=8:6.4 wuxiu: wuguang=10:8=6.4:5.12 Measured values for Piao-shi bianzhong, as above, coincide very well with the new interpretation. But as for the thickness the description in the next part of the text "Kao-gong-ji," where one reads pro- portion of thickness to size, does not coincide with Piao-shi bianzhong.

4. MEASUREMENT OF FREQUENCIES The musical pitches of a- and b-tones of a zhong bell are determined by resonance frequencies in (2, 0)a and (2, 0)b modes vibration. (The name of vibration mode is same as in the analysis by Rossing et a1.5)) Though we know little about the playing technique of zhong bells, it is not difficult for us to produce two tones separately as a single note. Resonance frequencies are measured by the fol- lowing method. The bell is struck by a wooden mallet, and produced sound is caught by a microphon sustained near a side of the bell opposite to the struck position, and recorded in a tape recorder through pulse code modulation. Then the waveform of the sound is put into a micro-computer and a

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

be compared under the same temperature con-

dition. The frequencies and musical pitches rel-

ative to the lowest tone of the 12 bells in Kyoto , put into the same temperature as 30•Ž, are listed in Table 2.

5. SIZE-FREQUENCY RELATION7)

Based on the proportionality in many parts of

Piao-shi bianzhong, we may expect a certain relation-

ship between size and frequency.

Here let us start from an analogy to the bending

vibration frequency of a thin hemispherical shell,

whose fundamental frequency is directly propor-

tional to the thickness of a shell and inversely pro-

portional to the squared diameter of the sphere.8) The zhong bell has much more complicated shape

than a hemispherical shell, and the zhong's wall is not so thin. But it is not absurd to assume at the Fig. 2 Size-frequency relation in Piao-shi first step of approximation that the frequency of bianzhong. Two curves represent the em-

a- or b- tone of each bell is proportional to its thick- pirical for mula, and a dot denotes

ness and inversely to its tip-to-tip interval squared. measured value for an individual bell.

Hence this simple relation is expressed as;

(1) strictly designated by a musical rule. where f is for frequency and h is for thickness and l denotes the tip-to-tip interval. The coefficient c, 6. TUNING AND TONAL SYSTEM which reflects the elastic properties of the material The size-frequency relation suggests an accurate and factors arising from the asymmetrical shape of technology of planning and casting, but they had a zhong bell, is assumed different for a- and b-tone. another technique, that is, tuning. If the tip-to-tip The thickness of the zhong bell is not uniform and interval be changed in 2 mm, for example, the pitch very difficult to measure owing to the rust clinging of the tone should be changed in 40 cents for the to the inside wall. Because the mass of a bell is largest bell or in 100 cents for the smallest. Such a nearly proportional not to the cubic but to the square big pitch deviation should not be allowed by the of 1, the thickness seems to be constant. original purpose of the set bell. Then the expression (1) is reduced to a simpler To adjust the pitches of the bells on some pre- form as ; indicated notes lastly, individual bell had to be (2) tuned by ear. Infact, at the 8 positions on the inside where c' includes the factor of thickness. wall near the mouth one can observe a cannal cut In Fig. 2l to f relation is given. Two curves rep- for tuning, but the tuning method is fully unknown resent the expression (2) with the coefficients c', at the present stage. determined by the least square method, as; It is an interesting subject to make clear an ancient c'=113 for a-tones, tonal scale in China by means of the measured frequencies. In order to attain the purpose, however, c'=137 for b-tones, we encounter a difficult question how the pitch have where f is measured in kHz and l in cm. The dot changed for long ages especially by the rusting in the with a horizontal short bar in the figure denotes the soil. But we cannot say anything unless we suppose measured value with the measurement error of l. that the pitches have not been changed to such an It is probable that the ancient bell makers had extent as entirely impossible to estimate the original used a certain formula like this to plan the individ- pitches if no special sign of damage is observed. ual sizes of the bells, whose pitches should had been The second question is whether the original set of

302 J. TAKAHASHI: SIZE-FREQUENCY RELATION IN CHINESE BELLS

bells had been tuned well or not, because all the sets (let the lowest be 0 cent), 200, 400, 500, 700, 900, had not been tuned well. It is reasonable to think 1,100 cents, but no tone appears at 100, 300, 600, 800 that only the set which had been actually performed cents. This agrees with a well-known Chinese hep- was tuned well and it is worth detailed investigation. tatonic scale "qisheng" (Žµãß), and we can conclude

The octave relationship and the musical interval Piao-shi bianzhong was cast for a certain scale, almost between the a- and b-tones serve for us to check equivalent to F# major in modern music. preserved condition and to trace tuned state of the ACKNOWLEDGEMENTS original set.

The octave relationship in Piao-shi set seems to The author wish to express his special thanks to holds well, as the interval between No. 1 and No. 6, SEN-OKU HAKKO KAN museum for the active

No. 3 and No. 7, and, No. 7 and No. 11 are in support to measure its invaluable bells, and to

1,200•}10 cents for both a- and b-tones. Less correct archeologist Hidenori Okamura for his kindness to octave relationship is found between No. 4 and No. 8, read ƒÌ3, which is mainly owing to his discoveries.

No. 5 and No. 9, and, No. 8 and No. 12. For these REFERENCES bells at least either a- or b-tone seems to be changed 1) Detailed holographic investigations on the vibration in 30 cents or so. Between No. 2 and No. 10 there mode corresponding to each tone are given in; Ma are 2 octaves for both a- and b-tones, and another bell Chengyuan, "Two-pitch bronze bells of the Shang- whose pitches come to the midpoint between them Zhou period," KAOGU-XUEBAO No. 1, 131-146 is expected. The ROM2-bell looks very likely to be (1981) (in Chinese). 2) Y. Sahara, "A set of Piao-shi Zhong," Bull. SEN- put here. We have no acoustical information where OKU HAKKO KAN 1, 65-93 (1984) (in Japanese). pitches of the ROM1-bell should be placed, but it is 3) H. Okamura, "Plan and structure of Chime bells- suggested by some music theoretical considerations Archaeological research of Piao-shi Bian Zhong," that the a-tone of the ROM 1 comes near to the b-tone Bull. SEN-OKU HAKKO KAN 3, 33-49 (1986) of No. 3 and the b-tone of the ROM1 near to the (in Japanese). a-tone of No. 5.9) 4) D. Dohrenwend, In a letter from the Royal Ontario

For No. 1 and No. 6 bells, musical interval be- Musium to SEN-OKU HAKKO KAN.

tween the a- and b-tones is 400 and 405 cents, which 5) T. D. Rossing, D. S. Hampton, B. E. Richardson, H. J. Sathoff, and A. Lehr, "Vibrational modes of is good major third. For Nos. 2, 3, 4, 5, 7, 9, 11, Chinese two-tone bells," J. Acoust. Soc. Am. 83, and, 12, the interval is 304, 315, 313, 293, 318, 311, 369-373 (1988). 313, and, 300 cents respectively, which we can 6) For detailed discussion see: J. Takahashi, "Pitch regard as minor third. But for No. 8 and 10, the determination of Zhe-diao-Zhong," Bull. SEN- interval is 351 cents both, which we cannot judge OKU HAKKO KAN 5, 42-44 (1988) (in Japanese).

whether major or minor third, but the latter is pref- 7) J. Takahashi, "Relation between pitch and size of Piao-shi Zhong," Bull. SEN-OKU HAKKO KAN erable to the music theory.9) 3, 50-59 (1986) (in Japanese). When we bring all the 24 measured tones into a 8) J. W. S. Rayleigh, The Theory of Sound (Macmillan musical scale by octave shift and assign to the nearest and Co., Ltd., London, 1894), p. 417 ff. among the 12 notes given by the equally tempered 9) T. Asahara, "Musical theory on ancient Chinese scale (a similar tonal system was established in bells," THE TOHO GAKUHO (J. Orient. Stud.)

China of that time), we find tones at the positions 0 No. 59, 63-123 (1987) (in Japanese).

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