Chinese cyclic tunings in late antiquity

@ university or ILLINOIS PRESS Me. MAN, Gr Society for Ethnomusicology Chinese Cyclic Tunings in Late Antiquity Author(s): Ernest G. McClain and Ming Shui Hung Source: Ethnomusicology, Vol. 23, No. 2 (May, 1979), pp. 205-224 Published by: University of Illinois Press on behalf of Society for Ethnomusicology Stable URL: http://www jstor.org/stable/85 1462 Accessed: 24/10/2010 04:43 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www .jstor.org/action/showPublisher?publisherCodesillinois. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@ jstor.org. Society for Ethnomusicology and University of Illinois Press are collaborating with JSTOR to digitize, preserve and extend access to Ethnomusicology. http://www jstor.org

Society for Ethnomusicology Author(s): Ernest G. McClain and Ming Shui Hung Source: Ethnomusicology, Vol. 23, No. 2 (May, 1979), pp. 205-224 Published by: University of Illinois Press on behalf of Society for Ethnomusicology Stable URL: http://www.jstor.org/stable/851462 Accessed: 24/10/2010 04:43 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=illinois. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Society for Ethnomusicology and University of Illinois Press are collaborating with JSTOR to digitize, preserve and extend access to Ethnomusicology. http://www.jstor.org

Ernest G. McClain with translations by Ming Shui Hung B.C. and the fifth the first A.D. certain century century Between of Chinese tuning theory were more than a thousandyears aspects ahead of similardevelopmentsin the West. The cycle of fifthswas studied to the limit of 60 and then 360 tones, and arithmeticapproximationsto equal temperament were developed. The simplicity and accuracy of Chinese methods, not fully appreciatedin the East, have been hidden from the West by the languagebarrier. Here is a complete translationof the 60-divisiontuningof ChingFang (-78 to -37) togetherwith an analysisof his procedure.Fromhis model, and in accordance with the instructionsthat survived, I reconstructthe 360-division tuning of Ch'ien Lo-chih (fl. +415 to +455), showing how simply it could have been achieved, and how it may be viewed as the natural culminationof the process ChingFang set in motion. I then suggest how the emerginginterest in temperament,accessible by a formula derived from ChingFang, may have helpedto motivateCh'ien'sextended study of the tone field. In the work of Chinese acousticianswe witness a remarkablebalance between theory and practice, between the pursuitof absolute perfection on the one hand, and the acceptance of commonsense simplificationon the other. HISTORICALBACKGROUND Throughouthistory the Chinese scale has remainedbasically pentatonic, on the model C D E G A, with its modal variations, tuned as a sequence of falling fourths and rising fifths. The first 12 tones of the extended series gave their names to the months. In some historicalperiods various auxiliarytones were also recognized,andritualtranspositions coordinated the scale with the calendar (Nakaseko 1957:147-172).The sequence of fifths 2:3 and fourths3:4 was sometimesextendedto as many as 5 x 12 = 60 tones, or 64 tones (providingfor 12 pentatonictransposi? 1979Society for Ethnomusicology 0014-1836/79/2302-0205$0.60

tions), or 84 tones (12 diatonic transpositions),and eventually, theoretically, to 360 tones. Successive dynasties establishedtheir own pitch standardsand freely revised or discardedthe calendricalritual,yet maintained a strongly conservative bias. Successive tones were generatedby the perfect fifths and fourths of "Pythagorean"tuning, and new pitch standards varied over the range of a musical sixth (Chuang1963:54-55). Ritualtranspositioncoupledwith a fondnessfor hugeorchestrasof winds, strings, and tuned bells gave Chinese acousticaltheory the same impetus toward a 12-tone simplificationthat chromaticmodulationsand triadic harmonies of the 17th century fostered in the West. Chinese theorists were far ahead of us in grapplingwith the problemsof temperament,but Chinese conservatism consistently doomed every innovation, however brilliant, that threatenedthe purity of perfect fourths and fifths. The oldest survivingChinese tuning,that by Kuan-tzufor the first 5 tones (recordedca. 3rd c. B.C.), establishesthe patternfollowed by later tunings. He alternatelyadds and subtracts? of successive tone-values (pipe and/or stringlengths)to producethe fallingfourthsand risingfifths of Pythagoreantuning (Nakaseko 1957:147-150): schema numbers Chinese "solmization" C 81 kung G 108 chih D 72 shang A 96 yii E 64 chiao In the Confucian ritual tradition,the tone now known as Huang-chung and translatedas "Yellow Bell" was tonic (kung)at the wintersolstice in the 1lth month: When Huang-chungis assigned the numericalvalue of 81, conceived as a pitch-pipelength of 8.1 inches (Chinese subdivisions normally being decimal), or as 9 inches divided into 9ths, as duringthe Chou dynasty, only 5 tones can be designatedaccuratelyby successive additions and subtractionsof ? withoutthe furtheruse of fractions.The Kuan-tzu tuning is thus in "smallest integers." Odd-numberedtones in the tuning sequence are yang (male) and even-numberedtones are yin (female) irrespective of whethergenerationbegins "downward,"as this one does, or "upward," as later tuningsdo. The 3 yang tones in this first set (C D E) and the 2 yin tones (G A) show a double implicationfor the "male" prime number3 and "female" primenumber2, which generate the tuning. The complete cycle of 12tones (lii) is firstgiven in the Huai Nan Tzu, the book composed for Liu An, Prince of Huai Nan, ca. - 122, both in 6-digit precise values and in 2-digitapproximationswhose derivationremains controversial(see Table I). The accuratedesignationfor the cycle of 12 tones must obviously begin on 311 = 177,147 to avoid fractions.

TABLE 1 PitchNumbersin the HuaiNan Tzu 1. 177,147= 81 C IIth month -? = 118,098 = 54 +? = 157,464 = 72 = 104,976 = 48 Huang-chung 2. 3. 4. G D A Lin-chung T'ai-ts'ou Nan-lii 6th month Ist month 8th month +1/3= 139,968= 64 -? = 93,312 43* +1/3= 124,413m57 +1/3= 165,888m76 - ? = 110,592* 51 E B F1 C; G Ku-hsi Ying-chung Jui-pin Ta-lii I-tse 3rdmonth 10thmonth 5thmonth 12thmonth 7thmonth D4 A; E; Chia-chung 2nd month -? = 98,304 45 +V = 131,072m60 5. 6. 7. 8. 9. 10. 11. 12. +V? = 147,456 m68 Wu-yi Chung-li 9thmonth 4thmonth *The originalHuai Nan Tzu value of 42 was correctedto 43 by PrinceChuTsai-ytiin the + 16th c. Ching Fang will latergeneratehis 60 tones fromthis same base, his last 48 necessarily being approximations,for he also avoids fractionsin his basic calculations. From our equal temperamentperspective, this tuning by perfect fourths and fifths producesa spiralratherthana cycle, for thereis a cumulative excess of 2 cents per interval.Chung-liiE$, the 12thtone in the sequence, is thus 22 cents sharp,and cannotlead to a cyclic returnto C. The mean deviationfor the set of 12tones, shown below in TableV, is almost 12 cents. A remarkablefeature of the accompanying2-digit approximations, when 42 is correctedto 43 as PrinceChuTsai-yiisuggests, is that their mean deviation from equal temperamentis only 7 cents (Kuttner 1975:176).The familiarPythagoreancomma of 531441:524288 between the 13thtone of such a series andthe originalreferencetone here acquires a Chinese simplificationto 81:80 (notice that 60 plus ? of itself gives 80), a value familiarin the West in quite a differenttonal context as the "syntonic comma" (approximately22 cents instead of 24). Two successive additionsfor Jui-pin F; and Ta-liiC0 compressoperationswithin an octave. Among the 2-digitnumbers,the risingfifths E-B, C;-GO,and D#-A# and the falling fourth B-F# are no longer "perfect" ratios of 3:2 and 4:3 respectively. Notice that when dividedby 3, the numbers64, 43, and 76 give remaindersof ?, which are apparentlydropped, while 68 gives a remainderof 2/3, which is roundedoff to the full unitso that2/ x 68 = 45. Whatpurpose these 2-digitnumbersserved is not clear, and thereis no agreement as to how they were achieved. They prove, however, that for at least a century before ChingFang the Chinesewere accustomedto working both with precise values and with approximations.His achievement can be understood, then, as a virtuoso variation on an ancient theme.

CHING FANG The 60-division tuning of Ching Fang (-78 to -37) and his own extensive commentary on it were recorded by Ssu-ma Piao three centuries later in the Hou Han Shu, the official history of the late Han dynasty (+25 to +219). Ssu-maPiao informsus that an imperialBureauof Music had been establishedby EmperorWu (-140 to -88), who "wanted to correct the music." Ching Fang received his appointmentto the Bureau from EmperorYiian (-48 to -33), after an examinationby Hsuang Cheng, "lecturer to the heir-apparent,and a certainofficial Chang, who was a censor" (Fan 1965:3000).The Early Han dynastichistorycontains a biographicalentry for Ching Fang written by Pan Ku, from whom we learn that Ching Fang was also an expert at divination, that from the hexagrams of the I Ching he could predict the weather-"always very accurately." His originalname, Li Fang, he changedalso accordingto a rule of divination. At the age of 41 he was executed by order of the emperor who had appointedhim-his headwas choppedoff in the market place-for "accusations against a high official" (Pan 1962:3160-67). Ching Fang has an honoredplace in the history of acousticaltheory as the first man to compute the greatly reducedcomma between the Ist and 54th tones in the spiral of fifths (Barbour1953:124).The simplicity and accuracy of his approximationtechniquecommandsattentionhere. His tuning is given in full in Table II, in scale order;moderntone names and the tuning order, or generationorder, have been addedto clarifyhis procedure. We do not translatehis tone names;beyondthe first 12in the tuning order, associated with the months, names appear to be merely convenient labels for the operations. That the whole constructionhas survived in nearly perfect condition is a tributeto the qualityof Chinese scholarship and to the care Ching Fang took to express results in three complementary ways: 6-digit shih ("substance") numbers are accompanied by 3-digit lii (pitch-pipe)lengths, includingsymbols for fractional remainders, and by 2-digit chun (long zither) string lengths, with exact fractions that serve as a cross-check against copyists' errorsin the shih and Iii numbers (Fan 1965:3003-14). Ching Fang's calculationis first carriedout in 6 digits, as an extension of the earlier series in the Huai Nan Tzu--correctedto keep the approximationsin tones 13 to 60 within a fractionof a cent of their true values in the spiral of fifths. These 6-digitshih numbersare then divided by 39 = 19,683to give pipe lengthsfor the 1ii(whose Huang-chungfundamental is 9 inches) and stringlengthsfor the chun(whose fundamentalis 9 feet), all subdivisions being decimal. The 3-digitlii numbersare practical for both pipes and strings. They are accurate to within 2 to 4 cents,

TABLEII The 60-DivisionTuningof ChingFang Scale order Name Shih (number) Lu (pipe) 1 Huang-chung 177,147 9 inches 2 54 Se-yi 176,7768.98+ 8.9+15,973/x* 3 13 Chih-shih 174,762 8.87+++ 8.8+15,516/x 4 25 * , Ping-sheng 172,410 8.76- 8.7+11,679/x 5 37 Fen-tung 170,089 8.64++ 8.6+8,152/x 6 49 j Chih-mo 167,800 8.52?++ 8.5+4,945/x 8 k Ta-1 165,888 8.43-- 8.4+5,508/x Fen-fou 163,654 8.31++ 8.3+2,851/x 161,452 8.21-- 8.2+514/x 1 7 Tuning order C C# Chun (string) 9 feet 8 20 9 32 10 44 Shao-ch'u 159,280 8.09++ 8+18,160/x 3 T'ai-ts'ou 157,464 8 8 11 D WMjlLing-yin 12 56 0 Wei-chih 157,134 7.98++ 7.9+16,383/x 13 15 Shih-hsi 155,344 7.89(+) 7.8+18,166/x 14 27 Ch'ii-ch'i 153,253 7.79-- 7.7+16,939/x 15 39 Sui-ch'i 151,190 7.68++ 7.6+15,992/x 16 51 Hsing-chin 149,156 7.58-- 7.5+5,335/x 17 D# 10 Chia-chung 147,456 7.49++ 7.4+18,018/x 18 22 145,470 7.39+ 7.3+17,841/x 19 34 143,513 7.29+ 7.2+17,954/x 20 46 141,582 7.19++ 7.1+18,327/x ++;8 +++; --; Remainders: K'ai-shih . Tsu-chia Cheng-nan +; 1(+). ++; *x = 19,683

TABLE II (continued) Scale order Name Tuning order 21 E 5 :9 Ku-hsi Shih (number) Li (pipe) 139,968 7.11+ Chun (string) 7.1+2,187/x 22 58 23 17 24 29•• Lu-shih 136,225 6.92+ 6.9+4,123/x 25 41 Hsing-shih 134,392 6.83-- 6.8+5,476/x 26 53 I-hsing 132,582 6.7311++ 6.7+7,059/x 4, Nan-shou 139,674 7.09+++ 7+18,930/x Pien-yi 138,084 7.01?++ 7+3,030/x 27 E#12 C , Chung-1ui131,0726.66-- 6.6+11,642/x 28 24 29 36 30 Nan-chung 129,308 6.57- 6.5+13,685/x Nei-fu 127,567 6.48+ 6.4+15,958/x 48 Wu-ying 125,850 6.39++ 6.3+18,471/x 7 Jui-pin 124,416 6.32+ 6.3+4,131/x 32 60 Nan-shih 124,154 6.31-- 6.3+1,511/x 33 19 Sheng-pien 122,741 6.23?++ 6.2+7,064/x 34 31 121,089 6.15+ 6.1+10,227/x 35 43 31 36 F# G 2 1 3~ Li-kung Chih-shih119,4606.07-- 6+13,620/x 4.c~ Lin-chung 118,098 6 6 37 55 Ch'ien-tai 117,851 5.99-- 5.9+17,213/x 38 14 ( Ch'ii-mieh 116,508 5.92-- 5.9+3,783/x 39 26 An-tu 114,940 5.84- 5.8+7,786/x 40 38 Kuei-chia 113,393 5.76+ 5.7+11,999/x 41 50 - Fou-yii 111,8675.68++5.6+16,422/x

TABLE II (continued) Scale Tuning order order Name Shih (number) Lii (pipe) Chun (string) 42 G# 9 I-tse 110,592 5.62-- 5.6+3,672/x 43 21 Chieh-hsing 109,103 5.54++ 5.5+8,465/x 44 33 Ch'ii-nan 107,635 5.46+++ 5.4+13,468/x 45 45 Fen-chi 106,187 Nan-1i 104,976 5.33++ 5.3+6,561/x 47 4 \ 57 N t Po-li 104,756 5.32++ 5.3+4,361/x 48 16 t4 Chieh-kung 103,563 5.26++ 5.2+12,114/x 49 28 Kuei-ch'i 102,169 5.19+ 5.1+17,857/x 50 40 Wei-mao 100,794 5.12+ 5.1+4,107/x 51 52 I-han 99,437 5.05++ 5+10,220/x Wu-yi 98,3044.99++ 5.9+1,857/x Pi-yen 96,980 4.93-- 4.9+5,333/x Lin-ch'i 95,675 4.86+ 4.8+11,966/x 46 A •' 4'1 A / y 52 A#11 • 5.39?++ 5.3+8,671/x 53 23 54 35 55 47 Ch'i-pao 94,388 4.79?++ 4.7+18,779/x 6 Ying-chung 93,312 4.74+ 4.7+8,019/x Fen-wu 93,116 4.73+ 4.7+6,059/x 56 B r44k 4• . 57 59 58 18 Ch'ih-nei 92,056 4.68-- 59 30 Wei-yu 90,817 60 42 Ch'ih-shih89,5954.55++4.5+10,215/x 4.61(+) 4.6+15,142/x 4.6+2,752/x meaning to within .01 inches on the pipes, and to within .1 inches on the 10 times longer strings. These lii numbers,however, are accompaniedby 7 symbols that increase accuracyby almostanotherdecimalplace, hence to within a fraction of a cent. Five symbols, translatedhere as +, (+), + +, 1/2+ +, and + + +, indicate the relative sizes of remainders that have been dropped. Two other symbols, translated as - and - -, indicate that remaindershave been raisedto the next largerdigit. The only errorsin the table are trivial ones concerningthese symbols for the remainders.1The large fractionalremaindersfor the 2-digitchun numbersare too awkward

for use on pipes or strings, but they are the essential bibliographic tool for verifying the shih numbers. The main significance of the construction can be found in the first three tones of Ching Fang's scale order. Chih-shih, third tone in the scale but 13th in the spiral of fifths, should differ from Huang-chung by the Pythagorean comma of 531,441:524,288, almost 24 cents. Ching Fang's shih approximation is accurate to within .01 cents, and his lii reduction shows this to be worth about 12 parts in 900 (9.00-8.87+++). Se-yii, second tone in the scale but 54th in the spiral of fifths, subdivides this comma, missing Huang-chung by only about 1/6 as much. To express accurately this new ratio of the so-called "54-comma" requires numbers with 26 digits; we are now involved with 353 and the nearest power of 2. It is here on Se-yii that we appreciate the power in Ching Fang's computational technique: shih lii Huang-chung 353 177,147 9.00 Se-yii 2"4 176,776 8.98+ The shih approximation is accurate to within .14 cents, phenomenal by any standard, and the lii reduction allows us to see this as less than 2 parts in 900. The "54-comma" formed first by Se-yii is duplicated between the 2nd and 55th tones and by all subsequent pairs. The complete set is rounded' off to 60 elements probably because of that number's hoary prestige as a basic calendrical unit. Ching Fang's table thus represents sequences of Pythagorean commas generated by successive 12th intervals in the spiral of fifths (see 1-13-25-37-49 starting from Huang-chung, or 8-20-32-44 from Ta-lii), with the last 7 tones producing "54-commas" within the first 7 Pythagorean commas. Note that each diatonic semitone (CO-D, D9 -E, etc.), or Greek leimma, contains 4 Pythagorean commas, while each chromatic semitone (C-Co, D-D#, etc.), or Greek apotome, contains 5 such commas. Philolaus taught the Greeks that the Pythagorean wholetone of 9:8 spanned about 9 commas; Chinese theorists could read the same lesson in Ching Fang's table. We see at a glance exactly how the whole spiral of fifths unfolds through wholetones, semitones, and Pythagorean commas to reach near agreement with the reference tone after 53 consecutive operations-as Mersenne, Kircher, and Mercator were to learn in the 17th century (Barbour 1953:124-25). Before analyzing Ching Fang's approximation technique, and in view of his historical importance, we translate in full the exposition that Ssuma Piao attributes to him at the time of his appointment to the Bureau of Music. Notice that Ching Fang does not claim originality. He explains the

generation process, links it to traditionalyin-yangdualism,the calendar, the I Ching, and, rather vaguely, to his weather forecasting, and then concludes with a description of the instrumenthe built for acoustical demonstrations. Our explanationsare in brackets []. I study, under Chiao Yen-shou, a magistrateof Hsiao-huang,the methodof 60 1ii.For shang [generationupwards]to give birthto hsia [generationdownwards], 3 gives birthto 2 [subtract?], andfor hsia to give birthto shang, 3 gives birthto 4 [add ?]. Yanggeneratingfrom below producesyin, and yin generatingfrom above producesyang, endingat Chung-liito completethe 12 ii. Chung-lii[E # ] generatingfrom above producesChih-shih[the first Pythagoreancomma],and Chih-shihproduces Ch'ii-mieh[the subsequentcomma, a fifth higher]."Up" and "down" give birth to each other and end with Nan-shih [a commaabove Jui-pin F ] to complete the 60 ii. 12 lii develop into 60 hiijust like 8 hexagrams develop into 64 hexagrams.Fu Hsi createdthe [Bookof] Changesto recordthe beginning of the yang-ch'i ["male force"] and made it into a rule [lii-fa] to establish a date for the sound of the arrivalof winterand of Huang-chungas kung ["do"], T'ai-ts'ou as shang ["re"], Ku-hsi as chiao ["mi"], Lin-chung as chih ["sol"], Nan-Iii as yii ["la"], Ying-chung [B] as pien-kung [transformed kung, an auxiliary tone], and Jui-pin [F 0 ] as pien-chih [transformedchih, a second auxiliarytone]. This is the originof the soundandch'i ["force"]andthe correct doctrineof the 5 tones [2 of the 7 namedbeingmerelyauxiliarytones]. Therefore each occupies a day; the rest of them follow in succession. Each in succession is adoptedas kung ["tonic"] for a particulardate. Shang and chiao follow the rule. TheBook of Rites states: "5 sounds, 6 Ii, and 12pipes each in succession become kung." Thatis whatI havejust said. The periodis dividedby 60 ii. Huang-chung starts from the arrivalof winter and appearsagain the following winter. This is how divinationcan predictyin and yang and cold or warm, wind or rain. Use this to examine differentsounds and to investigate whether they are highor low. Exceptfor the soundof trees or grass [i.e., except for the "white noise" of the wind?], everythingelse will be matched.This is what it means in YiiShu, which says, "lIa harmonizeswith sound." The bamboopipe cannot be used for tuning[evidenceof awarenessof the problem of "end-correction"?]and thereforeI makethe chun to determinethe number.The shape of the chun is like a se [zither],with a lengthof one chang [ten feet], and with 13 strings.Divide [the soundinglength]into 9 feet to correspond with the 9 inches of the ii of Huang-chung.Underneaththe centralstring mark with ts'un [inches] andfen [tenthsof inches] to measurethe 60 NI. Ssu-ma Piao concludes his introductionto the ChingFang tuningby observing that "his writingis not recordedin its entirety,thereforeI sum up the outline to supplementthe recordsin the formaldynastyhistory" (Fan 1965:3000-01). How did Ching Fang achieve his remarkableaccuracy? Since he began with 31 = 177,147, as in the Huai Nan Tzu, his first 12 tones are untroubled by fractionalremainders.In the alternatingadditionand subtraction of thirds that follows, however, only a few numbersare evenly divisible by 3, hence there are several remaindersof /? and k/. Reconstruction of the arithmeticshows that if Ching Fang had simply dropped all remainderswhile computingthe 6-digitshih numbers,32 of his 6-digit numberswould be 1 or 2 units smallerthanthey are-and his calculations

would still be accurateto withinabout .1 cents. His corrections,however they were made, improveaccuracyto withinabout .01 cents, far beyond the limit of perception. How did he achieve such near-perfection? I suggest that ChingFang trustedthe oscillationof additionand subtraction to balance the effects of positive and negative remaindersand allow most of them to be dropped.But he closely watchedthe unpredictable rhythmof remaindersof 2 and madecorrectionsat exactly 4 points, on tones numbered23, 34, 35 and 51 in the tuning(not scale!) order.The arithmetic suggests the following rationale: a) 3 successive remaindersof +2/ on tones 19, 21, and 23 of the tuning orderjustify raisingthe 23rdnumberby 2 digits (3 x 2 = 2), in the absence nearby of compensatingsubtractions. b) Tone 35 in the tuningorder is roundedto a full digit, from a +3 remainder,partly because there is also a +?/3on tone 36. c) At tones 34 and 51, a -2/ is roundedto a full digit duringthe subtractions,again in the absence of nearbycompensatingadditions. (The only other -2/ in the entire set is at tone 39, balanced by +21 on tone 38, hence both remaindersare dropped.) The effect of ChingFang's correctionsis to let his set matchthe uncorrected set (with all fractions dropped)on tones 1 through22, rise slightly above it until the momentaryagreementat tone 34, then rise slightly above it again until-aided by a subtractionon tone 51-there is a natural convergence on tone 56. Ching Fang could have improvedhis accuracy very slightly by other choices of corrections, but to no purpose.2His calculations as they stand display that harmonyof precisionand simplicity properly called elegance. If we reflect that 1 cent is approximatelythe ratio 1730:1731,we can see that Ching Fang's 6-digit numbers,approximately100 times larger, will safeguarddistinctions of about .01 cents-with optimalcorrections. That is more than enough accuracyfor acousticaltheory. In dividinghis 6-digit shih numbers by 39 = 19,683 to produce the 3-digit lii lengths, Ching Fang displayed his eminently practicalconcerns. In denying that the pitch-pipeswere accurateenoughfor acousticaltheory, and in making demonstrationson a 9-foot string, ChingFang proved his competence in the acoustics laboratory.But the significanceof ChingFang's work lies mainly in the insight latertheoristscould gainfromit. He lays the foundation for the first Chinese effort towards equal temperament. THE TEMPERAMENT OF HO CH'ENG-T'IEN (+ 370 TO +477) The earliest numericalapproximationto equal temperamentis that by Ho Ch'eng-t'ien, ca. +400, possibly inspired by Ching Fang's presen-

tation of the comma as about 12 parts in 900 (see Huang-chung9.00 versus Chih-shih8.87+++ in Table II). By the clever device of adding one digit (meaning .01 inches) to each successive lii length, Ho Ch'engt'ien avoided the commaand achievedthe firsttrulycyclic 12-tonetuning. The result is an "unequal"temperament,for his linearcorrectionsignore their own logarithmic implications, but the mean deviation from equal temperamentof only 2.2 cents makesthis "a remarkabletemperamentfor the time when it was constructed" (Barbour 1953:55-56and Kuttner 1975:173-6).Ho Ch'eng-t'ien'sprocedureis describedin the Sui Shu (History of the Sui Dynasty), compiled in the +7th century and edited in + 1024 during the T'ang dynasty (Wei 1973:389): Ho Ch'eng-t'ienstartedto establisha new ruleandfromChung-lii[E0 , the 12th tone in the tuningorder],went backto Huang-chung,completingthe cycle of 12 kung [insteadof arrivingon Chih-shih,a commahigher],The lengthfor Huangchung was 9 inches, T'ai-ts'ou 8.2 [not 8.00], Lin-chung6.1 [not 6.00], and Ying-chung4.79+ [not 4.74+]. Chung-liigives birthfromabove to 177,147[the 6-digit shih numberfor Huang-chung],completingthe cycle of 12. Neither Ho Ch'eng-t'ien's temperamentnor those of later theorists found favor in China. The perfectionof the fifth 3:2 legitimizedany dynasty's link with China'sgloriouspast when 1 was the numberof heaven, 2 of earth, and 3 of man, and the musical scale was the model for an harmonious society (Needham 1956:11,271). In such a context it was naturalto search for a cyclic returnin a moreextendedtone-fieldof fifths 3:2 and fourths 4:3, and Ho Ch'eng-t'ienhimself pointed the way in his book Li-fa Chih-i, ending with an interestingaccusation against Ching Fang (Wei 1973:389): Shang [generationupwards]and hsia [generationdownwards]give birthto each other. To subtractor add V3is the simpleand easy methodof the ancients.It is like the ancient calendar,which has 365-?4 degrees for the cycle of the sky [on oracle bones from the 2nd milleniumB.C.]. Laterpeople changedthe system in different ways. But ChingFang did not realize that and by this absurdmistake made it into 60. THE 360 DIVISIONTUNING The Sui Shu contains two accounts of the extension of ChingFang's tuning from 60 to 360 tones, and we shall translateboth accounts in full. The extension was first accomplishedby Ch'ien Lo-chih(Wei 1973:389): Ch'ien Lo-chih, a historianduringthe reignof Yiian-chia[+424 to 453] of the Sung [in SouthernDynasties], followed the Nan-shih of Ching Fang with 300 more tones endingwithAn-yiinof 4.4+ inches. Includingthe old tones, thereare a total of 360, one for each day. Eachfollows the successionkungchih [meaning the standardpentatonicorder].

The calculation was repeated in the following century by Shen Chung, described as a "scholar of the 5 [Confucian]classics," tutorfor the heirapparent,whose lecturesat the courtaboutthe 3 religions(Confucianism, Taoism, and Buddhism)were attendedby 2000Confucian-Taoistofficials (Ling-hu 1974:808-11).(The texts are silent on the relationshipbetween these two efforts.) Ch'ien Lo-chih of the Sung followed Ching Fang's Nan-shih with 300 more tones. Duringthe LiangDynasty [+502 to 556], a doctorate,Shen Chung,in his Chung Lii I [Discussion on musical rules] says: The I Chinghas 360 ts'e [diviningstraws]to meetthe numberof days of a cycle [year].This is the numberfor musicandthe calendar.Huai Nan Tzusays: "each luigives birthto 5 sounds, 12hi give birthto 60 sounds, and6 repetitionsgive birthto 360soundsto meet the number of days in a year. This numberfor musicandthe calendarreflectsthe way of heaven and earth." This has been the case since antiquity. [Shen]Chungcalculatedthe 360 luiwiththe originalnumberin the HuaiNan Tzu and the method of Ching Fang. The Sui Shu then names all 360 tones in scale order,in columnsheadedby the original 12 tones, and adds a few importantinterpretativecomments, but omits the numbers (Wei 1973:397-401). In Table III we convert the 360 Sui Shu names into numbers, arranged into similarcolumns (except for the reversalof rightand left)-by following clues gleaned from the materialtranslatedabove. Numbersare computed by alternatesubtractionsand additionsof 13--"the simple and easy method of the ancients"-beginning with ChingFang's Nan-shih = 124,154, 32nd in his list but 60th in his generative order. Since these 6-digit numbers are already generallyaccurateto within about .1 cents, we have left them uncorrected,simplydroppingall fractionalremainders instead of trying to compete with Ching Fang's even greateraccuracy. "Step numbers" added for the first column only will reveal at a glance that the table essentially consists of Ching Fang's originalsequences of Pythagorean commas (note underlinednumbers),easily recognized by their tone names, each one subdividednow by consecutive 54-commas.3 Note the long column for each chromaticsemitone(apotome),now containing 34 tone-numbers,and the shortercolumnfor each diatonicsemitone (leimma), containingonly 27 numbers.These totals are given in the Sui Shu, probablyto guardagainstcopyists' errors.One additionalnumber, the last tone-numberof the set, 88,472, has been placed in the last column, giving it a total of 28 tones. This 360thnumber-named An-yiin, meaning "peaceful circuit"-is the only one allowed to exceed the range of an octave from Huang-chung;when dividedby 39 = 19,683,in Ching Fang's manner,it gives the lii value of 4.4+ inches (moreprecisely4.494), exactly as we were told that Ch'ienLo-chihdiscovered.An-yiin is sharper

TABLE III The 360-DivisionOctaveof Ch'ienLo-chih(reconstruction) 54 107 160 213 266 319 13 6119 172 225 278 331 25 7-• 131 184 237 290 343 37 90 143 196 249 302 355 49 102 C Huangchung (34) 177147 176776* 176400 176028 175665 175298 174929 174762 174394 174026 173660 173302 172938 172573 172410 172045 171682 171324 170970 170609 170250 170089 169729 169370 169020 168668 168312 167957 167800 T~67442 208 261 314 166744 166397 166045 Step c# Ta-lui (27) 165888 165538 165189 164840 164501 164157 163810 163654 163309 162965 162624 162288 161946 161605 161452 161110 160770 160437 160104 159765 159429 159280 158940 158605 158277 157948 157613 D# Chiachung (27) 147456 147145 146834 146525 146224 145917 145609 145470 145164 144858 144554 144256 143952 143649 143513 143209 142906 142610 142314 142013 141714 141582 141280 140982 140690 140398 140101 148524 155 167089 *132582 D T'aits'ou (34) 157464 157134 156800 156469 156146 155821 155493 155344 155017 154690 154365 154046 153722 153398 153253 152929 152606 152288 151973 151653 151333 151190 150870 150552 150240 149928 149610 149296 149156 148837 148217 147909 147596 generates the E Ku-hsi (34) 139968 139674 139378 139084 138797 138508 138216 138084 137793 137502 137213 136930 136642 136354 136225 135937 135650 135368 135088 134802 134518 134392 134106 133824 133546 133269 132986 132708** 132582* 132300 E# Chunglu (27) 131072 130796 130520 130245 129977 129704 129430 129308 129034 128762 128493 128228 127957 127688 127567 127297 127028 126765 126501 126234 125968 125850 125582 125317 125058 124798 124534 F# Juipin (27) 124416 124154 123892 123630 123376 123118 122858 122741 122482 122224 121968 121716 121460 121204 121089 120833 120578 120328 120078 119824 119572 119460 1192 118954 118708 118461 118210 G# I-tse (27) 110592 110359 110126 109894 109668 109438 109207 109103 108873 108644 108416 108192 107964 107737 107635 107407 107180 106958 106736 106510 106286 106187 105960 105737 105518 105299 105076 111393 132021 111163 110932 110697 131749 131474 131197 53-comma at 176776 G Linchung (34) 118098 117851 117600 117352 117110 116866 116620 116508 T116263 116018 115774 115535 115292 115049 114940 114697 114455 114216 113980 113740 113500 113393 113153 112914 112680 112446 112208 111972 111867 11162T8 **132708 generates

than the octave of Huang-chungC = 4.5 inches by less than2 cents, that is, by only 1/12 of the Pythagoreancommaformedby Chih-shih,and by about half of the 54-commaformed by Se-yii. Two parentheticalcomments in the fifth column, for Ku-hsi E, call attention to the consecutive numbersthat generateSe-yii, the first of the 54-commas, and An-yiin, the almost perfect octave. The whole table is thus carefully presented to make clear the internallogic of the endlessly spiralingfifths, and the successive reductionsof the Pythagoreancomma throughthe 54-commato the infinitesimal360-comma.Since the Chinese had used a basic calendricalunit of 360 days for at least a millenium,it is easy to sympathize with Shen Chung's enthusiasm-in the +7th century-for linkingthis 360-divisiontuningto the reveredI Ching, causing embarrassmentto modern Chinese scholars less sympatheticwith numerical coincidences. The whole point of the constructionis missed, however, by those who only notice the coincidence with the numberof days in a schematic year. From internalevidence the table was clearly intended to complete the process that the ChingFang tuningsets in motion. Generation stops at the 360th tone not because that makes enough tones for the year, or enoughfor the weeks in a 5-yearcycle, but because on An-yiin, as its own literal meaning emphasizes and as the Sui Shu points out, we can see the cycle of fifths reachinga "near-conjunction" with the octave. The point is driven home by the reminderthat An-yiin has the 1ii value of 4.4+ inches, just beyond the octave at 4.5. The table solves a legitimate question in musical numbertheory: Where does the spiral of fifths come closest to the octave? The last two numbersin the last column, that for Ying-chungB, tell the tale: the number 88,658, generated as the 307thtone, fallsjust shortof the octave Huang-chungC' = 177,147/2,while An-yiinat 88,472 sightlyexceeds the octave, reducing the 54-comma. Western scholars, generallyunawareof ChingFang's computational method, have mistakenlysupposedthat he must have kept trackof exact fractional remainders,a moderatelyheroic task involvingup to 27 digits, and have therefore been misled into supposingthat an extension to 360 tones would have requireda super-humanpatience with grotesquefractions. Nothing could be furtherfromthe truth.Using ChingFang's6-digit approximations(without corrections), and at the casual pace of 2 or 3 subtractionsand additionsper minute,the additional300 tone-valuescan be computed in a couple of hours. An-yiinat the end of this process is actually correct to within .14 cents of its absolutelyperfect value-which would require calculations runningto 3359, a 172-digitnumber.If Ch'ien Lo-chih or Shen Chungused, say, two clerks doing the computations(to forestall errors)and one scribe to enter results in appropriatecolumnsas

they were read off to him (positions are known in advance from Ching Fang's table), they could have completed the 360-divisiontuningbefore the morning tea break. We are unwilling to stop here with this recital of facts, for further reflection on Ching Fang's tuningsuggests the possibilityof still another level of meaning in the extension to 360 tones. APPROXIMATIONS TO EQUAL TEMPERAMENT The work of Ho Ch'eng-t'ien proves that the 360-divisiontunings were developed within the context of a concern with temperamentand with perfectly cyclic tunings-and awarenessthatpurefifths3:2 and pure fourths 4:3 produce only spirals. Kuttnerhas shown that amongthe first 60 tones in the spiral of fifths there are 12 with a mean deviation from equal temperamentof only 4.7 cents, and thatamong360thereare 12with a mean deviation of only .76 cents, "so close to maximumtuning accuracy by the finest craftsmen that, from a practicalpoint of view, no improvement is possible or necessary" (1975:173).Both of Kuttner's tempered sets can be identifiedmerely by inspectingChingFang's table, and the second one provides a test of the accuracy of any 12-toneapproximation computed directly, like that of Ho Ch'eng-t'ien. I suggest that the following lines of thoughtwere accessible to any of the theorists we have discussed--once their curiosity was arousedby the question of temperament. The most importantfact in ChingFang'stableis the nearcoincidence between Huang-chungand Se-yii, 1st and 54th respectively,in the tuning sequence. Since there is an accumulativeexcess of 2 cents per interval over equal temperamentnorms and near agreementon the 54th tone, it follows logically that the first and last 6 tones of the tuning sequence, numbers 1-6and 48-53, will lie closest to equaltemperament.None can be further than about 2 x 6 = 12 cents from the desired values. To locate Kuttner's 12 best choices from among ChingFang's 60 candidates,take the first 6 (C G D A E B) as given, then take the last 6 (F0 C0 G# DOAt E#) one step earlier in his scale order. Note how this slightly lengthens stringand/orpipe measures,offsettingthe sharpness/shortness of the Pythagoreanvalues (see Table II). Now let us take an even closer look at ChingFang's ratios. Se-yii at 176,776, or 8.98+ inches, the 54th tone, is almost 1/6 of the distance between Huang-chung and Chih-shih,the Pythagoreancomma. Now the Pythagoreanwholetone of 9:8 has an excess of exactly 1/6 of this comma; hence-it follows logically-two tuning operations before Se-yii, at I-han

= 99,437, the 52nd tone, we possess a rathergood equal temperedA = B , a minor-seventh above Huang-chung C and a wholetone (200.29 cents) below its octave C'. (It is very slightly flat because Se-yii is not quite 1/6 of the comma sharp.) We can now see that successive 51st intervals in the spiral of fifths will generatethe 6 descendingtones of an equal temperedwholetone scale, A# GOF; E D C, the 6 yang tones. If the remaining6 yin tones are generatedfrom these in the usual manner(by addingor subtracting? of the yang values), no tone can possiblybe more than 2 cents sharperthan in equal temperament.The slight cumulative flatness in the yang sequence can actuallybe used to offset this characteristic sharpness: for tones 2 and 53, early in the yin series, substitutethe later tones 308 and 359, slightlyflatter,hence closer to equaltemperament (see Table IV). By this reasoningan acousticaltheoristcan identifythe 12 best approximationsto equal temperamentfrom Ching Fang's table before any of them are computed. Their values, when computed,are accurate enough to serve as a standardof reference(see Table V). Since the 359th tone is E; = F, we know thatthe 360this sharperthanthe octave by only the trivialexcess of the purefifth over a temperedfifth. The location of An-yiin, then, can be understoodperfectlywithoutactuallyperforming the last 300 calculations attributedto Ch'ien Lo-chih and Shen Chung. CONCLUSION Were Chinese theorists awareof all the meaningswe have readfrom Ching Fang's table? There is no evidence either way, but he was studied intensely for centuries by competentmen. At our distancein time, Ching Fang seems brilliant,but he claimedno originality("I study, underChiao Yen-shou, the method of 60 Lui"),his computationaldevices were foreshadowed in the Huai Nan Tzu, and Ho Ch'eng-t'iendid not hesitate to TABLE IV Equal Temperament Approximations Within the 360-Division Tuning yang tone operation tone yin C At G# F# E D [C 1 52 103 154 205 256 307] [-1/3 [+1/3 +V3 +1/3 -1 2 53 104 155 206 G]* E; =F]** Do C; B A G* (better) Eg =F** (better) -V3257

TABLE V Equal Temperament Approximations In Chinese Cyclic Tunings Huai Nan Tzu ca. - 122 12-Division Chinese Western Huang-chung Ta-li T'ai-ts'ou Chia-chung Ku-hsi Chung-liEu Jui-pin Lin-chung I-tse Nan-li Wu-yi Ying-chung C C D D E Ft G GO A Maximum deviation Total deviation Mean deviation (- 11) At B Ching Fang -78 to -37 60-Division Hou Han Shu NORM Cents 2-digit Value Cents Step Value Cents 0 113.69 203.91 317.60 407.82 521.51 611.77 701.96 815.64 905.87 1019.55 1109.78 81 76 72 68 64 60 57 54 51 48 45 43 0 110.31 203.91 302.86 407.82 519.55 608.35 701.96 800.91 905.87 1017.60 1096.30 1 49 3 51 5 53 48 2 50 4 52 6 177147 167800 157464 149156 139968 132582 125850 118098 111867 104976 99437 93312 0 93.85 203.91 297.75 407.82 501.67 591.89 701.96 795.80 905.87 999.71 1109.78 21.51 129.10 11.74 19.55 82.84 7.53

call him "absurd" for failingto follow traditionmore closely. The rising fifths and fallingfourths associated with subtractionsand additionsof 13, "the simple and easy methodof the ancients," have been documentedas a basic Babylonianlyre tuningin the 2ndmilleniumB.C. (Kilmer1976).In that very early period the Chinese were alreadycastingbronze bells and cutting sonorous stones in the same tuning, and with great accuracy (Kuttner 1964). We must apparentlyconcede the possibilitythat serious investigations of the extended spiralof fifths may have been carriedout far earlier in history than survivingdata can prove. Whateverthe early historical facts, which remain elusive, our study of Chinese acoustical theory reveals the elegance of simplemethods,and the greatstrengthof a continuous tradition.4 APPENDIX Further study of the 2-digit numbersin the Huai Nan Tzu suggests the following rationale: 42 = 14 x 3 45 = 15 x 3 48 = 16 x 3 51 = 17 x 3 54 = 18 x 3 57 = 19 x 3 60 = 20 x 3 or 15 x 4 64 = 16 x 4 68 = 17 x 4 72 = 18 x 4 76 = 19 x 4 [80 = 20 x 4 suppressed] 81 = 9 x 9 (traditionalbase) With all respect to ChuTsai-yti'sintention,he was probablyhistoricallywrongto correct42 to 43. The virtue of the Huai Nan Tzuseries is partlythat it succeeds in aproximatingan equal-temperedchromaticscale by simplearithmeticprogressions,semitonesof 14:15:16:17 being oversize, and 17:18:19:20being under-size.The Prince'simprovementmarsthe elegant and simplepatternof the original,forthe number43 does not fit this scheme. NOTES 1. It is easy to discover the exact valueof the remaindersof the 3-digitlii numbersby carryingout the division by 19,683two moredecimalplaces, but it is not clear what values Ching Fang's 7 symbols for the remaindersare intendedto representbecause errorshave intruded, thus makingcategoriesoverlap. I suggest the following: Symbol + Tones Comments Meaning 14 .0006 to .0020 Should possibly stop at .0015, with tone 34 changed

.0023 to .0040 Shouldpossiblybe .0019to .0026,withtone 59 changed ++ 16 V2++ +++ 6 3 13 .0012 to .0045 Shouldprobablystartat .0030,withtones 5, 15, 17, and 48 changedto + and tones 10,47, 51, and60 changedto (+). This set is most erratic. .0049 to .0059 Probablycorrect. .0062 to .0088 Overlapsnext category. .0062 to .0092 Tone 9, with a remainderof .0026, should be changed 3 .0093 to .0096 Probablycorrect (+) to ++ to (+). There is probablylittle pointin actuallymakingthese hair-splitting corrections.The modem editor of the Hou Han Shu reportsmanyvariantsin the liNmodifiersandchunremaindersin his various sources, and makes many corrections.He also reportsthe followingvariants among the shih numbers(Fan 1965:3017-24): 22 Nan-shou 43 Chieh-hsing 139,670 119,103 45 Fen-chi 57 Fen-wu 106,188 93,117 2. Courantsuggests that there are 8 errorsamongthe 6-digitshih numbersin Ching Fang's table, but Couranthimself computedthese numbersby carryingexact fractionsto the very end, althoughthese ran to 21 digits (1924:88).Such a grotesqueproceduremisses the whole point of ChingFang's elegance, buryingChinesecommonsense andaffectionfor simplicity under our modem passion for absoluteperfection,howeverirrelevantthat may be. Courantsucceeded in makingthe 360 divisiontuningseem impossiblydifficult,supposing that exact fractionswould similarlybe carriedto the end. My own studywas inspiredby Kuttner's personalconviction that the Chinese "devised some efficientmethodof cutting decimals without sacrificingaccuracy," and owes much to his data and personaladvice (1975:173).If Ching Fang had made his first correctionearlierand also had madeanother near the end he could have improveda few values very slightly-but to no purpose. 3. The 360-commaat An-yiin= 88,472has the value of 1.8453cents above the octave of Huang-chung, while the 307-commahas the value of 1.77 cents below it. Whereasthe 307-commafalls short of a cyclic repetition,the 360-commareducesthe 54-comma.(Proof: An-yiin 88,472 x 2 = 176,944, slightly larger than Se-yii = 176,776, but smaller than Huangchung.) ModernChinese studies of the 360-divisiontuningprojectAn-yiininto the wrong octave, thus missing its significance,and also ignorethe interestingquestionas to how the calculations were made (Liu 1948:332-366). 4. Edith Borroff, Fred Fisher and F. Joseph Smithdirectedmy attentionto various problematicaspects of the 360-divisiontuning,and my wife, Augusta,helpedwith translations and calculations. REFERENCESCITED Barbour, J. Murray 1951 Tuning and Temperament: A Historical Survey. East Lansing: Michigan State College Press. Chuang Pen-li 1963 Panpipes of Ancient China. Taipei: Academia Sinica Institute of Ethnology (MonographNo. 4). Courant, Maurice 1924 "Essai historiquesur la musiqueclassiquedes chinois," Encyclopediede la Musique et Dictionnaire de Conservatoire, edited by Lavignac. Paris: Delagrave, Part

Fan Yeh (comp.) 1965 Hou Han Shu. Peking:Chung-hua. Kilmer,Anne D., RichardL. Crocker,andRobertR. Brown 1976 Sounds from Silence: Recent Discoveries in Ancient Near Eastern Music. Berkeley: Bit Enki Publications. Kuttner, Fritz A. 1964 "The Music of China: A Short HistoricalSynopsis Incorporatingthe Resultsof Recent MusicologicalInvestigations,"Ethnomusicology8(2):121-127. 1975 "Prince Chu Tsai-yii's Life and Work," Ethnomusicology19(2):163-204. Ling-hu Te-fen (comp.) 1974 Chou Shu. Peking:Chung-hua[1971]. Liu Fu 1948 "Music chinoise: le developpementde l'echelle musicale chinoise de 5 a 360 tones," Bull de l'Universite l'Aurore3s 9:332-366,Oct. 1948, trans. by R. P. Tcheou. Nakaseko, Kazu 1957 "Symbolism in Ancient Chinese Music Theory," Journal of Music Theory 1(2):147-180. Needham, Joseph 1956 Science & Civilisation in China, vol. II. Cambridge: Cambridge Univ. Press. Pan Ku (comp.) 1962 Han Shu. Peking:Chung-hua. Wei Cheng (comp.) 1973 Sui Shu. Peking:Chung-hua.