A Modification of an Old Chinese Musical Scale and Its Comparison with Chu's Equal Tempered Scale Chen-To Tai Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109 Abstract An old Chinese method of forming a musical scale is modified and the result is compared with the famous equal tempered scale founded by Chu in 1584. Half of the notes are identical to the corresponding notes in Chu's scale and the other six notes differ only within ~0.22% or ~2 cents. The scale of C-major in just intonation is also included in the comparison. Our study can be viewed as a merger of two different techniques. A Review of the Equal Tempered Scale Created by Chu Tsai-Yu1 in 1584 It is now well known that the equal tempered musical scale was first formulated by the Chinese musicologist and calendar scholar Chu Tsai-Yu (1536-1610), a prince in the Ming Dynasty (1368-1628). The work was officially published in a book dated 1584, more than one century before the German musicologist, Andreas Werckmeister (1645 -1705), formulated the same scale in 1691 [1]. Chu's formulation [2] can be presented in the form = 2(13-n)/1 2.-. 13 (1) where lThe current Romanized spelling of his name is Zhu Zai-Yu, but the old spelling is well established, hence it is used here. RL-989 = RL-989

t1 = resonant length of a pipe or a string for the lowest tone, or the tonic. f13 = -' = resonant length of a similar pipe or a string for the first octave. For a string, its resonant length is the same as its physical length. For a pipe, its resonant length exceeds its physical length because of the sudden discontinuity of the pipe at the opening [Ref. 1, p. 388]. Chu's original design of the pipe took this factor into consideration. For our discussion it is simpler to deal with a normalized length scale defined by Ln = n I (2(13-n)/12) (2) 1 2 which is then applicable to pipes or strings of any length. Furthermore, for our study, it is more convenient to consider a frequency scale, commonly used in acoustics [3]. Since the resonant frequency is inversely proportional to the resonant length we define a normalized frequency scale in the form: 1n = - = 2 (2(n-13)/12) 2(n-l)/12 In terms of the actual frequencies within an octave Rn = (4) where fi =- the lowest frequency in an octave, f13 = 2fi =- the octave frequency or the first harmonic. We will still use the letters for the keys to denote the normalized frequencies. They are listed below: R1 R2 R3 R4 R5 R6 R7 R8 R9 RIO Rll R12 R13 C C# D D# E F F# G G# A A# B C where C denotes the octave of C, and the interval between adjacent tones is called a semitone. 2

It may be of interest to recapitulate the ingenious approach Chu used to calculate the complete scale. He first considered the middle note R7(F#), which is equal to 21/2, the square root of 2. He then took the square root of R- to obtain 21/4 or R4(D#). Finally. he took the cube root of R4 to obtain 21/12 or R2(C#). The last one is the ratio of any two adjacent notes or the semitone. Once the basic number 21/12, denoted by r, is found. the complete scale can be calculated either by a cumulative multiplication, Rn = rn-1, for n = 3.4, -. 13, or by a combination of the previous ones. For example, Ro = R4R7 = 29/12, etc. (5) In the 16th century the Chinese mathematicians were already familiar with the calculation of square roots and cube roots. Chu carried his calculations to 25 significant figures. It would be a very difficult exercise for us without a computer nowadays, which is why he called his creation a precise scale [2]. He used the Pythagorean theorem, known to the Chinese for more than two millennia, to demonstrate the square root of 2 using a right triangle with unit arm length. Actually, he considered 10 inches for each arm because he was dealing with the length scale. The values of Rn of Chu's scale are tabulated in Table I. The reciprocals of Rn correspond to the normalized resonant length scale Ln originally tabulated by Chu. For example, for the note F#, L7 = 0.7071.The other columns of that table will be covered in the subsequent sections. We conclude this section by quoting Sir Joseph Needham [4]: To China must certainly be accorded the honor of first mathematically formulating equal temperament. A less obvious but more precious gift may be concealed in the example of this retiring scholar who declined the princely rank to which he was heir in order that he might carry on his researches, believing that for him who understands the meaning of the Rites and Music all things are possible. Such was the faith which animated Chinese students of sound for more than two millennia. Needham's treatise discussed Chu's work in great detail, including some reproduction of the original figures. 3

Table I the Notes in A Comnarison of Different Scales n Notes Rn Sn Tn Jn 1 C 1 1 1 2 C# 1.0595 1.0687 1.0582 3 D 1.1225 1.1250 1.1225 9/8 = 1.1250 4 D# 1.1892 1.2014 1.1879 5 E 1.2599 1.2656 1.2599 5/4 = 1.250 6 F 1.3348 1.3515 1.3333 4/3 = 1.3333 7 F# 1.4142 1.4238 1.4142 8 G 1.4983 1.5 1.5 3/2 = 1.5 9 G# 1.5874 1.6018 1.5874 10 A 1.6818 1.6875 1.6837 5/3 = 1.6667 11 A# 1.7818 1.8020 1.7818 12 B 1.8877 1.8984 1.8899 15/8 = 1.8750 13 C 2 2.0273 2 Rn Sn in Jn 7'7 Equal Tempered Scale The Old Chinese Scale A Modified Up/Down Scale C - major in Just Intonation An Old Chinese Musical Scale A twelve note musical scale used to tune the lute and to make pipes and flutes existed in China in the 5th Century B.C. It was the imperfection of this scale that motivated Chu to create the equal tempered scale [2]. We will describe the formulation of this scale also based on the frequency scale instead of the resonant length scale originally used in China. The normalized frequency scale in this case will be denoted by Sn (S = Sino-, or Chinese) where n goes from 1 to 13. The scale is formulated by using two numbers: 3 3 P=-, and q = By starting from Si = 1, we create a sequence of numbers: l,p, pq, (pq)p, (pq)2 (pq)2p, (pq)3, (pq)3q, (pq)4, (pq)4p, (pq)5, (pq)5p, (pq)6 (6) (7) 4

Observe that the first seven numbers are found by multiplying J) and q alternatively. The multiplication by q is repeated from the 7th to the 8th. then the normal order follows again. Since p > 1, and q < 1, the process goes from Si 1 (C) to S8 = p(G) as shown graphically in Fig. 1. By multiplying p by q, we go backwards from S8 = p = 3/2 (G) to 33 = pq = 9/8(D). After p3q3, the next step is to repeat the q multiplication. Fig. 1 shows very clearly the process. In doing so, all the 13 notes have been covered to form a complete scale within an octave. As mentioned above, the method originally used 1/p = 2/3 and 1/q = 4/3 to build the resonant length scale for a pipe or a string. Since 2 1 4 1 - - =1 + (8) 3 3'3 3' the formulation was called the "1/3 minus/plus" method or an "up/down" principle in frequencies [Ref. 4, p. 221]. The numerical values of the sequence, listed in order of the frequency scale, are found in Table I in column Sn. One main defect is that the "octave" note ( C ) is not truly an octave. Prince Chu referred to this phenomenon as unreturnable [2]. Many notes are also not close to Rn. They are, in general, higher than Rn, by not more than 3%. However, the note Ss(G) is a perfect 5th, which is more pleasant than Rs from the point of view of harmony. Some musicians still consider the perfect 5th to be a desirable interval, particularly for string instrument players. This problem will be commented on again later. We now consider a modification of the old Chinese scale, the main topic of this article. A Modified Up/Down Scale The fact that the so-called octave note in the old Chinese scale deviates from 2 suggests that we pick another pair of numbers for p and q to force (pq)6 = 2. We let p remain the same and change q to a number k such that (pk)6 = 2 (9) then k = — 21/6/p= (21/6) 2/3 = 0.748308 *., which is smaller than 0.75, the value of q, as it should be. By following the same up-and-down process, we create a scale that will be denoted by Tn, that is, 5

SI S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S1, S13 I I I I I I 1 -- p pq -- 2 *Pq p2q2 3 2 p q 3 3 pq 3 4 p q 3_ _ 4 4__________ p q p3q6 4_________55 p q — 5 6 - - -_ - p qI I I i i 1- i p6q6 - I I I I I - I 11 I I I I I I I I C C# D D# E F F# G G# A A# B C' SI = C = 1 S8 = G = p 3 2 =1.5 S2= C =pq = 2187 =1.06787109 2048 S9 =G# = p4q4 65 = =1.601806641 4096 S3 =D = pq S4 = D# = p4q5 = 9 =1.125 8 19683 -=19683 =1.201354980 16384 Slo= A= p q= S=A#= p 5q5= S12= B = 73q2 = 27 16 = 1.6875 S5 = = p2q2 = 81 E = pq = =1.265625 64 59049 59049 1.802032471 32768 243 243= 1.8984375 128 S6 = F= p q =177 =1.351478571 131072 3 3 729 S =F p q = =- =1.423828125 512 S13= C= 6q6=531441 = 262144 2.02728603 Figure 1: A graphical presentation of the Old Chinese Scale, p = 3/2, q = 3/4. 6

T,: 1p. p4', p2k. (pk )2, (pk )2p. (pk')3 (pka )'3k.. (pk)6. The numerical values of these notes are tabulated in Table I under column T,. Observe that the odd numbered notes are identical to the corresponding notes in R, and the other six notes with even numbers for n differ from the equal tempered notes within ~0.2'7( or ~2 cents. One cent is equal to 21/1200. Each semitone in R, corresponds to 100 cents. To ordinary people, the ears can not detect such a difference. The modified Chinese scale is therefore practically the same as Chu's scale. One feature of this scale is the retention of the perfect fifth in the system. The graphical construction of this scale is the same as Fig. 1, with q replaced by k. We would like to point out that the steps involved in tuning keyboard instruments as described by Ellis, the translator of Helmholtz's book, in Sec. G of Appendix XX, p. 489, are the same as the sequence found in the Chinese up/down technique. Western musicologists apparently are not familiar with the Chinese music literature, hence that old technique was not mentioned. Our study provides a quite accurate mathematical foundation for the process. If we use two alternative numbers p = R8 = 27/12 = 1.4983, q = R5= 2-5/12 = 0.7492 we would cover the equal tempered scale exactly. Before closing, a few words must be said about the just intonation scale, a commonly used scale before the advancement of the equal tempered scale. Just Intonation The scale in just intonation, denoted here by Jn, is formulated by using three numbers designated as the major tone, the minor tone, and the semitone [3]. They are defined by 9 M = major tone = 8 10 m = minor tone = 9 16 s = semitone =1 15 Using these numbers a scale can be formulated by the sequence: 7

1. A 1. I. IrMms. f 112sl. lmnAI22s,,l31 2m..113nm2s2 Table II Notes in a Scale of Just Intonation 1 I 1.1 1.2 1.3 1.4 1.5 I I I 1I 1.6 1.7 1.8 1.9 -I I 2.0 L -,.- - L - - -. C D E F G A B C J1 J3 J5 J7 J8 Jlo J12 J --- - - - - - - - I 3 1 1 M Mm Mms 2 M ms M2m2s M3m2s M3m2s2 9 8 5 4 4 3 3 2 5 3 15 8 2 It happens that these numbers are all expressible in terms of fractions of simple numbers. They are tabulated in Table II, where we use the intervals in C-major as an illustration of the notes within one octave. Since there are many keynotes in the scales of just intonation hence many different notes. Olson [3] listed the frequencies in 15 major scales and 15 minor scales in his book. We have verified all of them except one note. In the Bb-minor scale (Ref. 3, Table 3.11), note Eb should be 316.6 Hz instead of 313.2 Hz. It was probably a misprint. There are altogether 36 distinct notes in the 30 scales. This is the reason why just intonation is not practical to be adopted for orchestral music and for transposition of a musical score. According to some musicians only in string ensembles, trios or quartets, the players like to tune their instruments to a particular key based on the just intonation. The two numbers 3/2 and 4/3 used in the old Chinese up/down scale forming correspond to the major 5th and the reciprocal of the major 4th. We should mention that the just intonation has certain inherent beautiful characteristics. For example, the major triads C, F, G, E, A, B G; C; D; bear the ratio 4: 5:6 8

that nakes them a very pleasant combination acoustically. Similarly, the minor triads in just intonation using A4 as the tonic A C. E: D. E, A E. G. B: bear the ratio 10:12: 15. We use a bar on a letter to denote the note in the next higher octave and an underline for a note in the lower octave. These relations are not found in the equal tempered scales. Table 3.1 in Olson's book contains two notes designated as Augmented fourth= - = 1.40625, 32 64 Diminished fifth = - = 1.42222. 45 He (lid not mention the origin of these two notes, particularly, the name(s) of the person(s) who proposed these ratios. In the opinion of this author, these two notes can be described by two simpler ratios, namely, 7 Aug. 4th=- = 1.4 5 10 Dim. 5th = 1.42857. 7 The fraction 7/5 was designated by Helmholtz [Ref. 1, p. 195] as the subminor 5th, but the fraction 10/7 was not in his list. Both pairs have the same geometrical mean value of V/2, that is (45 64 1/2 /2 32 45 and (7 1\ 1/2 /2 5 7 21/ 9

This is a small issue but it does put two more simple fractions into the scale. It is merely a remark for the serious musicologists to debate or to consider. Finally, we would like to comment on the ranking of the pleasantness of different tones in just intonation by Olson [Ref. 3. p. 39]. His rankings are listed in Table III. Table III. Ranking of Pleasantness by Olson[3] Note Value Ranking Unison 1 Octave 2 1st Major 5th 3/2 2nd Major 4th 4/3 3rd Major 3rd 5/4 4th Minor 3rd 6/5 5th Minor 6th 8/5 6th Major 6th 5/3 7th It seems that to rank the major 4th ahead of the major 3rd is debatable because if we play the triad CFG on a piano, which is not tuned to just intonation but quite close, the combined tone is not as pleasant as the major triad CEG. The ranking therefore involves a personal taste. In view of the listing of the fractions in Table II it may be of interest to identify more simple fractions with the established notes or tones in just intonation. The list is contained in Table IV. All these notes and tones are defined in [3], except those marked with an "*" which are found in Helmholtz's book [Ref. 1, p. 187]. The two notes marked with "t" are described in this paper. After the semitone, the notes are arranged in order of increasing normalized frequency. We have observed that nature has provided for us a 10

Table IV A List of Fractions and Their Corresponding Notes or Tones in Just Intonation Fraction: 1 1 2 1 3 2 4 3 5 4 6 5 Note or tone: Fraction: Note or tone: Fraction: Note or tone: Fraction: Note or tone: Unison, Octave, Major, Major, Major, Minor 5th 4th 3rd 3rd 7 6 8 7 9 8 10 91 11 15 10 14 subminor super major minor 3rd*, 2nd*, 2nd or, tone, major tone none 16 15 9 7 7 5 45 32 10 7 semitone; super major, subminor, 3rd* 5th* or mod. aug. 4tht aug., modified, 4th dim. Stht 64 45 8 5 5 3 7 4 16 9 15 8 dim. 5th, minor major subminor Grave major 6th, 6th, 7th*, minor 7th, 7th 11

merger of simple mathematics and beautiful music (AI.2) from the point of Xview of just intonation. The first four numbers (1/1. 2/1, 3/2,4 /3) in the list were recognized by the Greek philosopher Pythagoras (6th Century B.C.) as the prime numbers associated with music. The five numbers and their corresponding notes 1 9 D 3 5 (C).8(D) 4(E) - (G), (.). 1 8 4 2 3 were known to the Chinese in Confucius time (550-489 B.C.) [5]. They all have Chinese names such as Yellow Bell for C. Forest's Bell for G, South Tone for A. The just intonation was happily spoiled by Chu's superb equal tempered scale. However, his basic number 21/12 is also made of fractions in a superstructure of the octave (2) in the form 21/223 or [(21/2) /] In China, to execute a square root is called 'open a plane square' and for the cube root it is called 'open a solid cubic' or 'open a triple'. Chu never stated his theory in the form of expressions as we do. He stated his theory only by words then documented by eleven numbers with 25 significant figures. They are reproduced here in his honor. Table V. Prince Chu's Table of the Equal Tempered Scale C = R1 = 1 C# = t2 = 1.05946, 30943, 59295, 26456, 1825 D = R3 = 1.12246, 20483, 09372, 98143, 3533 D# = R4 = 1.18920, 71150, 02721, 06671, 7500 E = fL5 = 1.25992, 10498, 94873, 16476, 7211 F = 6e = 1.33483, 98541, 70034, 36483, 0832 F# 7= 7 = 1.41421, 35623, 73095, 04880, 1689 G = Rs = 1.49830, 70768, 76681, 49879, 9281 G# = R9 = 1.58740, 10519, 68199, 47475, 1706 A = Rio = 1.68179, 28305, 07429, 08606, 2251 A# = 1ll = 1.78179, 74362, 80678, 60948, 0452 B = i12 = 1.88774, 86253, 63386, 99328, 3826 C = '13 = 2 These numbers have been verified by a modern computer. Conclusion In this article we have reviewed the development of the equal tempered scale formulated by Prince Chu Tsai-yu, and the old Chinese '1/3 minus/plus' scale. A modified scale is 12

described here that preserves the technique of the old Chinese scale forming. but yields a result very close to the equal tempered scale. In practice, they are almost identical. \We elaborate some close relations between simple fractions and the notes in just intonation. The help which the author received from Prof. James 0. \Vilkes and Dr. F.B. Sleator is gratefully acknowledged. References [1] Helmholtz, Hermann, Sensations of Tone, Dover Publications, New York, 1954. [2] Chen, Wan-Nai, Studies on Chu Tsai-Yu, (in Chinese with an English summary), National Palace Museum Publication, Taiwan, 1992. [3] Olson, Harry F., Music, Physics, and Engineering, Second Edition, Dover Publications, New York, 1967. [4] Needham, Joseph, Science and Civilization in China, Vol. IV, Part I, p. 228, Cambridge University Press, 1962. [5] Wang, Guang-Qi, Chinese Music History, (in Chinese with a short translation of technical terms and names in Western languages), 1931; reprinted by Tai-Ping Book Publisher, Hong-Kong, 1962. 13