Musical Mathematics
on the art and science of acoustic instruments
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CHAPTER 11: WORLD TUNINGS
Part I: Chinese Music
Section 11.1
Of the texts that have survived, the earliest detailed description of Chinese tuning practice exists in a book entitled Lüshih ch’unch’iu (The Spring and Autumn of Lü Puwei), written c. 240 b.c.[1] According to this narrative, the semilegendary emperor Huang Ti (fl. c. 2700 b.c.) ordered music master Ling Lun to cast a set of sixty bells, and to tune them in twelve sets of five bells each. To accomplish this task, Ling Lun used the original 12tone scale, where each scale degree provided him with a fundamental tone from which to tune the original pentatonic scale. The oldest extant source that gives detailed calculations for this 12tone scale is a book by Ssuma Ch’ien (c. 145 b.c. – c. 87 b.c.) entitled Shih Chi (Records of the Historian), written c. 90 b.c.[2] Here, Ssuma Ch’ien cites the famous formula
* * ^{↓} * *
San fen sun i fa: Subtract and add onethird.[3]
* * ^{↑} * *
for the calculation of 12 pitch pipe lengths. An ancient pitch pipe consists of a tube closed at one end. One plays a pitch pipe like a panpipe by blowing across the open end of the tube. Ssuma Ch’ien’s formula works equally well for string lengths. According to Equation 7.18 and Equation 3.10, respectively, these two vibrating systems are very similar. The theoretical wavelength of the fundamental tone of a closed tube equals four times the length of the tube, and the theoretical wavelength of the fundamental tone of a string equals two times the length of the string. However, because the frequencies of tubes are affected by end corrections,[4] lip coverage at the embouchure hole,[5] and the strength of a player’s airstream,[6] the following discussion will focus on strings; the frequencies of the latter are easier to predict and control. Furthermore, since Ssuma Ch’ien based his calculations on a closed tube 8.1 units long,[7] and since fractional lengths are inconvenient, we will use an overall string length of 9 units. This length occurs in many historical texts, including a work entitled Lungyinkuan ch’inp’u by Wang Pinlu (1867–1921). Pinlu’s student, Hsü Lisun, edited and renamed this manuscript Meian ch’inp’u, and published it in 1931. Fredric Lieberman translated the latter in his book A Chinese Zither Tutor.[8] We will refer to this work throughout Sections 11.5–11.7.
Section 11.2
The ancient formula subtract and add onethird consists of two mathematical operations that have opposite effects. Subtraction of a string’s 1/3 length yields variable L(short), and addition of a string’s 1/3 length yields variable L(long). However, in the context of scale calculations, one may obtain identical results through the following operations of division and multiplication. (1) The act of subtracting 1/3 from a string’s overall length ( L ) is equivalent to dividing L by ancient length ratio 3/2, or multiplying L times modern length ratio 2/3:
(2) The act of adding 1/3 to a string’s decreased length is equivalent to multiplying L(short) by ancient length ratio 4/3:
(3) Similar to the first operation, the act of subtracting 1/3 from a string’s increased length is equivalent to dividing L(long) by ancient length ratio 3/2, or multiplying L(long) times modern length ratio 2/3:
For a string 9.0 units long, Table 11.1 shows that the Chinese upanddown principle of scale generation produces the following string lengths accurate to five decimal places.
To assure that these lengths produce tones within the span of an “octave,” or that all lengths are longer than 4.5 units, note two consecutive multiplications of ratio 4/3 near the center of Table 11.1. Joseph Needham describes this process of scale generation in the same context as discussed in Chapter 3, Section 18; namely, in ancient musical texts, mathematicians and musicians express scale degrees not as frequency ratios, but as length ratios:
* * ^{↓} * *
Before the idea of frequency existed, however, the same relation was expressed simply in terms of length, the length of a resonating agent multiplied by 2/3 being equivalent to the frequency multiplied by 3/2. The length of a zither string, then, multiplied by 2/3 gives a note which when struck is a perfect fifth higher than its fundamental. This is the first step (or lü) in a process which evolves an unending spiral of notes. The length of the resonating agent which sounds the perfect fifth is then multiplied by 4/3, the resulting note being a fourth below the perfect fifth . . .[9] (Bold italics mine.)
* * ^{↑} * *
Now, starting on C (or F), arrange these ancient length ratios (or frequency ratios) in the same order as shown in Chapter 10, Figure 16(a). With the exception of B#, or the ditonic comma,[10] ratio 531441/524288, Table 11.2 shows that the Chinese and Pythagorean progression of ratios are identical, which means that both sequences form an ascending spiral of “fifths.” (For more accurate results, recalculate the Chinese ratios to a greater number of decimal places.)
However, remember that this 12tone spiral of eleven ascending 3/2’s represents the basis of Chinese scale theory approximately 1500–2000 years before mathematicians in the West contemplated the musical possibilities of such a scale.[11] Finally, Table 11.3, Column 1, lists the 12 ratios; Column 2, the Chinese names of the scale degrees or lü; Column 3, the English translations of these names;[12] Column 4, the Chinese solmization of the five basic tones of the original pentatonic scale; Column 5, the note names starting on C; Column 6, the note names starting on F; and Column 7, the cent values of the 12 lü. Notice that the ratios of this pentatonic scale originate in the first five tones of the upanddown principle of scale generation; that is, 1/1 or do is kung, 3/2 or sol is chih, 9/8 or re is shang, 27/16 or la is yü, and 81/64 or mi is chiao. In diatonic order, these ratios are 1/1 or Huangchung is kung, 9/8 or T’aits’ou is shang, 81/64 or Kuhsien is chiao, 3/2 or Linchung is chih, and 27/16 or Nanlü is yü.
Section 11.3
The Chinese ch’in (or qin) is one of the most expressive musical instruments created by man. The fact that it has survived without significant modifications for over 3000 years[13] is a testament to its musical beauty and acoustic integrity. Performers on the ch’in produce three different kinds of sounds: open string tones, flageolet tones, and stopped tones. Moreover, the underlying mathematical organization of the ch’in categorizes these sounds into four distinct yet interdependent tuning systems: (1) the flageolet system, (2) the stopped hui integer system, (3) the open string pentatonic system, and (4) the stopped hui fraction system. Before we discuss these tuning systems in full detail, let us first examine the construction of this instrument.
In a book entitled The History of Musical Instruments, Curt Sachs (1881–1959) classifies the ch’in as a zither:
* * ^{↓} * *
A zither has no neck or yoke; the strings are stretched between the two ends of a body, whether this body is in the usual sense a resonator itself, or whether it requires an attached resonator.[14]
* * ^{↑} * *
The ch’in has a slightly tapered body approximately 4 ft. long, 8 in. wide, and 3 in. thick.[15] It is made from two different kinds of wood.[16] The top piece consists of a relatively soft wood called t’ung wood (paulownia imperialis). It has a slightly convex shape and acts as a soundboard against which the performer stops the strings. In contrast, the bottom piece consists of a relatively hard wood called tzu wood (tecoma radicanus). It is flat and gives the instrument structural rigidity. These two pieces of wood are jointed together on all four sides of the instrument. Consequently, an air chamber between the convex top and the flat bottom functions as a cavity resonator,[17] and amplifies the sounds of the strings. Two rectangular sound holes in the bottom piece allow the amplified acoustic energy to radiate into the surrounding air.
The ch’in has seven identically long strings of varying thickness, but no moveable bridges and no frets for the purposes of tuning. A permanent narrow bridge that spans the instrument’s width sits approximately 3 in. from the wide end. Between the bridge and this end, seven holes traverse vertically through the top and bottom pieces. The strings are threaded through the holes and tied to tuning pegs that push up against the bottom of the instrument. Because the pegs reside underneath the instrument, they are not visible to the casual observer. From here, the strings pass over the bridge and around the narrow end of the ch’in. This end acts as a nut. The strings are here tied to two large knobs that are fastened to the bottom piece.
Performers place the ch’in on a table so that the wide end with the bridge is near the right hand, and the narrow end is near the left hand. Although players use both hands to set the strings into motion, the right hand does most of the plucking, and the left hand, all of the stopping.
Section 11.4
(1) flageolet system. As discussed in Chapter 10, performers produce flageolet tones by lightly touching a string at the locations of nodes.[18] To find the nodes, ch’in players rely on flat circular reference points called hui. These markers are inlaid into the soundboard along the outer edge of the lowest string, or String I; this string is farthest from the performer. Turn to Table 11.4 and Figure 11.1, and note that the hui are numbered from right to left, or in a direction from bridge to nut. Each hui represents the division of a given string into an aliquot number of parts. Furthermore, notice that the arrangement of the hui constitutes a symmetric pattern with respect to hui 7, which marks the centers of all seven strings. The locations of hui 1 and 13 require string divisions into eight equal parts, of hui 2 and 12 into six equal parts, of hui 3 and 11 into five equal parts, of hui 4 and 10 into four equal parts, of hui 5 and 9 into three equal parts, and of hui 6 and 8 also into five equal parts. Table 11.4 gives modern length ratios and frequency ratios of the flageolet tuning system, and it lists the approximate Western tones for String I when tuned to C2. Finally, Table 11.4 also includes the aliquot parts, and the length and frequency ratios of theoretical hui 14; one needs the latter ratios to calculate the closest fraction and the closest ratio of shang, ratio 9/8, in Table 11.5. (See Note 25.)
(2) stopped hui integer system. At the locations of the hui, performers also play a series of stopped tones. Again, Table 11.4 gives the modern length ratios, frequency ratios, and approximate Western tones of String I. Observe that the first stopped tone at length ratio 7/8 sounds a “sharp major second,” or frequency ratio 8/7 [231.2 ¢].
Section 11.5
(3) open string pentatonic system. The ch’in has both an old and a new open string tuning system. This change in tuning was instigated during the Sung (a.d. 960–1279) and Yüan (a.d. 1279–1368) Dynasties, and codified during the Ming Dynasty (a.d. 1368–1644). For simplicity, Walter Kaufmann[19] describes the old as a preMing ch’in tuning, and the new as a postMing ch’in tuning. Figure 11.2 traces this transformation from the old to new tuning in the context of a parallel change from the original pentatonic scale to a new pentatonic scale.
Figure 11.2(a), Row 1, gives the approximate Western tones of the original pentatonic scale, shown here as a sequence of two identical scales, or a total of ten tones. Row 2 identifies nine exact interval ratios, and Row 3, five exact frequency ratios of the original scale. Row 4 shows that the first five frequency ratios of the old preMing ch’in tuning are the same as the original scale. Finally, Row 5 indicates that ch’in players tuned this scale on Strings I–V of their instruments (see Figure 11.1), so that kung is the first, or lowest string. This means that Strings VI and VII sound one “octave” above Strings I and II, respectively.
Figure 11.2(b) suggests that the new pentatonic scale in Figure 11.2(c) has its musical origins in the original pentatonic scale. In Figure 11.2(b), Row 1, observe that the first tone G is vertically aligned with the fourth tone G in Figure 11.2(a), Row 1. Consequently, note that Figure 11.2(b), Row 2, shows the same sequence of interval ratios as between G–G' in Figure 11.2(a), Row 2. Finally, Figure 11.2(b), Row 3, identifies five exact frequency ratios of the Gmode. Note carefully that the frequency ratios of the original scale and the Gmode are not identical. The original pentatonic scale — 1/19/881/643/227/16 — sounds a “large major third,” ratio 81/64, between the first and third scale degrees, whereas the Gmode — 1/19/84/33/227/16 — sounds a “fourth,” ratio 4/3, between the first and third degrees.
Figure 11.2(c), Row 1, illustrates the new postMing ch’in tuning in the context of a new pentatonic scale in C. Figure 11.2(c), Row 2, shows that the derivation of this scale is based on the interval ratios of the Gmode in Figure 11.2(b), Row 2. Consequently, the first six frequency ratios of the new postMing ch’in tuning in Figure 11.2(c), Row 3, are identical to the frequency ratios of the Gmode in Figure 11.2(b), Row 3. Finally, Figure 11.2(c), Row 4, shows that musicians shifted the tonic kung from C on String I to F on String III. This shift causes Strings I and II to sound one “octave” below Strings VI and VII, respectively. As discussed in Sections 11.7–11.8, kung on String III establishes the musical and mathematical basis of twelve mode tunings called tiao.
A comparison between Figure 11.2(a), Row 4, and Figure 11.2(c), Row 3, illustrates the differences between the old and new ch’in tunings. String III of the preMing tuning sounds chiao, or a “sharp major third,” ratio 81/64, above String I. In contrast, String III of the postMing tuning also sounds chiao, but in this case a “fourth,” ratio 4/3, above String I! Chinese musicians have acknowledged the dual identity of chiao for almost one thousand years. For example, the scholar Chu Hsi (a.d. 1130–1200) described the difference between the old and new ch’in tunings by noting:
* * ^{↓} * *
If the first string of the ch’in is tuned to the pitch huangchung [1/1] and sounds the tone kung, the third string should be tuned to the pitch kuhsien [81/64] so it can sound the tone chiao. Now present ch’in players all tune the third string not to kuhsein but to chunglü [4/3] using that pitch as chiao. It is done like this, but no one knows why . . .[20] (Bold italics, ratios in brackets mine.)
* * ^{↑} * *
Recall that Table 11.3 shows Chunglü not as ratio 4/3, but as ratio 177147/131072. To understand the origin of the 4/3identity, consider how the ch’in is tuned. Lieberman states that one tunes String I to [C2],[21] and then gives the following tuning sequence for the open strings of the ch’in.[22] In the illustration below, a progression of white notes represents the required scale degrees, and black notes indicate previously tuned degrees in the sequence. In the last step, one tunes String III as a “fifth,” or as a 3/2 interval, below String VI. This produces Chunglü as a “fourth,” or as frequency ratio 4/3, on the postMing ch’in.
In the last step, one tunes String III as a “fifth,” or as a 3/2 interval, below String VI. This produces Chunglü as a “fourth,” or as frequency ratio 4/3, on the postMing ch’in.
Section 11.6
(4) stopped hui fraction system. The musical possibilities of the tones designated by the hui markers are rather limited. On the C string, the flageolet tones produce 2/14/18/1 [C], 5/1 [E], and 3/16/1 [G]. The stopped tones extend this pattern by contributing two pitches, 5/45/2 [E], and 3/2 [G]. However, beyond this accretion, the stopped tones produce only four new pitches: 8/7 [D], 6/5 [Eb], 4/3 [F], and 5/3 [A]. Furthermore, observe that the stopped hui integer system does not indicate the location of the original pentatonic scale on the first string because ratios 9/8 [D], 81/64 [E], and 27/16 [A] are missing. To correct this deficiency, Chinese ch’in masters developed a unique technique for determining the approximate locations of unmarked tones. This technique consists of an invisible system of hui fractions. In his translation of the Meian ch’inp’u, Lieberman gives the following description of hui fractions:
* * ^{↓} * *
. . . fractions are used to specify points between two hui. The convention is to imagine that the distance between each pair of hui is divided into ten equal segments, or fen. The number y.x then denotes “x fen to the left of hui y.” Since the distance between pairs of hui varies, the fen do not have a fixed length.[23]
* * ^{↑} * *
In other words, one counts fen segments like hui markers from right to left.
Table 11.5 consists of three sections with six rows per section. Since all rows per section show identical categories, the following discussion does not refer to any specific section. In Figure 11.1, and in Table 11.5, Row 1, observe two different kinds of ratios associated with hui fractions. Ratios in regular typeface represent “primary tones,” or ratios required to play the original pentatonic scale on String I; and ratios in bold typeface represent “secondary tones,” or ratios required to play the original pentatonic scale on Strings II–VII. Due to the arbitrary division of the distances between hui into ten equal segments, notice that all hui fractions represent only approximate locations of the primary and secondary tones. Table 11.5, Rows 2 and 3, list hui fractions from two scholarly Chinese sources.[24] Row 4 gives the closest hui fractions required by the primary and secondary ratios. Except for two cases, 9/8–13.6, and 521/81–1.8, at least one fraction in Row 2 or Row 3 matches a fraction in Row 4.[25] Row 5 lists the actual ratios of the closest fractions in Row 4. Finally, Row 6 gives the cent differences that result when one subtracts the cent values of the ratios in Row 5 from the cent values of the ratios in Row 1. These differences vary from a minimum of –0.27 ¢ to a maximum of +21.51 ¢.
As an example of how to calculate a cent difference in Row 6, consider secondary ratio 16/9. In Figure 11.1, this ratio sounds kung [C3] on the open 9/8 string because 9/8 × 16/9 = 2/1; it sounds shang [D3] on the open 81/64 string because 81/64 × 16/9 = 9/4; and it sounds chih [G3] on the open 27/16 string because 27/16 × 16/9 = 3/1. According to Equation 3.32, on a ch’in string 1200.0 mm long, ancient length ratio 16/9 is located
from the bridge, or from the right end of the string. Now, to establish the location of fraction 7.6, begin by calculating the positions of hui 7 and hui 8. Observe that these two markers are located 600.0 mm and 720.0 mm, respectively, from the bridge:
These two measurements enable us to determine that six fen to the left of hui 7 equals a distance of 72.0 mm because
Consequently, fraction 7.6 is located 600.0 mm + 72.0 mm = 672.0 mm from the right end of the string. Finally, according to Equation 9.21, length ratio 16/9 has a cent value of
whereas hui fraction 7.6 has a cent value of
Note, therefore, a difference of 996.09 ¢ – 1003.80 ¢ = –7.71 ¢, which tells us that ratio 16/9 is a slightly lower tone than hui fraction 7.6.[26] However, since the next smaller fraction 7.5 produces a difference of –38.91 ¢, and since the next larger fraction 7.7 produces a difference of 22.93 ¢, we conclude that 7.6 represents the closest approximation with respect to ratio 16/9. Finally, verify this cent difference in three steps. (1) Reduce the length ratio of hui fraction 7.6 to lowest terms; (2) calculate the interval ratio between length ratios 25/14 and 16/9; and (3) compute the cent value of interval ratio 225/224:
At this point in the discussion, it is important to emphasize that the ch’in is a microtonal instrument.[27] Ch’in masters employ extremely complex playing techniques that include vibratos, sliding through intervals, and bending tones. Consequently, the above hui fraction analysis has only minimum musical value. Because the ch’in has no moveable bridges and no frets, virtuoso players have no problems adjusting the intonation of their instruments. In this respect, the subtle, intimate, and haunting musical beauty of the Chinese ch’in is very reminiscent of the Japanese shakuhachi, a large bamboo flute, which, due to its notched embouchure hole, also inspires many intricate microtonal inflections and ornamentations.[28]
[1](a) Kaufmann, W. (1976). Musical References in the Chinese Classics, p. 107. Detroit Monographs in Musicology, Detroit, Michigan.
(b) Needham, J. (1962). Science and Civilization in China, Volume 4, Part I, p. 172. Cambridge University Press, Cambridge, England.
[2](a) Kaufmann, W. (1967). Musical Notations of the Orient, p. 17. Indiana University Press, Bloomington, Indiana.
(b) Science and Civilization in China, Volume 4, Part I, p. 173.
[3](a) Musical References in the Chinese Classics, p. 147.
(b) Musical Notations of the Orient, p. 17.
[4]See Chapter 7, Section 12, and Chapter 8, Section 3.
[5]See Chapter 8, Section 7.
[6]Ibid.
[7]Science and Civilization in China, Volume 4, Part I, p. 175.
[8](a) Lieberman, F., Translator (1977). The Meian Ch’inp’u, edited by Hsü Lisun, pp. 186–187. In Lieberman’s The Chinese Long Zither Ch’in: A Study Based on the Meian Ch’inp’u. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.
(b) Lieberman, F., Translator (1983). The Meian Ch’inp’u, edited by Hsü Lisun, p. 18. In Lieberman’s A Chinese Zither Tutor. University of Washington Press, Seattle, Washington.
The dissertation is 851 pages long and includes everything contained in the tutor, which is only 172 pages long. However, because the illustrations and tables in the tutor are clearer and easier to understand, all Notes (except one) refer to Lieberman’s tutor. Readers who need detailed information should consult the dissertation.
[9]Science and Civilization in China, Volume 4, Part I, pp. 172–173.
[10]The ditonic comma, and its geometric division into 12 equal parts, plays a crucial role in Western tuning theory and practice because of the mathematical requirements of 12tone equal temperament. (See Chapter 10, Section 31.) Since musicians in China tune their instruments in just intonation, the enharmonic distinction between B# and the “octave” C' is not a determining factor. Consequently, the Chinese spiral of “fifths” simply ends on E#.
[11]This time span assumes that Chinese mathematicians defined the spiral of “fifths” in approximately 700–500 b.c., and that Europeans began implementing such a scale in approximately a.d. 1000–1300. Remember, the ancient Greeks did not contemplate 12tone scales. The socalled ‘comma of Pythagoras’ has nothing to do with the original discoveries attributed to Pythagoras. See Chapter 10, Section 22.
[12]Kuttner, F.A. (1965). A musicological interpretation of the twelve lüs in China’s traditional tone system. Journal of the Society for Ethnomusicology IX, No. 1, pp. 22–38.
[13](a) Sachs, C. (1940). The History of Musical Instruments, p. 185. W. W. Norton & Company, Inc., New York.
Sachs cites ch’in poetry from about 1100 b.c.
(b) Gulik, R.H., Translator (1941). Poetical Essay on the Lute, by Hsi K’ang (a.d. 223–262), pp. 51–70. In Gulik’s Hsi K’ang and His Poetical Essay on the Lute, Sophia University, Tokyo, Japan.
This essay recounts the ancient musical traditions of the ch’in.
[14]The History of Musical Instruments, p. 463.
[15]Apel, W., Editor (1944). Harvard Dictionary of Music, 2nd ed., pp. 170–171. Harvard University Press, Cambridge, Massachusetts, 1972.
[16]Musical Notations of the Orient, p. 275.
[17]See Chapter 7, Section 13.
[18]For a discussion on the acoustic properties of flageolet tones, and how to calculate the length ratios and frequency ratios of such tones based on standing wave patterns or mode shapes of vibrating strings, see Chapter 10, Section 50.
[19]Musical Notations of the Orient, p. 277.
[20]Lui, T. (1968). A short guide to ch’in, p. 184. Selected Reports I, No. 2, pp. 180–201. Publication of the Institute of Ethnomusicology of the University of California at Los Angeles.
[21]A Chinese Zither Tutor, p. 37.
[22]Ibid., p. 42.
[23]Ibid., p. 25.
[24](a) Ibid., pp. 26–27.
Figures 11 and 12 of Lieberman’s book show only hui fractions 8.5, 7.9, and 7.6 in Arabic numerals. The remaining hui fractions are depicted in Chinese characters.
(b) Wang, K. (1956). Chungkuo yin yueh shih, p. 26. Taipei, Formosa: Chung hua shu chu.
Although Wang’s book is in Chinese, the foldout on p. 26 gives hui fractions in Arabic numerals. In Table 11.5, I corrected 73.7, 72.2, and 70.8 to read 13.7, 12.2, and 10.8, respectively. Furthermore, I omitted Wang’s hui fraction 7.5 from Table 11.5, which he defines as the tone “h,” or “B natural.” This tone does not belong to the pentatonic scale on the ch’in.
[25]I calculated the relation 9/8–13.6 based on the logical assumption that theoretical hui 14 requires a string division into 10 aliquot parts. Such a division is consistent with the patterns in Table 11.4 because it would produce flageolet length ratio 1/10, flageolet frequency ratio 10/1, and the approximate flageolet tone E5. Furthermore, hui 14 also represents stopped length ratio 9/10, and stopped frequency ratio 10/9. So, for ratio 9/8, I used ancient length ratios 10/9 and 8/7 to calculate the closest hui fraction at 13.6, and the ratio of the closest fraction, 100/89.
[26]Due to complicated roundoff errors, I initially calculated all cent values in Table 11.5, Rows 1 and 5, to four decimal places. After subtraction, I then rounded the final cent differences in Row 6 to two decimal places.
[27](a) Sadie, S., Editor (1980). The New Grove Dictionary of Music and Musicians, Volume 4, p. 267. Macmillan Publishers Limited, London, England, 1995.
(b) The Chinese Long Zither Ch’in: A Study Based on the Meian Ch’inp’u, p. 11.
(c) Selected Reports I, No. 2, 1968, p. 183.
[28]For those readers interested in listening to ch’in music, I highly recommend a cassette tape entitled Music for the Ch’in Performed by Contemporary Masters of the Meian Tradition, compiled and edited by F. Lieberman. For a copy, write to the University of Washington Press, P.O. Box C50096, Seattle Washington, 98105. Throughout the tape, a given composition is performed by two or three ch’in masters, so that the listener has a chance to appreciate different interpretations of the same composition. The last two pieces on Side 2 are performed by Hsü Lisun.
