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 *Lü-shih ch’un-ch’iu* (*The
Spring and Autumn of Lü Pu-wei*), written *c*. 240
b.c.[1]
According to this narrative, the semi-legendary 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 12-tone 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 12-tone scale is a book by Ssu-ma Ch’ien (*c.
*145
b.c.
– *c. *87
b.c.) entitled *Shih Chi* (*Records
of the Historian*), written *c*. 90
b.c.[2]
Here, Ssu-ma Ch’ien cites the famous
formula

* * ^{↓} * *

*San fen sun i fa: Subtract and add
one-third*.[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. Ssu-ma
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 Ssu-ma 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 *
Lung-yin-kuan ch’in-p’u* by Wang Pin-lu (1867–1921). Pin-lu’s student, Hsü
Li-sun, edited and renamed this manuscript *Mei-an ch’in-p’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 one-third* 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 up-and-down 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

* * ^{↑} * *

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 12-tone 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 up-and-down 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 *Huang-chung* is *kung*,
9/8 or *T’ai-ts’ou* is *shang*,
81/64 or *Ku-hsien* is *chiao*, 3/2 or *Lin-chung* is *chih*,
and 27/16 or *Nan-lü* 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 pre-Ming *ch’in* tuning, and the new as a post-Ming
*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 pre-Ming *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 G-mode. Note carefully that
the frequency ratios of the original scale and the G-mode are* not*
identical. The original pentatonic scale — 1/1-9/8-81/64-3/2-27/16 — sounds a
“large major third,” ratio 81/64, between the first and third scale degrees,
whereas the G-mode — 1/1-9/8-4/3-3/2-27/16 — sounds a “fourth,” ratio 4/3,
between the first and third degrees.

Figure 11.2(c), Row 1,
illustrates the new post-Ming *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 G-mode in Figure
11.2(b), Row 2.
Consequently, the first six frequency ratios of the new post-Ming
*ch’in* tuning in* *
Figure 11.2(c),
Row 3, are identical to the frequency ratios of the G-mode 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 pre-Ming tuning sounds *chiao*, or a “sharp major third,”
ratio 81/64, above String I. In contrast, String III of the post-Ming 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 *huang-chung* [1/1] and sounds the tone *kung*, the third
string should be tuned to the pitch *ku-hsien* [81/64] so it can sound the
tone ** chiao**. Now present

* * ^{↑} * *

Recall that Table
11.3 shows *Chung*-*lü*
not as ratio 4/3, but as ratio 177147/131072. To understand the origin of the
4/3-identity, 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
*Chung-lü* as a “fourth,” or as frequency ratio 4/3, on the post-Ming *
ch’in.*

In the last step, one tunes String III as a
“fifth,” or as a 3/2
interval, below String VI. This produces *Chung*-*lü* as a “fourth,”
or as frequency ratio 4/3, on the post-Ming *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 2/1-4/1-8/1 [C],
5/1 [E], and 3/1-6/1 [G]. The stopped tones extend this pattern by contributing
two pitches, 5/4-5/2 [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

* * ^{↓} * *

. . . 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 Mei-an Ch’in-p’u*, edited by
Hsü Li-sun, pp. 186–187. In Lieberman’s *The Chinese Long Zither Ch’in:
A Study Based on the Mei-an Ch’in-p’u*. Ph.D. dissertation printed and
distributed by University Microfilms, Inc., Ann Arbor, Michigan.

(**b**) Lieberman, F.,
Translator (1983). *The Mei-an Ch’in-p’u*, edited by Hsü Li-sun, 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 12-tone 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 12-tone scales. The
so-called ‘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).
*Chung-kuo 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 round-off 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 Mei-an Ch’in-p’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 Mei-an Tradition, *compiled and edited by
F. Lieberman. For a copy, write to the University of Washington Press,
P.O. Box C-50096, 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 Li-sun.