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Tuning a Qin  /   Absolute pitch  /   Problems with just intontation tuning  /   rhythm  /   mode  /   glossary 首頁
Qin Tunings, Some Theoretical Concepts 1
 With Special Reference to Shen Qi Mi Pu
古琴定弦系統
一些理論概念

Introduction2

This article covers some theoretical concepts and mathematical calculations. For basic qin tuning none of this is necessary. One just follows instructions such as those given under Tuning a Qin. However, it is also interesting to consider the theoretical possibilities of various systems of musical tuning (temperaments).

The three most commen tuning systems are:

  1. Pythagorean tuning (sanfen sunyi:3 cycle of fifth tuning; compare meantone tuning4);
  2. Just intonation tuning;5
  3. Equal temperament tuning.6

Pythagorean tuning was clearly the standard for tuning the qin, but it seems most likely that notes actually played did not strictly follow this system: if nothing else, the common tendency to bend notes is evidence for this.7 Meanwhile, just intonation tuning was also clearly used, most specifically in harmonics played at the 3rd, 6th, 8th or 11th hui, something generally avoided in pieces developed during the Qing dynasty. As for equal temperament, although it seems likely that it was first theorized by Zhu Zaiyu, a prince who lived 1536 - ca. 1610, there is no evidence for its use in pre-modern Chinese music, let alone qin music. Specific mathematical differences between these three are discussed below in Comparing different tuning systems; the systems primarily considered for qin tuning, Pythagorean and just intonation, are compared in a footnote.8

Qin music today is largely pentatonic. And although SQMP is flavored with many other notes, its music still remains pentatonic at its core. Western and Chinese pentatonic scales are basically the same, as they are both the first five notes generated by going through the cycle of fifths (do so re la mi):

Table 1: Pentatonic scale


1. Numbers
5
6
1
2
3
 
5
6
1'
2'
3'
 
5'
6'
1"
2"
3"
2. Names
Zhi
Yu
gong
shang
jiao
 
zhi
yu
gong'
shang'
jiao'
 
zhi'
yu'
gong"
shang"
jiao"
3. Solfeggio
So
La
do
re
mi
 
so
la
do'
re'
mi'
 
so'
la'
do"
re"
mi"
4. Letters
G
A
c
d
e
 
g
a
c'
d'
e'
 
g'
a'
c"
d"
e"

The first two lines use Chinese terminology, the other two use Western; the first three give relative pitch relationships, the fourth implies absolute pitch, a common standard for today's a' above c' (middle c) being 440 vibrations per second (see details). Qin music is written in tablature (finger positions and stroke techniques), and the tuning varies, so my own transcriptions into staff notation maintain this sense of relativity. Thus, although on my recording Music Beyond Sound the first string is consistently tuned to about 58 vib/sec (approximately the Western A# two octaves below middle c), this is variously considered as Do, Re and So (depending on the mode), and my transcriptions notate it as C, D or G. Since the range of the qin is more than four octaves, making the lowest note two octaves below c this the only way to include all the notes without the addition of a lot of octave-change markings. And writing the music with c meaning do (rather than absolute pitch) means there are fewer accidentals in the transcription than there would be from any other method.

The qin (see illustration under Tuning a Qin) has seven strings, with the standard tuning being 5 6 1 2 3 5 6 , often played as 1 2 4 5 6 1 2 . The first string, with the lowest pitch, is the one furthest from the player; with silk strings it is often tuned to slightly above A = 55 vib/sec. Thus the note names in the charts and transcriptions, it must again be emphasized, indicate relative pitch only. The actual tuning depends on such variables as the size of the instrument and the consequent string length (varying from about 40 to 45" [100 to 115 cm]); the quality of the strings; and the climate where the instrument is played. With metal strings the tuning can be higher without the strings breaking.

Harmonics and the positioning of studs on a qin 9

The qin is marked with 13 hui (inlaid studs, usually flat) running along the far side of the first string and marking the harmonic nodes: when harmonics are played the string must vibrate on both sides of the place where the left hand lightly touches the string, which means a clear harmonic sound can be gained only if the ratio of the vibrating length of the string on one side of the finger to that of the vibrating length on the other side is a simple fraction. Thus the divisions are in half (1/2), half again (1/4 and 3/4), then half again (1/8, 7/8 -- 3/8 and 5/8 are omitted); in thirds (1/3 and 2/3), then this is halved (1/6, 5/6); and finally in fifths (1/5, 2/5, 3/5, 4/5).

The following chart shows the relative positions of the studs on the qin as the player faces the instrument. The fractions show the ratio of the string length plucked by the right hand (i.e., right of where the left hand touches the string) to the whole string length. The fractions are then shown divided into 120, to indicate more clearly their relative distances from each other (if the string were 120 cm long, the first stud would be at 15 cm, etc). The next row shows the resulting number of vibrations per second if a harmonic is played in these positions on a string with a basic (i.e., open string) resonance of 80 vib/sec; and the final row shows the equivalent note with 80 vib/sec taken as do (arbitrarily selected to simplify the letters and numbers, but quite close to what I use for my third string):

Table 2: Relative positions of studs on the qin

Stud number:
open
13
12
11
10
9
8
7
6
5
4
3
2
1
Fraction of string:
1
7/8
5/6
4/5
3/4
2/3
3/5
1/2
2/5
1/3
1/4
1/5
1/6
1/8
Division into 120:
120
105
100
96
90
80
72
60
48
40
30
24
20
15
Vibrating Frequency:
80
640
480
400
320
240
400
160
400
240
320
400
480
640
Relative note equivalent:
Do
do"
so'
mi'
do'
so
mi'
do
mi'
so
do'
mi'
so'
do"


1. Comparing different tuning systems

The basic tuning systems are Pythagorean, just intonation and equal temperament. As a preface to the following comments, here is a comparison of these three standard note relationships in terms of vibrations as well as fractions.

A.   Pythagorean note relationships

These relationships can now be used to generate notes according to what is usually called in English the Pythagorean system, which derives notes based on the cycle of fifths: the fifths interval results from the ratio of three over two, and the octave interval from two over one.

The cycle of fifths derives notes as follows (take x vib/sec = 1 , then multiply by 3/2):

1 (do) -- 3/2 (so) -- 9/4 (re') -- 27/8 (la') -- 81/16 (mi")

Bringing these within one octave by halving them whenever necessary changes this to:

1 (do) -- 3/2 (so) -- 9/8 (re) -- 27/16 (la) -- 81/64 (mi)

Ordering these in ascending sequence, gives:

1 (do) -- 9/8 (re) -- 81/64 (mi) -- 3/2 (so) -- 27/16 (la)

To extend the system downward two notes, divide sol and la by 1/2, thus lowering them each an octave. This gives the following seven note sequence, which is the standard qin tuning:

3/4 (So) -- 27/32 (La) -- 1 (do) -- 9/8 (re) -- 81/64 (mi) -- 3/2 (so) -- 27/16 (la)

If do' is 320 vibrations per second (Western d'#), the figures for this sequence are:

240 (So) -- 270 (La) -- 320 (do) -- 360 (re) -- 405 (mi) -- 480 (so) -- 540 (la)

This same result can be achieved by the Chinese sanfen sunyi sequence (see relevant footnote)

B.   Just intonation

Harmonic notes played at the third, sixth, eighth and 11th positions express mi by a simpler fraction: 5/4 (see Table 2: mi'/do' = 400/320 = 80/64 = 5/4). If la is a fifth above this (5/3, from 5/4 x 4/3, which is 80/48 instead of 81/48) the resulting scale is said to use just intonation. The above seven note pentatonic sequence is then:

3/4 (So) -- 5/6 (La) -- 1 (do) -- 9/8 (re) -- 5/4 (mi) -- 3/2 (so) -- 5/3 (la)

If do' is again 320 vibrations per second, the figures for this pentatonic sequence are:

240 (So) -- 266 2/3 (La) -- 320 (do) -- 360 (re) -- 400 (mi) -- 480 (so) -- 533 1/3 (la)

C.   Equal temperament

The twelve notes of the scale can also be organized into equal temperament tuning, by which all 12 semitones in an octave are generated using a logarithm, making them equidistant. The earliest known publication of this system was by Zhu Zaiyu in 1584, but it was in the West over the next several centuries that it was incorporated into the tuning of music instruments. Equal temperament semitones are measured as 100 cents, so the three relevant pentatonic scales are as follows:

Table 3: Comparing intervals in the three tuning systems

 
Do
 
Re
 
Mi
 
So
 
La
 
do
Pythagorean system
 
204
 
204
 
294
 
204
 
294
 
-- from Do
 
204
 
408
 
702
 
906
 
1200
 
Just intonation
 
204
 
182
 
316
 
182
 
316
 
-- from Do
 
204
 
386
 
702
 
884
 
1200
 
Equal temperament
 
200
 
200
 
300
 
200
 
300
 
-- from Do
 
200
 
400
 
700
 
900
 
1200
 


Equal temperament was never used on the qin: it is mentioned here by way of introducing cents. Also of note is the comma, which is the difference between a Pythagorean third and a just intonation third, typically 22 cents.

 
2. Tuning the qin

The standard method of tuning the qin leads naturally to the Pythagorean note ratios. The first step is usually to make the harmonic played on middle of the seventh string have the same pitch as the harmonic played on the 9th (or 5th) position of the fourth string. There is some disagreement about which note to change if the resulting notes are not the same. With good strings in a constant low humidity the strings hold their tuning quite well. In this case when re-tuning you might re-tune the top string first because its pitch is most likely to have changed since the last tuning. However, when the weather becomes more humid it is usually the thicker strings which change the most.

The following chart may give the impression that the starting place for generating the tuning of the seven strings is the third string, since that is given the ratio 1/1. This is simply done for ease of comparison and doesn't affect the relative relations. The chart also shows the first string tuned to 60 vib/sec, perhaps suggesting absolute pitch. Again it must be emphasized that it does not seem to have been part of the tradition to begin this sequence by testing against an absolute pitch (see further comment on Absolute pitch).

The full sequence for standard tuning is explained below, after the chart.

Table 4: Standard tuning (1st string played as sol; if A=440, the actual pitch is between B flat and B)

- note for line 5: 101 is actually 101.25; 506 is 506.25; and 304 is 303.75

String/pitch/fraction of do
open
13
12
11
10
9
8
7
6
5
4
3
2
1
1. (So [5]) = 3/4 (48/64)
60
 
 
300
240
180
 
120
 
re'
so'
ti'
 
 
2. (LA [6]) = 27/32 (54/64)
67.5
 
 
337.5
270
202.5
 
135
 
mi'
la'
d#"
 
 
3. (do [ 1 ]) = 1 (64/64)
80
 
 
400
320
240
 
160
 
so'
do"
mi"
 
 
4. (re [ 2 ]) = 9/8 (72/64)
90
 
 
450
360
270
 
180
 
la'
re"
fa"
 
 
5. (mi [ 3 ]) = 81/64 (81/64)
101
 
 
506
405
304
 
202.5
 
ti'
mi"
so#"
 
 
6. (so [ 5 ]) = 3/2 (96/64)
120
 
 
600
480
360
 
240
 
re"
so"
ti"
 
 
7. (la [ 6 ]) = 27/16 (108/64)
135
 
 
675
540
405
 
270
 
mi"
la"
do#'"
 


Intervals in cents: From each other:       204-294-204-204-294-204
from So:                 204-498-702-906-1200-1404

A harmonic at the ninth position is a fifth interval above one at the seventh position (row 7: 405/270 = 3/2), and the 10th position is a fourth interval above the ninth position (540/405 = 4/3 [down a fifth then up an octave is 2/3 x 2]). So harmonics at the seventh, ninth and 10th positions (or their mirrors at the fifth and fourth positions) can be used to determine string tuning precisely according to the Pythagorean note relationships:

  1. Seventh position on seventh string equals ninth position on fourth string,
    so the fourth string must be a fifth lower than the seventh string.
  2. Ninth position on seventh string equals 10th position on fifth string,
    so the fifth string must be a fourth lower than the seventh string.
  3. Ninth position on sixth string equals 10th position on fourth string,
    so the sixth string must be a fourth higher than the seventh string.
  4. Seventh position on sixth string equals ninth position on third string,
    so the third string must be a fifth lower than the sixth string.
  5. Seventh position on fifth string equals ninth position on second string,
    so the second string must be a fifth lower than the fifth string.
  6. Seventh position on fourth string equals ninth position on first string.
    so the first string must be a fifth lower than the fourth string.

Various other equivalencies between strings can then be used to check the tuning.
 

3. Incorporating just intonation intervals

Today harmonics at the just intonation studs (third, sixth, eighth and 11th) are avoided because of the dissonances which result when the Pythagorean and just intonation systems clash, but early qin music often uses harmonics in these positions. The Shen Qi Mi Pu piece Yi Lan (Flourishing Orchid), for example, actually begins with the apparent dissonance of (see Table 4) the 11th position on the first string [300 vib/sec] followed by the ninth position on the fifth string [303.75 vib/sec]). Although Table 4 has the first string as 5 (sol), since Yi Lan is in shang mode the first string is considered as 1 (do), and this is the main note in this mode. Thus 300Hz is a just intonation third over the main note, the latter is a Pythagorean third over the main note. How does one account for this?

When I first encountered such dissonances in my reconstructions I tried to see if they could be avoided by re-tuning the qin. The problem was that removing a dissonance in one place always brought a different one elsewhere. An account of such efforts is in a section called Problems with just intonation tuning, with especial reference to SQMP.

To sum up what is in the linked article, I have not yet found any qin pieces, except perhaps very short ones, in which I can use a form of just intonation tuning to avoid dissonances caused by harmonics played at the just intonation studs. This is not to say that such tuning was never used, only that I haven't found that in any one piece it succeeds in making all the sorts of harmonic note pairs described above (plus open strings) match. My own feeling is that, after in some cases making minor tuning adjustments so that clashes are minimized, these dissonances are really an interesting coloration of the sound

 
4. Indicating the pitch of stopped sounds: comparing the old and new systems

The present decimal system (see row New below), in use since the 17th century (see its origins), can indicate finger positions with mathematical precision: 7.6 and 7.9 mean respectively 6/10 and 9/10 of the distance between the 7th and 8th positions (hui). In theory this system can be used to indicate very precise tonal differences, e.g., 10.7 for a slightly sharpened E.

Since there is no indication in the literature or the tablature itself that this sort of precision was ever needed, the system used in SQMP (row Old) was in theory just as precise as the contemporary one. Thus the decimal position 7.6 was in SQMP almost always written 7 8, meaning "the correct place between the 7th and 8th positions"; 7.9 was rounded off to 8; and 7.3 was written 7-. When SQMP deviated from this system, using values found in row Alt., it was usually on slides (see explanation below the chart, since I could not fit all the various Alt figures within that one row).

Other early handbooks may be less precise.10 In particular, Zheyin Shizi Qinpu, although in pieces copied from SQMP it also follows that system, in pieces not in SQMP it normally used the values in row Alt., which can lead to much confusion. Thus, in pieces, or versions of pieces, occurring for the first time in Zheyin Shizi Qinpu, 7.6 may be written as 7 8, but it is more likely to be 8+ (above 8), 7 1/2, or even 7- (below 7), with perhaps all three occurring on the same page.

Table 5: Standard positions on a qin string tuned to C (items marked * have comments in the notes)

 
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
c
c#
d
d#
e
f
New
0 (13.9*)
(13.5*)
wai*
12.3
10.8
10
9.4
9
8.5
7.9
7.6
7.3
7
6.7
6.4
6.2
5.9
5.6
Old
0 (wai*)
wai*
wai*
12
11
10
9-10
9
8-9
8
7-8
7-
7
7+
6-7
6-
6
5-6
Alt
 
 
13*
 
 
 
9-*
 
8-*
 
7-*
7-8
 
6-7
6-*
6
6+
5-*

->
f#
g
g#
a
a#
b
c'
c'#
d'
d'#
e'
f'
f'#
g'
g'#
a'
a'#
b'
c"
->
5.3
5
4.8
4.6
4.4
4.2
4
3.7
3.4
3.2
2.9
2.6
2.3
2
1.8
1.6
1.4
1.2
1
->
5-
5
5+
4-5
4-5
4-
4
4+
3-4
3-
3
2-3
2-
2
2+
1-2
1-2
1-
1
->
5-6
 
 
4-*
4-*
 
*
 
 
 
 
 
 
 
 
 
 
 
 

Notes:

  1. in the modern repertoire wai is always 13.1; old handbooks usually call 13.1 wai, but may also call it 13, then use wai for 13.5 and even 13.9; but there is no consistency, so this is somewhat speculative.11 Sometimes a distinction between 13 and wai seems to occur in passages at 13.1 over several strings, with notes played by the 3rd finger using 13 and those by the 4th finger using wai. Could this be an example of a transcriber who may not have known about temperament or "correct" positions giving his impression of where the player seemed to put his fingers?
  2. "9-10" = "between 9 and 10" (9, 10 jian); likewise for "8-9", "7-8", "6-7", "5-6", and "4-5". SQMP quite consistently uses these Old positions as indicated, but other early handbooks (Zheyin in particular), generally use an Alternate figure, only one example of which is given in row Alt. Thus, in addition to expressing the modern 9.4 as "9-" (below 9, or 9 xia), it also indiscriminately uses "10+" (above 10, or 10 shang) and "9 1/2" (half 9, or 9 ban), all to mean this same position. The same variants are available for all the other "belows" on this line. Sometimes 8+ is even used for New position 7.9.
  3. All these three alternatives (as well as the Old figures) can also be found for positions between 8 and 9, 7 and 8, 6 and 7, 5 and 6 and 4 and 5.
  4. Positions higher than 4.0 are extremely rare, except when playing harmonics.
  5. Although SQMP is generally rather consistent about using the Old system, see, for example, Gao Shan, measure 33 of my SQMP transcription, where 8+ clearly means 7.9, but plain 8 is used for 7.9 in the rest of the piece.
  6. "7-" = "below (i.e., to the left of) 7" (7 xia);
  7. "7+" = "above (i.e., to the right of) 7" (7 shang);
  8. Ban (half), is particularly popular with sliding sounds. For example, pieces in which the modern 7.6 is written as 7-8 for fixed positions will often for slides write 7 ban.
  9. It is perhaps the inconsistent usage of the Old and Alt systems which led to the change from the potentially precise Old system to the more potentially precise New decimal system.
Assuming this table refers to the 3rd string, tuned to C, and that the 4th string is D, then G is 9.0 on the 3rd string, 10.0 on the 4th. Using the "now" figures, 10.8 on the 3rd string is Pythagorean E. If, as in the article Problems with just intonation tuning, Table 4, the 3rd string is tightened so that the harmonic at 11 becomes the Pythagorean E, then the stopped G on the 3rd string must be adjusted down to about 9.1 or 9.2 to give the same note as stopped G (10.0) on the 4th string. The modern system in theory has the precision to indicate this, but not the old system: "above" and "below" were almost exclusively reserved to indicate half tones, not commas.

In addition, the apparent precision of any of these systems can be somewhat misleading. For example, in actual play if you sound the octave harmonic by lightly touching the 7th position (7.0), then right away push the finger down to form a stopped sound you will find that the pitch is slightly sharp, the degree of sharpness varying with the height of the bridge and the tension of the strings. Thus, it would be very confusing if the persons writing the tablature judged finger positions by eye rather according to theory. Although that was generally not done, my experience is that corrections to tablature may have occasionally been done by eye, sometimes leading to confusion.

 
5. Influence of the transcription process on tuning

Did qin tablature describe the way someone played a piece or prescribe the way one ought to play it? The apparent precision of writing 7.9 to indicate a position implies a somewhat idealized prescription because, for the reason given in the previous paragraph, the actual stopped position is usually somewhere between 7.9 and 8.0. Traditionally one didn't learn a piece from tablature, one learned from a teacher, perhaps referring to the tablature when the teacher wasn't present. Since melodies are more easily remembered than finger strokes and positions, the function of the tablature was to remind the player of certain details, not to directly indicate a melody.

As for the Pythagorean versus the just intonation third, although the position 10.8 may imply a precise Pythagorean third, the position one actually plays is usually closer to 10.9, and so the old 11 is not in turn a precise indication of the just intonation third. At the 11th position the difference is usually very small, but at the 6th position, where 6.0 should be the just intonation third and 5.9 the Pythagorean third, on an instrument with a relatively high bridge or using somewhat tight stringing the actual position to achieve a Pythagorean third may even be as low (to the left) as 6.1.

Old qin tablature is generally thought to depict the actual performance of a piece by a master. Either the initial transcriber might indicate, or a later editor might revise, a position based on where they played the note on their own instrument rather than on the theoretically correct positions. This also brings up two other issues: qin construction and the origin of the melodies.

Qin makers did not always succeed in placing the studs precisely in accord with the harmonic divisions. A transcriber or editor making reference to a qin with studs placed imprecisely might give a position that was only correct on this particular instrument. And an editor in particular might make a change in one part of a document without making sure it was consistent with the other parts.

Some qin music originates with a qin player him- or herself, some was adapted from melodies already in existence. Someone adapting (or creating) a melody might find that only a tuning which includes some dissonances would allow the melody to be set to the qin. Efforts might then be made at the sorts of re-tuning described above -- with the same problems -- but the results published anyway. To understand this fully one should see if a melody like Yilan can "improved" by playing it in a way which avoids harmonics at the just intonation studs.

 
6. Standard and non-standard qin tuning

"Standard" tuning is 5 6 1 2 3 5 6 (also considered as 1 2 4 5 6 1 2 ) regardless of whether it is Pythagorean or just intonation. "Non-standard" tunings are derived by altering the tuning of one or more strings by a half tone, except in manshang mode, which lowers the second string a whole tone.

Today almost all qin pieces use standard tuning, the main exceptions being two melodies that use ruibin tuning, in which the fifth string is raised a half tone. In constrast, early qin music had a great variety of different tunings. For example, Shen Qi Mi Pu (1425) used seven non-standard tunings, Xilutang Qintong (1525) used thirteen.

The qin tunings found in Shen Qi Mi Pu are outlined in a chart at the top of Modality in Early Ming Qin Tablature. Xilutang Qintong tunings are outlined in a separate chart.

 
7. Conclusions

The above discussion has concerned tuning, not mode. In SQMP the word diao is used for both, and writers are not always clear in their distinction between the two. For standard tuning there are different modes, distinguished by their main and secondary notes. In SQMP non-standard tunings seem also to define the mode (with the exception stated for huangzhong). In some other Ming dynasty handbooks more than one mode seems available for some non-standard tunings.

On the issue of the Pythagorean system versus just intonation tuning one must keep in mind the distinction between qin tuning and qin play. Did a player who tuned the qin according to just intonation or the Pythagorean system use these intervals consistently throughout the piece? The old way of transcribing qin music does not have the precision to delineate this, and there are no classical writings I know of directly addressing this issue. In addition I don't know of any modern studies done to determine if modern qin players, in addition to using Pythagorean tuning, also generally follow Pythagorean intervals throughout.

As a qin player whose repertoire is almost exclusively my own reconstructions from 15th and 16th century handbooks (about 120 melodies so far) I have not waited for definite answers on either tuning or intonation before proceeding, instead proceeding according to my own assumptions. (See Historically Informed Qin Performance for a discussion of my approach: trying to copy my "teachers" [the tablatures] as faithfully as possible, making conscious changes only after recording my understanding of their music.)

My relevant assumption here has been that tuning followed the Pythagorean system with perhaps slight adjustments if an interval played at the just intonation studs was too harsh. I appreciate most of these sounds for their special color, though I also think they sometimes may have resulted from music being transferred from other instruments without regard to the resulting dissonances.

 
Footnotes (Shorthand references are explained on a separate page)

1. This page concerns mathematical relationships between positions on the seven qin strings. Basically it explains why the tuning methods (定弦法子) described under Tuning a qin result in the desired relative tuning (相對定音).
(Return)

2. Background: Modern standard note frequencies in Hertz (Hz [= vib/sec])
Note the following approximate modern concert pitch levels of a chromatic scale beginning one octave below A = 440 Hz. Some qin players today insist that the first (lowest) string on a qin should be tuned to this modern concert C (262 Hz on this chart). However, there is no historical evidence to support this claim.

Pitch A A# B C C# D D# E F F# G G# A
Hertz 220 233 247 262 277 294 311 330 349 370 392 415 440

In standard tuning I generally tune my lowest string to between modern A# and B. This is about as high as they can be tuned without causing the strings regularly to break. With silk strings this can be a bit higher when the humidity is low; strings break more easily in high humidity, especially when the humidity is constantly changing. The actual pitch also depends on the quality of the silk strings, but often the first string has a pitch of around 60 Hz, which lies between the modern A sharp and B. 60 Hz is also a figure convenient for the discussion here. This makes the sixth string 120 Hz and the harmonic played on the seventh position (another octave higher) 240 Hz.

On a separate page there is some discussion of how this affects playing with other instruments.
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3. "Sanfen sunyi" method compared to "Pythagorean" method (「三分損益法」相比於「畢達哥拉斯法」)
Sanfen sunyi literally means "thirds-divide subtract-add"; the Pythagorean method is also called the "cycle of fifths method". However, both of these systems are based on the same mathematical relationships: 2/1 for the octave and 3/2 for the fifth interval; and both seem to give the same overall results. The mathematical relationships themselves can be considered in terms either of frequency (vibrations per second) or of wave length: the results are reversed (2/1 gives an octave higher, 1/2 gives an octave lower), giving in one case ascending pitch and the other descending, but the relations remain the same. As will be seen below, there might also be differences in starting point, but in practical terms once again the results are the same.

The earliest statement of sanfen sunyi is said to be the following from the book of Guanzi, Chapter 58 (it follows a discussion of note associations; translation from W. Allyn Rickett, Guanzi; Princeton U. Press, p. 263):

凡將起五音,凡首,先主一而三之。四開以合九九,以是生黃鐘小素之首以成宮,三分而益之以一,為百有八,為徵,不無有三分而去其乘,適足,以是生商,有三分而復於其所,以是成羽,有三分去其乘,適足,以是成角。

To create the sounds of the five-note scale, first take the primary unit and multiply it by three. Carried out four times, this will amount to a combination of nine times nine (or eighty-one), thereby establishing the pitch of the huang zhong 黃鍾 (鐘) tube in the lesser su 小素 scale and its gong note. Adding one-third (27) to make 108 creates the zhi note. Subtracting one-third (36) results in the appropriate number (72) for producing the shang note. Adding one-third (24) is the means to achieve the yu note (96). Subtracting one-third (32) results in the appropriate number (64) for achieving the jiao note.

This can be summarized as follows:

81=gong, 108=zhi, 72=shang, 96=yu, 64=jiao.

Two things should be pointed out about this derivation. One is that it is generally considered to be a later insert into the original book; the other is that it begins its calculations with yi (add), not sun. A similar calculation by sunyi instead of yisun can be summarized as follows:

81=gong, 54=zhi, 64=jiao, 72=shang, 48=yu.

The only real difference between these two is that in the first one gong is the middle pitch of the series, in the latter one gong is the lowest (i.e., longest in wave length).

Another way of stating this sanfen sunyi sequence is by practions, as follows:

1 x 2/3 = 2/3; x 4/3 = 8/9; x 2/3 = 16/27; x 4/3 = 64/81

Multiplying this sequence (1 -- 2/3 -- 8/9 -- 16/27 -- 64/81) by 405 gives:

405 - 270 -- 360 -- 240 -- 320

Rearranging this in ascending order gives:

240 -- 270 -- 320 -- 360 -- 405

which is identical to the first five notes of the sequence resulting from the Pythagorean calculations, summarized in Section 1.A. by giving the following sequence, based on do being 320 vibrations per second (approximately Western d#):

240 (So) -- 270 (La) -- 320 (do) -- 360 (re) -- 405 (mi) -- 480 (so) -- 540 (la)

In sum, whichever way pitch is analyzed, "Pythagorean tuning" (or 五度相生律 "cycle of fifth tuning") seems to be no different from sanfen sunyi. Some aspects of the processes used may be a bit different but the end results seem to be the same. There is some evidence that the Chinese were aware of note derivation through the cycle of fifths earlier than was the Greek philosopher Pythagoras (6th. BCE). Perhaps this is one reason that some analysts of Chinese music do not like to use the term "Pythagorean tuning". It is not clear to me on what basis some people claim that the two systems give different results.

For a Chinese comparison of sanfen sunyi with Pythagorean tunings see 三分損益法與五度相生律之比較. The sanfen sunyi calculations for note origins are said to date back at least to book of Guanzi 中文, Chapter 58:

There is a complete translation of Guanzi by W. Allyn Rickett, Princeton University Press.
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4. Meantone tuning (sometimes written "mean tone"; 中庸全音調律法)
Another tuning system, often used in early Western music, follows what is called meantone temperament (中庸全音律), in particular quarter-comma meantone. This and the other tunings described here all try to resolve the tuning/intonation problems that occur when playing harmonic or polyphonic music. As a soloist who in the traditional repertoire would never play more than two notes simultaneously, the qin player did not usually have to deal with such issues.
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5. Just intonation tuning (純律定音)
For just intonation (sometimes also called natural tuning) the Chinese term is 純律 chunlü ("pure tones").
Section 1.B. outlines its basics.
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6. Equal temperament tuning (平均律定音)
"Equal temperament" is translated literally, 平均律 pinjun lü or 十二平均律 shi'er pingjun lü. Equal temperament, though apparently discovered by the Ming prince
Zhu Zaiyu, was never used on the qin. However, Section 1.C. has some explanation.
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7. Actual intonation used in early qin melodies
If one
compares the old and new systems, it is easy to see that the lack of precision in the old system would have prevented precise indications of mictrotones; on the other hand, until modern times there seems to have been nothing written about the potential for the decimal system to indicate microtones. Thus Tse Chun Yan's analysis of variant intonations, as with my own comments, must depend exclusively on examining the tablature itself, not on reading theoretical documents.
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8. Differences between Pythagorean and just intonation tuning
The following chart shows the mathematical differences between notes organized according to the Pythagorean relations and according to just intonation; as mentioned above, equal temperament was not used on the qin. Do (1) can be any note; the number of vibrations per second (v/s) of any note on the chart is then the number resulting from multiplying the number of v/s for do by the fraction indicated. The Adjustments row shows how much mi, la and ti are lowered to give the simpler fractions of just intonation.

Note
Do
Re
Mi
Fa
Sol
La
Ti
Do'
Pythagorean
1
9/8
81/64
4/3
3/2
27/16
(81/48)
243/128
2
Adjustments
 
 
80/64
 
 
80/48
240/128
 
Just Intonation
1
9/8
5/4
4/3
3/2
5/3
15/8
2

Both the Pythorean and just intonation systems reflect the apparent fact that the human ear finds consonance in sounds which are related to each other by simple mathematical relationships. The simplest of these are 2/1 and 3/2. The ratio of vibrations per second necessary to form what we call an octave is 2/1, and for a fifth is 3/2. Thus if A (the "main note": fundamental note, here called tonic) is 440 vibrations per second (v/s), A' (A up one octave) will be 880 v/s and E (a fifth higher) will be 660 v/s. B (a fifth higher than E) will then be 990 v/s; to bring this E back into the octave of A to A' you divide the vibrations by half, giving 445 v/s.

Related to this, to the human ear the closest sound to, for example, an A is not a G sharp or B flat but A an octave higher or lower; the next closest sound is a fifth. This may explain why in both Western music and Chinese music when the tonal center changes it most often seems to change to the fifth (see the summary chart in the article on mode). In fact, these intervals seem fundemental to almost all music systems around the word, the major exceptions perhaps being tuned-percussion ensembles such as those in Southeast Asia.

The cycle of fifths takes this ratio of tonic to fifth and extends it through the octave. Thus, if you make C the tonic, then start the cycle at F (a fifth below C), the first seven notes will appear in the sequence F to C to G to D to A to E to B . Adjusting notes which have gone outside the octave C to C', and putting them in order, gives for the first five notes C D E G A, the common pentatonic (five note) scale; adding the last two notes makes the sequence C D E F G A B, the common heptatonic (seven tone) scale, also called diatonic because it includes half tones.

The "Pythagorean" line in the chart above gives the relationship of notes derived through the cycle of fifths. The sequence in Pythogorean order would be as follows

Note
fa
Fa
Do
Sol
Re'
Re
La
Mi'
Mi
Ti
Fractions
2/3
4/3
1
3/2
9/4
9/8
27/16
81/32
81/64
243/128

As can be seen, some of the fractions are very complex. For this and other reasons the ratios were changed to the closest simple fractions, as shown on the chart line Just Intonation. Still later (the Ming dynasty prince Zhu Zaiyu was apparently the first to work this out) the notes were adjusted so they increased one by one according to a logarithm which made them equal intervals from each other. In the West this equal temperament made it possible for large numbers of instruments to play together on harmonic melodies which often changed key. (See also Comparing different tuning systems.)
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9. "Harmonics" and "Overtones"
In both English and Chinese there is some inconsistency in the use of these terms. Thus in the Harvard Concise Dictionary of Music (1978) the word "harmonics" has a similar definition to the one given here, but the word "harmonic" is said to be the same as what are here called "overtones". Here only two terms are used, as follows:

  1. "Harmonics": in music practice a sound produced by lightly touching a string while either bowing or plucking it is what is here called a "harmonic".
  2. "Overtones": in acoustic theory every musical sound has a fundamental pitch: for example, the fundamental pitch of the note called "modern concert A" is said to have a frequency of 440 Hz (vibrations per second). However, this frequency actually describes a pure tone that can only be produced scientifically. In nature each note is colored by a series of extra frequencies that are related in some manner to the fundamental. These extra frequencies are what are here called the "overtones".

The main reasons for the confusion between these terms are probably as follows. First, the two concepts are related mathematically but are used differently in different contexts; this is briefly discussed below. Second, precise definitions of overtones divide them into two types, "harmonic overtones" and "inharmonic overtones". An additional factor may perhaps be the fact that although on the qin extended passages in harmonics are very common, in Western music such passages are quite rare.

Harmonics and overtones: a brief discussion of their mathematical relationships.

  1. Harmonics
    In music a harmonic is produced by lightly touching a string while either bowing or plucking it. With a stopped sound only the part of the string between where it is stopped and the side where it is plucked vibrates. With a harmonic the whole string vibrates. This means that to produce a harmonic the string can only be touched in those places where doing so still allows the whole string to vibrate. These places are the "harmonic nodes", located in places that divide the string into simple fractions. The 13 studs (hui) on a qin mark the places that divide the string into the following fractions of the overall string length: 1/2, 1/3, 1/4, 1/5, 1/6, and 1/8. The markers at 2/3, 3/4, 2/5, 3/5, 4/5, 5/6 and 7/8 duplicate the above sounds. Thus, harmonics played at 2/3 produce the same pitch as those played at 1/3: in both cases the string is vibrating in 3 equal segments ("loops"); likewise 3/4 is the same as 1/4, 7/8 the same as 1/8, and 2/5, 3/5 and 4/5 all the same as 1/5. Note that the basic (fundamental) sound is 1: the sound of the open string.
  2. Overtones
    As mentioned above, in nature each note consists of a fundamental frequency plus a series of extra frequencies that are related in some manner to the fundamental. For practical reasons these extra frequencies are here simply called "overtones". As described more precisely in the Wikipedia article "
    Overtone",

    An 'overtone' is a partial (a "partial wave" or "constituent frequency") that can be either a harmonic or an inharmonic. A harmonic is an integer multiple of the fundamental frequency. An inharmonic overtone is a non-integer multiple of a fundamental frequency.

    By this definition the "first harmonic" is related to the fundamental by the multiple 1x1, so they are the same. The second harmonic is 2x1, third harmonic 3x1 and so forth. This mathematical relationship of the harmonic overtones to the fundamental sound is same as the mathematical relationship of the string divisions that produce the harmonics played on a qin. This can be seen by inverting the six fractions listed above as defining the divisions of the string that produce harmonics: 2/1, 3/1, 4/1, 5/1, 6/1 and 8/1 (note that the other seven fractions duplicate these six). Overtones whose frequencies have mathematical relationships other than these are called "inharmonic". The difference between the harmonics played on a qin and what are called "harmonic overtones" is that the qin harmonics are discreet sounds, while the harmonic overtones all occur together in one note.
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10. The main problem with the old system was that it was often used imprecisely and there were inconsistencies. These problems seem to increase towards the end of the Ming dynasty. Thus, for example, the handbook Zhenchuan Zhengzong Qinpu usually indicates all the intermediate finger positions with the word "half" (半 ban). If one assumes the melodies are idiomatically similar to those described in earlier tablature, then this is not a problem. However, it also makes it difficult to do research into such issues as whether there were in fact modal developments taking place at that time, or if there were regional (or personal) differences in intonation. (It should be noted that the decimal system can only be useful for this if it used precisely according to theory; in this regard see a comment above.
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11. Wai, 13.1 and 13
There is a clear example near the end of Section 2 of Shuixian Qu (my transcription m.67).
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See Modality in Early Ming Qin Tablature,
or return to the Guqin ToC or to Analysis.