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Tuning a Qin   /   Qin Tunings   /   Modality in Early Ming Qin Tablature 首頁
Problems with just intonation tuning
  with especial reference to SQMP 1
古琴純律調弦法問題
 

Some people have suggested using just intonation tunings to avoid dissonces in some early qin melodies.2 The following is an account of problems met in trying to avoid such dissonances by re-tuning the qin. It mentions all the ways I can think of to try to apply just intonation tuning to the qin.

As a reminder, the modern concert tuning system generally has A=440 vib/sec (Hertz/Hz), yielding a scale with the following approximate pitch levels:

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

It was pointed out in Qin Tunings that there is no such pitch standard for the qin, On my instruments, using standard tuning (Sol La do re mi sol la, also considered as Do Re Fa Sol La do re), the lowest string is often tuned to somewhere between B flat (or A sharp) and B. For ease of comparison this is taken here to be 60 Hz (raised two octaves makes it 240 Hz, between A# and B on the chart above). However, in the following discussion, in order to avoid use of fractions, these frequences are again multiplied by four, in effect raised two more octaves.

Yilan (SQMP, Folio II), begins with an apparent dissonance: a note played in the 11th position on the first string (300 Hz; on the chart below considered as 1200) is followed by a note on the ninth position/5th string (303.75 Hz, here considered as 1215). To bring these into alignment one can re-tune the fifth string lower one comma (from the 81/64 of standard tuning to the just intonation interval 80/64) so that the open strings sound as follows:

Table 1: Tuning with fifth string lowered one comma from Standard Tuning
                (partial just intonation tuning for open strings; for actual string frequencies divide by 4)
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)
240
 
 
1200
960
720
 
480
 
re'
so'
ti'
 
 
2. (La [6]) = 27/32 (54/64)
270
 
 
1350
1080
810
 
540
 
mi'
la'
d#"
 
 
3. (do [1]) = 1 (64/64)
320
 
 
1600
1280
960
 
640
 
so'
do"
mi"
 
 
4. (re [2]) = 9/8 (72/64)
360
 
 
1800
1440
1080
 
720
 
la'
re"
fa"
 
 
5. (mi [3]) = 5/4 (80/64)
400
 
 
2000
1600
1200
 
800
 
ti'
mi"
so#"
 
 
6. (so [5]) = 3/2 (96/64)
480
 
 
2400
1920
1440
 
960
 
re"
so"
ti"
 
 
7. (la [6]) = 27/16 (108/64)
540
 
 
2700
2160
1620
 
1080
 
mi"
la"
do#'
"
 


Intervals in cents:

From each other:       204-294-204-182-316-204
from So:                 204-498-702-884-1200-1404

Now, however, the 10th position/5th string (1600; see the 4th and 5th phrases of Yilan, Section 1) and the 9th position/7th string (1620; 5th phrase) are no longer in unison.

This can be resolved by re-tuning la as well as mi (2nd, 5th and 7th strings in all) down a comma, giving standard just intonation tuning for all the open strings, as follows :

Table 2: Standard tuning with 2nd and 7th as well as 5th string lowered one comma
            (complete just intonation tuning for open strings; for actual frequencies divide by 4)

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)
240
 
 
1200
960
720
 
480
 
re'
so'
ti'
 
 
2. (La [6]) = 5/6 (53.3/64)
267
 
 
1333
1067
800
 
533
 
mi'
la'
d#"
 
 
3. (do [1]) = 1 (64/64)
320
 
 
1600
1280
960
 
640
 
so'
do"
mi"
 
 
4. (re [2]) = 9/8 (72/64)
360
 
 
1800
1440
1080
 
720
 
la'
re"
fa"
 
 
5. (mi [3]) = 5/4 (80/64)
400
 
 
2000
1600
1200
 
800
 
ti'
mi"
so#"
 
 
6. (so [5]) = 3/2 (96/64)
480
 
 
2400
1920
1440
 
960
 
re"
so"
ti"
 
 
7. (la [6]) = 5/3 (106.7/64)
533
 
 
2667
2133
1600
 
1067
 
mi"
la"
do#'
"
 


Intervals in cents:

From each other:       182-316-204-182-316-182
from So:                 182-498-702-884-1200-1382

Now, however, one has conflicts between the 9th position/4th string (1080, Yilan, Section 1, phrase 6) and the 10th position/2nd string (1067, same phrase).

Next one can try to resolve this by also re-tuning re (the 4th string) down a comma, as follows:

Table 3: Standard tuning with 2nd, 4th, 5th and 7th strings lowered one comma
            (partial just intonation tuning for open strings; for actual frequencies divide by 4)

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)
240
 
 
1200
960
720
 
480
 
re'
so'
ti'
 
 
2. (La [6]) = 5/6 (53.3/64)
267
 
 
1333
1067
800
 
533
 
mi'
la'
d#"
 
 
3. (do [1]) = 1 (64/64)
320
 
 
1600
1280
960
 
640
 
so'
do"
mi"
 
 
4. (re [2]) = 9/8 (72/64)
356
 
 
1778
1422
1067
 
711
 
la'
re"
fa"
 
 
5. (mi [3]) = 5/4 (80/64)
400
 
 
2000
1600
1200
 
800
 
ti'
mi"
so#"
 
 
6. (so [5]) = 3/2 (96/64)
480
 
 
2400
1920
1440
 
960
 
re"
so"
ti"
 
 
7. (la [6]) = 5/3 (106.7/64)
533
 
 
2667
2133
1600
 
1067
 
mi"
la"
do#'
"
 


Intervals in cents:

From each other:       182-316-182-204-316-182
from So:                 182-498-680-884-1200-1382

Now there are new conflicts, for example 10th position/4th string (1422) and the 9th position/6th string (1440), which both occur in Yilan, Section 1, phrase 1.

We can solve this latter dissonance by also lowering the 6th string and its octave the 1st string, but this means lowering six of the seven strings, and it is easier instead simply to raise the remaining string, the 3rd. The result of this re-tuning is as follows.

Table 4: Standard tuning with the 3rd string raised one comma.
            (partial just intonation tuning for open strings; for actual frequencies divide by 4)

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)
240
 
 
1200
960
720
 
480
 
re'
so'
ti'
 
 
2. (La [6]) = 5/6 (53.3/64)
270
 
 
1350
1080
810
 
540
 
mi'
la'
d#"
 
 
3. (do [1]) = 1 (64/64)
324
 
 
1620
1296
972
 
648
 
so'
do"
mi"
 
 
4. (re [2]) = 9/8 (72/64)
360
 
 
1800
1440
1080
 
720
 
la'
re"
fa"
 
 
5. (mi [3]) = 5/4 (80/64)
405
 
 
2025
1620
1215
 
810
 
ti'
mi"
so#"
 
 
6. (so [5]) = 3/2 (96/64)
480
 
 
2400
1920
1440
 
960
 
re"
so"
ti"
 
 
7. (la [6]) = 5/3 (106.7/64)
540
 
 
2700
2160
1620
 
1080
 
mi"
la"
do#'
"
 


Intervals in cents:

From each other:       204-316-182-204-294-204
from So:                204-520-702-906-1200-1404

Now we have eliminated the latest dissonance, but we have re-created the same dissonance we started out with, between the 11th position/1st string (1200) and the 9th position/5th string (1215).

Going through this process with other pieces produces similar results. Here is a step-by-step listing of the attempts with Yilan as well as three other pieces:

Table 5: Conflicts resolved and generated through re-tuning by commas

Piece/Section
phrase
table
Conflicts (pos = position; str = string)
Yilan, Section 1
 
 
 
 
1
4 & 5
6
1
1
4
5
6
7
8
11th pos/1st str (1200) & 9th pos/5th str (1215)
10th pos/5th str (1600) & 9th pos/7th str (1620)
9th pos/4th str (1080) & 10th pos/2nd str (1067)
10th pos/4th str (1422) & 9th pos/6th str (1440)
Same as in Table 4
Tianfeng Huanpei, Section 2
 
 
 
 
3
3
4
1
3
4
5
6
7
8
11th pos/1st str (1200) & 9th pos/5th str (1215)
10th pos/5th str (1600) & 9th pos/7th str (1620)
9th pos/4th str (1080) & 10th pos/2nd str (1067)
10th pos/4th str (1422) & 9th pos/6th str (1440)
Same as in Table 1
Gufeng Cao
(No sections, so reference
is to double page [a=left;
b=right] and line of the
original SQMP tablature
21a6 & 5
21a1
20b10 & 21a1
21a1
21a7 & 6
4
5
6
7
8
11th pos/3rd str (1600) & 9th pos/2nd str (810)
7th pos/5th str ( (800) & 9th pos/7th str (1620)
9th pos/4th str (1080) & 10th pos/2nd str (1067)
10th pos/4th str (1422) & 9th pos/6th str (1440)
12th pos/3rd str (1944) & 10th pos/6th str (1920)
Bai Xue, Section 5
 
 
 
 
3
3
6 & 7
5
6 & 7
4
5
6
7
8
11th pos/3rd str (1600) & 10th pos/5th str (1620)
12th pos/7th str (3240) & 10th pos/5th str (1600)
10th pos/7th str (2133) & 12th pos/4th str (2160)
12th pos/6th str (2880) & 10th pos/4th str (1422)
12th pos/3rd str (1944) & 10th pos/6th str (1920)


Still other pieces produce similar conflicts, and 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 all such dissonances. 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.

 
Footnotes (Shorthand references are explained on a
separate page)

1. This page was originally written as an appendix to Qin Tunings.
(Return)

2. Using tuning to avoid dissonances
Others have argued for just intonation using only theoretical rationale, without regard to the fact that it may cause its own dissonances. Usually this is part of an attempt to explain what seem to me simply to be inconsistences in the tablature. One example would be stating that there is a difference in Ming tablature between "below 9" and "above 10". From my experience, this argument requires the assumption that the tuning basis of the instrument changes in the middle of a piece. This does not mean retuning the instrument, but saying that "below 9" is using one mode where the mathematical calculations lead to a slightly higher note, whereas "above 10" means that the mode has changed and so the interval as calculated forms a lower note. Such arguments seem to require the assumption that the music was written by music theorists rather than qin players. My explanation is usually connected to the fact that many melodies were copied a number of times over the years. From one copy to the next some notes or passages have usually changed. Where the piece has not changed, the writer may simply copy what was written before. Where the piece has changed, the writer may use a different way (e.g., "below 9" instead of "above 10") to indicate the same note.

If I am ever able to derive any logic from the complex modal arguments I will certainly make some adjustments to what I have written about this subject so far.
(Return)

Go to the related article Tuning the qin
or return to the Guqin ToC or to analysis.