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| Tuning a Qin / Qin Tunings / Modality in Early Ming Qin Tablature | 首頁 |
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"Problems" with just intonation tuning
with especial reference to SQMP 1 |
古琴純律調弦法問題
|
All intonation systems have inherent problems when applied to actual music. Here, for example, are some of these from among the three best-known systems.
| Pitch | A | A# | B | C | C# | D | D# | E | F | F# | G | G# | A' | C' | |
| Just intn ratios A | 1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2 | |||||||
| Just intn ratios C | 5/6 | 1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2 | ||||||
| Hertz (Hz) | 220 | 233 | 247 | 262 | 277 | 294 | 311 | 330 | 349 | 370 | 392 | 415 | 440 | 524 |
As mentioned above, using a just intonation system when playing qin may at times require changing the finger positions used, in particular if the tonal center changes within a piece. An example of this can be seen in the yu mode melody from 1525, Chun Si. Here, if just intonation follows the tonal centers consistently, when la is the tonal center (1) then re (higher by a fourth) will have a higher frequency than do by a ratio of 4/3; and mi (higher again but by a second) will be higher than re by a ratio of 9/8 (4/3 ÷ 3/2). However, if it is felt that do has become the tonal center (1) then re will be higher than do by 9/8 but mi will be higher than re by 10/9 (9/8 ÷ 5/4) instead of 9/8: the finger will need to be in a slightly different place on the qin. There is some further comment about how this may affect the written finger positions in this footnote about hui markers .
Keep in mind that the previous paragraph concerned theory, showing how strict calculations can change note positions. Where this occurs in a melody such as the one just mentioned, players quite likely will make adjustments by following their ears and changing the finger positions according to their taste (and/or skill). Unisons and octaves usually remain just that, but for other notes players may make a variety of adjustments according to taste. Bad adjustments will result in the music seeming dissonant and/or out of tune (at least to some people); good ones will make the music more rich and colorful (again, depending on taste).
This gets to the heart of this intonation "problem". It has been argued tha writers in ancient times were very concerned with this issued. For better or for worse, I have not studied this in detail as I have not seen how it is related to my main interest, which is trying to do historically informed performances of old guqin music. It has been pointed out that the decimal system for telling players where to stop the string when playing allows for very stating very precise intonation. However, there is very little evidence this was ever done, and in fact this decimal system seems not to have been introduced until around the 17th century, when it was very gradually introduced (details).
Perhaps a particular point can be made about the fact that the most obvious cases of mixed intonation occur when the passage is in harmonics. Here the problem can only be "resolved" by re-tuning the qin, whereas with stopped sounds very slight adjustmentss can be made to affect the intonation. As just mentioned, this is something that can be indicated by the modern (i.e., after 1600) method of indicating finger positions, but was that ever done?
Perhaps it is more likely that the good qin players, especially the creative ones who made new pieces, were not concerned with the practical implications of music theory. Today when players playing from old tablature find a "strange" sound they tend to change it, or perhaps just play it quickly.
Why not instead think, What a strange sound - very creative! Why not emphasize that?
Studying an example of the "problem":
measure 1 of
Yi Lan
Some people have suggested that by changing some of the Pythagorean tuning ratios into just intonation tuning ratios one can avoid dissonances that seem to occur 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 harmonic notes on the qin.
As was pointed out in Qin Tunings, there is no absolute pitch standard for qin tuning. 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.
The first two notes of the melody
Yi Lan
(SQMP, Folio II), are just intonation mi followed by Pythagorean mi: a dissonance that would be unacceptable to the standards of Western music of the
Common Practice Period. When I was first working on my reconstruction of this melody I too thought it could be a mistake, but eventually I found that if I play it right, it feels like a beautiful expression of the richness possible when playing the guqin with silk strings. This feeling was reinforced when I found a number of other pieces which show a same or similar phenomenon.3
As can be see by looking at the Yi Lan tablature and also the chart below, the melody begins with a note played in the 11th position on the first string (300 Hz; on the chart below multipied by 4 to 1200 so as to avoid large fractions) followed by a note on the ninth position/5th string (303.75 Hz, here considered as 1215). Should we try to bring these into alignment one by re-tuning the fifth string lower one comma (from the 81/64 of standard tuning to the just intonation interval 80/64)? This would change the open strings sounds of Table 1 (standard tuning) to become as shown in Table 2. (This argument then continues to this conclusion at Table 5):
Table 1: Standard Tuning
(Pythagorean tuning for the open strings; for actual string frequencies divide by 4 [q.v.])
| String/pitch/fraction of do | ||||||||||||||
| 1. (So [5]) = 3/4 (48/64) | ||||||||||||||
| 2. (La [6]) = 27/32 (54/64) | ||||||||||||||
| 3. (do [1]) = 1 (64/64) | ||||||||||||||
| 4. (re [2]) = 9/8 (72/64) | ||||||||||||||
| 5. (mi [3]) = 5/4 (80/64) | ||||||||||||||
| 6. (so [5]) = 3/2 (96/64) | ||||||||||||||
| 7. (la [6]) = 27/16 (108/64) | " |
From each other: 204-294-204-204-294-204
from So:
204-498-702-906-1200-1404
Table 2: Tuning with fifth string lowered one comma from
Standard Tuning
(partial just intonation tuning for open strings; for actual string frequencies divide by 4 [see standard])
| String/pitch/fraction of do | ||||||||||||||
| 1. (So [5]) = 3/4 (48/64) | ||||||||||||||
| 2. (La [6]) = 27/32 (54/64) | ||||||||||||||
| 3. (do [1]) = 1 (64/64) | ||||||||||||||
| 4. (re [2]) = 9/8 (72/64) | ||||||||||||||
| 5. (mi [3]) = 5/4 (80/64) | ||||||||||||||
| 6. (so [5]) = 3/2 (96/64) | ||||||||||||||
| 7. (la [6]) = 27/16 (108/64) | " |
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 Yi Lan, 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 3: 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 [see standard])
| String/pitch/fraction of do | ||||||||||||||
| 1. (So [5]) = 3/4 (48/64) | ||||||||||||||
| 2. (La [6]) = 5/6 (53.3/64) | ||||||||||||||
| 3. (do [1]) = 1 (64/64) | ||||||||||||||
| 4. (re [2]) = 9/8 (72/64) | ||||||||||||||
| 5. (mi [3]) = 5/4 (80/64) | ||||||||||||||
| 6. (so [5]) = 3/2 (96/64) | ||||||||||||||
| 7. (la [6]) = 5/3 (106.7/64) | " |
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, Yi Lan, 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 4: 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 [see standard])
| String/pitch/fraction of do | ||||||||||||||
| 1. (So [5]) = 3/4 (48/64) | ||||||||||||||
| 2. (La [6]) = 5/6 (53.3/64) | ||||||||||||||
| 3. (do [1]) = 1 (64/64) | ||||||||||||||
| 4. (re [2]) = 9/8 (72/64) | ||||||||||||||
| 5. (mi [3]) = 5/4 (80/64) | ||||||||||||||
| 6. (so [5]) = 3/2 (96/64) | ||||||||||||||
| 7. (la [6]) = 5/3 (106.7/64) | " |
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 Yi Lan, 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 5: Standard tuning with the 3rd string raised one comma.
(partial just intonation tuning for open strings; for actual frequencies divide by 4 [see standard])
divide by 4)
| String/pitch/fraction of do | ||||||||||||||
| 1. (So [5]) = 3/4 (48/64) | ||||||||||||||
| 2. (La [6]) = 5/6 (53.3/64) | ||||||||||||||
| 3. (do [1]) = 1 (64/64) | ||||||||||||||
| 4. (re [2]) = 9/8 (72/64) | ||||||||||||||
| 5. (mi [3]) = 5/4 (80/64) | ||||||||||||||
| 6. (so [5]) = 3/2 (96/64) | ||||||||||||||
| 7. (la [6]) = 5/3 (106.7/64) | " |
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 Yi Lan as well as
three other pieces:
Table 6: Conflicts resolved and generated through re-tuning by
commas
| Piece/Section | Conflicts (pos = position; str = string) | ||
| Yi Lan, Section 1
|
4 & 5 6 1 1 |
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 4 1 3 |
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 |
21a1 20b10 & 21a1 21a1 21a7 & 6 |
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 6 & 7 5 6 & 7 |
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) |
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)
3.
Other melodies with both just intonation and Pythagorean thirds
An incomplete list includes 1425
Meihua Sannong (Section 5),
Liezi Yu Feng (Section 5) and Xiao Xiang Shui Yun (Section 1) plus 1525 Shen Ren Chang (Section 6) and Feng Qiu Huang (Section 8).
(Return)
Go to the related article Tuning the qin
or return to the Guqin ToC
or to analysis.