• Pythagorean (sanfen sunyi)
  This system is based on the cycle of fifths, whereby pitches are raised by fifths (multiples of 3/2) then kept within the same octave by dividing by 2. However, if carried out through all 12 semitones to the higher octave the result is not the desired 2/1 octave but a note that is higher by about 23 cents (Wiki). This difference is called the "Pythagorean comma". Here the aim is usually to figure out where this comma can be put so that it sounds least intrusive.
• Just intonation (Wiki)
  This system deals with the Pythagorean comma by increasing or decreasing from the Pythagorean intervals in such a way as to give much simpler fractions throughout the diatonic (or even twelve-note) system. A basic problem here is that the resulting unequal intervals between notes may require changing the finger positions when playing a piece. This is discussed further below as well as in this discussion of stopped positions in terms of hui markers.
• Equal temperament (Wiki)
  Equal temperament tries to solve the problem by making all semitones equidistant. The chart in this equal temperament footnote shows the first problem: deciding if you should tune the strings themselves to equal temperament (never known to have been done) or, having tuned the strings to the Pythagorean system, tune the stopped sounds on the string to equal temperament? In any case, the result would unavoidably make many stopped sounds different from harmonics sounds; it thus seems an unlikely choice.

Before going further, here are the just intonation ratios for a diatonic (seven note) scale plus, as a guideline, the approximate pitch levels for 12 notes in the modern concert tuning system based on A=440 vib/sec (Hertz/Hz), using equal temperament (the missing just intonation pitches were omitted because they can have a variety of interpretations):

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).

What does one do, however, if such a mixed intonation occurs in harmonic notes, which cannot be changed so easily?

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 example used here is Yi Lan (SQMP, Folio II), but there are many other pieces which could be discussed as well.³ Yi Lan, however, is particularly interesting because it begins with such 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#'	"

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 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#'	"

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#'	"

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#'	"

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

Piece/Section	phrase	table	Conflicts (pos = position; str = string)
Yi Lan, 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)

 
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.