Tuning The Chinese Guqin

In this chapter, we're going to look at the tuning of a Chinese zither called the qin or guqin ("ancient" qin). Qin and guqin are pronounced with a "ch" sound for the "q". The website SilkQin.com, authored by John Thompson, taught me everything I know about it. 

The guqin is a long box with long strings, traditionally made of silk. The guqin has markers on the neck called hui. Sometimes they're flat inlays and sometimes they protrude as studs. The hui are placed at the positions of various harmonic nodes. They're usually made of jade, gold, or sea shell.

Let's talk about harmonics and harmonic nodes. If you touch a string lightly at a hui and then pluck the string, the plucked string can vibrate on both sides of your lightly touching finger, but with a limited set of vibrational modes compared to a full length / open string. This is in contrast to playing "stopped" notes, where your finger presses the string all the way down to the instrument body. When you play a stopped note, the string's length is effectively shortened and the new fundamental frequency is proportional to the inverse of the string length, just as it was for the open string. Every place that you press a stopped note will give you a different sound on that string. In contrast, there are multiple places you can play the same harmonic sound on a string - if you press a string lightly at 1/5, 2/5, 3/5, or 4/5 of its length, you're going to get the 5th harmonic at all of them, which has a frequency five times the fundamental frequency, i.e. five times the open string frequency. If you press lightly at 1/4, 2/4, 3/4 of the string length, the outer two nodes will give you the fourth harmonic, but the middle node will have the 2nd harmonic as its loudest spectral component (though the fourth harmonic contributes to the sound too). You can just reduce the fraction as look at the denominator to figure out the harmonic.

The guqin has 13 hui markers along its neck. Guqin players don't play harmonics nearly as much as they play stopped notes; the hui are mostly a way to guide you on where to place your fingers for stopped notes - like halfway between the 6th and 7th hui, for example. The player doesn't sit at the center of the strings of the instrument - more to their right, and the hui are numbered from right to left, so that the most leftward hui for the player is the 13th.

The hui are placed at [1/8, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 7/8] of the string length. The harmonics associated with these are simply the denominators of those fractions: 

[8, 6, 5, 4, 3, 5,) 2, (5, 3, 4, 5, 6, 8].

These numbers are symmetric around the half-way point, but the pattern doesn't steadily decrease and then increase due to two fifth harmonic nodes straddling the octave.

There are hui at almost all the multiples of 1/5-, and 1/6-, and 1/8-times the open string length, except not at 3/8 and 5/8. I don't know why not. This set happens to also cover multiples of 1/2, 1/3, and 1/4 string lengths.

If you were to play stopped notes at each of the hui, with your left hand fretting and your right hand plucking, you would get frequency ratios as

[8/7, 6/5, 5/4, 4/3, 3/2, 5/3, 2/1, 5/2, 3/1, 4/1, 5/1, 6/1, 8/1]

over the fundamental, with 8/7 at hui 13 and 8/1 at hui 1. These ratios are all 5-limit except for the 8/7, which is 7-limit. I don't think 8/7 is actually used much in Chinese guqin music as a stopped note, it's just easy to put a hui there since that's where a strong harmonic can be found.

So if the hui aren't identical to the fingerings used for stopped notes, what actual fingerings are used, and what are their frequency ratios?

On one one page of SilkQin, called "Taiyin Daquanji 1", Thompson shares a historic source that gives poetic names for the tones at various positions along the neck. Thompson then converts these positions to "decimal" notation. In decimal notation, e.g. 7.3 is between hui 7 and hui 8, and it is in particular 0.3 of the way between them. The hui aren't evenly spaced, so it takes a little bit of figuring to turn this into a string length, but we'll have to do the figuring if we want to know the actual string ratios and frequency ratios. Here are the decimals given by Thompson:

    [7.3, 7.6, 7.9, 8.5, 9.0, 9.4, 10.0, 10.8, 12.3, 13.1, 13.5]

I don't know how he come up with these numbers - like the original Chinese will give a poetic name for the 11th hui, but he gives 10.8 for the decimal notation. I assume that he knows where guqin players actually put their fingers in modern times, and he thinks that 11 was a shorthand for 10.8 in the past, but I wouldn't mind a little more discussion.

Let's just do conversions first. Here are approximate frequency ratios for all of those decimal-notation neck positions, as well as ratios for the hui from 7 to 13.

13.5 : 112c ~ m2 # 16/15

13.1 : 207c ~ AcM2 # 9/8

13 : 231c ~ SpM2 8/7

12.3 : 290c ~ Grm3 # 32/27

12 : 316c ~ m3 # 6/5

11 : 386c ~ M3 # 5/4

10.8 : 408c ~ AcM3 81/64

10 : 498c ~ P4 # 4/3

9.4 : 617c ~ SpA4 # 10/7

9 : 702c ~ P5 # 3/2

8.5 : 791c ~ Grm6 # 128/81

8 : 884c ~ M6 # 5/3

7.9 : 913c ~ AcM6 # 27/16

7.6 : 1004c ~ Grm7 # 16/9?

7.3 : 1099c ~ M7 # 15/8?

7 : 1200c ~ P8 # 2/1

Many of these are dead on, and I think all but two near the end (annotated with question marks) are within 4 cents. Maybe decimals aren't so precise further from hui 13. The Grm7 is off by 8 cents, and the M7 is off by 11 cents. Closer ratios would be 25/14 and 17/9 respectively, but that's pretty suspicious when all the other intervals besides the tritone (and hui 13 with a stopped note frequency ratio of 8/7) are obviously 5-limit. 

If we only look at the decimal-notation string locations from the Taiyin Daquanji page, and not all of the integer hui, then we knock out hui 7, 8, 12, and 13, which are [P8, M6, m3, SpM2]. That removes a few 5-limit ratios, but we still have 5-limit m2, M3, and M7. So it's not like they're putting markers on 5-limit harmonics but then only playing 3-limit stopped notes. It looks like Thompson is actually advocating for 5-limit tuning - or his source was, perhaps, if the decimals are a good representation. Which is cool.

In https://silkqin.com/08anal/tunings.htm#decimal, Thompson gives a few more neck positions / string length ratios in decimal-notation. These include 13.9 as a possible alternative to 13.1 and 13.5, as well as many decimal-notations numbers below 7, which the Taiyin Daquanji page didn't have.

[1, 1.2, 1.4, 1.6, 1.8, 2, 2.3, 2.6, 2.9, 3, 3.2, 3.4, 3.7, 4, 4.2, 4.4, 4.6, 4.8, 5, 5.3, 5.6, 5.9, 6, 6.2, 6.4, 6.7, 7, 7.3, 7.6, 7.9, 8, 8.5, 9, 9.4, 10, 10.8, 11, 12, 12.3, 13.5, 13.9]

The 13.9 decimal at the end is an interesting case. It's very close to the far left end, and may sometimes be used as an alternative to an open string. It's about 22 cents, which is the size for a Pythagorean or Syntonic comma. Want to calcualte it by hand?

Our decimal is 13.9. The 13th hui is at 1/8 of the string length. We want to move from hui 13 toward the next hui by 1/9 of the remaining distance, but 13 is our last hui, so presumably we just go from hui 13 to the very end of the string, giving a string ratio of

    (1/8) - ((1/8) * (9/10)) = 1/80

or a frequency ratio of 80/79 @ 22 cents, instead of the syntonic comma at 81/80. I think that's obviously the syntonic comma as represented to the available precision, and indeed it's only 0.3 cents off.

The [6.7 to 4.2] decimals are the same as the [3.4 to 1.2] decimals, just displaced by octaves. Here's the first of those two sets, octave reduced to fit in [0 to 1200 cents]:

6.7 : 107c ~ m2 # 16/15

6.4 : 221c ~ DeAcA2 # 25/22 (or 17 cents sharp of AcM2 # 9/8)

6.2 : 302c ~ Grm3 # 32/27 (but 8 cents too sharp)

5.9 : 415c ~ AcM3 # (but 7 cents too sharp)

5.6 : 506c ~ P4 # 4/3 (but 8 cents too sharp)

5.3 : 601c ~ ExA4 # 17/12

4.8 : 791c ~ Grm6 # 128/81

4.6 : 884c ~ M6 # 5/3

4.4 : 983c ~ Hbm7 # 30/17 (or 13 cents flat of Grm7 # 16/9)

4.2 : 1088c ~ M7 # 15/8

There are some weird cent values in here - they're not all obviously 3-limit or 5-limit. They're also not far enough away to be obviously something else. Probably we just need more decimal places at certain neck positions in order to zero-in on certain frequency ratios. To check this, we could calculate frequency ratios for each possible decimal position, i.e. if a stopped note played at position 6.4 isn't quite tuned to a just AcM2, we could see how 6.3 and 6.5 compare - maybe 6.4 is the closest you can get with one decimal place.

...

Let's finish by discussing string tuning. There are seven strings on a guqin. Thompson describes the standard tuning in terms of how one string relates to another that's fretted at a certain hui, like the open 7th string should match the 4th string fretted at hui 10, and hui 10 is 4/3 over the open string frequency. If you follow all of these references, you get frequency ratios for the strings, low to high, as

[1/1, 9/8, 4/3, 3/2, 27/16, 2/1, 9/4]

If we call the low string C and use Pythagorean pitch names, this is

[C, D, F, G, A, C, D]

with the last two notes being an octave above the first two. Thompson says that this tuning dominates the modern repertoire, but that there were other tunings historically that differed from this by the adjustment of a string or two strings by a half step.

Thompson also describes tunings that differ from the standard Pythagorean tuning above by adjusting strings by syntonic commas, and also mentions that 12 tone equal temperament was invented in the Ming dynasty, specifically in 1584 by Zhu Zaiyu. So apparently there is a vast history of tuning theory here that we've only scratched. But this has still been an enlightening read.

In short, in the modern era, the quqin is played with 5-limit stopped-note fingerings, sometimes over a Pythagorean string tuning and sometimes over 5-limit string tunings, but there's plenty of scholarship to be done about specific historic tunings and intonations.