I want to learn about historic chinese tunings. We're going to look at the Qin or Guqin ("ancient stringed instrument/zither"). Qin and guqin are pronounced with a "ch" sound for the q. The instrument used to have silk strings, but now people sometimes use steel and nylon. I'm learning about it from the site silkqin.com, authored by John Thompson. I've looked at this occasionally for years, but I usually bounce off of it. I find that I can't learn from it simply by reading it - one has to work through it. But I'm finally working through it now.
The guqin has markers on the neck called hui. I think they're usually flat inlays but they can also protrude as studs - maybe cheap instruments have them painted on. They're placed at the postions of various harmonic nodes. They're usually made of jade, or 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 fretboard. 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 mulitple places you can play the same harmonic sound on a string - if you press a string lightly at 1/5, 2/5, 3/5, 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 for the middle node, instead the loudest sound is the 2nd harmonic (though the fourth harmonic contributes to the sound too).
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 - 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, it's just easy to put a hui there since that's where a strong harmonic can be found.
So what are the actual fingerings used, and what are their frequency ratios? I don't know. Still learning.
Let's start with the strings. 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 hui, like the open 7th string should match the 4th strig 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 by a half step.
He also described several tunings that differ from the standard Pythagorean tuning above by adjusting strings by a Pythagorean comma (though I think he means a syntonic comma in several places (which is basically audibly indistinguishable from the Pythagorean one), since that's what needed for the arithmetic to work out).
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Okay, onto fingerings. One one page of SilkQin, "Taiyin Daquanji 1: Folio 1C : Miscellaneous Qin Information (I/48a-49a) 2", 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 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 that's how we'll figure out the actual frequency ratios.
[7.3, 7.6, 7.9, 8.5, 9.0, 9.4, 10.0, 10.8, 12.3, 13.1, 13.5]
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