Cents and Non-cents
Western music is fast because it's not in tune - Terry Riley
Guqin Hui and the Harmonic Series
So in this blog I'm going to discuss the 13 markers found on the guqin that in Chinese are called "hui" 徽. This will include discussion of the acoustic theory from physics behind the hui and discussion of the known history of hui. The hui are typically translated as "studs" which I think is a terrible translation because "studs" to me stick out. Perhaps "markers" would be better and in fact closer to the original meaning. However I will just stick to the Mandarin pronunciation as "hui" (if English speaker - pronounce as whey in curds and whey). The hui are inlaid into the surface above string 1 (see guqin picture below) and are usually mother of pearl although other substances including jade and gold have been used. One other thing - this discussion is free of traditional guqin scholar pitch pipe theory. You can view it either as being based entirely on "maker" design (by unknown guqin makers or designers from the Han dynasty or before) and also based on the theory that those designers understood the integer math behind just intonation.
The hui have two functions: 1. they mark harmonic positions and 2. they are used as part of an X, Y coordinate system in guqin tablature for helping players play notes. The Y part of the system is the open strings from 1 to 7. The hui are the long X axis part of the system to tell you where to press down notes or to play harmonics. Note that by "harmonic" I mean overtone which is the conventional meaning of the term amongst musicians in the English language. I do not mean the physics term that allows you to call an open string a "harmonic". So when I say the first harmonic I mean the 1st overtone (at an octave), 2nd harmonic, the 2nd overtone (a perfect fifth) etc. In terms of lengthwise left-hand fingering position, hui tell you where to put your left hand fingers to play a note. The left hand goes to a hui (or inter-hui) position and the right hand plucks one of the seven strings. Here we are really only interested in the harmonics aspect and not in the notion of pressed notes on the gugin. A hui we go.
To start with feel free to spend some time looking at: wikipedia on harmonic series
So go and twang string 1 on the guqin which we shall give its "modern" name of C2 at frequency 65.4 Hertz (HZ) assuming 440 HZ for "concert A" and that we are simply picking out C2 at 65.4 hertz because it is convenient. Note that C2 on a piano is an absolute pitch name at this time and place because we can figure out a known number of vibrations to go with it (Hertz). However it may also be used in the same sense as a relative note name a la DO RE ME SO LA. I will try and be careful about that if and when it matters. It doesn't matter here because string 1 is assumed at this point to be all by itself - or put another way we are looking at notes generated by string 1 on string 1. I am just going to proclaim that string 1 played open is "C" and if you have it tuned to Bb - no worries. It would not matter if it was A1 for that matter or C at 65.2 HZ. What will matter is interval degrees as for example string 1 hui 5 played as a harmonic is a perfect fifth. This is because the harmonic played at that note is a "G" above the open string which is a "C". So I will use note names a la C G etc.
When we talk about the notion of a harmonic series we are saying that string 1 as C when plucked may be viewed as a composite consisting of a fundamental wave (for example C2) plus overtones at equal integer divisions of that fundamental at various ratios. E.g., the 1st overtone (2nd partial) is at a 2 to 1 ratio (2:1) and can be viewed as being two wavelets half the size of the fundamental that are "waving" at the same time that the fundamental is vibrating. The all important 2nd overtone is at a ratio of 3:1. This is the perfect 5th. The third overtone is at a ratio of 4:1 and is a 2nd octave above the fundamental. The 4th overtone is a major third. The fifth overtone is a perfect fifth that is an octave higher than the 2nd overtone. The sixth overtone is a minor 7th (not marked on the guqin). The 7th overtone is yet another octave that is 3 octaves higher than the fundamental. Overtones go on forever at an increasing integer ratio. However in the real world they all have different strengths and some of the lower overtones are stronger than others. After a while of course the higher overtones are too weak to register at all on the ear.
Note this nice picture (purloined from wikipedia). It shows the 1st 8 "partials" (a partial is the fundamental and any other overtone related to it as generated when we twang our open string). Our picture does not show the 8th partial (1/8) but just imagine one more wave cluster with 7 nodes and 8 wavelets at the bottom of the picture. Note how the partials are increasing in frequency and note where the node points are as for example with the 1/2 specimen the node is at the halfway point.
|Figure 1: Harmonic overtone series - nodes/wave patterns|
So basically the first 8 partials (7 in the above picture) have the following integer ratios, and interval values (Table 1 below). We will assume string 1 is C so we can even give them note names. So for example if string 1 is C then the 1/5 ratio is a major 3rd and we can call it E. If you are paying attention and have not fallen asleep, note that you cannot play this particular E on a normal piano using the keys (you can play it on a guqin). Hopefully that statement has gotten your attention. Also of course you may have figured out that the node positions in the harmonic overtone series picture (barring 1/7) mark hui positions on the guqin.
Table 1: ratios and partial intervals and hui position
1/1 unison (C2) (open string as fundamental of tone)
1/2 first octave (1st overtone) (C3) (hui 7)
1/3 fifth (octave+fifth actually) (G3) (hui 5 and hui 9)
1/4 2nd higher octave (C4) (hui 4 and hui 10)
1/5 major 3rd (E4) (hui 8, 6, 3, 11)
1/6 fifth at 2 octaves higher (G4) (hui 2 and hui 12)
1/7 minor 7th (Bb4) (not found as marker on guqin)
1/8 3rd higher octave (C5) (hui 1 and hui 13)
We can take the above list of partials and state that it applies to all the guqin strings. So for example, if we say string 2 is D, all the intervals are the same. We would have the fundamental as D2, and then the first octave higher would be D3, the perfect fifth would be A, etc. For string 6 since the fundamental is C3, all the note names would simply be an octave higher than string 1.
There are lots of things you can infer about this glimmer of physics from the acoustics of music. Different strengths of the overtones is what gives us "tone color" and so if string 1 is silk or string 1 is nylon-metal as a guqin string they don't sound the same. Notes pressed at different places on the top even if the same pitch don't sound the same either (different wood excitation). This is because their overtones have different strengths. Of course if you played the same C on a piano it doesn't sound the same either. Overtones are what gives us tone color.
Let's take a look at a rather common picture though in terms of guqin - which is some "random" (maybe not) guqin and its 13 mother of pearl markers aka hui.
So this is a picture of the famous Tang dynasty guqin named "Jiu Xiao Huan Pei" (I happen to own a fake copy of it - so this might be a picture of the fake). Keep in mind that this guqin is on the order of 1300 years old (or 20 years old if it's a fake). Now let's consider why there are 13 hui.
When you are a guqin maker and you go to install the hui you measure slightly outside the string path of the 1st string based on making integer divisions of the total string length between the inside of the longyin (on the left in the picture) which is like a guitar nut and the inside of the yueshan (right in picture) which is like a guitar bridge. We can say the string path is the distance between the nut and the bridge. So the 1st hui maker division can be considered to be hui 7 which is right in the middle. It's divide by 2 and is supposed to go at the halfway point. The hui are numbered from 1 to 13 from right to left. Hui 1 is closest to the bridge (yueshan). You install hui at any position along that path that include integer divisions of 2/3/4/5/6/8. If there is a hui in a position (like hui 7) and it is also included in divide by 4 and divide by 8 -- well you don't install it again. Once is plenty. So let's have a nice hui installation table as follows (hui, integer division, interval value based on the fundamental):
hui 1 - divide by 8 - 3rd octave
hui 2 - divide by 6 - 2nd fifth
hui 3 - divide by 5 - major 3rd
hui 4 - divide by 4 - 2nd octave
hui 5 - divide by 3 - perfect 5th
hui 6 - divide by 5 - major 3rd
hui 7 - divide by 2 - octave
hui 8 - divide by 5 - major 3rd
hui 9 - divide by 6 - perfect 5th
hui 10 - divide by 4 - 2nd octave
hui 11 - divide by 5 - major 3rd
hui 12 - divide by 6 - 2nd fifth
hui 13 - divide by 8 - 3rd octave
One thing to point out about this is the hui are symmetrical from 1-7 and 7-13. E.g. hui pairings like 1 and 13, 2 and 12, 3 and 11, etc. are the same note. Now this symmetry and the number of hui and the position of where you play the harmonics are all implicit in the overtone wavelet picture in Figure 1. For example hui 7 is placed at the node between the 2 wavelets. When you lightly touch there you are canceling out the fundamental and our ears hear the 1st overtone (2:1 ratio) which is an octave higher. Hui 3, 6, 8, and 11 which are based on divide by 5 are all placed at the 4 nodes between the 5 "wavelet" markers or put another way, at positions that are 1/5th of the way between nut and bridge. These 4 notes are all major thirds. They are all the same note because they are the same divide by 5 ratio. On the other hand, hui 13 (and 1) are the 3rd octave because they are a 1/8 ratio. When you play one of those you have canceled out the divide by 2 and divide by 4 octaves. That note at the 1/8 ratio is not going to be the note at the 1/2 ratio (the 1st octave) because the first octave requires half of the string to vibrate. So all the hui cancel some partials out. A simple example is that divide by 5 is not going to allow divide by 2 to vibrate.
So let me point a few things out that are perhaps beginning to be understood at this point:
1. the hui on the guqin are basically positioned at the lowest 6/7 harmonic partial positions (neglecting the fundamental which is the open string).
2. Of the 1st 8 - the only partial that is skipped is divide by 7 which would add 6 more hui and produce a lot of minor 7ths that were of no interest to the designers of the guqin (or at least its hui). It might have crowded things up a bit and made things confusing.
3. why are there 13? Neglecting duplications the 2nd partial (1/2) gives us one hui, 1/3 gives us 2, 1/4 gives us 2, 1/5 gives us 4, 1/6 gives us 2, and 1/8 gives us 2. 1+2+2+4+2+2 = 13. Of course if we put in the 1/7 partials we would have 19 hui but never mind. If we left out the major 3rds we would have 9 hui. Of course there is an entire cosmological aspect of this like the notion that the 13 hui represent the lunar calendar months. Perhaps this is a reason for including the 5th partial? I will revisit this subject later. The bottom line is that the 13 hui are pretty solidly embedded in both the top and in fundamental acoustical theory.
4. if you are sad that the 1/7 minor 7th didn't make the cut, you can always find them for yourself. For example there is one around hui 4.4 on the guqin. There are 6 of them by definition. Of more interest is the 1/5 division which gives us the major third. It did make the cut. The line had to be drawn somewhere.
5. A certain party has pointed out that the divide by 8 fractions of 3/8 and 5/8 do not have hui. This is true. Perhaps this might be regarded as a clue that 13 hui was deemed enough or that putting hui in at 3/8 and 5/8 would have been "one (well two) too many". If we explain all the possible divide by 8 hui notes then 2/8, 4/8, 6/8 are all lower octaves and are dealt with by divide by 4 and divide by 2 already. 1/8 and 7/8 (which have hui) give us octave notes that are 3 octaves higher than the fundamental. One of these two notes (3/8 and 5/8) can be found about hui 5.7 or so which if you say play hui 1 first - you should be able to find it on your own.
So the hui on the guqin are based on a fundamental acoustical physics property - harmonic intervals. We don't know when they were invented but we do know that this is a solid principle of acoustical physics.
Because the hui markers are placed at the harmonic interval junctions for pitches including the following intervals (based on the idea that the pitch and the fundamental make a two-tone interval): octave, perfect fifth, major third, we can make the assertion that
all the harmonics found on the 7 strings therein are just intonation notes.
And there are 13 * 7 of them for 91 in all. 91. That is a lot of notes. Of course an overall theme of why the guqin is designed the way it is seems to be to have a lot of notes that are the same note more or less but have different tone colors for contrasting "two notes - different timbre". A great number of harmonics helps with this design notion.
Just intonation is defined as notes created by small integer ratios of the note in question and the fundamental. Just intonation is merely another way of looking at the result of the harmonic overtone series. So one could make a scale out of notes produced by just intonation. The set of harmonics on the guqin based on the 13 hui give us 3 notes (unison, perfect fifth, major third). So before when we said we had a major third (4 of them actually) - these are just major thirds. They are not equal temperament major thirds. They are also not pythagorean major thirds either. They are just major thirds.
The wikipedia page on just intonation provides more information: wikipedia: just intonation. In fact there is a dynamite table on that page that shows just intonation ratios for a C scale that we will come back to later (in the next blog post about guqin tuning).
One thing worth understanding is that just intonation is in some sense historically the dominant intonation idea for instruments and voices in most human cultures. This is because the 2 overtones of octave and perfect fifth are fundamental in music and so often make up the idea of tonic and dominant in melody. Equal temperament may have been invented in China by a Ming prince but it was used in European classical music (say more and more from roughly 1700 on) to solve various tuning problems with keyed instruments like "spinets" and eventually pianos. These "problems" more or less came out because of the use of triads (1/3/5) relationships in European classical music which from a global perspective is an aspect of European classical music from Renaissance times on. Of course it's an aspect of world music at this point given the ubiquity of equal temperament instruments like pianos and guitars. On the other hand vocal music and violin players (and guqin players) don't have any frets to fret us in. And even in classical music in Europe there was great resistance to ET for long periods of time especially by violin players. Many felt that the just major 3rd sounds much better than the ET equivalent of it (386 cents versus 400 cents).
Harmonic note choices and the just major third
If we stop for a moment and think about the design choices for notes for the 13 hui, you observe that you have a grand total of 3 kinds of notes -- all just intonation by definition:
- octaves of the fundamental including one octave above, 2 2 octaves higher, and 2 3 octaves higher.
- perfect fifths, 2 at a perfect fifth above the octave, and 2 more an octave higher than that.
- 4 just major thirds.
- string 4/hui 7,
- string 1/hui 8,
- string 6/hui 7.
You just sounded out a major triad that is basically G E C (an inverted C major chord). It should sound pretty good. This is a pretty western thing to do because it's a 3-note triad broken chord. It's not a very guqin/Chinese melody sort of thing. But it's a western music thing of 1700 or so because of the just major 3rd, and the just perfect fifth. You cannot play that chord on a piano tuned via equal temperament. Many classical western musicians were and are still in love with the just major 3rd and hopefully it sounds good to you. No worries - I probably haven't ruined your ears for equal temperament music.
- string 1/hui 8, and
- string 5/hui 9.
So the first one is a just major 3rd at 386 cents,and the second is a perfect 5th that in theory is around 408 cents when paired with string 1 to become a major 3rd. Ironically they are both Es. Welcome to just intonation (or perhaps the pythagorean extension of just intonation). Just intonation is a wonderful place where G# and Ab are not necessarily the same note. In Equal temperament they ARE the same note. To some extent I have to wonder if our ears are brainwashed in this regard by ET and we have a hard time as a result with this sort of thing. Lighten up. When you listen to these 2 notes you can note that they are NOT the same note, but yet they would fall in similar places in a scale setting. I suspect but of course cannot prove that to someone like the 13th century Song guqin master Mao Minzong (毛敏中) -- he would feel that the diversity therein was a good thing. And he never had any ET tuning in his ears ever.
What we know about the design of hui in the guqin
So somehow artisans in China in the BCE era had some very good empirical knowledge of a fundamental music acoustics principle. And some of them took that principle and applied it to the guqin -- marking out 6 of the 1st seven partials in the harmonic series is truly remarkable and was not an accident.
So of course we can ask questions like the most basic of all: just when did this kind of qin appear? or what were the stages in its development? Or the question we shall attempt to address in this section which is just when do we seem to have any documentation or pictures of hui as harmonic markers? A good starting answer to those questions is this: unfortunately nobody knows precisely.
We do however have some written statements that can shed some light on the matter. Joseph Needham in Science and Civilization of China, Volume 4-1 Physics and Physical Technology (Cambridge, 1961) has a long and interesting section on early acoustics in China. He points out a couple of things that are salient to our discussion. His main thesis is that in the Middle East in general and in Babylonia in particular ideas of the fundamental ratios for octaves, perfect fifths, and perfect fourths were invented and then transmitted outward both to Greece and to China. In China the idea of generating a 12 tone "gamut" based on the cycle of fifths was developed from the notion of generating the next perfect fifth via a measurement system that involved taking a length of bamboo (a silk string works and is in fact easier to use) and multiplying it by 2/3rds or 4/3rds to get the next fifth. This is in fact one way to do so-called pythagorean cycle of fifths - we'll explain it more in the next blog. You can get the same result via frequencies by multiplying a fundamental by a 3/2 ratio or you can do it with measurement with ratios based on thirds. This is known as the pythagorean cycle of fifths or if we are talking about a scale based on it, pythagorean tuning. Knowledge of the just perfect fifth is necessary as a precursor for the pythagorean cycle. The book known as the Lushichunqiu (呂氏春秋) is a text of approximately of 239 BCE. It clearly defines how to do the pythagorean cycle of fifths generation for a gamut of 12 pitches (yellow bell generates forest bell etc., etc.). So this gives us a hard date for knowledge of just intonation and its application to the pythagorean cycle. Of course the original knowledge of these ideas came before - and possibly long before.
You might read John Thompson's web page on guqin origins or at least look at the nice picture in the top at the right on that page. See: John Thompson on Qin Origins. So archaeology has been kind enough to pull several instruments out of tombs that date roughly from the 5th to 3rd centuries BCE. These graves are in regions associated with the Chu State 楚國 of Warring States times. Note from the pictures that there are no hui and actually it looks like you couldn't press the strings down to the surface to play a note either. If these instruments are qin, then they are qin ancestors or possibly cousins. They could also be some sort of zither that died out or a Chu state instrument about which we do not know much. In any case since we can't be sure that they are qin, we can't cite them as authoritative in terms of just when hui appear (or do not appear).
The HuainanziThere is a book called the Huainanzi presented to the Han Emperor Liu Che better known as Han Wudi in 139 BCE by his cousin Liu An. So that date can be presumed to be a rough "publication" date although it may be that the good news was that the presentation managed to get the book into the Han Imperial Library (therefore it survived). Liu An is presumed to be the editor of the book. It is quite a long book and is more or less a Daoist manual of how to run the state. It has quite recently been translated in full into English as The Huainanzi - A Guide to the Theory and Practice of Government in Early Han China, Columbia University, 2010, translated by John S. Major, Sarah A Queen, and others. Some think that the title basically means "Master of Huainan" (Liu An) because Huainan is a place name, and Liu An was its King at the time. This would make the title similar to other books like Mengzi (Master Meng) and Zhuangzi (Master Zhuang).
The 19th section is called 脩務訓 (translated by Major as Cultivating Effort) and in overview is supposed to be on the subject of why the ruler must put necessary effort into his work and how to argue or refute a point during an oral debate. The passage that is of interest to us is as follows:
The translation on p.778-779:
Now in the case of a blind person, his eyes cannot distinguish day from night or differentiate white from black; nevertheless when he grasps the qin and plucks the strings, triply plucking and double pressing (footnote 38), touching and plucking, pulling and releasing, his hands are like a blur, and he never misses a string. If we tried to get someone who had never played the qin to do this, though possessing the clear sight of Li Zhu or the nimble fingers of Jue Duo, it would be as if he could neither contract nor extend a finger. What is the reason for this? Such things are made possible only through repeated practice so they become habitual.
Our footnote (38) is devoted to the four character phrase in the middle of this section - 參彈複徽 which literally means "3 plays (plucks), back to the hui". Here is the text: Cantan, triply plucking (the strings) and fuhui, doubly pressing (the frets) refer to the movements of the player's right and left hands, respectively. We are grateful to Bo Lawergren and Yuan Jung-ping (private communication) for their help with the technical terminology of the passage.
Kudos to the Sinologists for asking Qin people about this. Kudos to the Qin people for helping out.
Some have doubted that hui in the above passage is referring to our beloved 13 harmonic series markers. There is a dictionary definition for hui of "threads/tassels" as well as "beauty" and "marks/badges/logos". I would observe that in general one has to be careful of dictionary definitions with old texts simply because context is so important in Classical Chinese. Old dictionaries don't necessarily have all the meanings of a term. And context may be of crucial importance sometimes in helping you to explicate a word. One old meaning of this term is more or less "badge" which can be understood to mean "mark". 13 marks is quite reasonable.
More to the point the entire chapter and this passage in particular are about "striving" in the sense that you have to practice your art to get it right. In this case we have the obvious contrast of the blind qin player who is a total master and is more or less "shredding his qin" (to put it in up-to-date lingo). His hands are like a blur and he never misses a string. I don't see how in the middle of this episode told as an object lesson it would make sense for the blind master to muck with the tassels underneath the qin (if there are any). Also the 4 character phrase can be interpreted as referring to both the right hand (3 plucks) and to the left hand "back to the hui" as a nice balanced phrase. One can take "hui" to be referring to the left hand playing positions on a qin after all as that is a major function.
So it seems highly likely that hui here are referring to the 13 markers and we have at least a date of 140 BCE to go on.
Xi Kang and his Qin Fu poem:Another early mention of hui is in Xi Kang's Qin Fu （琴賦）poem. Xi Kang of course was one of the Seven Sages of the Bamboo Grove, a qin player, the reputed composer of at least the early stages of the guqin piece: Guanglingsan （廣陵散) and a Daoist philosopher. He lived in the Three Kingdom period post the Han Dynasty and his dates are said to be 223-262 CE. Fu poems were popular in the Han period and were often long (for poems) and written in praise of some topic, place, or things with lots of detail and different points of view. Xi Kang's Qin Fu thus might be translated as a "Rhapsody on the Qin".
One source of the poem is the Zhao Ming Wen Xuan 昭明文選 compiled by the Liang Prince Xiao Tong in the early 500s CE. The Wen Xuan is a poetry collection which covers poetry from 250 BCE to about 500 CE and is an important source for pre-Tang poetry. I have consulted the two translations of Qin Fu in English and an additional translation courtesy of a dual language edition (Classical Chinese, colloquial Chinese) of the Wen Xuan from Taiwan.
The two English translations are: 1. Hsi K'ang and his Poetical Essay on the Lute by R. H. Van Gulik, published in Japan in 1969, and 2. David Knechtges'(truth in advertising - took a class from him long ago) monumental stab at translating all of the Wen Xuan, in this case Wen Xuan or Selections of Refined Literature, Volume III pp. 279-303. So having consulted three translations - it turns out that they all agree on "studs" - that is, in the two instances in the text where hui seem to mean those MOP (mother of pearl) markers on the top of a guqin. In total there are three uses of the character 徽 in Xi Kang's poem. One of them is clearly not the marker meaning and we will ignore it. That leaves us with two candidates which show up in some fairly simple text.
Literally this means: strings take Yuanke's silk, hui take Zhong Mountain's jade.
Knechtges on p. 287 translates this as: The strings are made of Yuan Ke's silk, for the studs they use Mount Zhong jade. A footnote points out that Yuan Ke was a silk growing savant and Mount Zhong refers (maybe) to the Kunlun mountains. Unlikely that tassels were made of jade.
Literally this phrase is: strings long, therefore hui cry. Van Gulik on p. 114 translates this (a bit strangely) as: As the strings are long, each can give the entire scale. His sense agrees with Knechtges p. 299 though who says: The strings are long and thus the studs can be used to sound the notes. Although of course this is a poem and thus either or both of the two functions of the hui (position for pressed notes or as harmonics) might apply. The Taiwan translation interestingly declared that the hui cry sub-phrase refers to the idea of making the harmonics sing out.
Pictures at an Exhibition
Although we can't say that the hui are shown in anything like their true mathematical nature - at least one can observe them in the picture.
|Xi Kang playing qin|