Monday, April 10, 2017

guqin open strings and pythagorean tuning

Introduction


In this essay I will first "define terms" and explain what pythagorean tuning means as well as equal temperament.  Then we will look into the history of pythagorean tuning in China.  After that we will discuss harmonic-based tuning on the guqin and explain why it is actually pythagorean.   Finally we analyze what pythagorean tuning seems to mean for the guqin in terms of its open strings and will look into various questions like:  "is using an Equal temperament electronic tuner a bad idea or just a handy use for a chromatic tuner?"
  What isn't going to be covered here is pressed notes on pressed strings which will be covered briefly (one hopes) in the next blog outing.

Some Terminology and Things Pythagorean (or Huangdi-ian)

It might be a good idea to read the previous blog on the design of the guqin hui before you read this one. The basic reasons for that are to understand the note relationships and just intonation aspects of the 13 hui system.  We need to understand in particular the just perfect fifth at 702 cents.  We also defined the notion of cents in the previous blog.   See this link.

Before we go on in this blog and having defined Just Intonation in the previous blog as a basic artifact of both human melody (everywhere) and acoustical physics - there are two other notions of tuning that we should define:

 pythagorean temperament/tuning or the pythagorean cycle of fifths.


For now briefly the cycle of fifths is a very old idea but it boils down to this:  take the harmonic interval value for the just perfect fifth which is a vibrational ratio of 3/2 to the unison itself.   (Given C,  produce G).   So if you have a note at 65.2 hz you multiply it by 3/2 to produce a perfect fifth at 97.8  hz  and you then take that new note (G) and repeat multiplying by 3/2 to produce the next fifth (D) etc.  So D is G's perfect fifth. From D we get A, A to E, etc. etc. Until you get back around to F and then attempt to produce C again (this fails).  Note that this measurement scheme is based on sound frequency (absolute value in hertz as one way to do it) and is not based on length (inches, millimeters, furlongs).   3/2 means the perfect fifth vibrates 3 times to the unison's 2 times.  Also the gamut produced is rather strung out over multiple octaves.   It is possible however to use a very simple calculation to reposition a note - you simple double or halve the frequency to move a particular note an octave up or down.


Needham in his essay on ancient Chinese acoustics  (Volume 4-1 of Science and Civilization) points out that measurement via length came before understanding of frequency (as manifested in cycles per second as an example).   And presumably sometime long ago in the Middle East (assume Babylon as a possibility) knowledge of three fundamental lengths came into being that are crucial to just intonation.   For this discussion and since we are talking about China - assume bamboo pipes although this discussion applies to string lengths as well.


1. The octave could be produced by taking a  length N and cutting it in half or doubling it to produce a  higher or lower octave.  So 2*N and N * 1/2 are both ways to generate octaves.  A shorter length is a higher pitch value.


2. The perfect fifth could be produced by taking length N and measuring out 2/3rds of it (2/3 * N).  A shorter pipe produces a higher sound so if we start with C and length 3/3 then G at length 2/3 has a higher sound.   String lengths work in the same way.   So for example press your guqin down on string 6 at hui 7 and sound that (we shall call it C) and then move your pressing finger to hui 5 and play that (we shall call that G).   The C note consists of the string from hui 7 to the yueshan/bridge in terms of vibration.   Moving to hui 5 means the string is shorter by 1/3rd hence "2/3rds".   The G in question is higher than the C in the next octave up.    Note that perfect fourths and perfect fifths are complementary (inverses in a sense).


3. We can also generate a perfect fifth by multiplying by 4/3s.   In this case the resulting note is actually a perfect fourth (down) from the starting note.   So in this case back to the qin and play string 6 at hui 7 and then play string 6 at hui 9.   The string 6 hui 9 position has produced a G with a longer string and hence a lower note from the starting C.


The ancient Chinese system for generating pythagorean pitches called "sanfensunyi" is based on the perfect fifth (or fourth) measurement idea as we shall see later.


One very important item to notice here is that in terms of cents (1200s of an octave), the just perfect fifth is 702 cents. The Equal temperament fifth is 700 cents so the just fifth is slightly sharper.   Since the first generated fifth from a unison in the pythagorean system of fifth generation IS the just fifth, the 1st pythagorean fifth is at 702 cents too. So the underlying pythagorean fifth/note cycle is dependent on using the just fifth over and over again.  C begets G which in turn begets D, etc.  So the second fifth (D) is 4 cents away from the ET D assuming C is the basis, or 204 cents to 200 cents for ET assuming C as the basis for the interval.    As we generate pythagorean fifths - if we compared them to their equal temperament note equivalents - as we go "out" -- every new fifth is off by +2 cents assuming you are doing this:  C G D A E B F# C# G#(or Ab) Eb Bb F.   If we try to generate C2 by spinning our cycle out one more time - our new C will be 24 cents beyond the C produced by simply doubling.  The 24 cent error is called the pythagorean comma.   This horrified the ancients and even though they used pythagorean "notes" both East and West -- they wisely ignored the pythagorean "octave" and stuck with doubling for octaves.  An F that is produced by 11 generation cycles is actually 22 cents in total variance as it is one step shy of the false generated octave (hence shy 2 cents).   As it is possible to generate "down" as well as generate up using just perfect fourths -- if so you are are producing notes by moving the other direction from C (as in F is C's perfect fourth,  Bb is F's perfect fourth etc.). These notes once again have a -2 cent deviation when compared to ET.   So in that case you might spin out as follows:  C F Bb Eb.   In that case our F would be at 498 cents compared to an ET perfect fourth at 500 cents.  In summary assuming you choose a center pitch you can spin up by perfect fifths (C G D A E etc) or you can go down by perfect fourths (C F Bb Eb).  What you cannot do though is generate a reasonable octave since the starting C results in C+THECOMMA at the end of the generation.  Also note that the F if reached by going up almost to C in terms of generation is not the same note by far as the F produced by going down a perfect fourth.   It is quite common for just intonation derived pitches that are on paper the "same" to not be the same exact pitch.


So once you have the set of 12 fifths you can put them all in the same octave and claim you have a 12 tone scale or a 7 tone or 5 tone scale using some of the generated notes.   A pythagorean scale of 7 tones is not that unusual and has been used in Europe in the past (say around 1400 in choral music) and has certainly been used in China possibly in quite early times.   One can point out that the first five fifths if said to be C G D A E can be rearranged as C D E G A - hence a five tone pentatonic scale (do re mi so la).   Now ask yourself this question: given that scale if we want to add a perfect fourth to it (F) do we go down from C (2 cent deviation) or do we go up 11 times with a 22 cent deviation?


By the way a major third in pythagorean tuning is sharper (408) than an equal temperament major third (400) which is sharper by a lot compared to a just intonation major third (386).   Presumably you noticed that the just perfect fifth used in pythagorean tuning is 702 cents to the ET 700 cents.   Keep 702 cents per fifth in mind though.  Also note that pythagorean major thirds are there in the open strings of the guqin in standard tuning - as for example string 3 and string 5 (F and A).   

 Equal temperament


The second idea is based on a 2000 year mystery quest for somehow fixing up the problems in the pythagorean pitch cycle. One could claim that problems were perceived by theory people as opposed to musicians but in the West at least some musicians wanted keyboard instruments that could allow you to change keys without having to have 21 keys per octave on your piano. Other musicians liked the just perfect third a lot in triad chords and musicians and composers in Europe generated an entire universe of temperaments that are neither pythagorean (a possible starting point) or equal temperament (hadn't got there yet).  In many cases they were trying to play chords in multiple keys but some triad combinations (in terms of perfect fifths or perfect thirds) would be problematic.   In China literati types wanted to solve the problem because they were horrified from a cosmology point of view. This problem became a more or less well known mathematical/philosophical dilemma with various attempted ad hoc solutions.   A Ming Prince in the 17th century by the name Zhu Zaiyu solved the age-old math puzzle by pointing out that if you took C at some pitch and multiplied each of the 12 half-steps from C1 to C2 say at a value of 2 ** 12 (1.05946) you would end up precisely back at the correct octave without veering off into the dreaded pythagorean 24 cent mistake.  This meant all the semitones in a 12 tone scale were equidistant from each other.  A wag might point out that this means that all the non-octave notes are slightly wrong.  Chinese musicians (and others) did not care (at the time).   But this made keyboard makers and some keyboard users in Europe starting in the late 1600s  happy *eventually*.   Basically you take every pitch in a 12-tone scale and fiddle with it so the only thing that matches by definition with Just Intonation is the Octave.   Thus we have Equal Temperament (because every half-step is tempered (mucked with) by the same amount.   Of course the only thing we can say about Equal Temperament is that the guqin has nothing to do with it.   We will come back to ET though later with the question:  if you use a chromatic ET electronic tuner on the guqin - have you committed a tuning disaster?  


 Another thing to point out is that the notion of cents is an ET idea.   An octave with 12 steps is considered to have 100 cents per half/step in it.   So there are by definition 1200 cents per octave.   A Just Intonation major third can thus be said to be at 386 cents (by calculation) and an ET major third by definition is 400 cents.   When we give cents measurements we are usually doing it to allow us to compare various intervals in tuning systems.   For example the just perfect fifth is 702 cents (comparing fifth to unison of course).   The ET fifth is 700 cents so is just a little flat in comparison. The pythagorean major third is about 408 cents and if you hear it next to the Just Intonation version at 386 cents you may cover your ears (sharp!).   Most people planet wide take ET as the right thing at this point but in truth a fair number of European musicians who were having fun with mucking with systems based on Just Intonation more directly (e.g, comma meantone) were slow to be convinced. Apparently an early acoustical scientist did a musicological experiment in the 1800s that showed that a bunch of competent piano tuners were NOT tuning pianos to equal temperament.   A lot of traditional music systems as in India and China never bothered.   A small but growing number of western classical music advocates have been going to to pre-ET systems for baroque music (Historically Informed Music).  It turns out there are so very many pre-ET systems.  So  ET should not be regarded as "the right thing".  It is just "another thing".   Up with Vallotti!  (a temperament named after - you get one guess.   Vallotti!).  If you have an unfretted instrument you may be producing just intervals from time to time anyway.   

You might wish to read this page - Kyle Gann on some historical tunings.   The section on pythagorean tuning is appropriate for us.   

What about the Chinese circle of fifths?

Having introduced the subject of the pythagorean cycle of fifths we might as well explain it from the Chinese historical point of view.   Not to get ahead of myself on this or anything but calling this the pythagorean circle of fifths is more or less totally ethnocentric (typical of western classical music theory though).   東方皇(帝).
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 discovered 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 fifth via measured ratios.  When these ideas were used in the middle east or Greece in toto or in part (or not at all) is hard to say.   Gamut just means a set of notes related in some way (eg,. generated by the pythagorean generation process) but not necessarily claimed to be a scale.   It is also possible that just intonation and the pythagorean cycle of fifths based on it were independently discovered in Greece and in China.  

He also discusses the founding myth of Chinese pitch theory which is found in the Lu Shi Chun Qiu 呂氏春秋 of around 239 BCE which in paraphrase gives the story of Huang Di (the legendary sage emperor of 2700 BCE) sending his music minister Ling Lun off to the west (of China). Ling Lun has a magical interaction with phoenixes that leads him to establish a 12 tone gamut made up of Bamboo pipes.  When Ling Lun returned with his pitch pipe system, Huang Di further ordered him to use it to make a set of tuned bells.  Needham points out that Ling Lun as a name could be said to mean "music rule".   (pp. 178-180).   This 12 tone gamut is based on using the just perfect fifth to generate one fifth after another in a particular way.   

Needham then suggests that you would keep a set of bamboo pitch pipes as a musical standard if you will and then one can create a similar set of strings for the same pitches.   The string set can be used as an engineering guide for making metal bell sets which might have been made crudely by ear at one point but over time were made more accurately and certainly reflected the cycle of fifths (notes generated on a 3/2 fifth ratio) and the more fundamental set of just intonation notes (a la perfect 5th, perfect 4th, and the octave).  Keep in mind that a set of metal bells was something deemed necessary at a rich Kingly noble court in early China and presumably many a lesser court as well including princes and wealthy nobles.  The bells were integral to a courtly musical ensemble.   And in general producing a set of metal bells was something a maker would want to get right because of the time and expense needed to make them.   And of course you wanted to make your customer very very happy or dire things might happen if said customer was a unhappy bigwig.



Bell set of Marquis Yi from circa 400 BCE


So the Lushichunqiu of 238 BCE has the following utterance:



黃鐘生林鐘,林鐘生太蔟,太蔟生南呂,南呂生姑洗,姑洗生應鐘,應鐘生蕤賓,蕤賓生大呂,大呂生夷則,夷則生夾鐘,夾鐘生無射,無射生仲呂。三分所生,益之一分以上生;三分所生,去其一分以下生。黃鐘、大呂、太蔟、夾鐘、姑洗、仲呂、蕤賓為上,林鐘、夷則、南呂、無射、應鐘為下。

The text here even though ancient is not hard to read if you understand two things:  1. there are 12 two-syllable nouns that are to be viewed as pitch names although originally they were bell names in bell sets.  This includes huangzhong, linzhong, taicu and the like.  2. this is a very clear exposition of just how you generate the cycle of fifths in pythagorean terms.  The 12 bell names (pitch names) can be found here for reference: 12 Chinese pitch names

Roughly the text is in 3 parts.   In the first part we are told that the note Huangzhong 黃種 generates Linzhong 林種.   Then the note Linzhong generates Taicu 太蔟, etc.  The names go out for 11 sets of paired generations ending with 無射生仲呂 - 
Wuyi generates Zhonglu and stops before it gets back around to Huangzhong again.


In the second section we are told that we use three part divisions to make the sequence.   And superior generation means to add in a 1/3rd section (meaning by 4/3rds) and that inferior generation means to subtract a 1/3rd section (meaning by 2/3rds).   We are then told there are thus two sets of bell pitches,  6 of which are superior and 5 of which are inferior.    The last part just lists the 6 names in the superior set and the 5 names in the inferior set.


We call this generation system in Chinese: sanfensunyi 三分損益 which literally means "3 divisions decrease add".   This is a measurement-based system.  (Too early for HZ).    The basic idea is visualize a pipe (or a string) and divide it up into 3 parts.   If you take 1 part away (multiply by 2/3) you produce a fifth that is higher in pitch.  The system starts with inferior generation so C produces a G that is higher.  
If you add 1/3rd to multiply G by 4/3 you also produce a fifth.  Actually you are going down in pitch because the length is longer.   Hence you are actually producing a perfect fourth *down* but that's the same thing as a perfect fifth "up".  So we overlook the going down by a 4th part and say this is still a fifth.   This time however the fifth is lower in value. So Linzhong (G) generates Taicu (D) which is down a perfect fourth (but it is still higher than Huangzhong and in fact is a major 2nd higher than Huangzhong).The reason for alternating the generation of fifths is to keep all the notes in the same octave. So the 1st two generation phases have produced a C, a G (that is higher) and a D that is lower than the G (so it is still in the same octave).   I'm going to skip the rest of the process but you should have the idea at this point.   You end up with 11 notes that you could claim are a pythagorean chromatic scale (according to Needham that wasn't the intent).   And in China they understood that the horrible comma wasn't going to be the path to the next octave.   We get there by doubling our starting units.

The result is a series of 12 tones called 十二 (shierlu) which we can cleverly translate as the 12 tones.    Earlier Chinese orchestras with bell sets were not playing compositions by Schoenberg.  So it's a pitch starter set and one function quite literally in the bell orchestra is to use one of these pitches as a starting pitch.   Tune to Huangzhong!   We can use these tones to produce a scale of say 5 or 7 tones.   It is common these days in Chinese to assume Huangzhong corresponds to C in some way although of course the notion of "C" as a pitch name is hardly as old as the notion of Huangzhong as a pitch name.  More to the point this system is pythagorean with every perfect fifth by definition a just perfect fifth at 702 cents.

So in summary in this section we have the legend of Ling Lun sent off to the "west" who brings home a pitch system to be used in bell making.   And we have a nice written algorithm from 250 BCE or thereabouts that outlines how to make your 12 tone bell gamut.    We can say the pythagorean series and a 5 tone pentatonic scale were thus known and documented by 250 BCE but they could have been known long before.


Needham (pp. 184-5) makes an interesting assertion that if you are tuning a bell by a bamboo pitch pipe you are still relying on your ear to get it right although things are much better than before when you had no pitch pipes to go on.  The rough idea is that you make an instrument with very long strings that are correctly tuned so that when a bell's pitch is correct it will sound itself in resonance.  Thus one more or less has a string tuner with long strings that can be used to more precisely tune the all important and expensive bell sets.


Our other point is that somehow we in the west have managed to assign the pythagorean tuning

to  the Greek savant Pythagoras.   Pythagoras left no writings so pinning the invention of this tuning on him is a lot like the guqin tradition of claiming that Li Bai wrote Guanshanyue (the guqin music not the poem).    In fact there seems to be no clear indication of just when or how pythagorean tuning cropped up in medieval Europe although it clearly did so.  This may have had something to do with Ptolemy (200 CE) and his musings on music or Bothieus (500 CE) but in any case just how and why pythagorean tuning came into use in Europe is not clear in its origins. Given that the Chinese version of it is available from 250 BCE perhaps it is fair to say that Ling Lun was robbed.   A nice discussion of  pythagorean tuning in Europe is available on-line.    From a history of science viewpoint it would be fair to call "pythagorean tuning" "yellow bell tuning" (Huangzhong means Yellow Bell literally).   The Yellow Bell cycle of fifths has a nice "ring" to it.  

Guqin homework (pressed pitches)


So given the cycle of fifths generation system in China described above - might we somehow use it to test our understand of it in terms of string 1 on the qin? (If you don't have a qin - you can skip this part).   So for example let's use the length of string 1 and assume it is C, and then generate G,  D, and A and having done so this might shed some light on how the traditional yellow bell generation system works.   Instead of using the Chinese bell names for pitches (yellow bell 黃種)I'm going to stick to C, G etc. String 6 is assumed to be C.   We will do this on string 6 to avoid the wrapped string density problem but the pitches so generated are (more or less) the same pitches as with string 1.   In fact, they should match the open strings as well since both open strings and pythagorean pressed pitches are all the same set of pitches.


We could measure our string lengths but the hui already tell us about divide by third distances.


1. C generates G so we need to divide our string up into thirds.   Remember way back when we did hui generation? The 2 lower perfect fifth hui are hui 9  and hui 5.  If we start with the entire string 6 as C we "generate" G at a 2/3rds position at hui 9.  Of course the string overall is now shorter by 1/3rd.    Note how it matches string 4 of course.   (2 Gs).   This was multiply by 2/3rds.  


The next measurement will use the length of G (from bridge to hui 9) as the starting point (3/3 divisions).


2. G generates D so we take G's length and multiply it by 4/3.   If the total string length 
is 713 16ths total, then this note is found at 633 16ths from the yueshan bridge.  So where is that?  It's not a hui position but it is a note you know very well if you are experienced at all.   Hui 13 would be positioned at 623 16ths so this is slightly outside hui 13 towards the nut.  We of course call it 十三外. Play it and note how it should be the same as string 2 open.   D is C's major second of course.

3. D generates A by taking the value for D and multiplying it by 2/3rds.   This note is found at 422 16ths from the bridge.   Trust me on this one.   Basically our note is found at finger position hui 7.9.


Guess what - you just generated 4 (6 actually given the two octaves) of the guqin open string notes.


Question:  how to find the F (the fourth) on string 6?  This is our one missing note.  Finding it is a useful skill because with the G string you need to find two pitches that are fifths that come before G. In this case we have the magic ratio 3/2 for finding a perfect fourth.   We already have this measurement from the hui as hui 4, 7, and 10 divide the top playing path into four quarters.   Therefore if from the bridge to hui 7 is 2/2 (two quarters), then we get 3/2 by adding in the distance to hui 10.   Thus if we press at hui 10 we get F.   We will talk about this more in the next blog about pressed pitches.   Note that we cannot generate this pitch with sanfensunyi (you can't multiply by 3 * 3 * 3 etc and get a denominator of 4).   


Now of course we do not notate pitches this way in our traditional fingering system.  We use the relative fen system (divide each hui up into 10 parts and if necessary 100 parts).  It is nicely relative to the length of guqin which is not standard.  Of course hui positions are based on a fixed ratio system starting with the non absolute distance between bridge (yueshan) and nut (longyin).   As e.g., hui 7 is always half that distance.   Now having dealt with sanfensunyi we can go back to laying out pitches courtesy of just perfect fifths.  

traditional guqin tuning mostly via harmonics



According to the Yuguzhaiqinpu of around 1850 there are three tuning methods that can be used with the guqin: 1.  open strings, 2. pressed strings, and 3. harmonic-based tuning.   I'm going to mostly dismiss the first two.

Getting the open strings right is non-trivial and in any case somewhat dicey in a world where our ears are accustomed to ET.  (This makes me wonder about "perfect pitch" since does that mean the person claiming it learned ET measurements?   If you played a just major 3rd - would they tell you that was wrong?).  


Pressed strings is also a bit dicey simply because it may be good for a rough check but it isn't as precise as harmonic-based tuning and furthermore based on a previous blog there are some issues raised by inherent problems in string thickness/density with lower strings.     The problem being that a precise hui position for string 1 in particular may not match the hui itself and the true note position varies more the closer you get to the yueshan.   This is a practical engineering problem true of lower strings and  90 degree windings on fretless instruments.   We shall neglect that problem here.  However without analyzing the system too much in detail one can point out two salient facts about the system.  For example, pressed string tuning could start with the following sequence:


match open string 7 (declared correct) with pressed string 5 at hui 10.  Which allows you to tune string 5 to seven and then


match open string 7 (still correct) with pressed string 4 at hui 9.  This allows you to adjust string 4.


Now the music theory here is that hui 10 as a position for pressed strings produces just perfect fourths at all the pressed positions.   In other words for string 5 (an A) at hui 10 we have D (A's perfect fourth).   We match this to the open D string (string 7).   Hui 9 positions are perfect fifths.  So for string 4 as G, hui 10 pressed is its perfect fifth - D.   All the hui 10 positions are perfect fourths for their open strings.   All the hui 9 positions are perfect fifths for their open strings.  Out of all the notes produced this way there are two interesting notes that are not in standard tuning.   One is found at hui 10 for string 3 and this is Bb given string 3 as F.   The other is found at string 5 hui 9 and is E given string 5 as A.   This is why if you looked at the entire pressed note tuning pattern you have to use position 10.8 on string 3 to get an A pitch for matching against open string 5.   In the previous section we explained that hui 10 is at a 3/2 division which produces a perfect fourth.  We also explained that pressed notes at hui 9 are perfect fifths for the open string in question.   Let's go on to harmonic based tuning.


So that leaves us with a tuning system that we might just as well have found in Changan during the Tang dynasty.  I'm going to present this system on the side of the hui closer to the bridge on the right hand side of hui 7.   Of course this system is symmetric and can be done on either side of hui 7.   I personally feel it is easier to do it on the bridge side because you can see what you are doing and can perhaps use your eyes more than your ears to guide the "touches" as your ears are needed for pitch discrimination.  Lots of people do it on the other side though.   It doesn't matter.   Roughly the system goes as follows:


1. first we have to assume that one string (say string 1) is pitched correctly and we use some external means to do that.   This might consist of an electronic tuner, a pitch fork, a pitch pipe, or a small flute you whittled out of bamboo (if it's Changan in 709).   Traditionally we set string 1 to a pitch that is high enough that it sounds ok but not so high that string 7 will break (this has been an important rule of thumb with silk strings).   So let's just assume we set string 1 (lowest and most fat)  to a low C.   These are probably nylon-metal strings since traditionally silk strings might have string 1 set to A or Bb - but never mind the kind of strings.  You could be setting some other string to some other pitch but for the sake of argument we'll start with string 1.   Ok, we declare string 1 as accurate.    String 1 is in tune!   Unfortunately that's not impressive because we have six strings left.


There are two patterns that can be used which the author of the Yuguzhai more or less calls "skip one string"  (e.g., play strings 1 and 3 and skip 2,  play 2 and 4 and skip 3, etc.) or "skip two strings" (e.g., play strings 1 and 4 and skip over 2 and 3, etc.).   So in all cases you pair up two notes, and declare the first of the pair to be "correct" and tune up or down (using the pegs or finger tweaking) the 2nd note in the pairing.   So e.g., if string 1 is correct then you tune string 3 to match it.    You can separate this out and use the "skip 2 strings" method as a quick tuning check but it does not allow you to proceed on the basis of "correct string used to tune (assumed) incorrect string".   For that you need to tie together the two patterns: "skip one string" and "skip two strings".   This also allows you to double check strings as for example if we are tuning string 3,   we can tune it with string 1 and then check it with string 6 assuming string 6 is in tune.


2.   the skipping 2 strings harmonic tuning pattern is as follows:  match up the following harmonics (string/hui)


match  string 1/hui 5   to string 4/hui 7

match  string 2/hui 5   to string 5/hui 7
match  string 3/hui 5   to string 6/hui 7
match  string 4/hui 5   to string 7/hui 7

One often hears people doing this and/or the other pattern or some combination as fast as possible which is OK if you think the qin is in tune for a quick check.   It isn't ok if the qin is out of tune but it may reveal which strings are out of tune.


Note that musically assuming we are tuned to C D F G A c d,   in every case we are pairing a just perfect fifth for the 1st note with its octave equivalent on the 2nd note of the pair.   In other words,


             string 1/hui 5 is G (just 5th for string 1) paired with string 4/hui 7 (G at the octave above the fundamental)


The consequence here of tuning string 4 via harmonics to string 1 is that your tool to do so was the 702 cent just perfect fifth found at a known place on string 1.  So the G harmonic note on the C string was used to tune string 4.  Welcome to pythagorean tuning in action.  The two notes should be exactly the same in terms of pitch (HZ) with no beating.


So the other "skip 2" pairs in terms of notes are string 2 and 5 A/A, 3 and 6 C/C,  5 and 7 D/D.


As I said before this is NOT a careful check based on the idea of string 1 is right and all other strings are wrong.   We can say though that if you started with this pattern - string 1 can be used to correct 4, and then 4 can be used to correct 7.  String 2 can be used to tune string 5 and String 3 can be used to tune string 6 BUT how did string 2 and/or string 3 get into tune given that we started off by assuming only string 1 was in tune?   So this is why we need the "skip one string" pattern.   Both patterns tie things together.


3. The skipping 1 string pattern is as follows:  note that in this case we can't make this work for tuning string 5 from string 3 (as they don't match up in standard tuning).


match string 1/hui 4 to string 3/hui 5

match string 2/hui 4 to string 4/hui 5

match string 3/hui 4 to string 5/hui 5

                 BOGUS!   THIS ONE DOESN'T WORK (so leave it out but note that it is useful
                 for common tunings that rely on changing one string)

match string 4/hui 4 to string 6/hui 5

match string 5/hui 4 to string 7/hui 5

So the "skip one string" tuning patterns are based on the musical idea of matching the hui 4 harmonic notes which are octaves on the string in question to the just perfect fifths found at hui 5 of the strings in question.   So for example we have these notes:


           string 1/hui 4 is a 2nd octave up (string 1 is C so this note is an octave of C) --   to the string 3/hui 5 note which is a C as a perfect fifth on string 3 (F).   This allows us to tune string 3 from string 1.     So this is how we can tune string 3 and now we can use "skip by two" to tune string 6 as one way to tune string 6.   Note that the F here is *close* to the C as it directly generated in one step down with the pythagorean/just perfect fifth system.   Speaking in cents there is a 2 cent discrepancy.


           string 2/hui 4 is a 2nd octave up D to the string 4/hui 5 which is a D as a perfect fifth on string 4 (A).    Since 4 was correct already courtesy of tuning it to string 1 we can use it now to tune string 2.


We can use a similar scheme for tuning pairs string 4/hui 5 and string 6/hui 4 and  string 5/hui 5 and string 7/hui 4.   This gives us the tools needed via "skip one string" and also "skip two string" to get the harmonic-based tuning done.   String 5 can be tuned to the system once that string 2 is in tune and then we can use "skip two strings" to tune it via strung 2.   We can then use either string 5 or string 4 or for that matter string 1 (octave tuning) to tune string 7.


Of course the "ringer" (bad pun) is the string 3/hui 4 - string 5/hui 5 combination.   String 3 is not string 5's perfect fourth - but a major third instead.  String 3/hui 4 generates an F and string 5/hui 5 generates a E (A's perfect fifth)  There is a reason why they don't match by the way.   String 3 (as F) is actually the cycle of fifth's note that comes BEFORE string 1 logically.   Or put another way - F is C's just pefect fourth.   This mismatch is useful as it turns out because some alternate tunings require raising string 5 to match 3 (ruibin diao) or lowering string 3 to match 5 (linzhong diao as one name for the tuning).   If we raise string 5 as A by 1/2 step we get Bb which we can derive from string3/hui 4 (F produces Bb going the other way in the cycle of fifths).   In a similar manner from standard tuning we can lower string 3 by 1/2 step and thus turn the open F into an E.   This is done courtesy of the E tone at string 5/hui 5.


Typically the two highest strings (6 and 7) are octave doublings of strings 1 and 2 and this can either be done as outlined above say with "skip by two",  "skip by one", or simple octave comparisons.


If we put both of the "skip a string" methods together we can construct a mindful tuning as follows that allows you to tune all the strings together slowly with careful listening (and some practice).


start with string 1 and match

                       string 1/hui 5 to string 4/hui 7   (thus string 4 is in tune)  (C tunes G)
                       string 1/hui 4 to string 3/hui 5   (thus string 3 is in tune)  (C tunes F)

now take string 4 as correct and match

                        string 4/hui 4 to string 2/hui 5  (thus string 2 is in tune)   (G tunes D)
                        string 4/hui 5 to string 7/hui 7  (thus string 7 is in tune)   (G tunes the upper D)
                        string 4/hui 4 to string 6/hui 5  (thus string 6 is in tune)   (G tunes the upper C)

and finish things off (tune string 5) as follows:

                        string 2/hui 5 to string 5/hui 7 (D tunes A)

So tuning is done via just perfect fifths (or just perfect fourths as the case may be depending on circle of fifths direction).   If you tune G from C that is a perfect fifth.   If you tune F (string 3) based on string 1 that is a perfect fourth down (or from F's point of view a perfect fifth up).   The most complex tuning in standard tuning in this case is tuning string 5 from string 1 as you must first tune string 4, then string 2, then string 5.   C to G to D to A.


And what do we call tuning based on just perfect fifths running thru the cycle of fifths?   Well pythagorean tuning of course.  


analysis of guqin tuning

There are some potentially interesting things here to notice.   Of course the main one was just mentioned at the end of the previous section.   However there are some other questions that we might explore.


First of all let's make a table that compares in cents (or in cycle of fifth counts as for example C to G to D means 2 cycles out) so that we can compare just intonation, pythagorean tuning, and equal temperament.   In this table we are only going to worry about the 1st five guqin strings that at this point in time are said to be C D F G A and we will ignore strings 6 and 7 because they are only higher octaves of strings 1 and 2 respectively.   We however will add in a few common note values for strings 3 and 5 that may be used for other tunings.


just/pythagorean/ET comparison table 

string/note           just cents       c5 count     c5 cents      ET  cents
1/C                             0                0               0      0                 0  
2/D                             -                 2              4   204             200
3/F                         498                -1            -2   498            500
3/E                         386                 4             8   408             400
4/G                         702                 1            2   702              700
5/A                              -                 3            6   906              900
5/Bb                            -                -2           -4   996            1000

So the 1st column gives the string number and its usual note value or possibly a few alternate note values based on common tunings.  The second column gives us just notes in cents. These are intervals as for example E would be C's major third so a just version is 386 cents out of the octave. Some of these values will match pythagorean values but some will not (e.g., the major 3rd).   Note that you can compare these values to the ET cents values which by definition are aligned with 100s of cents.  So e.g., a major third is 4 half-steps therefore 400 cents.   The c5 count column is showing how many times we have to generate "forward" or "backwards" in the cycle of pythagorean fifths to get to a particular note.  So for example string 3 is down one fifth from C. String 2 is up two fifths - C to G to D.  The c5 cents value gives us a two tuple so 1st you get the cents discrepancy with ET and then the total value in cents for that pythagorean pitch.  So string 2 is 204 cents and +4 cents higher than ET.   String 3 is -2 cents down from the ET value with the pythagorean/just value of 498 cents.   ET cents of course gives us the cents for those notes in equal temperament.   There are 1200 cents *by definition* in an octave.   It is important to understand that every time we generate a fifth with the pythagorean system we add a two cent deviation when you are comparing the output to an ET equivalent note.   If we are going up we add +2 cents.   If going down we subtract 2 cents.


By the way note that pythagorean and ET octaves are the same with the just octave.   You don't mess with the just octave.


There are two kinds of deviations here in a sense.   One is found by comparing the pythagorean discrepancy produced by adding or subtracting 2 cents per cycle with its ET counterpart.  So if e.g, string 3 is tuned to E when we assume C as the basis for the interval - we have 408 cents compared to the ET version.   This is a discrepancy that some I suspect can hear.     The other one is that notes in the pythagorean "scale" can begin to have a lot of deviation in them themselves.

A simple example of this is that if start with C and go down a perfect fourth to F you have a 2 cent discrepancy.   OTOH if you go up 11 times (C to G to blah blah blah to F and back to C*) you are now 22 cents off with that F.   Played in contrast with other notes from this set - the 1st one will sound good and the 2nd one will sound like shite.  To you and probably to any ancient Egyptians hanging about even if they are wrapped up in themselves.    Possibly because of the harmonics being just by definition one should not wander too far afield with the pythagorean tuning.

Now what might we learn from this table:


1. there is a clear order here of related cycle of fifth notes:  to wit:  Bb->F->C->G->D->A->E.

True this does take in three different guqin tunings including "standard",  "ruibin" and "linzhong".
I think the latter two are the most common non-standard tunings.   The point is that they too are related to the "yellow bell tuning" process.

2. furthermore with any given guqin tuning the pitches are usually setup so that at maximum there is a limited amount of real estate covered by the tuning process itself.   So for example in standard mode you might argue that F->C->G->D->A gives you a 8 cent discrepancy from ET point of view.   Or you could claim with a straight face (I would do this) that F is a perfect 4th down from C and is tuned to C therefore string 5 as A  is the most deviant in terms of just or ET tonality for that matter from C's point of view at least.  However a 6 cent discrepancy with C is probably not noticeable except in a very subtle way.     I will ask the question below:   is there anything wrong with tuning your guqin with an ET chromatic tuner?   


3.  to be clear if you e.g., choose to tune string 5 up 1/2 step to Bb (thus ruibin diao) you have done the following:   you replaced the F C G D A gamut with Bb F C G D.   This did not widen the discrepancy.   It is still maximum 8 cents.  If you tuned string 3 down 1/2 step from  F to E you changed the gamut from F C G D A to Bb F C G D.   In both of these cases you still have 5 fifths that are close together.   There is in some sense a "conservation of pythagorean distance" principle in guqin tuning.   Even in the old tuning used in some Shenqimipu pieces like Lisao and Chu Ge the tuning can be said to be Eb Bb F C G d and the principle of closeness is still preserved even if the gamut is now 6 tones.  Although in that case Eb and d are 10 cents apart.   In this case Eb and d are a pythagorean major 7th.


There are also I think some major takeaways from the table.   For example,


Why are the first five strings  not tuned C D E G A (do re me so la)?


A cheap answer to this question is this - because the strings are tuned in a cycle of fifths pattern so that we maximize the number of octave/dominant fifth pairs.


 Which by definition are all just perfect fifth pairs as direct consequences of harmonic-based tuning.    This kind of open string setup maximizes the use of open strings as notes of consequence in guqin musical phrases.   This is a very big deal.   In case it has not dawned on you yet,  strings 1/4, 2/5, 3/6, and 4/7 are tonic/dominant fifth pairings.   I personally think this is an open and shut case with this little piece of evidence but you may need to be a composer to actually understand it.   Open strings have a tendency to be big movers and shakers in a melody and are often tonics or dominants even if a piece is modal.


There are other possible reasons including the notion that major 3rds are not all that important in a Chinese musical instrument invented on the order of 2000+ years ago.   Thirds in tri-tone chords are a bit of a European obsession and then only in the last 500 years or so.   A bias towards fourths and fifths and away from major thirds is certainly not that strange.   It is also perhaps a fundamental aspect of the monochord design of the guqin but we will return to that subject in a later blog.


It seems to be that there might be one more reason for this setup which is that making string 3 F gives string 1 as C a perfect fourth as well as a perfect fifth - both tuned just.  So strings 1, 3, and 4 make a nice set of just notes (tonic, perfect fourth, perfect fifth).   Note that the F in this case is produced by going DOWN from C as opposed to being the 11th cycle out.   In other words it is -2 cents in terms of discrepancy as opposed to 22 cents out.   It also allows us to tune string 5 to a Bb - going down a perfect fifth again and  thus preserving the closeness of the cycle of fifths gamut of notes (still 8 cents max).    


It is worth pointing out that the 8 cent discrepancy between string 1 as tonic and string 3 as pythagorean major third (408 cents compared to ET 400 cents) might have been felt to be jarring.  F as a perfect fourth down is still pure just intonation.   Perhaps having this discrepancy between strings 3 and 5 was more out of the way?


One can also point out that if the question is rephrased as "is string 1 yellow bell or is string 3 yellow bell"? --  string 3 isn't a lower pitch so given that in the actual pythagorean sanfensunyi algorithm "huang zhong" is the lowest pitch with all other tones in the gamut inside the next octave up - it would be fair to assert that string 1 is yellow bell in that regard.   Part and parcel of the same confusion might be that labeling string 1 "gong" (宮) was felt at some point to be wrong because string 3 can be seen as "do" as in a do re mi so la sense.   However if you take "gong" as meaning "the boss" (string) that is very fair as the lowest string does have the most gravitas in a melodic sense.

Does using a chromatic ET tuner cause any problems with guqin tuning?


My fear is that our modern ears are thoroughly infected with equal temperament as a sound and we simply cannot distinguish the subtle difference between a guqin tuned with some kind of electronic tuner to ET values and a guqin tuned with the traditional harmonic-based tuning approach.  It could very well be that if you discussed this problem with someone named Mozart (or his father the violinist) he might "justly" be able to distinguish the difference.   This is because in his time people were still tinkering with just intonation and the love of certain intervals like the just major 3rd and to some extent, the just perfect fifth were still dominant.   Unfortunately that isn't a test we can make.


We can say a few things though about it:


1.  By definition the harmonic based tuning approach is right.   Assuming you can do it competently. Certainly you can use your ET tuner to tune the starting string to some known pitch.   Or you can use the ET tuner to tune all of the strings and then do harmonic-based tuning to correct things a bit.


2. When we compare the pythagorean tuning paradigm on the guqin to an ET tuning paradigm we end up with an 8 cent discrepancy.   This 8 cent discrepancy is spread out amongst 5 strings however.   So for example, any given string pair of tonic and dominant (say string 1 used to tune string 4) is only a little off in terms of direct comparison of ET to just intonation.   I don't think I personally could discern a 2 cent deviation.  I think I can discern an 8 cent deviation but that isn't directly relevant here for the most part.   On the other hand, in standard tuning the major third difference between string 3 and string 5 (F to A say) is 408 cents to 400 cents and I believe I can hear the difference there.


Remember I'm not saying you can't use a tuner to produce an initial note for whatever string you use as the by definition "in tune" string.   But being able to do harmonic-based tuning well is certainly the mark of a more experienced guqin person.   Every guqin student I think should cultivate their skill with harmonic-based tuning.


OTOH let's be engineers for a minute.   I can think of two workarounds for using a tuner for all the strings (there may be more).


1. if you have a tuner that can only do ET but it shows cents -- assume you will use string 1 as your basis and then having set it to the ET value (which is only being used as a starting pitch),  then go ahead and tune string 5 (A) to A + 5 or so cents.   The observation here is that if you tune string 1 to C2 and then tune strings 3, 4, and 6 to F G and C3 - these are simply just intervals without being derived indirectly via the pythagorian circle of fifths.   This leaves us with string 2 (4 cents) and string 5 (6 cents) in terms of deviant strings.   I think you can then backfill string 2 and use harmonic-based tuning to tidy up or sanity check your work.


2. tuners for cell phones exist as apps that have pythagorean tuning as an option.   I have one called "instuner" that at the time of writing costs $4 US and does a good job of showing cents.  (It does a terrible job of preserving battery life though).   What it seems to be doing is setting the pitch claimed to be A to 440 HZ (or 110 HZ in our case) and then using just fifths to space the other pitches out appropriately which means pitches farther from A will be a few partial hertz lower than with ET.   For example, one can see that it claims string 1 should be tuned to 65.2 HZ (as opposed to the ET 65.4 HZ).  This is due to spreading via 702 cents for a fifth as opposed to the ET value of 700 cents.   I went over the strings and made a little table showing the differences in HZ values.  


string         pythagorean hz                ET hz

C                65.2                                 65.4
D                73.3                                 73.9
F                 86.9                                 87.3
G                97.8                                 97.9
A                110                                  110
C3              130.4                               130.8
D                146.6                               146.8

One can try several experiments if you have a tuner like "instuner" that shows cents as a digital display (as opposed to a needle showing cents in an analog fashion).  For example you can set two guqin side by side and using ET tune both strings 3 and 5 to say F and A, and then with the 2nd guqin, tune string 3 to the ET F and then tune A to ET A and then bump it higher 8 cents.   Of course you can't do this exactly but you can get pretty close.   I believe I can hear this difference.   You can also just try playing the two A strings by themselves to listen for any difference.


Another variation is to tune the A on the 2nd qin first to ET, and then try and raise it to 5, 10, 15, and 20 cents and compare it to the other A.   Or you could lower A to match G and then try raising it 5 cents at a time.  You might not be able to hear 5 cents as a variation but odds are high that you can hear the higher values.   20 cents is close to the 24 cent "comma" which is more or less roughly one quarter of a semi-tone anyway.  

Summary 

1.  the 13 hui (mother of pearl markers) mark harmonic positions which all produce just intonation notes including the just perfect fifth and octave of the open string as fundamental.   This is a per string function (one string at a time - not all seven as a system).   

2. the open strings of the guqin are tuned via the pythagorean cycle of fifths courtesy of the just intonation perfect fifth harmonic.   Just intonation perfect fifths as harmonics are very important to the overall design of the guqin as we derive accurate tuning from them.  Thus open strings are tuned to pythagorean tuning.   The harmonics per string become pythagorean too so for example on string 5, hui 7 is now an A that is "up" from string 1.   This may be obvious but it is worth pointing out.  


3. the pythagorean cycle of fifths system in China was fully operational by at least 250 BCE and by definition means that the fundamental measurement notions of octaves, just fifths, and just fourths were understood before that time.


4. open strings on the guqin can be viewed as pairings of tonic and dominant (as e.g., strings 1/4, 2/5, 3/6, 4/7) giving us a set of open tonic/dominants.   These are useful in the creation of melodies.


5. there seems to be a notion of conservation of fifths in the tuning systems for the guqin.   In general one does not pick a fifth that is far away from the other notes although to some extent you could argue that this closeness is an artifact of the pythagorean tuning system itself.   A few basic tunings when mushed together give us a gamut on the order of Eb Bb F C G D A E with the caveat that you more or less pick out five related fifths out of that list for open strings.  So standard tuning is F C G D A for note choices and ruibin diao is Bb F C G D for note choices (talking about the set of fifths - not the order of strings).

6. use of a chromatic ET tuner for tuning all the strings isn't a horrible thing to do as e.g., you might do it to get badly out of tune strings into reasonable shape.   However using harmonic-based tuning strictly speaking is accurate for pythagorean tuning.   And I believe I can hear the difference between F and A (strings 3/5) if tuned to Equal Temperament or to Pythagorean tuning.   The ET major third is 400 cents and the pythagorean "equivalent" is 408 cents.

In the next blog we will talk about pressed strings.