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appendix on bach articles
The articles on which I received reactions can be found on my pages:

Click here to read my article on Bach's Goldberg Variations.

Click here to read my article on Bach's 14th Invention.

I have been following the threads in some newsgroups about my articles.
I shall try to render them as completely as possible, as the discussions were quite interesting!

Tjako van Schie.

From the ""-newsgroup I extracted the following:

Mr. Sybrand Bakker wrote in reaction to the article:

I don't want to start yet another flame-war. However, having studied musicology myself, I did investigate number-symbolism and I soon learned we were going from interpretation to 'hinein-interpretieren' (this is not a favor to Michael Zapf) (who is another frequent poster in this group. Tjako), I simply don't know the English equivalent). The work of Kees van Houten and Marinus Kasbergen is a clear example where they have conducted this ad absurdum. There is no proof whatsoever that Bach knew the number-alphabet and has used it. Refer to Ruth Tatlow, Bach and the riddle of the number alphabet, Cambridge 1991. Your article about invention 14 is yet another example of the same kind of analysis. If you were right Bach must have computed every note. Of course he didn't. One of the problems with this kind of analysis is, usually it doesn't use manuscripts, but Urtext editions, and how dependable those can be, these are 'sources' the composer never has seen. Also Lutheran orthodoxy rejected kabbalism, and we know Bach was an orthodox (not a pietist) Lutheran. One of the main problems with analyses likes yours, those of van Houten/Kasbergen (and other, older musicologists, like Smend and Jansen) you can't scientifically prove Bach did actually use the kabbalist number alphabet. This means your complete analysis is founded on quicksand. As I said I don't want to start a flame war. My only intention was to show to others we need to be very very cautious by interpreting Bachs works in this manner.
Best regards, Sybrand Bakker

I replied to the newsgroup:

Dear Mr. Sybrand Bakker & others,
The nice thing about Bach & numbers is this kind of debate! I know the numerological theory is a hot and disputed item in Bach's music. You can of course argue that Bach did not use this kind of symbolism. On the other hand there are numerous examples of the use of numbers being a construction element in the entire history of mankind. The perfect North-South alignment of e.g. the pyramid of Cheops in Egypt is IMO a proof that older cultures had some knowledge which has been lost in later times. Besides, in my opinion, Bach was such a genius that it would not surprise me that he KNEW the alphabeth coding, that he knew Kabbalistic things. The perfect constructions we encounter every time again (not only the numerological ones) are at least a clear proof of his big mathematical sense of harmony. And IMO Bach DID compute every note, and with a big sense of beauty, like mathematicians can see the beauty in the discovery of an elegant new formula. In my earlier Goldbergs article (which people also can read) I tried to explain that Bach's knowledge of this 'esoterical' or 'metaphysical' things is not unlikely. But even when people believe Bach did NOT use numbers deliberately it doesn't take away the beauty of his music and of the structures he created! (By the way: the conclusion of Kees van Houten & Marinus Kasbergen that Bach predicted his own death goes beyond my sense of understanding too.) My plan is to collect all reactions on my articles(like the one of mr. Sybrand Bakker), and publicize them as a separate appendix on my site. Let people judge for themselves. To me the discoveries of these kind of beautiful coherences is just big fun, and proves (to me at least) that Bach is heavily underestimated being the most remarkable composer who ever lived!
Tjako van Schie

Mr. Sybrand Bakker replied to me:

No, I did not contend Bach was not using number symbolism. The examples you mention are fairly obvious. There are numerous examples in older music where the symbolism is also fairly obvious, ie at the surface at the music. Example In a renaissance mass the first Kyrie is almost always in triple meter, because the first invocation is to God the Father, the second one is in duple meter because that is to God the Son, and the third one is triple again because that is to God the Holy Spirit. Also, almost all sections of the Gloria and the Credo referring to Christ are in duple meter. Where I start having difficulty is all examples where the symbolism is not obvious from the structure. In the chorus 'Herr, bin ich's' where this question is being sung 11 times, the symbolism is obvious. In the aria 'Ich will bei meinem Jesum wachen', where the second chorus 10 times responds 'So schlaefen unsre Suende', the number 10 has been explained as: a) referring to the Ten Commandments b) the number of disciples being present in the garden Gethsemane. For me, the first explanation is far-fetched, and the second a bit more plausible. In the recitative sung by the evangelist immediately after the death of Jesus, the passage about the earthquake can be divided into three parts. The number of 16ths (or 32ths?) notes has been stated to refer to the psalms mentioning earthquakes. This is already a bit far-fetched. Experiments have been conducted whether people are actually hearing inversions, retrograde movement and the like. The experiments (described in Alan Walker studies in Musical Analysis) have show positive proof. For me any number symbolism where the musical structure has been lying on a procrustean bed, is far-fetched. I also think (and I know there is no proof for that) if all those examples of number symbolism are correct, Bach must have been sketching quite alot more than we now are aware of. For me, this is a fascinating subject, but probably I am too much of a sceptic/cynic, to take for granted all examples of number symbolism people have 'found' in the music. There is a story about a Dutch poet, reading analyses of his own poems, he was amazed all the things people read in his poems, and he clearly didn't intended.
Sincerely, Sybrand Bakker

Mr. Zachary Uram wrote in reaction:

Hi Tjako! I also find this debate very interesting. I am still learning more of this myself. Are there any another good contemporary sources of info besides the book by Ruth Tatlow? (hers was written about 4 yrs ago?) (Tjako wrote: "And IMO Bach DID compute every note, and with a big sense of beauty, like mathematicians can see the beauty in the discovery of an elegant new formula.") That is fascinating idea. Bach probably could do this very fast in some type of mental process. Just as mathematicians can think through very complex arguments and instantly 'comopute' the results of what they are doing. (Tjako wrote: "In my earlier Goldbergs article (which people also can read) I tried to explain that Bach's knowledge of this 'esoterical' or 'metaphysical' things is not unlikely. But even when people believe Bach did NOT use numbers deliberately it doesn't take away the beauty of his music and of the structures he created! (By the way: the conclusion of Kees van Houten & Marinus Kasbergen that Bach predicted his own death goes beyond my sense of understanding too.) Bach is heavily underestimated being the most remarkable composer who ever lived!") Amen! Bach is best! woo hoo :)
regards, Zach

Mr. Zachary Uram wrote in reaction to Mr. Sybrand Bakker:

Hi Sybrand, This is very interesting. Yes Bach used symbolism on many different levels heavily in the Mass in B Minor. I was wondering any references for good rigorous analysis of symbolism in the B Minor mass as well as in the St. John and St. Matthew Passions.
Regards Zach

Mr. Sybrand Bakker wrote in reaction to this:

Hi Zach, Of course my post was compiled from various references. For the Matthew Passion the most rigorous treatment could be in Martin Jansen, 'Bachs Zahlensymbolik, an seinen Passionen untersucht', Bach Jahrbuch 34 (1937), p.98-117. I never could get hold of it, though many researchers are referring to this. I know Kees van Houten has written a book about the Matthew Passion (in Dutch). I did read it and found it interesting. It doesn't go that far as 'Bach en het getal'. The problem here is he evidently didn't find a publisher, and he has published it himself. I remember finding details about it using the Yo Tomita bibliography website. Then, not about number symbolism in particular, but about allegory in general, but very very interesting (and probably the most close to Bachs intentions) is of course Eric Chafe Tonal allegory in the vocal music of J.S. Bach. It discusses both passions in detail, and he is also trying to link the music to the theology of that time.
Regards, Sybrand Bakker

Mr. Ioannis Galidakis wrote:

It is not a flame war. I also am impressed by the controversiality of such hidden information. Some of the pieces I've looked at, namely the CBED fuge in WTC I (article on this fugue can be found on: ,Tjako) do indeed show *some* evidence that there is tremendous amounts of info hidden in the music. As far as Tjako's analysis goes, I think I will agree with the evidence he presents. I have also analyzed the CBED fugue and there appear to be 3 interwoven numbers (1,2,3), which taken literally, of course mean nothing. *But*, taking into account several other facts about Bach, such as the fact that he MIGHT have had knowledge of some Kaballistics and that he often played signature tricks, it would *appear* that it is not a coincidence. On the other hand, being a Mathematician myself, find the notion "it is a coincidence" preposterous. It simply is impossible (or has a very low probability) for those patterns in the CBED fugue to have been created without at least partial knowledge of some elementary math. But then again, presenting the numerical evidence in the CBED fugue, is a bit above from what I hoped. It's like the Chicken and the egg problem. For me it is a pardox, yet, the evidence that I have seen so far, point towards the fact that Bach, although highly religious, had a certain affinity for numbers as well. I think if someone sits patiently and examines "The Art of Fugue" has a very high chance of finding more such tricks. As far as Bach being a devout Lutheran, I think we have exhausted this topic many times. Although he indeed intended most of his music to be religious, it appears to me that his genious was way above petty dogmas and typological adherings (in response to "he was a Lutheran so he could not apply Kabalistics in his music). The man's mind was way above the simple devout and pious religious man. He appears to be much more knowledgable than any other composer. The circumstances led him to depend heavily on the religious status quo of his times, but I think that he could easily incorporate many number tricks, as he was a master of harmony, so adding a few bells and whistles here and there, would be for him trivial. There also appears to be a certain "shift" during his later years, with him "signaturising" many later pieces and presenting a gradual pointer to "self" and not to God. That shift, may very well be an honest attempt to hide his messages (which point to the ego instead of God) from the church elders at this time. If this is the case, then Bach clearly shows us two things: 1)He was not interested in showing any evidence to the elders, because the repercussions might have been severe. 2)He provides many "clues" to the listener, as a way to show that perhaps "self-reference" provides a solution to the existentialism problem, albeit in a clever and concealed way, which would *have* to remain concealed, from fear of the church authorities. In any case, whatever the case might be, it appears that Bach is carrying with him a very clear message: That in order for the listener to "understand" his message, he must walk upwards in the stair of "initiation", where initiation here means knowledge of internal details and wisdom in general. The walking upwards has two paths: The first is God, the second is self (ego). Both are allowed, so I think he provides with examples from both sides. There are other interesting coincidences as well. I don't know if any of you have read the book "Godel, Escher, Bach, an eternal golden braid" by Hofstader, where he presents an analogy between Godel's incompleteness theorem and various other sectors coming from biology and Math. All I know, is that continued playing of Bach on all my CD players's, have destroyed them all eventually: My big CD player went bad, my computer CD went bad and another smaller CD player has gone bad as well. :*) I do make use of his music at least 10 hours a day, but I find it a bit strange to say the least. (Very similar to Hofstader's example of a disk which, when played, destroys itself through its own resonance frequencies, because it describes itself) This may be a grand coincidence of course, but I think that he did in fact compute every note. It was for him probably easy. So it does not sound inplausible.
Cheers Ioannis Galidakis

Mr. John Mrgummer wrote:

Well, this is a most interesting field of debate, and one of which I know nothing at all. It seems to me unlikely that Bach would have used all this number symbolism for no reason. It strikes me as odd that if he did indeed do all this tricky stuff with numbers, we have found no sketches, no writings, not even a faint trace of his work in this field apart from in his music. My father is a mathematician, and I spoke to him about this business after reading a book called the Bible Code and how the Bible held within it a code predicting many uncanny events. My father said it sounded unlikely, and suspected it was purely coincidence; the reason being that if you take any book of such a length and with so many translations, there will be, if you look hard enough, some pattern to it that happens to coincide with some parts of history. Take Tolkien's Lord of the Rings, if one looked for long enough, one could find a secret code that predicted everything. I does not mean it is deliberate. Anyway, back to the point. I don't know whether or not this sybolism is true and original, whether it was deliberate or coincidental. I do, however, think it is very interesting to study, to see what we can come up with in Bach's music. It may be that great music does, by neccessity almost, have in it a certain mathematical beauty. A fugue may naturally be a profound numeralogical symbol. I don't know, and I would think it is for each person to decide for himself (or herself).
Cheers John

Mr. Ioannis Galidakis wrote:

There also appears to be a certain "shift" during his later years, with him "signaturising" many later pieces and presenting a gradual pointer to "self" and not to God. That shift, may very well be an honest attempt to hide his messages (which point to the ego instead of God) from the church elders at this time. If this is the case, then Bach clearly shows us two things: 1)He was not interested in showing any evidence to the elders, because the repercussions might have been severe. 2)He provides many "clues" to the listener, as a way to show that perhaps "self-reference" provides a solution to the existentialism problem, albeit in a clever and concealed way, which would *have* to remain concealed, from fear of the church authorities.

Mr. Zachary Uram wrote in reaction to this:

This is absolute pure speculation on your point, and weak at that. Some just can't reconcile that Bach had great devout faith (NOT religion) and was also as you mentioned very willing to incorporate new ideas and material but this applied to his musical life. You are making some VERY strong comments about Bach's mindset. It is evident Bach had very strong devote faith in Christ and one only has to see the touching chorale he dedicated on his deathbed. I'll speculate that Bach would have firmly rebuffed you on your 'existentialist' ideas. As a Christian Bach emphasized Christ and NOT the self. Some just cannot accept that Bach was a devote Christian. Bach's faith increased as he got older not decreased. Bach knew he was getting sick and he wanted to see that works such as WTC and others were published and also Bach wanted to arrange his music and tidy up some loose ends so to speak. This in no way means he was moving away from God and towards his self/ego.
Regards, Zach

Mr. Zachary Uram also wrote in reaction to Mr. Ioannis Galidakis:

Geia sas Ioannis. Can you explain more of this concept of 'initiate'? Ok I see more what you are saying now. I thought you were saying Bach's faith was ebbing in deference to his ego/self. In Christianity though the goal is actually to be dead to oneself but alive in Christ. This doesn't mean one doesn't have their own mind but it means they are seeking God's Will always and allowing Him to direct their actions. But yes we still have choices to make. We need faith, not religion. And faith without works is dead. So living faith works reflect the divine spirit in the believer and also their submission to a greater Will than their own ego-based temporal desires. It's one thing to find discordianism 'interesting' and I agree your interest doesn't mean you have lost your faith. But if you accept Discordian doctrine which is in direct opposition to Christian truths then there is a problem. Sorry if I was a bit outspoken, I feel very defensive of Bach sometimes hehe
Regards Zach

Mr. Ioannis Galidakis wrote in reaction to this:

Hi Zach, and others. After a long absence i am back to my favourite newsgroup :*) Ok, what I meant by "initiate", has a very wide definition, and I suppose references to it will not be found in any modern biographies of JSB. (I don't know about the book Tjako mentions, I think it has not been translated to English yet, so I haven't read it) An "initiate" would be a person who through certain procedures (official or personal) has gone beyond the level of "religious symbolism" either through personal efforts, or through certain teaching by somebody else, and can go one level further and analyze a "symbol" to its consituent meanings. The reason I mentioned the masons, is because (at least through the words of an old friend who was a mason and subsequently left) they do go through this process where the candidate is allowed to take certain tests which determine his future "understanding" of the symbols. I personally find such "official" procedures a bit extreme and I do not condone them one way or another. Yet, there is a group of people who may have gone through this understanding without adhereing to some official procedure, through personal efforts. I suppose it is pure speculation on my part, apart from the evidence that shows up in Bach's music, but the masons do use exactly such symbolism as a means of some sort of communication/recognition between them. It would be far fetched to claim that Bach was a mason, (however note that many prominent past figures were originally masons and at the same time highly religious) but it appears as though Bach needs to communicate some sort of information, if all this number stuff is, in fact, based on reality. I have no problem with that either way and I'd rather concentrate on Bach's religious principles but finding consistently hidden number messages in his music is, at least, surprising. I agree with your observations as far as faith is concerned, yet one clarification needs be made on this: Clearly, from the stuff that I've read so far about JSB, the man was indeed humble, as there are virtually no quotes by him, exhalting his works. *But*, I also believe that the man DID have an objective idea of his genious, as extrapolated sometimes by either his fights with the local church authorities, or by simple secondary quotes, such as "The King loves his flute more than the music". Also, before his subsequent fight with one of the flutists where he told the flutist that "you'd make a better "shoemaker" than a flutist", and subsequenltly fought with him out on the street afterwards with the two men pulling their little swords and finally Bach getting a minor jail term (I am not quoting exactly please correct if I am wrong) Also, Bach was no angel in keeping his obligations with the students. He, many times, completely ignored his contract and avoided teaching Latin to the kids, contrary to many repremendations by his church employees. Now, such incidents are simply too little evidence to conclude anything, but it appears as though Bach had a primary purpose, and secondary distractions left him completely oblivious. That, by no means means that he had no faith, but it also shows IMO, that he would not back off even an inch, when his regular primary purpose was in danger of being annulled. I think that when Anna Magdallena writes in her notebook: "This man is made to create beautiful music and sooth people's souls and NOT to be concerned with petty fights with them...", she pretty much has grabbed the general idea that Bach had for himself about his mission. I personally have no trouble reconciling such a view with a Christian view, but I'd conclude, at least, that his persona was very much different from what we call a modern day Christian, such as me or you, or anybody who tries to adhere, say, to some Christian principle by completely annulling the ego or, say, keeping some spcific dogma in mind. Just my 2c
Ioannis Galidakis

From the "rec.puzzles"-newsgroup I received also some quite interesting reactions:

"Mythman" wrote in reaction to my article:

Tjako, I have read your article and here are my thoughts... (Note: I am not a musician of any kind) There are two ways to analyze this material as I see it: First, pretend that it was written by someone else and find out how accurately the musical analysis matches. Second, take something that was written by someone else and find out how accurately it matches Bach. The First Analysis: The title: "Invention 14"... 14 can be written as 2+12... or 2 M's as in "Myth Man" This Inventio has 20 bars, which can be split as 17+3 = RC = Roman Catholic (which I am). 20 bars, 2 parts = 20x2 = 12+1+19+8 + 12+1+19+8 = math+math; my majors in undergraduate and graduate school. The list of similarities continues for quite some time. The Second Analysis: This may be difficult for you as you seem intimately familiar with Bach's work, but if you could find someone who isn't and convince them to do the work for you, you can find better evidence. My first impression is that _most_ of the numerology associated with this work is pure coincidence. Bach may have put in one or two references deliberately, but it is doubtful that so much has been deliberately put in.
Mythman (who apparently helped write Invention 14)

I replied to Mythman:

Hello Myth/math/man, I can understand your point, as it is the argument used by many people against the 'deliberate' theory. But I have to find the first work written by Bach in which he does not do things like I mentioned in my articles. On the other hand: in other compositions made by other composers I seldom find these kind of matching numbers. It seems to me that Bach, who was a master in Maths too due partly to his Gymnasium education in the ARS RHETORICA and who probably had a deep knowledge of Kabbalistic principles, did deliberately hide so many things in his works. I think Bach simply used numbers and number relations and proportions as the constructional elements in his 'buildings'. But I am aware of the potential errors I can make in these analyses. It just struck me that things fitted so well. Even if I am not scientific, I just like playing around with the idea. To me it is also another way to experience the richness of Bach's music.
Tjako van Schie

"Thomas" replied to this:

Hi Tjako, In that case, wouldn't you consider concentrating primarily on his fugues? I think fugues are the strictest compositions in mathematical sense. And therefore his organ music could be the richest ground where your investigations might be more fruitful. Can I add it here, that apart from rhythm, melody, and harmony, there is a derivative of tunes and themes in the form of the direction of changing. Say, there is a popular favourite, Toccata in d minor: tata taaaa, taradara daaa taaaa. (This "notation" of scat nonsense does tell something of the rhythm, by the way) Now, writing: Start tone-down-up-down-down-down-down-down-up. This notation, on the other hand, does not tell us anything about the rhythm, nor about the exact tones. It only tells the relative change of each subsequent musical sound in the theme. It is similar to a derivative function in math, which shows the tendency of its primitive function. It does not show the values of the primitive function, those are shown on that, itself. The derivative shows only the tendencies: up, down, steady. There is a book, Directory of Tunes and Musical Themes, by Denys Parsons. He uses the same notation like this: *dudd dddu. Repeating sounds are marked with "r". Thus, Albinoni's Adagio: *dddd rdudd ddrdu dudud u...etc. The starting tune is always an asterisk, thereafter each change - note, not tone - is marked by "d" for down "r" for repeat and "u" for up. They are grouped in fives for easy reading. These three letters are also in order in the alphabet, and so, sorting a list of this notation is ideally easy. This system is used for looking up the composer whenever we forgot, but still remember the tune. I think you may find it useful in your searching for patterns.

I replied to this:

Thank you, Thomas, on your info. The Bach fugues are in fact very strict forms and there has been much analysis on them. The inventions however are quite strict as well, and probably are used by Bach as exercises for the big works. (See the preface he wrote in the autograph.) The nice thing is that this up-down technique has been subject of my analysis in some works by Bach. The disadvantage is that with 'your' system of 'd' and 'u' some information is lost on how much up and how much down a note goes. I sometimes use a similar notation: in halfsteps up and down, like * d2 u2 d2 d2 d1 d2 d1 u1 =your famous Bach organ toccata d-minor, or in diatonical scalesteps: * d1 u1 d1 d1 d1 d1 d1 u1 (the same theme) (By the way : this theme consists of 9 notes (*du ddd ddu) which might be 3x3 too: the 3 start notes *du are copied in the 3 end notes *du, and in between the descending scale fragment *dd. It MIGHT be a symbol of trinity as well, where God sends his son down to earth ?) These kind of structures/patterns are clearly visible. The meaning they have is another thing, and there one might guess wrong. But when some structures are so obvious, I presume that Bach deliberately and not subconscious, gives some deeper sense of meaning to his music. Bach composes layer on layer many times! I composed a piece myself once, in which I used the interval up and down technique on the famous Bach theme. (I.e. e.g. 2 up 1 down 3 up 8 down: e.g. c-d-c#-e-g#) This was even before I discovered that Bach already used this same technique in a.o. his Goldbergs, in order to write his signature on a very particular spot. (See my goldbergs article: the graph of the strange loop in var. 25.) He does the same also in the a minor Partita at the end of the 1st movement. Also it is a known thing that Bach uses e.g. the Cross-motiv a lot on special places: *dud or *udu. Like the theme of fugue c# minor(WTK I). this theme (*d2 u4 d2) is almost a replica of his Name-theme: B-A-C-H =* d2 u3 d2. In diatonical notations the themes are both *dud. It is an interesting way of comparing and valuing things one might discover.
Tjako van Schie

Mr. Rich Grise replied "Thomas" with this quite humourous reaction:

Maybe it doesn't tell you, but My Goodness! I didn't know which was "Toccata in D Minor," but you've actually transmitted it! In US, it's referred to as "The Captain Nemo Song," as that's what Captain Nemo was playing in the movie, "20,000 Leagues Under the Sea," which came out some time in the 60's or so. Amazing! Reading your description, when I realized which composition you're referring to, I actually got the tinglies! (you know the feeling - sometimes called "gooseflesh" or "Goosebumps") Cool! Thanks!

And then this thread went quite off topic as you can imagine... so I won't publish the rest here.

In the "rec.musical.classical.recordings"-newsgroup the serious debate was also interesting:

Mr. Thomas Wood wrote in reaction to my article:

If EVERY number is symbolic of SOMETHING, then every number and every permutation has symbolic significance. I'd be more impressed if, as a control, you demonstrated how some pieces by Bach (or by others) do NOT contain kabbalistic/numerological secrets.
Tom Wood

I replied to this:

Dear Mr. Thomas Wood & others, The truth is that every numer had it's own symbolic meaning in the Kabbalists opinion. What is interesting here is how Bach uses combinations of numbers to point out some interesting symbolic meanings. The same statement you gave about numbers could be said about the English alphabet. Every letter has a certain look, without specific meaning. But the combinations give them meaning in the form of words, sentences etc. And about the control: there are many other pieces mainly written by other composers in which the analysis on numerical symbolism lead to complete nonsense results, in which there are no direct relations. In many works of Bach however, analysis leads to surprising results. I too try to find works by Bach in which there are no relevant symbolic things. I have to find the first piece yet!
Tjako van Schie

Mr. Thomas Wood also wrote:

...and please account for the fact that, when Bach originally composed the Inventions, the one in Bb was #8, not #14....
Tom Wood

I replied:

Yes, but Bach deliberately changed the order of the Inventions later. Also to match them with the Symfoniae! An architect has the right to make revisions to his original schemes, when he can build a better building in that way! In the Goldbergs he used an earlier stand-alone piece from Anna Magdalena's Little Note Book (the Aria), a piece he wrote years earlier. Than much later he wrote the set of 30 variations on his own piece. In the St. Matthews Passion he put extra arias and left other things out, and revised his own work to achieve more balance and harmony. He even rewrote whole pieces! But deliberately, in order to create even more harmony!
Tjako van Schie

Mr. Thomas Wood replied:

But my point is: how can the number 14 be a significant number in the Bb invention when, at the time it was composed, it was not in any way associated with the number 14? If I had time, I would chose a piece by Cecile Chaminade or Philip Glass or Manuel Arriaga and analyze all the the note-by-note numerological significance in their music. By your methods, it would be found. But I'm not going to do it, because it's a waste of time.
Tom Wood

I replied:

Spoken about waste of time... Is making or listening to music also a waste of time? When children are playing they waste time? Is research in the fields of musicology a waste of time? Is reading my postings a waste of time? And replying? etc. I would love to see an analysis from your hand on e.g. Cecile Chaminade. By the way, I think Bach did not write the B-flat Invention in order to put in on the 14th place later on. But as the piece is in B-flat, it contains the 1st letter of his name (i.e. german B) in the key signature already. And he probably has 'signed' this piece already with the used symbols 14=BACH and 20=XX, BEFORE putting it where it is today. Probably Bach then realized that this would be a nice piece to put on the 14th place! Besides I discovered similar symbols in other inventions already. One example which struck me too as there are 2 possible ways of looking musically to the structures in one piece is Inventio 8: Inventio 8 has 34 bars. These can be divided on their MUSICAL structure in: A. {FIRST OPTION} 11 bars (to the small ending in bar 12 in c-major) 14 bars (developement) 9 bars (analogue to the end of the first 11 bars, ending in F-major) These 14 (=BACH) bars stand apart, when we look at the material. The first 11 and last 9 can be taken together to form 20(=XX or ICH bars.(they are quite cyclic too: if you would start on the -9- group in B-flat (!) you could play on to bar 12 ending in C.) In the -14- bar group there are by the way 20 accidentals! And remember the 11 group can be divided MUSICALLY into: 3(=C) bars opening theme in F 8(=H) bars sequences+cadenza to C If taken together with the 9(=I) bar group on the end one might see a clear signature:"Ich Bach". B.{SECOND OPTION} Another possible MUSICAL division is: The piece opens with 14 bars of thematic material. Suddenly in bar 15 the E-flat and F=sharp cause a change in the progression! This gives a direct division of 14(=BACH)+20=(XX) bars. And now there are 10(=X) accidentals in the 14 bar group! We even can split them in 1(=God) x B-flat and 9(=3x3 Holy Trinity) x B-normal and 0 x #. I presume that Bach placed this invention on the 8th place, as 8 was the symbol of completion. (Like in the octave: the high note of an octave is the 1st harmonic and 'next cycle of vibrations' of the low one). Also interesting is that the theme of Inventio 8 is built around an octave: first in 3 ascending jumps up: f-a {third}/ f-c {fifth}/ f-f {octave} (and 3+5=8!) then in 3 descending tetrachords with keynotes f c a f. Then 3x3 motives of 16th notes. (each motiv = 1+3) These are just a few things I think are in this piece. And there are many other interesting symbolic things in this invention to find. I have been analyzing it and tried to work out a scheme for the events in this Invention, as also here the mere beauty of the patterns strikes me. And when something beautiful strikes me I don't think it's a waste of time! There are two ways of thinking about how Bach did it, if he did it deliberately: 1. The musical structures were founded on his knowledge of number encryptions 2. The number and symbol insertions were founded on the basic musical structures. I think these 2 possibilities were interwoven so well in the compositional style of Bach, as he was a great 'structure builder' AND a great musician at the same time. (Or did he waste his time writeing such beautiful music ?) :-)

Regards, Tjako van Schie

Mr. Johan van Veen wrote: It's an established fact that numbers have a special meaning in Bach's music. But to what extent is that characteristic for Bach? What about his contemporaries? No one - as far as I know - writes about symbolism in works of Telemann, Graupner, Fasch and so on. Is it not there? Maybe something of that would be discovered if the amount of energy and time put into the research of Bach's music would be spent at carefully studying the works of his contemporaries. Only then we will know if he is unique in this respect. Are you aware of any studies of that kind? Or does somebody else know something about that?
Johan van Veen

I replied: Of some composers the use of symbolism is quite known. e.g. Mozart, be it not a late baroque composer, being a freemasonner (spelled correct?), uses the number 3 a lot when he refers to the freemasonnery symbols. (e.g. the famous triple chord in Die Zauberflöte.) In the Requiem there are also some symbolic things to find. I don't know who has investigated this in more detail however. Bach's contempories also could have used these kind of symbolic numbers from time to time. Especially in church music it was quite common practice to use the number 3 to point to the Holy Trinity. And the devil was often depicted with low notes, and God with high ones. In the works of Heinrich Schütz there are many symbolic places too. The reason that this is less known is that ,I suppose, they could not reach the high level of skill that Bach supposedly applied, and this could be the reason why this is less mentioned in the sources I know of. The musicians of that time all were responding to the new intellectual climate, though maybe less directly and no doubt less consciously as the philosophers like Locke and Hume. Late-baroque composers also favoured formal plans which allowed them to express themselves in not only musical, but more rhetorical ways. Remember, the studies of both Music and Mathematics were part of the same study of the Ars Rhetorica. We can detect an ambition to stake out the field of music, as it were, and fill it in systematically- an ambition based on the conviction that all the elements of music could be investigated, encompassed and controlled at man's will. A similar conviction underlay the bold efforts of scientists, philosophers, technicians and craftsmen of that time. It was the time of a new confidence in human knowledge and the evolution of modern ideas. The religious experience shifted from the dominance of the institute 'church' to the relation between human and God. In Late-Baroque we often can see representations with symbols. Look at the paintings of e.g. Dührer. But also in works from composers like Händel, Telemann, Vivaldi, Gabriëli, and earlier: Josquin Du Prés, Ockegem, Monteverdi and others we see a lot of textual representations in their music, in which sometimes numbers play their role. But none of them used number symbolic as much and as clever as Bach, I presume. Another thing is that Bach had the advantage of his name, which could be encoded in notenames. But your question is a good and interesting one. Should be an item for further research.
Tjako van Schie

Furthermore I received some personal mail, which I just shall not publish here for privacy reasons.
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Tjako van Schie