A closer look at the Goldberg Variations
(by Tjako van Schie, pianist)
Introduction The time of Bach Origin of the Goldbergs Structure, number & proportion More symbols Means of variation Conclusion
Literature Graph of Bass notes Graph of structure Graph of "strange loop" Goldbergs on CD Fermatas
||The Bach monogram (You can diagonally read J S B in it from the upper left to the lower right corner.)
The great Dutch pianist Dirk Schäfer once said about Bach: "The more perfect the work, the less the quantity of those to whom it reveils itself completely. In despite of exhibitions, reviews, yes seeming popularisations the Art Work is and remains the property of solitude. A perfect contrapuntist certainly shall be able to analyze Bachs works technically and, being musically talented, also musically -But Bach is more than a miraculous contrapuntist. He is deeply devoted, not according to a certain formula (which isn't devotion after all), but devoted 'in the spirit', the spirit of unrestricted love for the supernatural powers and for Mankind. -He bears the characteristics of the Protestant idea in the metaphysical meaning - (like Händel that of Catholicism?). Bach is already a gifted style artist by surrendering his personality to divinity, the resignation in suffering, like Beethoven does in his later works. The spiritual powers of Bach subliminates above the style period from which it originated, while possessing the seeds that leads to re-creation again and again. -His creations span all style epochs by spiritual power. Geniuses are here in the first place to prove by their lives, that all beauty and noble things they give are happening outside their persona. The will to register and making tangible of this mystical power originates from the lack of self denial. The more we can approach this higher power the more she shall enrich us. Beauty, or call it the ultimate love comes to us when we give ourselves totally to her, when we are willing to die for her: the accelerated process of mental development of the chosens."
Well, this was an elaborate introduction to this article about Bach's Goldberg-Variations. It makes you feel humble. How difficult is it to capture Bach, and his music in words! A number of items which I am going to mention have been extensively descripted by others already, but it might be interesting for you as my reader to look closer to some things I discovered. I do hope this article can lead you to a better understanding of the work, that's why.
The time of Bach
We go to Dresden, 1742. About 250 years ago, in the middle of the late baroque era. Society and religion are characterized by big contrasts: people live with full religious devotion on one side, but on the other hand we see a strong growing rationalism. In everyday life we see humility collide with the urge to develop the individual.
There is a growing interest in science. The Ars Rhetorica is the base of many art disciplines, both in religious and secular areas. Big geniuses like Galileo, Kepler and Newton were busy laying the foundations of our modern astronomy, mechanics and physics. We look at the birth of the Encyclopedia Brittanica and for the first time see the development of precise scientific gadgets. Leibnitz and Newton worked on integral calculus, which formes a basis for our modern mathematics.Van Leeuwenhoek developed a microscope and observes protozoa. Clocks and time counting machines reach a very high level of perfection.
In this light in Baroque the music itself becomes a time measuring instrument. Music becomes the art of 'organized sound in time', and is taught according to 'scientific methods' as an almost mathematical part of the Ars Rhetorica. A composition had to comply to some criteria, like the rules of these, antithese, synthese: composing became putting musical arguments in the right place!
We are not astonished therefore that Bach, who took lessons in Ars Retorica during his gymnasium time, is always and in every possible way looking for perfect unity of contents, proportions, numbers and shape. Especially in Bach we see a syntesis of all 'ancient' forms and styles, in which there is the confidence that every musical element could be examined and controlled systematically. Even the human emotions could be abstracted in music. By means of abstraction the music could even easily have larger impact on listeners than the emotions of real life.
The melodies in those days as a result of that tend to extreme complexity, ornamental welth and density. Each curl has another curl attached to it and on that curl there is another little curl ... In this way you often hear very long ongoing spinning lines, which can be interwoven with other lines with genius in counterpoint.
The Baroque conception is: Music reflects a large scale of human emotions, but also is a life time philosophy that unites the humble man with the magnitude of the Cosmos. Bach is a skilled craftsman with only one goal: to serve. He never shall compliment himself openly but he is very aware of his own genius. But small voices also reach out to God, so Bach regularly makes small encrypted annotations in his music to his person. As an example: Bach writes in the preface of the Goldbergs that he wrote 'some variations' ("Einige Vartiationen"), suggesting that it is just a bunch of accidentally collected pieces. In this way he obscures the brilliantly planned Large-Scale-Architecture. We shall now take a closer look to this 'Aria mit verschiedene Veränderungen', as the original title of the Goldbergs is.
Origin of the Goldberg-Variations
There are many articles about the origin of the Goldberg-Variations, but here a short summary. Bach supposedly got an assignment from count Kayserling to write a work that could give distraction and diversion during his many sleepless nights. This music should be played in an adjacent room by the harpsichordist of the count, Goldberg. As Bach was a protégé of the Dresdener count, who was a fanatic music lover and arranged many business affairs for Bach, he could not resign from this job, although at first he was a bit reluctant to write variations anyway because of the repetitive harmonical structure of variations. Another story tells that Goldberg was a student from Bach's son Wilhelm Friedemann and that at a certain moment Wilhelm Friedemann was unsecure about himself in this teaching because of the tremendous talent of Goldberg. Wilhelm asked Dad (J.S.) to take over Goldberg as a student, because Dad was a much better teacher and pedagogue. Bach supposedly wrote the 30 Variations on the Aria (which already existed earlier, there is a copy from 1725 in the Anna Magdalena Notenbüchlein) as exercise stuff for Goldberg.
Probably both stories contain partial truth. The idea of 'music as a medicine' does not sound crazy. The concept of a healthy mind in a ditto body arose in this same period. And why should music that affects the inner mind not also have the power to affect the outer body and to make physical complaints more bearable? (An analoge reasoning is practised today in as well psychosomatic complaints as in homeopathy, which states a gradual difference between body and mind; and who is nowadays not aware of the calming (eg. 'new age') or exciting (eg. 'house') function of certain kinds of music to the body?)
Structure, number and proportion
Bach's view of the world and his faith were based upon a unity between man and his creator. The microworld of human experience was the mirror image of the macroworld of the creation (-compare this with the approach of infinity of mathematician Mandelbrot or of the Dutch painter Escher). And this unity could be represented or symbolized in simple proportions and patterns. In this case also one could use numbers and numeric relations. In the cabbalistics, which Bach certainly had knowledge of through the group called 'Rozencreutzer' , the 'number' was not primarily used as an expression of quantities, but each number had a certain power which orginated from the relation between things and fundamentals in nature of which they form an expression and to which they refer.
For sake of a correct interpretation of this number symbolism I shall give a brief summary of the properties of numbers according to the cabbalists.
0 represents infinity, the frontierless essence, source of all things, the universal egg, the solar system as a whole, therefrom the universal, the going around, but also 0 is the universal paradox, the infinite large and infinite small, the circle of infinity, but also the centerpoint, the undividable small atom.
1 is symbol of perfection, the positive, active principle, that gave birth to the entire universe; revelation and unity of the Cosmos.
2 is duality of the creation: heaven-earth, mind-matter, light-darkness, moment-eternity, positive-negative, good-evil, active-passive, male-female, profits-losses etc. It is the number of the composition and division of parts, unifying and splitting, the concept of revelation and obsurity. The deliberate outspoken and the silent inner logic.
3 is symbol of the Holy Trinity, (Father, Son & Holy Spirit). Life, substance & intelligence. Power, matter and conscience. Creation, maintainance and resolvation. Birth, life, death. The family: father, mother and child. The thinker, the thought and the subject of the thinking. The past, the present and the future.
4 is the number of reality and concretion, the matter of the universe. The cube or quadrat, the second power. Physical laws, logic, intellect, knowledge, reasoning. The cross, intersection, division, ordering, classification.
5 represents expansion, or spanning, understanding, intelligence, judgement, increment, fertility, propagation. Justice, mowing and harvest. Reproduction of the ego in the material world.
6 means cooperation. Marriage, interweaving, a piece of a chain, relation, mutual influence, counterbalance.. Also harmony, peace, redemption, the interaction between the mental and physical (Compare with the Jewish Star, a triangle pointing upwards and one pointing downwards.)
7 is the number of fullfilment. Time and space, duration, distance. Age, decay, death or persistence. The seven eras, the days of the week, the Seven Seals, the priciples of mankind, the seven notes of the scale. The perfect human being. The circular course of evolution, wisdom, balance, equality, quietness.
8 is the number of solution. The sublimating of the natural in the spiritual, inhaling followed by expiring. Giniality and inventions. Revolution. Deformations and eccentricity, capriciousness.
9 stands for the number of rebirth. A new birth, ecclesiasticism, expansion of senses, presentiments. Dreams, clairvoyance. Reformation, nebulousness, pulsation, rhythm. The stretching, travelling, distancing.
Out of these basic numbers by composition new numbers can be formed. Eg. 10 is the synthesis of 0 and 1, of cosmos and creating force. (But also of eg. the 10 Commandments, and the Roman figure X, which often is used as a pointer to Christ.)
The ancient Greek musicologist and mathematician Pythagoras (about 580-500 BC) computed all intervals as proportions between frequencies. The purest was the proportion of 1:1. In notes this is the prime interval. Thus 2:1 becomes the octave, or the base note mirrored on a higher plan. (Eg. an octave a'-a'' = 440 Hz: 880 Hz = 1:2). According the same principle 2:3 forms the quint, 3:4 the quart, and in that way Pythagoras computed a numerical relation for every note-to-note proportion. The more complex the proportion, the more away from perfect consonance.
Bach uses a similar proportional principle in the Goldbergs, but now he applied it to the form. There are 32 pieces. (Aria, 30 variations, and the Aria da capo). These can be divided in different ways. The most obvious is the division in two parts: the second half of the whole piece starts with an Ouverture (=opening). Thus we get two parts with 16 pieces each, so in proportion 1:1. This macro architecture corresponds with the inner or micro architecture of each single variation. Every variation is built around 32 Bass notes (fig.below) which serve as the harmonical fundament.
And each variation consists of two halves which share the duration proportion 1:1. (32 or 16 bars for the first halve, 32 or 16 bars for the second). Within each halve of the variations there are 16 Bass notes interwoven. (The Ouverture is an exeption: there we see 16 slow bars in the first halve with 16 Bass notes, and 32 in the second with the remaining 16 Bass notes: so proportion 1:2)
Click here for a scheme drawing of the Goldbergs
In each variation there is the binary principle (=Forma Bipartita). Harmonically this is accomplished by starting on the tonica G and by closing down on the dominant D. The second halve of each variation leads us back again from D to G. In schedule: G->D | D->G. (See also fig) This leads to a symmetrical architecture, with in the middle the harmonical culmination point. (One may compare this to a tower on a church, which serves as a pointer up to God.)
The 32 variations also have a "church tower" in the big structure: Variation 15 ends on a high single 'heavenly'-high d. And variation 16 (the Ouverture as 'opening' to God) starts with a triple G-major (3=divine trinity...). The first G major is a massive chord, the second G an upclimbing G-major scale and the third G a rhythmical and vitally punctuated descending G-major melodical chord.
The 3 as trinity appears also in a second layer of architecture, marked by 9 CANONS. Each third variation is a canon in which two solo voices walk or run after each other. A more free base line accompanies these voices. (By the way: writing complex canons was almost a scientific job, because each voice has to 'sound well' with the other one! Bach also wrote riddle canons for special situations or festivities, which could be 'solved' given one voice together with some cryptic description. Bach also wrote 14 (=2+1+3+8 = BACH) separate 'riddle'-canons BWV 1076 on the first 8 bass notes of the Goldberg Variations).
Let's take a look at the numbering and the building of the canons. The first canon (var. 3, canon all'unisuono) is a 'normal' canon at the unison (like we know from childhood) in which the second voice enters on the same pitch as the first one. Var. 6 (canon alla secundo) is a canon at the second: here voice 2. enters a note higher than voice 1.
So in every next canon the second voice enters on a bigger interval. The canon at the third: Var. 9 (NB. third=3 and 9=3x3), canon at the fourth: Var. 12 and so on until the very last canon, the canon at the ninth: Var. 27(=3x3x3). In this canon there is going on something special: the free composed bass voice has disappeared. There are only left the two canon voices which follow each other on the distance of a none. Like symbolically the Earth-character of such a lower bass voice is not longer necessary: there is resignment from the old, there is rebirth. The canon at the octave, Var. 24, which precedes the canon at the ninth, has in its own concluded a little 'cycle', because the octave was 1:2, we are on the same note, but in a higer octave; we have been 'sublimated'.
In the series of canons there is also a form of symmetry: canon 4 and canon 5 (together 9 =3x3 by the way) both are in motu contrario, they walk mirrored. They form hereby the the heart of the first 8 canons. (3 in front, 3 following)
In taking another structural division we only look at the variations. The 30 variations themselves too are builded up progressivily and not only by the division in 10 groups of 3(!) because every third variation is a canon, but in the larger scale as well. This gets even more clear if we look at variations with a sub-title. (fig)
Variation 10 (=X=cross=Christ) is named Fughetta. This forms the closing piece of the 9 previous variations and as it be the synthesis of Aria ("Var.0") with the following 9 variations. Here also lays a division in 10 + 20 variations (=1:2), because the fugue is placed commonly at the end of a suite of pieces. (We also see that variation 20 (=XX) too has some remarkable properties: in this fastspinning variation two voices are continuously 'crossing' each other as well in a rhythmical and melodical manner.)
An other form of symbolism is hidden in variations with subtitles. Most of the variations do not have a sub-title. Everytime when Bach writes additional titles, we may assume a possible extra meaning.
Variation 7 for example has the subtitle "al tempo di Giga". The number 7 represents the principles of creation and also for moral behaviour and commandments. We do not be surprised that in this piece we can hear a symbolized representation of the Biblical 10 Commandments in a remarkable ascending little fast loop, sounding like a raised little finger. The symbolism goes even beyond, if we almost literally hear the eight commandment (Thou shall not give false testimony..): the loop points down here!)
It is a known fact that Bach regularly autographed his works with hidden signatures in the form of B=2, A=1, C=3, H=8. So BACH=2+1+3+8 (=14). Sometimes however we also see J.S.Bach=9+18+14=41 (14 in reverse!) or J.S.B.=9+18+2=29, but 2138 is the most common and typical Bach-related number.
We shall illustrate this with some examples: In the Goldbergs there are 32 pieces. The last "Aria da capo" is not written down in notes, but only mentioned as "Aria da capo". Actually Bach wrote down only 31 pieces. Of those 31 there are 3 variations in minor keys (g-minor, var. 15, 21, 25) and 28 in G-major. This leads to: 1 (=A) (not written variation) 3 (=C) (minor key variations) 28(=BH) (other variations) (This is depicted elaborate in the book 'Bach and the number', in which are highlighted also many other intersting things!)
Variation 15 has the subtitle "andante". It is the first minor key variation after 14 !(=2+1+3+8) major key variations and also the last variation of the 1st halve of the Goldbergs. (Close to God <> churchtower <> individual devotion). Furthermore in cabbalistics 15 equals the P. And 'P' represents the Latin 'Pisces'=Fish=symbol for Christ.
Variation 16 is the Ouverture: The opening piece of the second part. There shall follow 14(!) variations after this one. Also remarkable is that the division of the whole GV in two parts makes that between correspondenting parts in the 1st and 2nd part every time exactly 14(!) variations are spanned. (Eg. between var. 7 and 22 are 14 variations, but also eg. between 1 and 16, or 10 and 25).
Variation 22 also has a subtitle, namely "Alla Breve". The letter X=22 in the 'Bach-alphabet' is another symbol for Christ. In this variation the Bass notes are very recognizable.Also there are exactly 14 crosses (#) in notation: a pointer to Bach's devotion to Christ?
Variation 25 (adagio) has the strange almost atonal atmosphere of a minor key cantilene after the first cycle of the 8 triplevoiced canons (remember var 24=3x8). Also in 25 there is a suspicious place. For a long time I have been pondering about why Bach suddenly switches to triolas instead of 16ths and 32nds notes in bar 7. I assume that this is done to mark his signature. (See figure below.) A strange searching little loop, which has almost no relation to the rest of the variation! The signature is cleverly hidden in as well 'Bach-notes' (German b=H and German b-flat=B) as in the number of notes and also in the number of half tone distances between the notes. (Besides maybe there is also a symbolic link to the number 12 here. There are 11 triola notes and 1 with binding to the next. Maybe a symbol of the twelve Apostels, with Judas (the traitor, so just a bit different from the rest) as the last one.) Between the 6th and 7th note we can see 4 up, 1 down also as a weaker signature of J.S. Bach(9+18+14)
Variation 30 (=3 x 10, and completion of the cycle) is a Quodlibet. Its place is where we would expect to find a canon at the tenth according to the canons series, however Bach had something better in mind, also, because it is the very last variation. A Quodlibet is a piece of music, in which literally we hear 'beloved songs'. Traditionally the manufacturing of a quodlibet was an entertaining thing amongst music lovers: make up different popular tunes that can sound all together at the same time. Bach ingeniously interweaves two known popular songs in this Quodlibet (one of them more or less as an canon at the octave, the other one more or less as a canon at the fifth, and all of that at once!) with the base line, which contains the material of the 32 Bass notes. And the choice of these songs are not randomly made! Because what was beloved mostly by Bach? He secretly points to Christ again.
The first song in the Quodlibet is "Ich bin so lang' nicht bei dir gewest, rück her, rück her" (=I was away so long from you, come back, come back) , a seemingly innocent love song, which one knew in Bach's days mainly as a dancing melody ("Kehraus"), that often concluded a dancing party evening. But the text refers also to the "Aria", which has sounded such a long time ago and has to rerturn back (and it does later as "da capo"), and to Christ, who shall return on Earth. (And if we look at the "Aria" as 'variation 0' the similarity is obvious. The creating principle ("Aria") is expected back to close the circle.)
The second song is "Kraut und Rüben haben mich vertrieben, hätt' die Mutter Fleisch gekauft, so war'ich langer blieben" (freely translated as: 'bad food' has caused me to leave, but if Mother would have bought meat I had decided to stay longer), a say goodbye song. (In Holland we know it as two child songs: "Zagen zagen wiele-wiele-wagen" and "Sinterklaasje bonne-bonne-bonne".) Here 'Kraut und Rüben'(=Garbage) refers on first sight to the silly Variations, which have chased away 'mich' (=the "Aria"). Here too not only Bach makes a joke about his own writings and inventions, but maybe also points to a deeper sense of understanding: Bach depicts the difficulty to keep a steady spiritual faith, and to remain devoted despite of all ecclesiastical opposition (which Bach experiences a.o. as cantor) and different opinions (and the rise of newer world philosophies and scientific thinking methods).
Means of Variation
Bach uses many divergent methods to vary. By the way what is a 'variation' anyhow ? A proper definition to me seems: "The repetition of the same thing, but then with a difference." Bach varies as well on micro scale and on macro scale. Let's take a look at this 'small-scale variation', to find out how everything grows out of a single seed.
The Aria, for example. (Bach wrote it earlier than the rest of the variations, but he probably chose it as the fundamental of the Goldbergs because it is in itself a variation work!) The Aria starts on a 'g'. What can you do with it? First possibility is: just repeat it. So now we have 2 'g'-s behind each other. Where is the variation? (In the accompaniing notes of the left hand: at first a 'g' too, then a 'b'!) After these two g's in our melody you could place another 'g', but then you repeat a repetition, so that does not have enough variation in it. We now take just a higher note, an 'a'. Now we can return back to 'g'but we already have two of them, so we continue to go to the 'b' (but with another rhythm this time: 8th with point and 16th, otherwise there is not enough variation!) If we would step further and go to 'c' now, we have a 'step up' which repeats, so then back to this 'a'. For the first time now we have this descending step. After all these small steps it gets time to hurry a bit. We take a slightly bigger step to the 'd' below. But this jump should not be too blunt, so we fill it with a third in between and two little 'small notes'. (The first 'small note' (g) is passing by, the second (e) is holding back to this long 'd'.) So, we have a nice theme now and made some progress. This theme can be repeated, of course. We do it, but now we start on a lower 'g', and we add some more grace notes (also the accompaniment uses some different chords now, however everything still looks alike).
Etcetera. Every little element of the theme is elaborated in a detailed way right from scratch, like a miniature stone in a big mosaic. The "Aria" hence is not accidentially the opening piece of a series of variations, but has an almost infinite large number of 'mini-variations' enclosed in it.
Then we hear variation 1. Because count Kayserling was from Polish origin, Bach chooses as a tribute to the count to open with a Polonaise style rhythm. Also he opens the 'polonaise' with the last 3 notes of the "Aria" (g-f#-g) and from here developes scale like figurations, which ascend to heaven with bravoura and descend afterwards to the Earth again. Where the Aria had a quiet and almost improvising nature, now the 1st variation sounds strictly and rhythmically. A big contrast which demands a solution.
Variation 2 is therefor more relaxing. Now we open again with the same 3 notes (g-f#-g) but now we put them in the base line, meanwhile the 2 upper voices communicate and imitate each other like in a trio sonata. Actually variation 2 is an accompanied duet. (=2!)
In this way every next variation bears elements from the previous one, thus affirming that these are variations, but everytime we see enough contrast as well.
For the rest Bach varies his forms (some are dances, others almost inventions or cantilenes), the character, direction of voices, metre (each new variation has a different metre from the previous one), density (sometimes only a few notes per beat, then on other occasions a lot of them), recognizability of the bass notes (sometimes very clearly to be heared, on other occasions hidden in the middle voices or divided amongst voices), playing techniques (sometimes simple, sometimes very complex with crossing hands and crossing voices or fast-against-slow), etude-like-elements, instrumentation (some variations are concstructed more instrumentally, others remind one of a singing choir, and others are like orchestral or like a soloist singer).
To highlight all facets is (especially in the context of an article) impossible. For those who want to learn more about the Goldbergs I gladly refer to the literature and to the music itself! As a performing pianist I have played the Goldbergs many times in recitals. Besides I made a cd-recording of the work. Everytime again it shows that the audiences are higly impressed by this piece (and not only because of my playing, I assure you!). The work is like a big diamond, polished by a super-diamond-polisher, in which every time again a certain face can be admired or looked at. The more often you hear the piece, the more it becomes clear, but also the more miraculous stuff you can discover in it! To end with Dirk Schäfer: "Art is the direct revelation of truth."
Hans Brands Buys - J.S. Bach, 48 preludia
Casper Höweler - Inleiding tot de muziekgeschiedenis (Introduction in musical history)
Sepharial - De kabbala der getallen (The cabbala of numbers)
Peter Schat - De toonklok (The tone clock)
Rolf Dammann (very interesting! but in German) - J.S. Bach's Goldberg-Variationen
Kees van Houten/Marinus Kasbergen - Bach en het getal (Bach and the number)
Dirk Schäfer - Het klavier (The piano)
Joseph Kerman - Listen
Geoffrey Hindley - Larousse encyclopedia of music
Thanks to Daan van den Berg for his many hints on correcting my English translation.
See also: articles invention fermatas
You can order Tjako van Schie's Goldbergs CD from this page