I've been listening to this particular piece since I heard my pianist father play it while I was a small child. It sounds just as good and affects me the same now that I'm 97!
To my ear what makes this piece perfect is the economy of means. No other piece of music conveys so much so simply. Perhaps Bach's G Maj. Prelude to his Cello Suite #1 approaches this piece in perfection. I would love to see your analysis of this piece. Both pieces touch the core of my being deeply and profoundly.
Thank you for the intriguing analysis! It inspires me as a composer. You prefaced the video by saying “It’s a pretty bold claim to call a piece of music absolutely perfect”, and I think it IS a bold claim when you don’t define what you mean by “absolutely perfect” and acknowledge that there are other possible definitions. In the conclusion you implicitly defined it by saying that there isn’t really anything you could add or remove to improve the piece or without it falling apart, and I think that’s a totally valid definition of “perfect”-but not the only one! For example, while I like the piece at hand, I enjoy others significantly more because I find them more beautiful and interesting to listen to. Just something to think about. It does not really take away from the overall quality of the analysis. (:
Very interesting analysis. There is more to Bach’s tuning than the cryptic curlicues in on the title page. If you further analyze Prelude I in C you will find that EVERY note of the 12 tone scale from C to C is used in the piece. I have read that it has been speculated that this first “perfect” piece was also Bach’s “tuning” piece to see if the harpsichord was properly tuned.
Bach wrote this wonderful piece of music at age 37. Think of it, what looks like a piece for children is actually a piece that requires extensive knowledge and deph. Only one who is already a masterful composer can write something like this.
37 is a special number, as you'll hear in the 73rd episode of Big Bang Theory. When you multiply it by 3, 6, 9, 12, etc., it yields 111, 222, 333, 444 etc. It has other nifty properties that are covered in the show and which I've forgotten. Although in the show he's touting 73 as the special number, everything Sheldon says applies to 37 as 73's mirror image.
@@genesisPiano Wow, that makes it even more fascinating. A perfect piece, mathematically speaking, composed at a special age by the number. Meanwhile, at the same age, Beethoven and Chopin were busy composing complex works of high difficulty, you know appasionatas full of octave passages, forttisimos, crescendos, and such things. Bach was more concerned in writting a piece that, although simple to execute, resonate perfectly with the universe’s perfection.
@@shadowjuan2 Bach designed this pieces to teach his sons the art of playing keyboard in a progressive way rather than being showcase works. He composed complexed works for organ earlier in life. These works by way, were part of both Beethoven and Chopin education on how to play the piano.
Wrong. Bach published WTC in 1722, when he was 37, but many of the pieces in WTC date from much earlier. WTC is a compilation, not a newly composed work. Nothing in WTC "looks like a piece for children."
You yourself are my golden mean. What fascinates me the more is that most of the explanation goes over my head but admire your savvy approach of all this. Somehow this makes me have the same goosebumps l have when l listen to PACHELBEL’s canon and some Gregorian chants sung in ancient abbeys. Wow !!! I’m blown away. My hat off to you !!!
Bach is my guy. I play this piece on the guitar. I found very original and interesting details in your interpretation, like the sutil ascelerandos and those fortissimo basses almost at the end. I really enjoyed your analysis of it and sure it has enriched the way I understand this piece. Thank you!
I agree with you. I’m in the process of learning all the preludes and fugues from both books and there are gems of extraordinary beauty throughout, but none match the beauty and clarity of the opening prelude in book one. As far as the golden mean I would add that the shape of the motif consists of five notes rising and then three notes rising for a total of 8 sixteenth notes per measure where 3, 5, and 8 are proportionally close to .618. And then Bach’s use of the major 6th between G2 and E3 in the lower register has a very striking effect at key places in the harmonic journey that is reaching home. Some argue that the major sixth is the most beautiful interval and it is proportionally close to the golden mean. I never considered that the golden mean underlies the structure of the entire piece, but there it is neatly packed like the seeds of a sunflower and even turning back on itself as spirals do. That’s really extraordinary. I have to wonder if Bach did that consciously. Public education in Eisenach was very rigorous but I’m not aware that Bach’s interest in math went beyond the rudiments of ordinary operations.. Either way, the presence of Phi is everywhere in this prelude in an unforced, very natural way.
@@danielkrenz - thank you. I make it enjoyable by reading the scores away from the instrument and imagining how it would ideally sound in terms of the tempi, dynamics, and phrasing shapes that I like. I jump around a lot so as not to get in a rut with any one in particular. And then I took a cue from the elder Bach himself by hand copying them on manuscript paper sometimes with colored pencils which even if applied somewhat indiscriminately has the effect of enlivening the score. And I’ve decided that when I’m at the stage of perfecting the performance I’ll leave the opening prelude and fugue for last since they are the best and easiest to play, and I don’t want them to sound cliché.
it's also a canon harmonically if you carefully analyse it. I remember seeing one in a book on canons and showing how Bach shaped the harmonic discourse canonically. Astounding piece of music.
Wow! This does enhance my understanding--and appreciation of Bach's music. So much of this is new to me. I wish I had more time to dig deeper! Thank you, thank you for this presentation!
Thanks for an illuminating presentation. Herman Hesse was asked if he knew a perfect piece of writing. He said he didn't, but that he knew a perfect piece of music: the Ave Verum Corpus of Mozart.
I agree this prelude is beautiful. I love Bach. I don´t know all of his works of course but among those that I know - my favorite is Seufzer Tränen kummer Not
Very nice presentation. I also downloaded and read your professor’s paper. Excellent! I created a similar paper in the early 1980s regarding Bach’s Gigue from Partita No. 2 in d minor, from the Partitas and Sonatas for Solo Violin. One listen, and you will hear it. I recommend the recordings of those works by Henryk Szeryng, who seemed to understand the concept of Concinnity, bringing out the golden mean points in his performance. It turns out that major major 7ths aren’t as rare as you might think in Bach - appearing quite often in his preludes and allemandes, and other such works - but they are quite often associated with something close to a Golden Mean marker. I say “close to” rather than “they are” because mathematically they aren’t exact. And are we measuring bar lines? Attacks? Releases? Somewhere in a sustained note? Does it work when a section repeats? Or maybe only when it repeats? It may be imprecise, but the work is strengthened by the presence of these markers nonetheless. I was particularly impressed with Mike Linton’s discussion of how the Golden Mean is often overlaid upon artworks in not-very-convincing ways. Wishful thinking. But it’s an odd proportion, and in the real world we deal with whole numbers, so sometimes the work can only suggest the Golden Mean, not precisely outline it. Sometimes the subject has to smile before the golden mean is achieved. (A joke based on Linton’s discussion of the GM in faces) It was altogether a very enjoyable discussion.
Bach was famously an avid numerologist. In no way do I think that Bach intended for this piece to be played any other way than front to back, but I am just highlighting some of the musical characteristics.
Wonderful breakdown and analysis of a beautiful, beautiful piece. I am of the belief that of the great composers, or great song writers, have the simplistic ( God-like or supreme design ) golden rule in their very being and soul. They do not analyze their work and iterate through the creation of a piece, rather it flows from their mind through the quill and ink onto the paper. I also believe they see/saw the world connected through the golden ratio.
It has been in use, but it was still rare. And for the case of Frescobaldi that chord would come across as a natural extension of the counterpoint. But what I find interesting is that Bach used it functionally and very intentionally.
That would be cool! I don't think he was as interested in mathematical relations as Bach, but his music is interesting. Would love to see a breakdown of any of his music.
I agree with this. Actually, IMO, the C Major scale is the most beautiful scale of the modern temperament. Some of the most memorable compositions of the most famous composers are in C major.
I think Bach was thinking contrapuntally throughout the piece rather than as block chord progressions. The chords are the result of the interacting parts.
I would think that the voice leading on the bar from the diminished 7 of V to that ofI is permissible according to counterpoint rules, as the two enharmonically are closely related (although I am not an expert on Fuxian counterpoint). Think of a triad followed by dim 7 of IV. Similar thing; one can be seen as an extension of the other. He would not want a G in the bass there because it would anticipate and neutralise the effect of the long dominant pedal which arises next. My view is that the symmetric properties of the piece arose organically; I don't think it was consciously done. It gets back to the fundamental point Adorno makes in the most important book in musicology: The New Philosophy of Music. Music must come from the object itself, not the subject. (And this is a major criticism of Schenkerian Theory.) A songwriter would understand this: a good song sounds like it wrote itself.
@@danielkrenz It is a discussion that we can not really have here. There are a number lessons and various interpretations. One -- not mainstream -- is from Denis Zhdanov which gave me a lot, but together with the Fugue. The pieces of Bach invite for interpretation. Gould is not necessarily the only authority.. )
Good stuff. I'm unfamiliar with the terminology where we call maj7 chords "major major 7th", dom7 "major minor 7", etc. The one chord type I would normally refer to that way is minor major 7. It all makes sense, but my question is whether that particular terminology is more widely used than I'm aware of. Perhaps outside the jazz geek realm I live in ...? It would be handy to know if that's the case. Cheers.
The reason I called those chords that way was to (hopefully) clear up any confusion. I know if I say Maj7 to a jazz friend they will know that it is a Major triad with a major 7th. But if I say Maj7 to some classically trained friends there has been some confusion on what that actually specifies. So I was trying to bridge that terminology gap. I don't know if there is a perfect way to do it though.
Hey, check out what that character van B. does in the golden section of the 1st mvmt of the 8th, and how "the silence" takes you home. Everything Bach did was perfect, even his "parallel 5ths" in the incomplete A of F variation. As GG said, Bach would've corrected it, but do composition students and composers out there see the incredible composition lesson in these unfinished last notes? Well...... DO YA !!!???
Thank you for this video! I just didnt understand one point, how the major7th chord relates to the goldent ratio, why this chord at this points, whats the meaning of it?
Thank you. I've always felt a forwward tug witb this Prelude and how it turns toward the pedal point with the increasing dissonances before the release. As to the Golden mean. It can be mostly debunked by skeptics but the fact of the matter is, it's not arithmetically precise but is precise geometrically which is area and space. I believe that Bach had studied rhetoric and numerology and I seem to remember someone saying or writing this was what made his music- especially his fugues so compelling. The last lpoint I would want to make is general. beware of simplicity especially with Bach.
I don’t think that playing the piece mathematically is necessarily the best strategy for musical production. But certainly the math helps the overall understanding.
While the golden mean idea is intriguing, and wouldn't be incongruent from Bach's scope of learning, I seriously doubt that listeners would be consciously aware of or touched by its presence. The real genius of the piece is how Bach can take the basic boring key of C and stuff it chock full of grace, poise, soaring interest, tension, and finally release by species harmony. All keyboard instruments of Bach's time could not express dynamics as organically as the modern piano, it was always block dynamics. Most expression then had to come primarily from harmony, articulation, ornamentation. IMHO the reason why Bach's music transcribes so well to so many different instruments and the zoo of timbres is because Bach was fluent in species harmony like nobody before him or since him.
Yes, the first Prelude is perfect in its way. But there are all kinds of perfection. I would say that Josquin's _Ave Maria_ is equally, but differently, perfect. _Suum cuique._ Cheers from overcast Vienna, Scott
The additional measure in Schwencke and Czerny that you cite is first attested in Forkel and also appears in Hoffmeister (and Gounod leaves it in). Labeling it "Bach's original" is questionable, since it is in none of the surviving autographs. Wilhelm Friedmann's Klavierbüchlein, for instance, lacks it. I'm confused about your measure numbering where you mention the inserted measure (and immediately jump-cut the video to the two following measures!) but I see that it is measure 23 in the analyzed score that you present. What's most daring about it, if it's authentic, is that it's a complete diminished ninth - and best analyzed that way: vii°[4 2 add10] That jarring C in the B-C-D tone cluster isn't a mere passing tone, it's harmonic color in its own right - a minor triad on iv (which we've seen already in the piece) overlaid on a diminished triad (or diminished seventh) on the leading tone, The iv proceeds by natural voice leading to the vii° in the next measure, and the vii waits for it there, with the new tone of G in the bass changing the whole from vii[6] to V[7]. On the whole, that progression seems at once too daring and too perfect for anyone but Bach to have attempted in 1722, so I suspect that Forkel transcribed the composer's final intention. In that case, the fact that there's an earlier version, with a different measure count and lacking this particular bit of drama suggests that the Golden Ratio hypothesis is coincidental, even though it's tempting given Bach's fondness for numerology! In any case, it's surely a much more sophisticated piece than most people give it credit for - they tend to dismiss it as an exercise for the early grades. Truly a beautiful cameo!
I based my analysis off of this off of the Wagener-Volkmann autograph from 1732 which is the same as the 1725 Gerber copy. The Forkel copy that I am aware of is only 24 measures long. The Friedmann Klaviebüchlein that you mentioned is similar to the Forkel.
Sorry Daniel, Well tempered tuning is much closer to Equal Temperament. That is why it is called Well meaning you can play in all keys with some sounding smoother and others sounding less smooth. Meantone tuning is designed to sound as smooth as possible in about 5 keys and horrific in all the rest.
My understating was that Equal Temperament (as we understand it) wasn’t in widespread use in Germany until the late 18th century where as a loose “Mean Tone” was in use. It is still unclear precisely how Bach wanted this tuned, so maybe those loops on the title page are the instructions!
Sigh, this argument again huh? Well Tempered was a broad category, not a rigid specification. It was unique to whatever instrument that Bach could meld with a tuning hammer, and was different depending on the inharmonicity of the strings among many other acoustic factors. Don’t overthink it.
*_«In general, the interference equation can be used to measure resonant amplitudes for any musical interval under any temperament or octave division. This equation tells us that minimum resonance occurs at the fourth root of an octave (or square root of twelve) while maximum resonance occurs at the cube root of half an octave. Taken together, these results offer clear evidence that harmonic interference balances naturally around 12 as the most rational and harmonic number possible.»_* - *_«We find here the most amazing thing. The arithmetic mean converges toward PI, or mathematical constant π ≈ 3.14159, located in the middle of the curve. We further find this point in the distribution curve to be equal to Unity (or 1) when the domain value X = 12. This is significant because twelve is the square root of 144, the value shared by both harmonic and Fibonacci series in a 12-step octave. Squaring each of the table values and dividing by twelve confirms that 12.02383 ≈ 12 is the point of balance between foreground and background._* *_The significance of twelve as a point of balance in the octave interference pattern is proven further by plugging it into the equation, confirming the curve height equal to Unity at the octave. But even more significant than this is the fact that plugging the square root of twelve into the equation results in the amplitude y = 5.0666. Care to guess what this number represents?_* *_It is none other than the y-axis amplitude for the golden ratio in an octave. Yes, the square root of twelve in the Gaussian interference pattern occurs precisely at Φ, right in the “cracks between the keys” of a major 3rd and minor 3rd in an octave. Just like the dense lattice region between a major 6th and minor 6th, the infinite golden ratio also provides an anti-harmonic proportion in the lower half of an octave. This occurs naturally at the square root of 12 (or fourth root of 144) in a 12-step octave._* *_No matter how you do the math, both harmonic and Fibonacci series reach a harmonic balance with one another at n=12 and an anti-harmonic dead zone at n=√12. Division of the octave by twelve (not eleven, nineteen or any other number) is revealed here as a completely natural pattern produced by linear harmonics that are curved in pitch space by Fibonacci proportions as they converge to Φ. Could Gioseffo Zarlino’s decision to divide the octave into twelve steps have involved some knowledge of this simple relation between harmonics and the Fibonacci series?»_* - *_«As a surprising correspondence between music and math, this little trick reveals the Pythagorean comma accurate to 3 decimal places. More amazing still, if we recalculate using the un-rounded arithmetic mean 12.02383 found earlier in place of 12, we obtain a slightly better estimate for the Pythagorean comma good to 4 decimal places. This bizarre associative property in the interference equation using the anti-harmonic golden ratio location of n=√12 proves the golden ratio is a physical property in the natural harmonic series and not some kind of error or “evil” in nature as portrayed by the Church. Vibration needs room to resonate in space and the Pythagorean comma created by the golden ratio appears to be just the right amount of room needed.»_*
I am unsure about this. There is no written evidence (other than the music itself) that Bach wrote this to conform to the Golden Ratio. But we know that he was aware of it, and was famously obsessed with numbers and their meanings.
It's entirely possible that Bach himself could not have told you just HOW he did it. Let's just say that, however he did it, Bach knew what he was doing.
Maurice Ravel called Claude Debussy's "Prelude to the Afternoon of a Faun", "the only perfect piece of music". I know there are other perfect compositions. I nominate the first movement of Beethoven's "Moonlight" Sonata.
This one was tricky. But if we divide the bar into 16th notes and count when the dissonance happens within the bar 21.63 was close enough for me. It is a fair point.
I fell in love with this prelude as a little child, when I first heard Nana Mouskouri sing the Gounod version of Ave Maria: th-cam.com/video/ckQC7_ywFb4/w-d-xo.html
La famosa teoría de la perfectibilidad de Bach. Un impostor de cabo a rabo. Una figura absolutamente menor en su época llevada a la categoría de mito por sobre encima de varios compositores contemporáneo muy superiores a él , pero olvidados. La pieza que muestran en primer término es de Anna, su segunda esposa, algo absolutamente comprobado por la historiografia y la grafología. Qué patético resulta oír como los sordos elevan al séptimo cielo su música. Es una experiencia harto irritante soportarla.
Merci beaucoup for this. I'm still struggling to learn this piece, since I started the piano only after my retirement. I'm trying to develop music and a melody in my head, and I find this the most difficult task since I never learned to play music with someone, I never harmonized with someone in my life. I loved your lecture, especially your discussion of the greek work, Chiro. I worked as a Chiropractor, practicing with my hands to alleviate the aches and pains that pianists and other humans suffered. But I have a major orhopedic problem with my shoulder, and I live with a FURS syndrome. So I adapted my techiques to continue to work, and I have a video of exercises. The low back exercises take 30 seconds, but need to be done every day. And there is a section for the shoulders, arms and hands to help with tensions in the arms and strength in the fingers. th-cam.com/video/tK9sc0ckSpw/w-d-xo.html
The Indian model might have the most mathematically perfect face, but it's definitely not the most beautiful. There is nothing special about her face, and perhaps that's because perfection can be boring. The most beautiful faces tend to have at least one feature that is striking because it enhances the face despite not conforming to conventional beauty standards. For example, things like moles, wrinkles, exceptionally large eyes, or a very large, square jaw can easily make a face unattractive, but they can also make a face strikingly beautiful - on the right person.
Unfortunatley, you have said almost nothing about the music itself. In other words, Bach could have thrown a bunch of darts at a score but put a major 7th where you point to them, then that would be a perfect piece of music. Ridiculous.
In no way do I claim that a major 7th placed appropriately makes a piece perfect. But Bach chose that harmony for very specific places and this is an investigation into that.
No, I think he made his argument. He talks about the simplicity of the piece and yet at the same time the complexity of the patterns he found within it, like the descending 4ths and 5ths being a palindrome. Watch the video again.
@@danielkrenz Yes, indeed, Bach chose that harmony for a specific reason but certainly NOT to outline a Golden Section. He did so as a demonstration (as is the whole piece) as an element of the rules of funtional harmony. I would enjoy, as would others here, I'm sure, you're analysis of the harmonc structure of the piece and how it demontrates those rules and how it creates a meaningful whole. I'm sure you would come up with some interesting points and this will point towards Bach's genius, which is sublime, and cannot ultimately be put into words or theoretical analysis anyway!
@@josephciolino2865 I do not think that Bach created this prelude specifically to reflect the Golden Mean, yet the music does reflect the properties. The harmonic analysis in the video is my own and there are many other structural points that Bach uses the harmony to reflect. I also agree that there is something ineffable about this prelude. It is so great that it has obviously captured the attention of countless people, and all of the analyses and descriptions of the piece fall short ultimately.
The more you understand it the more you appricate it and the more magical it becomes.
So true!
I've been listening to this particular piece since I heard my pianist father play it while I was a small child.
It sounds just as good and affects me the same now that I'm 97!
Great video Daniel, probably the best I've seen on Bach and the Golden Mean. Also, Mike Linton rules!
To my ear what makes this piece perfect is the economy of means. No other piece of music conveys so much so simply. Perhaps Bach's G Maj. Prelude to his Cello Suite #1 approaches this piece in perfection. I would love to see your analysis of this piece. Both pieces touch the core of my being deeply and profoundly.
I agree completely!
Thank you for the intriguing analysis! It inspires me as a composer.
You prefaced the video by saying “It’s a pretty bold claim to call a piece of music absolutely perfect”, and I think it IS a bold claim when you don’t define what you mean by “absolutely perfect” and acknowledge that there are other possible definitions. In the conclusion you implicitly defined it by saying that there isn’t really anything you could add or remove to improve the piece or without it falling apart, and I think that’s a totally valid definition of “perfect”-but not the only one! For example, while I like the piece at hand, I enjoy others significantly more because I find them more beautiful and interesting to listen to.
Just something to think about. It does not really take away from the overall quality of the analysis. (:
Very interesting analysis. There is more to Bach’s tuning than the cryptic curlicues in on the title page. If you further analyze Prelude I in C you will find that EVERY note of the 12 tone scale from C to C is used in the piece. I have read that it has been speculated that this first “perfect” piece was also Bach’s “tuning” piece to see if the harpsichord was properly tuned.
Bach wrote this wonderful piece of music at age 37. Think of it, what looks like a piece for children is actually a piece that requires extensive knowledge and deph. Only one who is already a masterful composer can write something like this.
37 is a special number, as you'll hear in the 73rd episode of Big Bang Theory. When you multiply it by 3, 6, 9, 12, etc., it yields 111, 222, 333, 444 etc. It has other nifty properties that are covered in the show and which I've forgotten. Although in the show he's touting 73 as the special number, everything Sheldon says applies to 37 as 73's mirror image.
@@genesisPiano Wow, that makes it even more fascinating. A perfect piece, mathematically speaking, composed at a special age by the number.
Meanwhile, at the same age, Beethoven and Chopin were busy composing complex works of high difficulty, you know appasionatas full of octave passages, forttisimos, crescendos, and such things. Bach was more concerned in writting a piece that, although simple to execute, resonate perfectly with the universe’s perfection.
@@shadowjuan2 Bach designed this pieces to teach his sons the art of playing keyboard in a progressive way rather than being showcase works. He composed complexed works for organ earlier in life. These works by way, were part of both Beethoven and Chopin education on how to play the piano.
Wrong. Bach published WTC in 1722, when he was 37, but many of the pieces in WTC date from much earlier. WTC is a compilation, not a newly composed work. Nothing in WTC "looks like a piece for children."
You yourself are my golden mean.
What fascinates me the more is that most of the explanation goes over my head but admire your savvy approach of all this.
Somehow this makes me have the same goosebumps l have when l listen to PACHELBEL’s canon and some Gregorian chants sung in ancient abbeys. Wow !!!
I’m blown away.
My hat off to you !!!
Always a good day when Mr. Krenz drops a new vid.
Really enjoyed this as an adult beginner learning this piece. Great video!
Excellent analysis. Thank you.
Bach is my guy. I play this piece on the guitar. I found very original and interesting details in your interpretation, like the sutil ascelerandos and those fortissimo basses almost at the end. I really enjoyed your analysis of it and sure it has enriched the way I understand this piece. Thank you!
I agree with you. I’m in the process of learning all the preludes and fugues from both books and there are gems of extraordinary beauty throughout, but none match the beauty and clarity of the opening prelude in book one. As far as the golden mean I would add that the shape of the motif consists of five notes rising and then three notes rising for a total of 8 sixteenth notes per measure where 3, 5, and 8 are proportionally close to .618. And then Bach’s use of the major 6th between G2 and E3 in the lower register has a very striking effect at key places in the harmonic journey that is reaching home. Some argue that the major sixth is the most beautiful interval and it is proportionally close to the golden mean.
I never considered that the golden mean underlies the structure of the entire piece, but there it is neatly packed like the seeds of a sunflower and even turning back on itself as spirals do. That’s really extraordinary. I have to wonder if Bach did that consciously. Public education in Eisenach was very rigorous but I’m not aware that Bach’s interest in math went beyond the rudiments of ordinary operations.. Either way, the presence of Phi is everywhere in this prelude in an unforced, very natural way.
Good luck on learning both books, that is a massive and rewarding undertaking.
@@danielkrenz - thank you. I make it enjoyable by reading the scores away from the instrument and imagining how it would ideally sound in terms of the tempi, dynamics, and phrasing shapes that I like. I jump around a lot so as not to get in a rut with any one in particular. And then I took a cue from the elder Bach himself by hand copying them on manuscript paper sometimes with colored pencils which even if applied somewhat indiscriminately has the effect of enlivening the score. And I’ve decided that when I’m at the stage of perfecting the performance I’ll leave the opening prelude and fugue for last since they are the best and easiest to play, and I don’t want them to sound cliché.
it's also a canon harmonically if you carefully analyse it. I remember seeing one in a book on canons and showing how Bach shaped the harmonic discourse canonically. Astounding piece of music.
The book is The Technique of Canon, by Hugo Norden. It takes the Taneyev approach.
Wow! This does enhance my understanding--and appreciation of Bach's music. So much of this is new to me. I wish I had more time to dig deeper! Thank you, thank you for this presentation!
Thanks for an illuminating presentation. Herman Hesse was asked if he knew a perfect piece of writing. He said he didn't, but that he knew a perfect piece of music: the Ave Verum Corpus of Mozart.
That is a fascinating story, I hadn’t heard that before. To be fair, Ave Verum Corpus is a pretty fantastic piece!
Inspires me to include this ratio in one of my pieces. Thanks for a great video!
I agree this prelude is beautiful. I love Bach. I don´t know all of his works of course but among those that I know - my favorite is Seufzer Tränen kummer Not
I love the ornaments you did in the last bar. I copy them for myself
Wow!!! Thank you for sharing this!!'
Well done, interesting video!
Very nice presentation. I also downloaded and read your professor’s paper. Excellent! I created a similar paper in the early 1980s regarding Bach’s Gigue from Partita No. 2 in d minor, from the Partitas and Sonatas for Solo Violin. One listen, and you will hear it. I recommend the recordings of those works by Henryk Szeryng, who seemed to understand the concept of Concinnity, bringing out the golden mean points in his performance. It turns out that major major 7ths aren’t as rare as you might think in Bach - appearing quite often in his preludes and allemandes, and other such works - but they are quite often associated with something close to a Golden Mean marker. I say “close to” rather than “they are” because mathematically they aren’t exact. And are we measuring bar lines? Attacks? Releases? Somewhere in a sustained note? Does it work when a section repeats? Or maybe only when it repeats? It may be imprecise, but the work is strengthened by the presence of these markers nonetheless. I was particularly impressed with Mike Linton’s discussion of how the Golden Mean is often overlaid upon artworks in not-very-convincing ways. Wishful thinking. But it’s an odd proportion, and in the real world we deal with whole numbers, so sometimes the work can only suggest the Golden Mean, not precisely outline it. Sometimes the subject has to smile before the golden mean is achieved. (A joke based on Linton’s discussion of the GM in faces) It was altogether a very enjoyable discussion.
Wonderful!
This is numerology. The spiral can't go backwards like that because we don't play music backwards.
Bach's crab canon, John Luther Adams' Become Ocean?
Bach was famously an avid numerologist. In no way do I think that Bach intended for this piece to be played any other way than front to back, but I am just highlighting some of the musical characteristics.
Can't you have a declining golden ratio?
Wonderful breakdown and analysis of a beautiful, beautiful piece. I am of the belief that of the great composers, or great song writers, have the simplistic ( God-like or supreme design ) golden rule in their very being and soul. They do not analyze their work and iterate through the creation of a piece, rather it flows from their mind through the quill and ink onto the paper. I also believe they see/saw the world connected through the golden ratio.
The quickest 1-D search algorithm uses the golden mean...
the major 7th chord was used all throughout in the baroque period. even frescobaldi (early 17th century) uses it a lot...
It has been in use, but it was still rare. And for the case of Frescobaldi that chord would come across as a natural extension of the counterpoint. But what I find interesting is that Bach used it functionally and very intentionally.
I enjoyed that, cheers. Would love to see an analysis of John Dowland's 'Frog Galliard'
I don’t know this piece, but I will check it out!
That would be cool! I don't think he was as interested in mathematical relations as Bach, but his music is interesting. Would love to see a breakdown of any of his music.
I agree with this. Actually, IMO, the C Major scale is the most beautiful scale of the modern temperament. Some of the most memorable compositions of the most famous composers are in C major.
In C.
All scales are similar in modern temperament mate
I think Bach was thinking contrapuntally throughout the piece rather than as block chord progressions. The chords are the result of the interacting parts.
The irony being that the key feature of Bach’s music is Change which is the very anti-thesis of Perfect. It certainly aims for perfection.
I would think that the voice leading on the bar from the diminished 7 of V to that ofI is permissible according to counterpoint rules, as the two enharmonically are closely related (although I am not an expert on Fuxian counterpoint). Think of a triad followed by dim 7 of IV. Similar thing; one can be seen as an extension of the other. He would not want a G in the bass there because it would anticipate and neutralise the effect of the long dominant pedal which arises next. My view is that the symmetric properties of the piece arose organically; I don't think it was consciously done. It gets back to the fundamental point Adorno makes in the most important book in musicology: The New Philosophy of Music. Music must come from the object itself, not the subject. (And this is a major criticism of Schenkerian Theory.) A songwriter would understand this: a good song sounds like it wrote itself.
Awesome 👍
Question about pedal. I tend to use it as little as possible with Bach. Originally it is harpsichord anyway. But -- as people like.mmh..
I have heard wide ranging recordings of this prelude specifically. Some go slow with lots of pedal. And Gould does it with half Legato half Staccato!
@@danielkrenz It is a discussion that we can not really have here. There are a number lessons and various interpretations. One -- not mainstream -- is from Denis Zhdanov which gave me a lot, but together with the Fugue. The pieces of Bach invite for interpretation. Gould is not necessarily the only authority.. )
Great
Good stuff. I'm unfamiliar with the terminology where we call maj7 chords "major major 7th", dom7 "major minor 7", etc. The one chord type I would normally refer to that way is minor major 7. It all makes sense, but my question is whether that particular terminology is more widely used than I'm aware of. Perhaps outside the jazz geek realm I live in ...? It would be handy to know if that's the case. Cheers.
The reason I called those chords that way was to (hopefully) clear up any confusion. I know if I say Maj7 to a jazz friend they will know that it is a Major triad with a major 7th. But if I say Maj7 to some classically trained friends there has been some confusion on what that actually specifies. So I was trying to bridge that terminology gap. I don't know if there is a perfect way to do it though.
@@danielkrenz very interesting Daniel. That was my question precisely. I'll sub up and see you in there. Cheers again.
And Fibanacci sequence was not known to JSB - I assume? Great video - thank you!
Hey, check out what that character van B. does in the golden section of the 1st mvmt of the 8th, and how "the silence" takes you home.
Everything Bach did was perfect, even his "parallel 5ths" in the incomplete A of F variation. As GG said, Bach would've corrected it, but do composition students and composers out there see the incredible composition lesson in these unfinished last notes?
Well...... DO YA !!!???
Thank you for this video! I just didnt understand one point, how the major7th chord relates to the goldent ratio, why this chord at this points, whats the meaning of it?
That is placed at “golden mean” points in the piece.
Thank you. I've always felt a forwward tug witb this Prelude and how it turns toward the pedal point with the increasing dissonances before the release. As to the Golden mean. It can be mostly debunked by skeptics but the fact of the matter is, it's not arithmetically precise but is precise geometrically which is area and space. I believe that Bach had studied rhetoric and numerology and I seem to remember someone saying or writing this was what made his music- especially his fugues so compelling.
The last lpoint I would want to make is general. beware of simplicity especially with Bach.
This "in" my favorit piece to play. Love the mathematical analysis. Thanks! Do you play music mathmatically as well? Just curious.
I don’t think that playing the piece mathematically is necessarily the best strategy for musical production. But certainly the math helps the overall understanding.
@@danielkrenz Good. I find that mathematical playing lacks in expressivemes. :)
34, not 24! (7:50)
Great catch! My bad!
While the golden mean idea is intriguing, and wouldn't be incongruent from Bach's scope of learning, I seriously doubt that listeners would be consciously aware of or touched by its presence. The real genius of the piece is how Bach can take the basic boring key of C and stuff it chock full of grace, poise, soaring interest, tension, and finally release by species harmony. All keyboard instruments of Bach's time could not express dynamics as organically as the modern piano, it was always block dynamics. Most expression then had to come primarily from harmony, articulation, ornamentation. IMHO the reason why Bach's music transcribes so well to so many different instruments and the zoo of timbres is because Bach was fluent in species harmony like nobody before him or since him.
Yes, the first Prelude is perfect in its way. But there are all kinds of perfection. I would say that Josquin's _Ave Maria_ is equally, but differently, perfect.
_Suum cuique._ Cheers from overcast Vienna, Scott
The additional measure in Schwencke and Czerny that you cite is first attested in Forkel and also appears in Hoffmeister (and Gounod leaves it in). Labeling it "Bach's original" is questionable, since it is in none of the surviving autographs. Wilhelm Friedmann's Klavierbüchlein, for instance, lacks it.
I'm confused about your measure numbering where you mention the inserted measure (and immediately jump-cut the video to the two following measures!) but I see that it is measure 23 in the analyzed score that you present. What's most daring about it, if it's authentic, is that it's a complete diminished ninth - and best analyzed that way: vii°[4 2 add10] That jarring C in the B-C-D tone cluster isn't a mere passing tone, it's harmonic color in its own right - a minor triad on iv (which we've seen already in the piece) overlaid on a diminished triad (or diminished seventh) on the leading tone, The iv proceeds by natural voice leading to the vii° in the next measure, and the vii waits for it there, with the new tone of G in the bass changing the whole from vii[6] to V[7].
On the whole, that progression seems at once too daring and too perfect for anyone but Bach to have attempted in 1722, so I suspect that Forkel transcribed the composer's final intention. In that case, the fact that there's an earlier version, with a different measure count and lacking this particular bit of drama suggests that the Golden Ratio hypothesis is coincidental, even though it's tempting given Bach's fondness for numerology!
In any case, it's surely a much more sophisticated piece than most people give it credit for - they tend to dismiss it as an exercise for the early grades. Truly a beautiful cameo!
I based my analysis off of this off of the Wagener-Volkmann autograph from 1732 which is the same as the 1725 Gerber copy. The Forkel copy that I am aware of is only 24 measures long. The Friedmann Klaviebüchlein that you mentioned is similar to the Forkel.
Sorry Daniel, Well tempered tuning is much closer to Equal Temperament. That is why it is called Well meaning you can play in all keys with some sounding smoother and others sounding less smooth. Meantone tuning is designed to sound as smooth as possible in about 5 keys and horrific in all the rest.
My understating was that Equal Temperament (as we understand it) wasn’t in widespread use in Germany until the late 18th century where as a loose “Mean Tone” was in use. It is still unclear precisely how Bach wanted this tuned, so maybe those loops on the title page are the instructions!
Sigh, this argument again huh? Well Tempered was a broad category, not a rigid specification. It was unique to whatever instrument that Bach could meld with a tuning hammer, and was different depending on the inharmonicity of the strings among many other acoustic factors. Don’t overthink it.
*_«In general, the interference equation can be used to measure resonant amplitudes for any musical interval under any temperament or octave division. This equation tells us that minimum resonance occurs at the fourth root of an octave (or square root of twelve) while maximum resonance occurs at the cube root of half an octave. Taken together, these results offer clear evidence that harmonic interference balances naturally around 12 as the most rational and harmonic number possible.»_*
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*_«We find here the most amazing thing. The arithmetic mean converges toward PI, or mathematical constant π ≈ 3.14159, located in the middle of the curve. We further find this point in the distribution curve to be equal to Unity (or 1) when the domain value X = 12. This is significant because twelve is the square root of 144, the value shared by both harmonic and Fibonacci series in a 12-step octave. Squaring each of the table values and dividing by twelve confirms that 12.02383 ≈ 12 is the point of balance between foreground and background._*
*_The significance of twelve as a point of balance in the octave interference pattern is proven further by plugging it into the equation, confirming the curve height equal to Unity at the octave. But even more significant than this is the fact that plugging the square root of twelve into the equation results in the amplitude y = 5.0666. Care to guess what this number represents?_*
*_It is none other than the y-axis amplitude for the golden ratio in an octave. Yes, the square root of twelve in the Gaussian interference pattern occurs precisely at Φ, right in the “cracks between the keys” of a major 3rd and minor 3rd in an octave. Just like the dense lattice region between a major 6th and minor 6th, the infinite golden ratio also provides an anti-harmonic proportion in the lower half of an octave. This occurs naturally at the square root of 12 (or fourth root of 144) in a 12-step octave._*
*_No matter how you do the math, both harmonic and Fibonacci series reach a harmonic balance with one another at n=12 and an anti-harmonic dead zone at n=√12. Division of the octave by twelve (not eleven, nineteen or any other number) is revealed here as a completely natural pattern produced by linear harmonics that are curved in pitch space by Fibonacci proportions as they converge to Φ. Could Gioseffo Zarlino’s decision to divide the octave into twelve steps have involved some knowledge of this simple relation between harmonics and the Fibonacci series?»_*
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*_«As a surprising correspondence between music and math, this little trick reveals the Pythagorean comma accurate to 3 decimal places. More amazing still, if we recalculate using the un-rounded arithmetic mean 12.02383 found earlier in place of 12, we obtain a slightly better estimate for the Pythagorean comma good to 4 decimal places. This bizarre associative property in the interference equation using the anti-harmonic golden ratio location of n=√12 proves the golden ratio is a physical property in the natural harmonic series and not some kind of error or “evil” in nature as portrayed by the Church. Vibration needs room to resonate in space and the Pythagorean comma created by the golden ratio appears to be just the right amount of room needed.»_*
Gram to codziennie w domu .
Did bach purposely write it to conform to the golden ratio, or did it just happen that way?
I am unsure about this. There is no written evidence (other than the music itself) that Bach wrote this to conform to the Golden Ratio. But we know that he was aware of it, and was famously obsessed with numbers and their meanings.
It's entirely possible that Bach himself could not have told you just HOW he did it. Let's just say that, however he did it, Bach knew what he was doing.
I have always said that if Bach had been a physicist, he would have rivaled Newton.
Maurice Ravel called Claude Debussy's "Prelude to the Afternoon of a Faun", "the only perfect piece of music". I know there are other perfect compositions. I nominate the first movement of Beethoven's "Moonlight" Sonata.
That is an interesting quote especially coming from Ravel. Faun is a pretty darn great piece of music…
Muy interesante!!!
0:03
Me: yes, Echoes by Pink Floyd
Does that tune have the ratio, or are you just putting it forward as a perfect piece for other reasons?
What about the fugue?
For another video perhaps!
Why didn't you round up 21.63 to 22?
This one was tricky. But if we divide the bar into 16th notes and count when the dissonance happens within the bar 21.63 was close enough for me. It is a fair point.
Webern Op. 21 also perfect
You are the first person to actually name another piece!
I fell in love with this prelude as a little child, when I first heard Nana Mouskouri sing the Gounod version of Ave
Maria: th-cam.com/video/ckQC7_ywFb4/w-d-xo.html
Ever heard of the art of fugue? That’s perfecter
I’ve got a feeling that he has indeed heard of the Art of Fugue.
Bach's prelude in C = western harmony in a nutshell.
La famosa teoría de la perfectibilidad de Bach. Un impostor de cabo a rabo. Una figura absolutamente menor en su época llevada a la categoría de mito por sobre encima de varios compositores contemporáneo muy superiores a él , pero olvidados. La pieza que muestran en primer término es de Anna, su segunda esposa, algo absolutamente comprobado por la historiografia y la grafología. Qué patético resulta oír como los sordos elevan al séptimo cielo su música. Es una experiencia harto irritante soportarla.
The pulse is not even; of all works this meter and tempo should be regular
Merci beaucoup for this. I'm still struggling to learn this piece, since I started the piano only after my retirement. I'm trying to develop music and a melody in my head, and I find this the most difficult task since I never learned to play music with someone, I never harmonized with someone in my life.
I loved your lecture, especially your discussion of the greek work, Chiro. I worked as a Chiropractor, practicing with my hands to alleviate the aches and pains that pianists and other humans suffered. But I have a major orhopedic problem with my shoulder, and I live with a FURS syndrome. So I adapted my techiques to continue to work, and I have a video of exercises. The low back exercises take 30 seconds, but need to be done every day. And there is a section for the shoulders, arms and hands to help with tensions in the arms and strength in the fingers.
th-cam.com/video/tK9sc0ckSpw/w-d-xo.html
As fascinating as this is I do wonder if you aremetephoricly just slapping a spiral on music in the way you say people do to art
The Indian model might have the most mathematically perfect face, but it's definitely not the most beautiful. There is nothing special about her face, and perhaps that's because perfection can be boring.
The most beautiful faces tend to have at least one feature that is striking because it enhances the face despite not conforming to conventional beauty standards. For example, things like moles, wrinkles, exceptionally large eyes, or a very large, square jaw can easily make a face unattractive, but they can also make a face strikingly beautiful - on the right person.
subbers
Several factual errors in the presentation.
Unfortunatley, you have said almost nothing about the music itself. In other words, Bach could have thrown a bunch of darts at a score but put a major 7th where you point to them, then that would be a perfect piece of music. Ridiculous.
In no way do I claim that a major 7th placed appropriately makes a piece perfect. But Bach chose that harmony for very specific places and this is an investigation into that.
No, I think he made his argument. He talks about the simplicity of the piece and yet at the same time the complexity of the patterns he found within it, like the descending 4ths and 5ths being a palindrome. Watch the video again.
@@danielkrenz Yes, indeed, Bach chose that harmony for a specific reason but certainly NOT to outline a Golden Section. He did so as a demonstration (as is the whole piece) as an element of the rules of funtional harmony. I would enjoy, as would others here, I'm sure, you're analysis of the harmonc structure of the piece and how it demontrates those rules and how it creates a meaningful whole. I'm sure you would come up with some interesting points and this will point towards Bach's genius, which is sublime, and cannot ultimately be put into words or theoretical analysis anyway!
@@josephciolino2865 I do not think that Bach created this prelude specifically to reflect the Golden Mean, yet the music does reflect the properties. The harmonic analysis in the video is my own and there are many other structural points that Bach uses the harmony to reflect. I also agree that there is something ineffable about this prelude. It is so great that it has obviously captured the attention of countless people, and all of the analyses and descriptions of the piece fall short ultimately.
Just another person trying to make money on TH-cam. I have been playing Bach for 55 years and am still learning new things from his music.
Ha, I haven’t made a cent off of TH-cam. This is just a passion project to share insights about the music that I have gained from my teachers.
How is he making money off this, exactly? Some people will criticize anything and everything without giving it a fair shake.
This thumbnail is so cringe dude