I am reading the book "Gödel, Escher, Bach", in which the author tries to present concepts in mathematics, drawings and music as instances of a certain "pattern", that he calls "strange loops". He makes several comments about Bach's music, in particular, as the title suggests. So, he talks about the fugues, sonatas and partitas, modulation, themes played against themselves, how Bach manages to get two or more musical lines going simultaneously and so on.

But I don't understand a thing about these musical concepts. I have tried to listen on YouTube to some of the compositions he mentions in the book, but my stupid brain only hears a stream of notes flowing. I can't recognize the patterns he talks about, nor can I grasp, for example, the fact the the "Canon per Tonos" rises successively until it reaches the key C again.

So how can I learn more about these musical concepts, so as to better follow the book's main ideas? To make this question narrower, I'm not asking for general references to learn about music, but something very specific to the content of the book, so that I don't miss important information (and fun) from his exposition.

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    G, E, B is one of my favourite books. I've read it several times and always find new things I hadn't spotted before! There are quite a few musical ideas (as you say, mostly related to musical structures) mentioned in the book; if I get a chance I'll post an answer with info about each of these later. In the meantime, it is worth noting that, although Bach was a great exponent of the musical forms mentioned in G, E, B, other composers have used these forms too... Commented Jul 19, 2014 at 16:08
  • Thanks, everyone! All the answers are really great and I will be following your recommendations. Having already done some of the things you suggested, I can see a bit more clearly now the patterns, voices, themes etc. Commented Jul 20, 2014 at 12:49

8 Answers 8


As someone who has no formal training in music whatsoever but who fell in love with Beethoven and Bach upon hearing them, I discovered that my visual senses are much better at picking out patterns than my auditory senses. Here is a rendition of Bach's great fugue BWV 542 which shows it all to you while sacrificing none of the auditory pleasures derived from hearing this work.

Sometimes just visually looking at what's being played is all you need. The patterns are just so obvious when you hear them AND see them. In this case you'll see what a fugue is and a few of the things Bach is doing with the main melody. You will quite literally see the individual voices. I recommend playing this a few times and focusing on one voice so that you can see and hear what it is doing.

Here is his Fugue BWV 565. In general I recommend any MIDI viewing program and then you can get free MIDIs (for example here), open them up in your favorite MIDI viewer and view the piano roll. There are plenty of paid ones of course but the one I like is called Synthesia which has a pretty good free version and it does everything I need it for. And here is a sample of what Synthesia does. I am still ridiculously slow at sight reading but using MIDI viewers, I can now play my favorite pieces from Bach, Beethoven, Chopin, and Mozart for my own pleasure.


Tough question about an amazing book. "Hearing" the fact that that particular canon continues rising in key is more difficult, and I think it's more interesting just to know that fact. Hearing the sort of patterns Hofstadter discusses is far more important to your enjoyment of the book. I would get a recording of Bach's Well-Tempered Clavier, and listen specifically to the fugues (the even numbered tracks). Listen to the same one over and over until you can hear the way that a melody is presented entirely by itself in the beginning. The next thing that happens will be a copy of the same melody (usually lower) even as the original melody continues along. Really take time with the beginning of the fugue, listening to the two voices mirroring each other.

Now listen to the whole thing, keeping an ear out for subsequent entrances of the original melody. Listen especially for the pattern of the first few notes to come back many times. Some fugues only have two voices, some have as many as 5, that refers to how many distinct melodic streams are happening at the height of the piece. But even in 2-voice fugues, you should be able to hear that original melody in many incarnations after you train yourself to hear it.

Ideally, I would get the score of whatever fugue you focus on, even if you don't read music, you might be able to see the patterns (higher symbols generally mean higher pitches). If you can get to a keyboard and learn enough to at least plunk out the basic melody, that would help to.

As with a lot of GEB, sometimes you have to go with the flow trusting that understanding will develop as you continue and, most likely, reread.

  • +1 I couldn't agree with your last paragraph more, Pat! I still haven't entirely got my head around the Incompleteness Theorem. Commented Jul 19, 2014 at 16:11
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    I eventually read some of the books on the Incompleteness Theorem recommended in the GEB bibliography in order to increase my understanding. Gödel's Proof by James Newman was particularly helpful, though not as engaging or user-friendly as Hofstader. I think it's one of those things that you have to read many different angles of before you really start to grasp it. I still have to refresh my comprehension from time to time. Commented Jul 19, 2014 at 19:43

A key to following the polyphonic music of Bach is to be able to single out the individual voices. You will be very much helped by having a score at hand. In a fugue, the theme is presented in one voice, then repeated on the dominant with slightly modified intervals (but same rhythm and pitch profile) in another voice, then introduced in the tonic in a third voice and so on. A good pianist will bring out these entries, but seeing them in the score will most likely aid your understanding of what goes on.

Keeping track of modulations throughout a piece can be quite challenging even for experienced musicians, but in the score these different tonal regions can be easily identified.

By the way, I think the ideas about self reference and Gödel's incompleteness theorem are much harder to grasp for most readers of GEB, but still the book is just as enjoyable.


The easiest way to understand the concepts in GEB is to understand basic music reading. Look at the notes only as letters for music.

A good intro to reading music reading can be found in this link: "The Basics of Reading Music" by Kevin Meixner


I agree with Kevin's answer. You need to understand music to understand those concepts. I also read GEB last year but at that time I couldn't understand it. But now I have learnt how to read music, what are scales and keys, etc. This time I was able to understand it more properly.

You should visit this website: http://www.musictheory.net/lessons .


I agree with luser droog's answer.

My understanding of the book (which I did not finish) was that there is an overall theme of language, which is then illustrated through the three artists. While the letters in each of these languages are based on natural, measurable, physical phenomena (projections to 2 dimensions in the visual language of Escher, mathematical elements in the metaphysics of Gödel, and musical tones in the language of Bach), the languages themselves (i.e. the "grammars") are artificial.

The specific artists are not drawn out merely to indulge in a cult of personality, but because each were innovators in developing the modern features of their respective language. I suspect you may have guessed where I'm going with this.

To get the deeper understanding that you're after, you have to read the scores. If you don't read staff notation, learn it. I don't mean to be curt or rude, but there is to my knowledge no other way.


Using sheet music is a great idea, but, in this modern computer age, you can go one better and use digital sheet music. I don't just mean a PDF or any similar format that is essentially just computerized scan of the original paper, but actual digital files designed for sheet music programs.

While the music will not sound as good as if played by actual musicians, you will have the advantage of an indicator showing you exactly where you are in the music. For example, if you know what a key change looks like, you can see exactly when it changes and hear what that sounds like. And you can visually see the repeated patterns and know when they are being played. If you look into more advanced options, you can even discover how to mute an part or parts temporarily.

Since Bach is in the public domain, I would suggest using the free program MuseScore and using their website to look for the piece in question. Probably someone has uploaded it. For example, here is the "Canon per Tonos."


Youtube has some good videos that might help:

canon per tonos

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