Part 3 -- The Ten Puzzle Canons of the Musical Offering



Bach offers ten puzzle canons to you in his Musical Offering, all employing the Royal theme. That is, you have to seek how to play the canon. Bach provides you with a kind of musical orientation race map, to help you on your way. His hints can be found in the form of:

1. The title

A. “a 2,” or “a 4”:

Refers to the number of voices in canonical relationship with each other. For example, “a 2,” indicates two voices played as a canon with each other, or “a 4,” four voices in the canon.

In addition, there is usually a separate voice played simultaneously, which is not in canonical relationship with any other voice. A canon with “a 2” in the title may have 3 voices – two played as a canon with each other, and a third voice accompanying, as a free counterpoint voice.

In some, the Royal theme is the subject of the canon, and another voice plays an independent counterpoint. In others, the counterpoint voice is the subject of the canon, and the Royal theme accompanies, as an independent voice.

B. The title may contain a hint, in Latin, as to the type of transformation principle to be used.

2. Symbols

Clefs of different types, and in different positions, indicating the placement and directionality of the voices.

The clefs used are:


G-clef, or treble clef: Used to indicate the descant (high) range of notes. The line in the middle of the clef’s spiral indicates the note “g” above middle-C.

F-clef or bass clef : Used to indicate the bass (low) range of notes. The line in middle of the two dots is “F” below middle-C.

C-clef or moveable clef, or alto/viola clef: Can be used anywhere. The line in the middle of the two curves indicates the note “C” in any octave, and on any line. Ignore the term baritone clef, but notice that the line in the middle of the C-clef, which has been moved, is "C". 


   (For now, the symbol " % " is used to represent this)   Symbol meaning to "start here." Placed somewhere above the first voice, to indicate that a new voice should begin, when the first voice arrives there.

The object of this musical pedagogical, is for you to discover the types of change, transformation principles, or contrapuntal techniques Bach has chosen in order to play the finished canon.


  When referring to two voices in canonical relationship to each other, the first voice is called the “leader,” and the second, “the follower.”

Format for the following pedagogical workshop:

1. Each of the canons is first shown in Bach’s enigmatic puzzle form.


2. A series of questions are posed to help you solve the puzzles. Try to figure them out by yourself before looking at the solutions.


3. The solutions are presented.


4. Perform the same transformations with geometrical shapes.


5. Play the canon. Many of the canons have MP3 files with some of the voices. You can add the missing voices and perform the completed solutions. (Note that the music contained in the MP3 files is tuned to C=256.) You can also write out the new voice(s) beforehand.


6. Invent your own musical examples, using the same principles of change. Those who really want to try to relive Bach’s method are encouraged to perform the same transformations upon simple self-invented musical motives.


7. Investigate the musical paradoxes contained in the completed canon. How do these particular transformations set up musical problems that the composer has to solve (including: the creation and resolution of dissonances;  does the "mode" change with the introduction of the new voice, and how (see note on modes at the end of Part 2);  how the placement of what would be the equivalent of vocal register shifts changes; any other changes you are able to discover)? How do they enable the composer to join the battle with the resulting musical conflicts (and emerge victorious)? 


8 .How is the "comma" reflected in the performance of each canon? (For the Pythagorian comma, see note on modes at end of Part 2.) The music heard in the mp3 files are performed with a piano and flute -- two fixed-note instruments, but you can substitute stringed instruments and human voices, to use the canons as experiments to see what happens when a second, or third voice is added. What happens to the intonation of the first voice when being played/sung alone, or together with one or two other voices? How do the contrapuntal relationships effect the pitches of the different voices? How is the performance of the single independent voice subsumed by the musical idea of the whole?


Have fun!  





P.S.: A pdf-version of the notes for the entire Musical Offering is found at the end of Part 4 Conclusion. It includes the enigma canons written out in the usual musical notation as part 1, but try writing them out yourself first. It also has the notes to the solutions as part 2, but don't peak!


For further reading: 

See the end of Part 4 Conclusion for a reference to LaRouche-associated David Shavin's article on  The Musical Offering, which can be read after you try to figure out the puzzle canons yourself.




Thanks to :



Elsebeth M. Thing: flute

Janus Kramer Møller:  animations

Poul E. Rasmussen and Tom Gillesberg: homepage help


Lyndon LaRouche for inspiration to find a means to relive the discoveries of the great masters of the past.

David Shavin, Jonathan Tennenbaum, Sergej Strid and Mindy Pechanuk for discussing some of the musical ideas.


Timothy A. Smith, and his homepage “Canons of the Musical Offering,” for the pictures of the puzzle canons shown here. (reference at the end of Part 4.) 

Yo Tomita: Bach monogram watermark




 To canon 1