## An Analysis of Fugue #4 from Well Tempered Clavier's Book I

### Version 1.2 of 20/2/2011-3:00 p.m.

Take fugue number 4 from the Well Tempered Clavier. The famous "sea-bed" fugue, in C-sharp minor. It has 5 voices. The main theme is C#,B#,E,D#, thus the name. A fugue's theme is always followed by a sub theme, the accompaniment to the theme in another voice, called countersubject. The seabed fugue has 3 counter subjects, 2 of which are of major importance, because they play the role of separate themes later in the fugue. The Master starts the fugue with cbed, develops a one page fugue with the main theme, and then he starts combining the main theme with the second and third counter subjects, mimicking a fugue with 3 themes (cbed, counter2,counter3).

Take some graph paper and count the bars in the fugue. Now decide on arbitrary height for marking active vs inactive on all three themes. I.e. at height 12 mark with pencil all the bars where the main theme (cbed) is active. At height 8, mark the bars where countersubject2 is active, and finally at height 4 mark the bars where countersubject3 is active. Now label each subject as:

theme: 1, countersubject2: 2, countersubject3: 3

Now break the graph into segments, according to how the three themes combine with each other. Bars 0-16 are combination [1], since only the main theme is active there.

Bars 17-19 are [0] since nothing is active there.

Bars 43-46 for example are [1,2]. Ok, here's the whole pattern throughout the fugue excluding bar numbers:

Block 1
[1]
[0]
[1]
[0]
[1]

Block 2
[1,2]
[3]
[1,2]

Block 3
[2]

Block 4
[1,2,3]
[2,3]
[1,2,3]
[2,3]
[1,2,3]

Block 5
[2,3]

Block 6
[3]
[2,3]
[1,2,3]
[2,3]
[1,2,3]
[2,3]
[3]

Block 7
[1]
[1,3]
[1,2,3]
[2,3]
[1...]    (from subsequent block)

Block 8
[1,3]
[3]

Block 9
[1,3]
[3]

Block 10
[0]

Block 11
[1]
[1,3]

Now look at the symmetries that show up. All blocks are either symmetric with respect to their center pattern, or are pyramids, or both.

Moreover, all possible combinations of the digits 1,2,3 can be found in some pattern.

To be more exact, all possible subsets of the set {1,2,3} appear. The archetype {1,2,3} is a very important pattern in Christian doctrine.

The combinatorics discipline was barely developed at the time of Bach as it was mainly developed by Pascal and Fermat. It appears to be that Bach was quite knowledgeable about the basic issues. Nobody still knows if he actually intended those patterns to be included in his music or he just composed and those things just showed up.