In the two preceding sections (books written in 1960 and in 1978), we dealt only with music of irreproachable rhythmic structure. At no time was there question of changing anything in the music itself, only possibly changing the way in which it was set to paper (as in the Chopin and the Brahms). The problem now will be much deeper, affecting the music itself, more specifically its rhythmic structure, its container. This might not be for the weak of heart, and a certain discretion is required before deciding to dive in. The cultural critic of a major newspaper wrote an essay in which he explores "the inner life of music and mathematics", and in an exploration of "melodic echos and tensions", quotes the theme of Bach's Eb minor fugue (from Book I of the Well-tempered Clavier).

The Problem

The subject of the fugue is quoted here as it appears in its first entry, in the central voice (Alto-Tenor) of the of the 3-voice fugue. The author of the book clearly identifies the two phrases of the subject, A and B, the second acting as an "echo" of the first. Each phrase occupies the length of 3 s, a bar-and-a-half of Common Time (4/4). The first phrase cadences (with a Feminine Rhyme) on beat 1 of the second bar, and the second phrase cadences (with a Masculine Rhyme) on beat 3 of the third bar. None of this evidently ternary activity seems to preoccupy the author in the least, and its presence in a binary 4/4 container passes completely unnoticed, not only by the author but by all the critics who lavished praise on the book.

Click on the music to hear the sound.

The Solution

As far as the subject of the fugue is concerned, the solution is very much the same as it was at the beginning of the Brahms song. Giving it the footsies treatment (for each ), we would get L-T-R for each phrase of the subject. In other words, the subject is clearly in 3/2 time starting on beat 2. Note that EACH phrase starts on beat 2 and ends on beat 1 of the next bar, in accordance with Pick-up Association which is normal at this larger level. Since we are limiting ourselves to Bars and Bar-lines on this page, we will not go into the Ternary Transformation of the 3/2 bars, nor the syncopations in this theme.

One Container For The Whole Fugue ?

If the opening subject is in 3/2 time, should each appearance of the subject also be in 3/2 time ? should each appearance of the subject also start on beat 2 of a 3/2 bar ? should the whole fugue be in 3/2/ time, or should the fugue alternate between sections in 3/2 and sections in 4/4 ? (a little like the Brahms song). This is where the going gets rough - the answers aren't easy.

We have above, in 4/4, the original version of the opening exposition with its three entries of the subject, and a short "divertimento" (interlude) between the second and third entries of the subject.

We have above a suggestion for the 3/2 version of the opening exposition, with its three entries of the subject, all starting on beat 2, as well as the short "divertimento" (interlude) between the second and third entries of the subject, two short (1-bar) canonic entries which also start on beat 2 (Alto-Tenor, then Soprano, in the third line of the musical text). Since the original version consists of 10 bars of 2/2 (20 s), and the 3/2 version consists of 8 bars of 3/2 (24 s), it was necessary to add 4 s in the 3/2 version. This suggestion is, of course, based on the large "IF" that the theme of the subject is in 3/2 time, and it should not be interpreted as a correction of any kind. The 3/2 structure of this version can only be adequately felt and appreciated by beating Ternary Footsies all the way through, starting on L(eft), beat 2 of the complete R-L-T, one beat per . You might also try beating Ternary Footsies through the original version, or beating Binary R-L Footsies all the way through, one beat per , or possibly alternating between Ternary and Binary.

Another modification to this fugue has been suggested in The Chromatic Minor Mode Revisited.

Back to the Unknown Container chapter page