## Art of Fugue: Stretto Fuguesand Canon at the Tenth

### Theme Developed By Other Means

In the next group of three fugues, Bach appears to suspend thematic transformation in order to establish a different process of variation involving stratification of the subject. Whereas the first four fugues contained no stretti, here Bach uses stretto to the extreme, demonstrating how a well-crafted subject can be made to "counterpoint" itself in endless ways. Bach gives us a simple, but elegant, model for stretto variation in the first four canons of his Fourteen on the Goldberg Ground, where he layers the subject with its retrograde, the inverted subject with its retrograde, the subject with its inversion, and the inverted subject with its inversion.

The subject (t8) of each stretto fugue is a slight variant of t5 from Contrapunctus III. Ever so slight, the variant is not without significance to the composer. For the first time in the Art of Fugue, the subject is comprised of fourteen pitches. Reason enough for Bach (B+A+C+H=14) to dwell upon it for awhile! In Contrapuntus V this motive is stated 22 times, half of them right-side-up and half upside-down (t9). Similarly, the 28 statements of t8 in Contrapuntus VI, are evenly divided between rectus and inversus. Of the 28 statements of t8 in Contrapuntus VII, 16 are right-side-up and 12 are inverted. The stretto fugues contain 75 statements of t8 or t9. Only Contrapunctus VII transforms the subject in two minor variants which we call t10 and t11 (below). It is as if, in coming to this last stretto fugue, the composer says, "I've shown you how the subject can counterpoint itself, now let's vary it some more."

• Contrapunctus V
This fugue has some tricks up its sleeve! The trick is to show how many ways the subject can be combined with itself in simultaneous counterpoint. Here the subject accompanies itself eleven times (22 statements). No two combinations are alike. Each stretto employs a different time interval or contrapuntal variation of the subject. Beat separations range from six to one. Some stretti are in contrary motion, others in canon. Only the stretti beginning in mm. 1 and 23 repeat; yet, even there, Bach employs an exchange of registers. At the conclusion Bach composes a hyperstretto in which right-side-up and upside-down versions of the subject appear with no separation at all. Click the blue boxes to hear combinations. Note: left-to-right orientation shows stretto timing...some entries change key (or mode) when they participate in a different pair.
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Notice how the stretti give this fugue atmospherics much different from those of the preceding simple fugues. The sound is one of sweet tenderness, with subjects caressing each other like lovers in a park. Subsequent fugues are also stratified but detailed analysis should not be necessary; they contain new tricks which shall occupy our attention.

• Contrapunctus VI a 4, in Stylo Francese
As a youth Johann Sebastian lived for three years in Lüneburg. This city was famous for its Ritteracademie where young noblemen studied the French language, social graces (fencing and dancing), and, most important, listened to French music. Young Bach was able, also, to visit the city of Celle, fifty miles to the south, where Duke Wilhelm of Brunswick-Lüneburg maintained a bevy of French musicians. It was probably in Celle where Sebastian first heard the music of Couperin, Lully, Muffat, Fischer, and Raison.

With its clipped dotted rhythms and flights of thirty-second notes, Contrapunctus VI represents a fusion of the French Mannier with solid north German counterpoint to produce a one-of-a-kind fugue in stile francese. The subject of this fugue consists of francophile versions of t8 and t9. But, for the first time in the Art of Fugue, Bach employs rhythmic proportions. Both versions of the subject are stated in normal values (half notes) and in diminution (quarter notes). The inversus (t8) and rectus (t9) forms of the subject are each stated 14 times. No two stretti are alike, and nearly every proportional or inversional permutation is represented.

• Contrapunctus VII a 4, per Augmentationem et Diminutionem

This fugue is about the subject accompanying itself in rhythmic proportions of double and half its normal values. The subject is stated four times in half notes (its normal pace), four times in whole notes (+), and twenty times in quarter notes (-). The fugue contains seven episodes in stretto (left). See if you can locate the following:

• ONE episode with the subject in diminution only
• ONE episode with the subject in augmentation and normal values
• TWO episodes with the subject in diminution and normal values
• THREE episodes with the subject in diminution and augmentation

If you have played the above segments (by clicking the graphic to the left), you will have noticed that the climax of the fugue comes in m. 35--the episode in which the subject is stated in its longest values. This is followed immediately by the episode with which it most contrasts--the one with the subject in diminution only.

Twelve times the subject is upright and sixteen times upside-down . The four statements in augmentation alternate rectus and inversus. Similarly, the first two statements of the normal subject (mm. 2 & 14) are upside-down, while the second two (m. 36 & 38) are right-side-up.
• Canon alla Decima in Contrapunto alla Terza
The "Canon at the Tenth with Counterpoint at the Third" comes after the stretto fugues and before the double/triple fugues. With ties across three of its four barlines, the subject of this canon (t12) represents the most syncopated transformation of the cycle. This canon is also one of the best examples in music literature of double counterpoint at the tenth. Please refer to the Canons of the Art of Fugue study for an analysis of this canon.