26 A COMPANION TO

of a series of three entries of x, with much of the effect of afugal exposition of a new subject. Then, in bars 66-70 we

.

{s /\

have the inversion in the lOth, B A in loth.2

A s-bar episode on the last quavers of /\ leads to the2combination /\ doubled in lower 6ths in S, A, over /\ in B.1 1

The resulting key is F major. The next episode develops anew figure, (y),

Ex. 16.

for 6 bars. Then, swinging from G minor to D minor we havethe combination /\ doubled in 3rds over /\, with the following2 1result :

{

s 'i' in 12thA /\ in lOth

1B/\

2

The next episode again introduces a new figure ~, with ~.This should be viewed as a whole; its character is independentof its optical resemblance to an ornamented figure of /\.1

Ex. 17.

~~c i r~ i-~F=1.JYet a new imitative scale-figure appears in bars 98-100. Aftera total length of 14 bars this episode reaches B~ major,where.(bars 103-6) we have the combination

{

s /\

T ~ upper 3rdB/\ .

1 '!"

The last episode fills the 8bars 107-14. Re-establishing (andtaking only 4 bars to re-establish) D minor, it leads to thefinal combination-

.

BACH'S 'ART OF FUGUE'

{

A ~in 12thT /\ in lOth

1

B/\2

with which the fugue ends almost abruptly. It is a wonderfulstudy in an almost modal mixture of tonality. Played quietlythroughout, its effect is as romantic as anything in Bach.Notice that the harmonic character is not only deliberatelysought, but is brought about mainly by the D-C in the loth.This is perhaps the only case where a definite harmonic effecthas ever been obtained from inversion in the loth apart fromits use in added 3rds.

In Contrapunctus XI Bach sets himself to work out a triple No.fugue on the inversion of the three subjects of Contrapunctus XI.VIII. The Berlinautograph puts it immediatelyafter that fugue;an arrangement which has its point. The notation of both fuguesin 2/4 time, with notes of half the present values, is anotherinteresting indication oftemP9, the inference being that Bachwas afraid of taking these fugues too fast. No. XI is the mostdifficult of all the aesthetically important parts of K. d. F., andis the better for a tempo appreciably slower than that ofNo. VIII.

The task Bach here sets himself is not strictly possible. Ofthe three themes the motto-theme is the only one that wasnaturally conceivedas invertible. The first theme of VIII canstill be construed when inverted, but is then obviously not aspontaneous idea. The second theme makes a series of suspen-sions that when inverted will resolveungrammatically upwardsinstead of downwards. Bach, instead of attempting to construethe resulting harshness, avoidsall difficultyby representing theinversion of 1\ by a similar theme which indeed climbs up2instead of down, but which resolves its discords respectably.

The student may make his own choice of a conventionfor the analytic signs of this fugue. I find it best to abandon allattempt to make the signs run through the whole of K. d. F.

27

Ii

"

28 A COMPANION TO

and I prefer to treat this fugue as an independent work,acknowledging the whole scheme only by once putting a starto the motto-theme.

We will accordingly disregard No. VIII and will mark theopening theme of No. XI as 1\. It is given an unbroken1exposition in the order A, S, T, B. Note that Bach does notspoil his answer by trying to make it tonal. That could be doneeither by making it go into the subdominant, or by stumblingover two D's in the first bar. Common sense forbids suchpedantries. The last bar of the theme gives rise to a 4-barepisode, after which S drifts once more into the subject, withcrowded imitations in the other parts. This concludes the firstsection with a full close.

Now A givesout 1\, an inversion of the 1\ in Contrapunctus2 1VIII. Bach, not being proud of the melodic result of thisinversion, conceals it beneath a rising chromatic scale, whichwe will callI\. This figure never rounds itself off as a defimtexsubject; but, with its inversion (promptly appearing in B), ithenceforth pervades the fugue and makes it typically chro-matic. With Bach this kind of chromatic texture is one of therecognizedspeciesoffugue. He had already used it in No. III:its presence in Frederick the Great's theme impels the fuguesin DasMusikalischeOpfer to fall into the same style, and it maybe recognized locally in the pair of new counter-subjects inthe second stretto of the great E major fugue, W. K. II, bars16-21.

At great leisure Bach makes his exposition of this chromaticcombination,whichisharmonically gorgeous.Three bars elapsebefore T answers; and an episode of one-bar steps in triplecounterpoint (Ep. 2) rotates for 6 bars before B has its turn.Then a 9-bar episode (Ep. 3) follows, on new lines (Ep. 2), atlast drifting into the steps of Ep. I. Then (bars S'5~-6o) Senters with 1\, but re-inverted into its rightful shape as in2No. VIII. This showshow sensitiveBach is to aesthetic values

BACH'S 'ART OF FUGUE' 29

in this fugue: he will not allow the grotesque inversion toappear more prominently than he can help.

Episode 4 carries on the permutations of Ep. 2 for 8 bars,

after which 0 (i.e. the right form of No. VIII 'i') enters in thebass and brings this section to a full close in A minor.

The third section consists of a regular exposition of ~( = ~of No. VIII and V of K. d.F.). The order is T (A minor),S (D mi.), B (answer), A (D mi.). Two bars, added to theseregular 16, make the third formal close in F.

The fourth section introduces 1\ of this fugue in combination3with 1\. As we have seen, 1\ cannot, for harmonic reasons, be2 3a true inversion of 1\ in No. VIII. In its Present form it is a

2 -

figure that can rise indefinitely and be doubled in 3rds and

6ths. Both T and A take advantage of this, while ~ (or ~ofNo. VIII) rises in the deep bass. (Some commentators see inthe alto of bar 91 an allusion to Bach's name. I cannot believeeither that Bach would first have anticipated this by the tenorin another key, or that he spelt his name Baccch.) A and Sanswer the two themes in bars 94i-7; after which S imitates anepisode on the rightful form of the figure of 1\ (as in 1\ of3 2No. VIII). This (Ep. S) lasts only for 4 bars, after which 1\1(accompanied by the 'rightful' quaver theme) enters in A inD minor. It is immediately followed by a new combination:

fA ~ (= 1\ of VIII)2 2

IT V (= 1\ of VIII)

x 1BV

The harmony is turned asideinto B b; and a sixth episodedevelops for 5 bars on the lines of Ep. 5, closing into the com-

bination{

S 'i' with 1\ in the bass. Episode 7 develops bothAI\ x2

versions of 1\ together with 1\ and Vx with great vigour for3 xIS bars, comingat the twelfth bar to the fourth formal close in

-,--

3° A COMPANION TOthis fugue. From this closein A minor the remaining 3 bars re-turn to D minor with the quaver figures rising in 6ths in a trueinversion over the descending figure in 3rds. Now (bars 133-6)

B gives ~ while S and A in 6ths have 0 inverted (i.e. thefigure of /\ in No. VIII). Then T, instead of answering B,2enters with /\ while S and A in 3rds have /\. Episode 82 3continues the line of Ep. 7 for 6 bars, closing into C major.This turns off into E minor, where we now have the combina-tion of the three themes intended to represent the inversion ofthe combination in No. VIII. In bars 146-9 the order is

{

S'iA/\3T/\1

in E minor. Episode 9 continues on the lines of Ep. 7. Butnow, instead of answering the triple counterpoint, Bach takesa new line and shows us the simultaneous combination of /\ and1

~that had appearedattheendofContrapunctus V. Bars 158-62. r 1

show the position { s X. After a 2-bar interlude this isAl

{T/\

answered in G minor by t , inverted in D-C at the lOth.BV

Then a short loth episode, still on the lines of Ep. 7 leads tothe triple counterpoint in the position

rA/\

lT~BA::1

immediately followed by ,;,.

rS~

lT~B/\

3

\ BACH'S 'ART OF FUGUE' 31

with which the fugue ends. It is a majestic and gorgeousmovement which improves on acquaintance. But Bach evi-dently places no reliance on its triple counterpoint. And heseems to be impelled to think seriously and methodically aboutthe rules by which a harmonic scheme can be totally inverted,having here proved by experience that you cannot safely inverta combination not made for the purpose. It is even possiblethat he had at first thought that this fugue would complete hisscheme; and that all that followsis the result of his disappoint-ment.

The next two fugues are tours deforce, being compositions No.that can be inverted note for note from beginning to end. For XIIpurposes of comparison Bach writes his Inversus under hisRectus, so that you can see their relation as in a mirror. Thedifficultyof such counterpoint can be overrated. Suspensions,as Contrapunctus XI shows, must either be avoided or (asBach was probably going to prove in the unfinished fugue, andas I prove in my conjectural finish) so contrived as to resolve'both ways. A 4th from the treble will become a 4th from thebass and must be treated accordingly. Contrapunctus XIII isturned inside out as well as inverted, so that Treble=Bass,Alto=Treble, and Bass=Alto. This complicates the resultsof 5ths and 4ths so much that they must be treated as in triplecounterpoint. Otherwise there are no special difficulties,and the composer only needs to watch the results of inver-sion without trusting only to rules of thumb, and he willachieve quite respectable music in both versions, though theremay be no very clear evidence why either has been writtenat all.

Contrapunctus XII is a smooth little fugue of thesimplest kind. The subject of K. d. F. is put into 3/2 timeand, in the Rectus, is exposed in the order B, T, A, S,with a counter-subject that does not survive the exposition.Only 3 bars intervene between the soprano entry andthe systematic exposition of a new variation of the theme.

Ii;

I..I

II

32 A COMPANION TO

Here are the two versions on one stave for purposes ofcomparison:

BACH'S 'ART OF FUGUE' 33

in Ex. 20 the total inversiO'n of (a) remains ,unchanged, while(b) inverts into (c).

S announces this variation in bar 21. It is answered by AinG minor in bar 26. After a 2-bar interlude T has it in Bb

(bar 32). From bar 21 onwards it has been given a stretto-likeeffectby close imitations in the other parts; but the fugue hasno really elaborate devices. Its only developed episode fills the6 bars from 36to 41, after which B enters with the varied themein D minor. A 4-bar episode separates this from the finalentry in the alto at bar 5°.

The Inversus of course needs no separate analysis. It must

put its answer into the subdominant, for A minor becomesG minor. Freedom in dealing with accidentals is essential: atotally invertible harmony that renders semitone for semitoneis (paceCherubini)a verypoor pedantry. Bach'sschemeis asfollows, with liberty as to accidentals:

~~~-"'r

Ex.I9.

And Bachmust have enjoyed a good laugh at the end of No. 12.Rust (Bach Ges.Jahr; xxv) believes that Bach intended 'thefugue which he had inverted note for note in all four parts' tobe the crown of the work. This is certainly not so; it is, for allits astonishing 'slickness' (there is no other word for it), a verysimple affair, the simplest since Nos. I and II.

Contrapunctus XIII is in a much higher order ofte~hnique No.and is a delightful piece of playful music. The last bar is the XIII.only one in which Inversusis obviously not an original concep-tion; and in the rest of the fugue the Inversus can have fewturns that could no( occur in the Rectus, for the fugue' is initself a fugue by inversion, like No. V, but without stretti. Itssubject is a brilliant comic variation of the filled-out K. d. F.

, theme, as the following paradigm shows:Ex. 2I.

~I~rr~~-:~~~~~~~-:.. ~iri"Li"~~-ji~

Though accidentals are handled freely, it is not permitted torepresent a diatonic semitone by a chromatic one: i. e. if twonotes are in alphabetical order in the Rectus (as Gb, F), theymust be in alphabetical order in the Inversus (as F, Gb).On the other hand, Bach allows two F's an 8ve, apart orseparated by a short rest, to be represented by an Fq and anF~ in the inversion.

The wholescheme has some curious properties; for example,

[~~~'=~r J--~I~~

34 A COMPANION TOIn the RectusA opens with V. At the end of the fourth bar B

answers with 1\; and 4 bars later S (which runs so high thatBach writes it in the treble clef instead of in the soprano usedelsewhere in K. d. F.) completes the exposition with V.Episode I arises, like all the others, out of the fourth bar of Vand I\. It fills7bars. The 1\ appears in A, followedby a 5-barepisode, in the last 2 bars of which S has 2 bars of 1\ in asubdominant aspect of B~. (In the Inversus this becomes adominant ofF.) But the real entry is in B (end of bar 28) withV in B~. This is answered by S with 1\ in the same key, andfollowed by A with V in G minor. The next episode leadsin 7 bars, with some emphasis, back to D minor. The Inversuscannot achieve the same emphasis, though the runningdiminished 7th in bar 46 cleverly provides a dominant chordfor both versions. At the end of bar 47 A (of Rectus) has 1\.Another episode leads in 8 bars to a rhetorical pause on thediminished 7th, the only form of dominant which remainsdominant when inverted. Two more bars lead to V in B,answered by the final entry of A in S.

This analysiswill of course apply to the Inversusby changingthe signs and substituting B for S, A for B, and S for A.

Versionfor Two Claviers.

Bach was so pleased with both Rectus and Inversus of thisfugue that he determined to make them playable. Totallyinvertible fugues cannot submit to the further restriction ,ofbeing playable by two hands. Even if the stretches could beso confined, the parts have to cross in ways that could neverbe made clear on one keyboard. Bach did not arrange Contra-punctuS XII; and its long-sustained notes do not suit keyboardmusic. It could be played,like mymirror':'fuguein Appendix C,by an inverted string-quartet of one violin, viola, iand twovioloncellos.

But Contrapunctus XIII is simply asking to be played ontwo keyboard-instruments. Now, the two players have four

1

i

I

BACH'S 'ART OF FUGUE' 3S

hands, and this fugue has only three parts. So Bach providesa fourth part throughout, in both Rectusand Inversus; and evenfills up the rests in the exposition and elsewhere, therebyshowing how little he knows or cares about the orthodoxtheory that the addition of an accompaniment to a fugue-exposition annub the fugue.

These free parts are the most astonishing tour deforce in thewhole of K. d. F. Every rigid contrapuntal device that is notmerely crazy is child's-play to the task of providing a fourthpart to such counterpoint as this lively twin-pair of fugues. Ofcourse the free parts do not join in the scheme of total inver-sion. Their object is to make both fugues practically enjoyableas pure works of art regardless of their structure. The resultingclavier-versions, then, are not additions to the scheme ofK. d. F. but a means of removingboth versions of Contrapunc-tus XIII into the world of independent music. The newBach-Gesellschaft edition makes three almost incredible blun-ders here. First, it numbers the pair as another Contrapunctus.It then prints them as mirror-fugues, with the free parts in thesame type as the rest-which obliterates the mirror-effect. Itfinally crushes all four crossing parts into a reduction on twostaves which is as unplayable as two simultaneous Liszt con-certos on one keyboard and as illegibleas two snapshots on oneplate.

. In the present edition it will be found that the small stavesand small print of the free parts do not interfere with themirror-effect of the score. And I myself had no idea of theartfulness of these parts until I so disentangled them. Inthe Rectus a free part actually seems to be able to share in thefugue-subject, because it changes place with an essential part.

The clavier-versions themselves have always been inade-quately edited in a point which here and in our pianoforteedition is put right. We all know how dangerous it is to defercrossing t's and dotting i's until one's task is otherwise finished.The crossingsand dottings of dotted quavers and semiquavers

II

II

I}

36 A COMPANION TOare far more troublesome; and Handel never could be botheredwith them. Yet we areconstandy afflictedby learned editorswho,. in preparing Handel for modern performance keepmeticulously to every distinction between

,mJ I mJ \ ,mJ I

when it is perfecdy obvious that Handel never meant anything

but l' inJ every time. But that is an irritating hindranceto fluent penmanship, and even Bach has not always thepatience to finish the last .strokesand dots of such rhythms.I see, for instance, no reason to suppose that in the Grave of

the C minor Clavier Partita the rhythm; m J is ever

intended, and I am sure that Bach always played and taught

9.m J throughout.'Now we know that Contrapunctus XIII was written before

the clavier versions. Bach could no more have written themfirst and then extracted the pure invertible fugues from the;mthan David could have first written Psalm cxix in English andthen extracted from it an alphabetical acrostic in Hebrew.That great Bach-scholar Rust himself loseshis way in compar-ing the two versions. In another way his usual acumen desertshim. He meticulously preserves Bach's oversight in bar!46,where one group in the Rectus has lost its dots and cross,es.Finding that the alto of Inversus slighdy disagrees in thisparticular, Rust levels the parts up in the wrong direction.There is no reason in music whythe prevailing rhythm shouldcease here, and plenty of reason in penmanship why it shouldcease here and anywhere when the writer is tired. Copy 'thispair of fugues in a hurry, especially in 2/4 time witlHwice asmany group-bars, and see how often you wish the dots andcrosses elsewhere!

The autograph of the clavier version is very close, very

BACH'S 'ART OF FUGUE' 37rapid, and very firm, with no signs of the imminent breakdownof Bach's eyesight. He has evidently just begun to put in somedots and crosses. To suppose that he did not intend to com-plete the process as in Contrapunctus XIII is to follow theexample of the monk who persisted in reading the non-existentword mumpsimus though everybody told him that it was amistake for sumpsimus.

The editor's obvious duty is to carry out Bach's manifestintention here; and it would be easy to satisfy all consciencesby citing the 57 places (I think I have counted right) whereBach has begun to put in the dots. But if any of my readers isa Mumpsimite I will not deprive him of the pleasure of inde-pendent res~arch in this matter.

r

The Canons.

\ AfterContrapunctusXIII the oldeditions(includingRust'sB.-G. volume) give the earlier version of No. X and call itContrapunctus XIV. Disregarding this, wecometo the Canons.I confess that lam not clear as to Bach's purpose in writingsuch long movements in strict canon. Technically the problemis neither new nor useful. It is no more difficult to carry on a2-part canon for 100 bars than to confine it to the length of afugue-subject. The canons in DasMusikalischeOpferare muchmore difficult and also more like spontaneous works of art,because they are, with three exceptions, counterpoints to theCantofermo of Frederick's theme. Of the exceptions the onlylong movement is the canonic fugue, written (though Bach,writing in enigmatic notation, does not say so) for flute,violin, and continuo. But there is no parallel elsewhere inBach for the easygoing canons of Die Kunst der Fuge.

The Berlin autograph contains two, the inverse canon byaugmentation and the canon in the 8ve. The latter it placesbefore the triple-counterpoint fugue No. VIII, and thissuggests that the canon in the 12th should precede the fuguein the 12th (No. IX) and the canon in the loth should precede

I

~J

38 A COMPANION TO

the fugue in the lOth (No. X). But Bach actually correctedthe proofs of Nos. I to XI and let them be printed in theirpresent order. We do not know whether he intended to writemore canons than these four; and we do not know what heintended to demonstrate by them. Evidendy those in the 12thand lOth illustrate the corresponding double counterpoints,because they are turned round so as to become canons in the8ve. The inverse canon by augmentation might appropriatelyprecede Contrapunctus VII. I confess to preferring its earlyversion in Appendix B. The inverse augmentation there isreasonably melodious. In the final version we can only admirethe beautiful counterpoint Bach builds on it as on an uncouthcantofermo. He is unhampered by any technical difficulty, forthe follower lags more .and more behind the leader, which canproceed as it likes, knowing that the consequences will notconcern it, unless the composer attempts something notattempted here, the difficult task of making the canon reallyperpetual by getting the leader to run round twice while theaugmentation goes once. This means that the leader mustcombine with both of the magnified halves of itself. In DasMusikalischeOpfer Bach achieves this over the cantofermo ofFrederick's theme, but only with extreme difficulty and withone actual (but remediable) oversight. Even the present easierproblem is not artistic: no human ear or memory can trace anaesthetic relation between the absurd bass of bars 45-6 and thetreble of bar 21. And all hope of defending the compositioJ:1.as beautiful counterpoint to a grotesque bass is dashed byBach's plan of inverting the whole canon in double counter-point.

And so the Daddy-Longlegs sprawls in the treble untilmusical sense reappears in the 4-bar coda. The other canonsare as good as such long unaccompanied canons can be,ithoughthey do not rival the equally long and exquisitely poetical F~minor slow movement of the A major sonata for cembalo andviolin. They are smoother than the canons in the Goldberg

'.

BACH'S 'ART OF FUGUE' 39Variations; but those canons, like the canonic minuet and trioin Mozart's C minor string quintet, or wind-octet, are themore enjoyable for being cut up into square sections. StrictCanon lends itself to the purposes of epigram as naturally asfugue lends itself to the purposes of architectural climax.

The most effectivecomposition in the present four canons isthe lively 9/16 canon in the 8ve. All four canons begin withgorgeous variations of the K. d. F. theme; and the 8ve canonuses it in the direct as well as the inverted form. It makes anamusing clavier-piece. The other canons would sound moreconvincing on pairs of instruments with good vocal tone. Thatin the 12th has the finest of all variations of our motto-theme,except that in the unfortunate inverse augmented canon, ofwhich the opening might have served for a great slow move-ment.

The canon in the lOth begins with a fine illustration of thecumulative effect of a long-limbed sequence rising in 3rds.Why did Bach not show, when he inverted the whole in thelOth from bar 44 onwards, how either or both leader or followercould be doubled in 3rds? The result would have been veryfine; but Bach has omitted to plan the small adjustments thatwould be necessary,sowe cannot suppose that he contemplatedit. However, his interest in this piece as a musical compositionleads him to provide it with a pause for an extempore cadenza;and the obvious duty of the cadenza will be to illustrate thispossibility. In the pianoforte edition of the canons the editorhas supplied a dutiful cadenza accordingly.

I have put the canon in the lOth after instead of before thatin the 12th, so as to follow Bach's precedence of counterpoints.And I have confined the tide Contr(lpunctusto the fugues.Bach, no doubt, would have called the canons 'tied-fugues'(fugae ligatae)as distinguished from recherchefugues (ricercare).So perhaps I am wrong in this detail. But further conjecturesabout these canons are useless; one guess is as good as another,and the topic is not important. We do not know whether

4° A COMPANION TOcanons in more than two parts were contemplated, orwhether the -Art of Canon was not going to take as muchspace as the rest of the Art of Fugue. Perhaps Bach mayhave expected that in the higher orders of canon he wouldrepeat his experience of the higher orders of fugue, and thatthe more intricate problems would solve themselves in themore expressive music. This is certainly the case outsideK. d. F. Very little extra restriction would be needed to bringthe whole Qui tollis of the B minor Mass into exact canonicform as a four-part vocal canon accompanied by a two-partinstrumental one. Evidently between the extant K. d. F.canons and such a movement there is as great a distance asthere is between Contrapunctus I and the unfinished quad-ruple Fugue.

Do the present four Canons show any feature of style whichin the Jirst place depends on their canonic form and in thesecond place contains the possibility of developing into thingslike the Qui tol/is of the B minor Mass?

I think there is such a feature, and that it may be found inMozart's C minor quintet, in the 'Hexen Menuet' of Haydn'sD minor quartet Op. 76, No.2, and in all the most serious aswell as the wittiest of canons. It consists in making a rhetoricalpoint of the way in which the leader pulls the follower after it,while the follower, in its turn, seemsto egg the leader on to dis-cover new ground to run over. Once begun the canon cannotstop; and this is either a fatal defect or an interesting qualiry.The extant K. d. F. canons make almost a joke of it; and thatis as far as a long unaccompanied canon can go.

No. We now come to the mystery of the unfinished work. HereXIV. conjecture is not at a loose end; the data are extraordinarily

definite.Bach left, in such connexionwith the manuscript o£K. d. F.

that his familyand first editors had no doubt of its relevance, anenormouS unfinished fugue. Before it breaks off it is alreadylonger than any other of Bach's fugues. It contains three

BACH'S 'ART OF FUGUE' 41

subjects, the last being based on the name-BACH in Germanmusical nomenclature where B is BDand H is Bq. Bach's last7 bars contain the first combination of the three themes. Theoriginal edition breaks off 7 bars earlier; why, nobody can say;any more than why it ignores Bach's proof-corrections andalters his significant special title to his last organ-chorale.

Rust and others, noting the absence of the K. d. F. theme,argued that this fugue does not belong to the work, and thatBach had really finished it, regarding the simple little 4-partmirror-fugue as the crown of the whole. Nottebohm, thedecipherer of Beethoven's sketch-books, settled that questionby discoveringthat theK. d. F. theme combines with the otherthree in a manner quite beyond the possibility of chance, thequaver tail of the theme even filling out a halt in the rhythm ofthe rest of the combination. Riemann accordingly -embodiesNottebohm's discovery in a short coda making no pretensionsto artistic composition.

Now what does the earliest tradition say? Mizler, writing in1754, only four years after Bach's death, says: 'His last illnessprevented him from completing according to plan the lastfugue but one, and from workingout the last, which was to con-tain 4 themes and to be inverted note for note continuously[nachgehendsJin all 4 parts' (apud B-G, XXV, p. xviii).

Rust says that this is impossible. Compositions like Con.trapunctus XII can have no suspended discords and cannot beextended to the length of a quadruple fugue. Bach must havemeant that that four-part fugue which he had already invertednote for note was to be the crown of the work. He must oftenhave said so in his last illness; and of course his wife anddaughters and the Is-year-old son Johann Christian, whoconstituted his household at the time, understood nothingabout such learned music and could not fail to spread errorsbroadcast when they tried to talk about it.

I see no reason to suppose that Mrs. Bach and the boyJohann Christian did not understand Bach's intentions quite

42 A COMPANION TO

as well as he expressed them. Mrs. Bach sang all her husband'ssoprano music, and wrote a musical hand which is hard to tellfrom his. Johann Christian was a better musician at 15 thanI was, and I should have had no difficultyat his age in under-standing whether Bach meant w4at Mizler said or what Rustthinks he meant. If I had been one of Bach's household at 15,he would probably have been exercising me and JohannChristian in score-playing with each fugue as soon as it waswritten. And Mizler's statement has a very different kind ofprecision from the picturesqueness of legends.

From this point the most interesting way to reach a con-clusion is by analysing the fugue as far as Bach finished it, andthen describing how its data lead to a finish on the linesfollowed by me, and how, in connexion with Mizler's state-ment, they indicate the project of another fugue on the lines ofmy mirror-fugue in Appendix C.

The mighty composition which I call Contrapunctus XIVbegins with a section II5 bars long, on one of the severestthe~es everinvellted. We may callit the cantofermo theme.The great qj: minor fugue, with its 3 subjects and its enormousstretto, is exactly the length of this one section.

The exposition is (as usual in K. d. F. and rather unusualelsewhere) quite uninterrupted, and in the order B, T, A, S,at the 6-bar intervals filled by the subject. Note the tohalanswer, with the repeated note which the dotted minim makespossible as it was not possible with the crotchets of Contra-punctus XI. There is no counter-subject; but a prevalentcrotchet figure

[/\

a r' ~~$:~ L- ,i"

is evidently derived from the first four notes of the subject, byfree diminution. Such derivations are valid according to theirimmediate context. These same four notes may be found later

~BACH'S 'ART OF FUGUE' 43

as parts of another theme with no reference to (a) in theirintention. Immediately after the exposition there is a strettoof B V and T 1\ at 3 bars. One bar after this A has V,followed after a bar's interlude by stretto 2. Here B precedesS with an allusion to the first 4notes, but the real stretto isbetween S /\ (tonal answer) and A 1\ (subject) at I bar.Immediately afterwards B has 1\ in F (approached from C).After a 7-bar episode on figure (a) T Venters in G minorfollowed by Stretto 3, which is the converse of Stretto 2, viz.B /\ (subject), answered tonally by S I\. A 2-bar interludeleads to Stretto 4, the inverse-contrary of Stretti I and 2,viz.: A V answered tonally at I bar by T V, in BD. After abar's interlude Stretto 5 appears in D minor: S 1\ answered at2 bars by A /\. An episode on a new line leads in 3t bars toStretto 6. This is in 3 parts, beginningin BD with B 1\answered at I bar by T, which, beginning with an inversion,changes its direction after the fourth note. At the third bar ofthe B subject, A 1\ enters in G minor, crossingover the soprano-to the confusion of earlyeditors. Their error is rectified in thenew Bach-Ges. edition. Stretto 7 immediately follows in Dminor. It is in 3parts and veryclose,consistingof a new answer-position of T /\, answered at half a bar by S V (with anornament in bar 101) and at 2 bars by A /\. After a 2-barinterlude there is a solitary final entry of B /\, accompaniedby (a) in dialogue above. Four more bars closethis great firstsection. Through the last chord runs the beginning of /\ in A.2This is a :richcoloratura theme filling 7 bars. At the eighth barS answers in the dominant, and Band T followregularly. Six

{S/\

bars after this exposition we have the combination B A' It1

rAAis answered after a 3-bar interlude by ~ T A in A minor.L 1When ~ is combined with other themes it is singularly difficultto manage in any other part than the bass, and Bach has to

44 A COMPANION TOdouble-dot its second note before he can find a proper bassfor it. An episode of 5 bars arises from the last figure of 1\.2

Ex. 22.

1\b

Then we have what looks like the D-C inversion of the pair

{SI\

in the 8ve. It is in F major T A. But it is a new combina-2tion; for 1\ enters a bar too late. A 6-bar episode leads toIanother new combination in G minor: B 1\ with A 1\ a bar2 I

late and in stretto with S 1\ (tonal answer moving to C minor)at I bar. Six more bars Iclose this section in G minor, andintroducing ~ as well as 1\.b

The third subject now enters. It begins with Bach's ownname B (D), A, C, H (=B~), to which it adds a turn whichmakes the theme a chromatic phrase in D minor. This isannounced by T, and answered in stretto by A. Its continuation

~ ~(x)

is not meant to remind us of (a). Senters 6 bars after thebeginning of A and is answered at 2 bars by B. Mter thefourth bar ofBthere is a 3-bar interlude on (x). The T en~ers.(I cannot agree with Busoni that the syncopated soprano ofbars 2II-13 is intended to allude to ~, which would be quiteirrelevant here unless it could be followed up.) T is answered

at 3 bars by A~. This inversion is not according to the scaleof Ex. 19, but on the following scheme:

Ex. 23.

t~ ~ ~ ~~=:

I)

BACH'S 'ART OF FUGUE' 45Then we have a close stretto of 1\ between S and B at the

3half-bar. The bass has to expand one note from a minim to a

semibreve. It then gives V: closing into C, where anotherclose stretto (1\ syncopated in T with A in normal rhythm3after 3 crotchets) returns to D minor. Three crowded bars,with something like a diminution of x, end the section with ahalf-close. .

The fO)lrth section now begins with the 3 subjects in thecombination

rA'i

lT~BI\

I

S freely imitates~. Bach's autograph breaks off with':> inthe tenor. What washis intention for the sequel? Three thingsare certain. First, the theme of K. d. F. was to make thecombination a quadruple counterpoint. Secondly, neither 1\

~ I

nor ~are in doublecounterpointat the 8ve with the others.Bach himself found it difficult (as we have seen) even to get/\ into an inner part in combination with 1\. In the deferredI 2

position of bars 169-74 it will combine with /\ as well as with31\ ' but will not then also combine with the fourth theme.2 '

Thirdly, the movement must not again be interrupted in orderto give the fourth subject a separate exposition. That would beafatal error of composition. The entry of 1\ has brought back2quaver-movement; we are certainly entering upon the finalsection of the work, and nothing must stop its flow. Moreover,to introduce the new theme as a surprise without interruptionis the one deviceof composition still in reserve for this juncture.Contrapunctus IX introduced it early, without interrupting thefugue by so much as a half-close. Slightly less certain thanthese 3 points is the main assumption underlying my composi-tion of the whole peroration. I am absolutely certain that Bach

46 . A COMPANION TOintended to invert all 4 themes in the quadruple counterpoint;and the only possible doubt is whether this was to happen inthe present fugue or in the other which Mizler says wasprojected. In 1\ the following notes2 .

Ex. 24.

~~show every sign of being constructed for inverse-contrarycounterpoint. Melodically they areartificial,though construableenough, and their evident purpose is to make accented discordsthat resolve both ways, downwards within one crotchet in theoriginal position, downwards at the minim when inverted. AndI think that the total inversions were to happen in this fugue,because it requires a peroration of at least 80 bars, and thisgives plenty of room for all the variety we can obtain from thenow incessant whirl of the quadruple counterpoint.

I begin by immediately answering Bach's three-part com-bination with an A minor entry of

is 1\~ A~ in 12th.LT~

There must be millions of chances in favour of 'the assumptionthat the treatment of 1\ in the 12th represents Bach's intention.

1 :

One theme may work accidentally in the 12th with one othCir,but hardly with two, and certainly not with three. .

An episode (on \I and ~) leads in 5 bars to an inverse-contrary combination in A minor:

{

SO

A¥ in 12th.BV

(Not being Busoni, I cannot make up Bach's mind to force 0into exact semitonic correspondence by putting F# against the

,!,.

r\I

t11

t

il

BACH'S 'ART OF FUGUE' 47

F qin O. Bach is not afraid of such collisions,as the wonderfulslow movement of the first Brandenburg concerto drasticallyshows; but he draws the line sharply at taking them by skip.)'

The fourth theme must not enter too soon, but we have nowdeferred it for 3 entries, and must prepare for it; Five bars on\Iand ~lead with emphatic steps back to D minor. The alto,made conspicuous by striking an 8ve with the soprano, nowdelivers the K. d. F. theme in the inverse-contrary 4-partcombination

{

S~AVTOBO

I The reader will expect some comment on Busoni's great FantasiaContrapuntistica, which contains the whole extant portion of Bach'sunfinished fugue, developed not to an end of its own but to the pur-poses of a much larger work. It seems unusual, even with acknow-ledgements, to absorb 238 bars (or more than 10 minutes) of pureBach into a modern composition; and I cannot work up any enthusiasmfor compositions or cadenzas that purport to review the progress ofmusic since classical times. Modern styles aspire to a purity of theirown: introduced into older styles they are mere impurities. Bach'sown style would be a ghastly impurity if introduced into a PalestrinaMass. With contrapuntal forms there is really neither interest nortechnical merit in merely taking advantage of modern possibilities aslicences. A genuinely modern polyphony requires modern material.

Of course Busom is able to combine the themes of this fugue casuallywith everything elsein K. de F., to make brilliant inexact stretti on ~,and, as observed above, to preserve the chromatic character of ~ evenwhere this means taking B~ by skip against Bj,.

Our present task is not to produce a review of musical progresssince Bach, but to follow the humbler and higher aim of carrying outwhat is discoverable of Bach's actual intentions.

Where my combinations are the same as Busom's this is becauseiI1those features Busoni was also working out Bach's own design. I didnot know the Fantasia Contrapuntistica when I worked out rriyconjectures. But if I had known Busoni's work I should have'had nomore scruple in coinciding with his true combinations than in copyingthe finished part of Bach's fugue. And of course I no more dispute hispriority than I dispute Nottebohm's or Riemann's.

48. A COMPANION TOThis is followed,.after a bar's interlude, by the direct com-bination

{

s ~in 12th.A/\

3T/\

2B/\

4

(It will be seen that 1- must be in the 12th when it is not in the

bass, and ~must be in the 12th when it is not in the soprano.)

A 3-bar interlude (still on ~and":') leads to the inverted com-bination

1

r~~T~B ~in 12th.

Another 3 bars (on another figure of /\) lead to A minor, with2the direct position

fs~AI\

3

lT~B1-

Now it is evident that both 1\ and ~ are, in spite of theircapacity for inversion in .the ~2th, most at ease in the bass.Accordingly B tends to have more than its share of them. Thisfact had better lie turned to rhetorical purpose; and so theclimax is brought about by the following steps in which /\moves up in the bass from a low dominant to a final tonic. 1

Ex. 25.291 295 300

..--- 305 310 '"

rJ.

BACH'S 'ART OF FUGUE' 49The quadruple combinationhaving finishedat bar 295,theupper parts continue first with ~ and":' in the soprano,.andthen with 3-part imitations of 1\. At bar 3°5 the final com-abination appears: .

SI\4

AI\2

TI\3

81\J.

and. on the final tonic pedal Bach's signature appears on the

top with the latter portions of ~and ~ in the middle parts.

II

Appendix.

Appendix A contains the chorale-prelude which Bachdictated on his death-bed. In harmonies of childlike simplicityand beauty, three lower parts treat eachphrase of the chorale infugue by inversion until the soprano gives the phrase in longnotes above. The reader should make his own analysis. Onlyfour letters are required, one for each phrase of the tune:capitals for the cantofermo in the soprano, and small lettersfor the other parts. The signs 1\ and V will be requiredthroughout, together with the sign = when figure (d) is givenin crotchets as well as in quavers.

The bars in which the cantofermo is present are transcribed,without the original florid ornamentation, from a setting with-out interludes in the Orgelbuchlein. Bach dictated the newinterludes on his death-bed in a room darkened to spare hissuffering eyes.

The original editors of K. d. F. gave this wonderful piece asa compensation for leaving the last fugue unfinished. Rust'sdescription of the manuscript in B.G. XXV. ii is a beautifullittle biographical essay.

Appendix.B is the earlier version of the canon by inverseaugmentation.

A COMPANION TO50

ConjecturalFinale.Appendix C is the final step in the vindication of Mizler.

He may have made a mistake and, hearing that the unfinishedfugue was to be on 4 invertible subjects, may have confusedthis with an invertible fugue on 4 subjects. That was my firstidea of what happened; but I can see no reason why Mizlershould not have been right. One does not from I1J.ereconfusiondraw Mizler's explicit distinction between an unfinished lastfugue but one and an unwritten last fugue. Rust's a prioriobjection is not evidence. Solvitur ambulando.

If this fugue existed, what would its themes be? Two ofthem we know; one must have been the K. d. F. theme andanother must have been B, A, C, H. I am sure that the fuguewasnot goingto be written on the four themes of the unfinishedfugue. Even if Bach had not intended already to use theinverse-contrary positions in that fugue, there would be nopoint in working out the same themes in a pair of mirror-fugues. Moreover, I cannot find two positions of that quad-ruple counterpoint that will go into exact mirror-relationshipto each other; though I confess that I have not the patiencefor an exhaustive search. But tonal accommodations, free playof joints, and inversioI}sin the 12th are of no avail to producemirror-fugues. Nor, on the other hand, do I believe that Bachfound his combinations by exhaustive mechanical search. Ithink that when he composed the unfinished fugue he beganby making a draft of exactly my bars 306-10 (except for thefree final joints), and thought how fine it would be if hillbasswas not under the usual necessity of avoiding complete chordswith the upper parts but could invert in the 12th and so havea firm Sth above it whenever the harmony would be the betterfor it. I am also certain that he found himself obliged (unlikeRiemann) to syncopate the third bar of the K. d. F. tlieme; andI should be astonished if he found that beyond this isolateddetail any other variation was workable. The detail looksanomalous on paper, but does not worry the ear at all; and, in

r

I

BACH'S 'ART OF FUGUE' 5I

spite of his uncanny schematic exactness, nobody wrote moretruly for the ear and more independently of the eye than-Bach.

Mizler's supposed last fugue, then, must have been on a newcombination. I have tried experiments with other themeswhose capacity has not been exhausted in K. d. F.

Contrapunctus X, for example, gives only one side of a matterthat has at least four; and, apart from that, it is hard to believe

that such an approximation as this could happen by accident.Ex.26.

:=::c:t: :~~~==~~~. ~ :' ,~*. -=4~ "",.e.J 1

which inverts into this:

'I

I

)

i="'~ - .J : ~~:::::=~r. n r

~;~ ~r~ ==' ~A41~ -""- =&c.

But here Bach's name must always clip its first note into a

52 A COMPANION TO

crotchet, because the beautiful suspension of the dotted minimagainst it will not invert. Yet the suspension is necessary toexplain the doubled 4th at *. And the theme of ContrapunctusX is compelled to drift into something very like the latter part of1\ in the unfinished fugue. This will never do. Nothing is2

more astonishing in K. d. F. than the complete absence ofunmotived recurrence of phrase or mannerism in 14 fuguesand 4 canons all on the same subject, in the same key, and inpure part-writing.

I do not claim to have exhausted the possibilities; but I thinkthe attempt unnecessary when the probability is that any finalfugue Bachhad in mind wasat least as different from the othersas they are from each other. Unless we regard a mirror-fugueas two fugues, the only fugues that are related as pairs are thecouples VI-VII and VIII-XI. This gives no ground forthiriking that the last fugue was to pair with any others. Andso I have proceeded on new lines.

First, a few more words must be devoted to the question oftotal inversion. As has already been said, its difficultyhas beengrossly overrated. And that difficulty is much lessened andthe resources greatly increased if the style is chromatic. Theextant mirror-fugues are diatonic. This creates a strong pre-sumption that the final fugue was to be chromatic. Nearly anychromatic harmony that proceeds (like Bach's) by semitones,will invert into something tolerable. For instance, here is.theTugendqualMatiffrom Wagner's unwritten Tantris und Salide,though some believe it to be the Abgeschiedene-Vielfrassweis'mentioned in Die Meistersinger(Ex. 28).

This becomes more familiar when inverted (Ex. 29). I amsorry for the Spohr-like inner part of the last bar; but pureWagner will not invert, any more than a random extract ofBach himself. . ' ; ; ;

It is no use speculating as to what can be done with totalinversion in post-Wagnerian harmony: where there are norules there are 1;10difficulties; and where there are arbitrary

rBACH'S 'ART OF FUGUE'

Ex.28.Langweilig und $Cham/os.--

53

--

-- ae..'"

........

..L-

t:\-- I" 1 "---'

f ===-t:\

Ex. 29-Langsam und $Chmachtend. ,.

....

,..

'------

, .1 r-----

~ ~.r~ .t. 'J ,'. ~~~. f'-" \V

Ij I" 1 --I ..J , ~r Ir':- r' , I

r br \V

systems you can get just as good another arbitrary system byturning your music upside down. Anything that sounds nobetter right-side up than upside down will do for an up-to-datepiece of totally invertible counterpoint.

54 A COMPANION TOBach wrote many things that Wagner would have thought

bold; such as the following episode from an unfinished fuguein C minor:

BACH'S 'ART'OF FUGUE' 55

Then, of course, there is the lheme of K. d. F., which I give inthe following rhythm:

b ~~~~ -F~and Bach's name:

But this is not Wagnerian; and Wagner's wonderful sugges-tions of remote tonality by means of long appoggiaturas would,I believe, have seemed to Bach like a new kind offalse relation.Some such effects often turned up in my first sketches forAppendix C; and a sufficiency of them might have made aconsistent style on Wagnerian lines. But, though I was at nopains here to keep to Bach's language I could not think fit tota~e the task so easily, and I have therefore eliminated every-thing that I could suppose would annoy Bach. For a shortchromatic fugue on-so rich a plan as is implied by the use of4 subjects the most natural style would be emotionally highlycharged; and so one of my original themes is declamatory.

Ex. 34.

~)~tl 3--t-d~~!~.~

Ex. 32. ,i"1\ r "'

fw~T~~iJt-irTt i.rg' ""-===-

There is no room for episodes; nor can we afford four separateexpositions. With every permissible means of compression thisfugue, if it is to be a composition in any proper sense of theword, will take nearly 100 slow bars. (It actually takes 96.)

The analysis of the Rectus is as follows:The exposition of /\ takes the order T, B, a bar's interlude,

~ 1

S, A (with a subdominant answer). After a 2-bar interlude B

enters with ~and T givesout the B, A, C, H, theme'i' Onthe last note begins the reverse combination, T 'i' with A O.Mter this S enters with /\, and the first subject is abandoned.2

Then B completes this second exposition by entering with O.After' a 3-bar interlude there is a close stretto, at intervals of

2 beats? with A ~, S 0, and B O. Three more bars bring thefirst main section of the fugue to a closein the tonic. Throughits last chord A announces a diminished version of the K. d. F.theme, answered in stretto, and by inversion, by T and S.Mter 4 bars of this the full-sized theme 1\ enters in B accom-3

panied by ~ and~. On its last note T answersit, and Sannounces /\. Again the last note overlaps with the combina-4

{s 1\

tion A A ' and this overlapsby yet another bar the com--4

Ex. 31.1\

~ i i ~~~~~-2~In order to present the Inversuswith better opportunities for

a convincing top-part, I base my normal quadruple counter-point on the inversion of this theme, so that it comes right-sideup in the Inversus, in places where the Rectus has the otherthree themes direct.

My other original theme is in the style of Bach's 3-partInvention (Sinfonia) in F minor:

!"

BACH'S 'ART OF FUGUE' 57after which the imitative 3-bar adagio close alludes to theinverted K. d. F. theme.

The complete scheme of K. d. F. would appear to be asfollows:

I. Four simple fugues, two upon /\ and two upon V; thelatter pair comprising (a) a chromatic fugue with a counter-subject and a variation; and (b) a study in highly developedepisodes.

II. Three stretto-fugues, one by inversion, one by diminu-tion and inversion, and one by augmentation, diminution, andinversion.

III. Three fugues in the principal orders of double counter-point, (a) a triple fugue in the 8ve; (b) a double fugue in the12th; (c) a double fugue in the loth (and, incidentally, in the12th).

IV. Studies in total inversion, (a) a fugue freely invertingthe triple counterpoint of the former triple fugue; (b) a strictlyinvertible simple fugue in 4 parts, with a variation of thesubject; (c) a strictly invertible 3-part fugue by contrarymotion, with the parts reversible in an inside-out order; andwith subsequently added free partsoutsidethe invertible scheme.

V. Two quadruple fugues, one with 4 invertible subjects ofwhich 1 is in the 12th; and the other a totally invertible fuguewith 4 subjects. (My effort at this also contains diminutionsand stretti.)

VI. Four canons which Bach seems to have thought ofattaching to the four fugues in the corresponding orders ofcounterpoint. He abandoned this intention if he ever had it,and we do not know whether these are all the canons in hisplan, or what their function is.

Does K. d. F. contain every kind of fugue that interestsBach? It is noticeable that he almost never professedly writesthe kind of double fugue that begins with two themes at once.The only mature examples I know are that at the end of thePassacaglia and the second movement of the C major Sonata

56

{A/\

bination T A' A close stretto (at one minim) on ~ then4

occupies all 4 parts for 5 bars, ending emphatically on thedominant. Then a second skirmish of stretto in diminution,

this time with 't and ~ as well as 'it and ~, leads in 5 bars

to the quadruple counterpoint. The first position is.

{

S 'i-AOT~BV

It is immediately answered by

jS'iA/\2

lT~BV

A 2-bar interlude is followed by

{

S~A/\1T/\3B/\2

to which S adds, by way of filling up the first bar,~. (Thisdiminution was the way out of a trivial difficulty. Often theone really unobtrusive way round an awkward harmonic corneris by a canonic device.)

Four more bars end in a climax on Bach's useful and imper-turbably invertible diminished 7th. Then, after a pause, 4 barsin broken rhythm lead to the final combination

I

s/\

A~IT/\

lBt

A COMPANION TO

58 A COMPANION TO BACH'S 'ART OF FUGUE'for two violins; and I can recall only one other example, veryearly and not important. On the other hand, he would havebeen quite justified in calling the main part of the ED Prelude,W. K., I. 7 (from bar 25 onwards) a double fugue, and theA major Prelude, W. K., 1. 17, and .the three-part F minorInvention triple fugues of this kind.

Bach's choruses often contain a type of fugue hardly possiblewithin the limits of four keyboard parts. We may call it around-fugue. In a chorus with free orchestral accompaniment,or a double-chorus motet, let one voice start a coloraturafugue-subject. Let it accompanythe next voicewith a counter.subject, and a second counter-subject, and a third, and so ontill the exposition is complete. Then let the chorus continue totreat the result like a round that oscillates between tonic anddominant. Two more entries will have made a big continuoussection; and, if this has reached the dominant or any other key,the orchestra may intervene with a ritornello, and the roundmay be resumed once more and finished in the tonic. Theform thus mapped out can exist with quite as solid an effect ifthe later counter-subjects are obliterated.

Such is the structure of the huge first Kyrie of the B minorMass, and (after the massive introductory sequences) of theEt in terra pax and Cum sancto spiritu. The great isolateddouble chorus Nun ist das Heil is on this plan and preserves asextuple counterpoint.

Figured chorale-fugues are also outside the scope of DieKunst der Fuge as their subjects change but do not combin~.

With these exceptions Die Kunst der Fuge is a completedemonstration of what Bach understood by the term Fugue asapplied to whole compositions and not merely as a kind oftexture used here and there. It cannot be too strongly insistedthat Die Kunst derFuge is, for by far the most part, nor~al key-board music dealing with the central elements of Bach's art, inhis latest and most perfect style, and with his fullest power offree composition on a large scale.

PART II

~

THE above analysis has dealt with K. d. F. from a singlepoint of view from which no wide digressionshave been made.But there are other points of view less central and thereforeless amenable to connected argument, but none the less insis-tent in their claims to attention. Some of them are discussedhere.

\

I. The Berlin Autograph in relation to the orderof the Fugues and Canons.

In Bach-Ges.XXV.I willbe founda list(reproducedin thenew B.-G.) of the readingspeculiarto the Berlinautograph.Without exceptionthey are details which Bach deliberatelyalteredafterwards;and in somecasesthe alterationsare madeon the autograph itself. One point makes orthographichistory, viz. the systematiccorrection,in Contrapunctus7,

of the old inaccurate notation fm (which so many modern

editors take for a triplet) into the accurate ~~, a pointexplicitly referred to in the final docket 'Corrigirt'. The stileFrancesewasbecoming old-fashioned,but Leopold Mozart hadnot yet persuaded musicians to adopt the double dot.. The autograph is not in open score and suggests no doubt

that the music was for the keyboard. The order of the piecesproves nothing except that the only admissible order is that ofthe printed edition as far as Contrapunctus XI. Probably thepieces stand in the autograph in the order in which they werecomposed. If this is so, it would seem that Bach started withthe intention that Contrapunctus I, on the direct subject,should be followed by the fugue on the inverted subject (nowContrapunctus III); but that the idea of the present Contra.punctus II interrupted his further plans. For ContrapunctusII, which stands third in the 'Berlin autograph', is manifestly