**賦格藝術** ( *Die Kunst der Fuge*, *The Art of Fugue*), BWV 1080，係德國音樂家巴哈晚年嘅一集對位音樂，一般認為係未寫完嘅。集曲大概喺1742開始寫。巴哈嘅第一版有十二支賦格曲同兩支卡農曲，來自1745年嘅抄本(*was copied in 1745*)。呢份抄本嘅標題係巴哈個 son-in-law Altnickol 加嘅 *Die Kunst der Fuga*。第二版本係渠死（1750年）後至出版。入面有十四支賦格曲同四支卡農曲。呢集曲表現咗巴哈掌握嘅對位技術。每支曲由簡單嘅主題攷妙發展成精密又富音樂感嘅結果。成套賦格藝術表現咗多種曲式同結構，好多音樂人當渠做係歐洲音樂傳統嘅一座頂峰。

背景
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巴哈係渠最後十年靜靜探索賦格同卡農。1742年 高德堡變奏曲 (BWV 988)出版，其中有九支嚴格(*strict*)嘅卡農曲（最近重揾到十四支附加嘅）；1747年音樂奉獻 (BWV 1079) 出版，其中有兩支精釆嘅*ricercari* （「*ricrecar*」 係賦格曲嘅前身）, 九支*cerebrally devised* 嘅各款卡農，同埋*Vom Himmel hoch* (BWV 769)嘅卡農變奏，都顯示巴哈越來越沈醉於對位音樂。好自然，巴哈喺同期又開始以「賦格藝術」(‘Die Kunst der Fuge’ BWV 1080)為題寫另一組卡農同埋賦格曲 ，其中每一支都用d小調，每一支都係同一主題嘅變奏。

近排嘅分析顯示：賦格藝術早響1742（即定稿前五年）已有初稿。值得討論構思同埋後來改寫嘅動機。呢個日子吻合 Johann Mattheson 係 *Der vollkommene Capellmeister* (1739)上討論賦格，其中Mattheson 列舉咗響 Angelo Berardi 嘅 *Documenti armonici*(1687年)入面嘅好多種對位曲式。巴哈可能喺1738年尾見過 Mattheson嘅初稿。可能巴哈被Mattheson 嘅挑戰－－寫有三主題嘅賦格曲－－激發，終於實現渠好可能好長期嘅構思。但咁唔係話我地真係欠Mattheson 好多感謝。就算Mattheson 唔講，巴哈都或會識 Berardi 嘅技法，因為巴哈自己都可能有一本*Documenti*；賦格藝術嘅 《對位 18》 用 Berardi 嘅 *Fuga reale*來開始，而 Mattheson 無講過呢支*Fuga reale*。

初版有十首賦格同兩首卡農曲，有可能係 Mizler Society 嘅關係，啓發咗巴哈去擴寫份作品到而家咁樣。Mizler Society 要求會員每年交一份科學報告，重要用音樂來寫。巴哈響渠參加時演示咗首 *卡農 Triplex* (BWV 1076) 同埋《卡農變奏曲》；第二年交咗份《音樂奉獻》。因高過六十五歲嘅會員唔使再交，好可能巴哈打算喺1749年6月交已經有啲睇頭嘅《賦格藝術》做最後一樣功課。

呢輯曲係1751年巴哈死後出版，其中最後一支賦格重未完成；翌年音樂理論人Friedrich Wilhelm Marpurg 加大咗前言，重新出版。呢本入面有好多唔屬於巴哈意願嘅修改，包括：

1. 加咗對位10 嘅一初期版本：正式版本入面，巴哈響開頭加咗廿三小節，亦修正咗好多道。
2. 改過啲卡農嘅次序，反映啲賦格由簡至細嘅排列（睇落去）。
3. 收尾支未完嘅賦格(Contrapunctus 18) 擺喺成組最尾，而唔係第十四高潮位。
4. 收錯咗兩組安排畀雙古鍵琴嘅三部鏡賦格，其實喺成組緊密嘅賦格同卡農入面冇佢哋嘅位，仲連次序都擺錯埋。
5. 收咗最後支管風琴曲 *Wenn wir in höchsten Nöten sein* 。

曲同埋譜
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巴哈開列(*open scoring*)咗賦格藝術嘅樂譜，每聲部都有自己嘅五線。雖然有多種證據話份作品係寫畀鍵盤嘅，仍然有啲流行嘅猜測，話佢嘅寫畀樂隊(*ensemble*)嘅，或者話佢係隻絶對音樂－－即係一種對位技法範式，而無諗住畀人奏。第一可能就梗唔係，因為啲賦格旋律線嘅音域多數都唔啱當時樂器玩，鍵盤就例外；但係絶對音樂嘅假設就無咁易排除。

Marpurg 喺 1752 版嘅前言實則無乜用，但呢版度加嘅料顯示，巴哈啲仔肯定佢哋係奏得嘅。可惜咁都唔係好肯定嘅，因為1751年趕住出，可能暗示，呢版唔一定係為音樂而音樂：巴哈唔係有錢佬，財產夾埋唔多，可能渠地真係好等錢使。既然已經做咗一大 噘（？guet），搞埋佢可以幫補下家計。之但係，好可能因為呢版唔係幾好賣，所以翌年就加大佢再版。 加送嘅曲雖然可以當做彌補最後支賦格嘅唔完整，但亦可能係用來吸引管風琴同第啲鍵盤手，令本嘢好賣啲。啲加料就唔關「係唔係絶對音樂」事。 之但係，如果巴哈譜賦格藝術係畀 Mizler Society嘅，咁佢肯定會揀同佢往年一樣嘅「開列(? open-score model)」款來寫。Both the Canonic Variations and the *Musical Offering’s* six-part ricercare (the provenance of which dictates its instrumentation) were published using the same open-score format to demonstrate the complexity of the counterpoint. By association, this helps to negate the absolute music speculation. 

睇吓組賦格曲，可以準確排好渠地嘅次序。第一組賦格嘅次序大致可用複雜度同主題發展來排好。 呢幾支係簡單賦格，each of which employs a single form of the subject: with the exception of Contrapunctus 4, which was added for the printed edition, these movements are close to the stile antico and are restrained in both harmonic and expressive content. Continuing in the same manner, Contrapunctus 5, combines two versions of the subject that have so been employed, those of Contrapunctus 1 (the original theme) and 3 (its inversion). Contrapunctus 6 bears the title ‘in stile francese’, and uses the emphatic dotted rhythms peculiar to the French ‘ouverture’. It also combines rectus and invertus versions of the subject in both original and diminished (halved) note values. Similarly, Contrapunctus 7 incorporates diminution with augmented (doubled) note values. Contrapunctus 9 and 10 form a pair of double fugues that involve invertible counterpoint at the twelfth and tenth respectively, each opening with new subjects that are later combined with the theme; in the case of the former in augmentation, and in 10 with a subject reminiscent of the inverted theme of Contrapunctus 6. Contrapunctus 8 and 11 are both triple fugues that use related secondary subjects, their only real differentiation being that 8 is in three voices. Contrapunctus 9 forms the final fugue of the existing autograph. From here on things become more complicated: the order of the four canons (Contrapunctus 12-15) has been discussed; all are two-part: the first being a simple canon in gigue style; the second and third, at the twelfth and tenth respectively, also involve invertible counterpoint and thus correspond directly with Contrapunctus 9 and 10; the final canon (15) combines its subject with one that is both augmented and inverted. The mirror fugues that follow (Contrapunctus 16 and 17) are in both their original form and in one in which the counterpoint is inverted completely. Contrapunctus 18 is 239-bar fragment that breaks off shortly after the exposition of the third subject (B-flat, A, C, B-natural, corresponding in German musical notation to the letters BACH). Given the nature of this subject, it is likely that Contrapunctus 18 was intended as the final fugue of the cycle. Its unfinished state may lead us to think that death stayed the composer’s pen. Even Carl Philip Emanuel seemed confused when, many years later, he appended to the manuscript ‘NB. Whilst working on this fugue, in which the Name BACH appears in the countersubject, the author died.’ The incompleteness of the fugue, however, does not necessarily suggest that this was what actually existed at the time. Since Bach would certainly have had to test the possibilities provided by the themes, it must be assumed that a draft version was extant. Therefore, it is possible that the work, as it has come down to us, is in a more incomplete state than it was when Bach died. 

The opportunity to speculate on what this draft version contained is too great to ignore. A number of scholars and musicians have provided us with their own completions, but few have relied on evidentiary material from the main autograph and early imprints for their workings. Initially, it would seem odd were Bach not to introduce to this fugue a fourth subject, especially when considering reports that this was his intention. However, the cycle’s titling suggests otherwise, since in the manuscript, only the autograph canons (nos. 12 and 15) carry any sort of nomenclature, and these are in Latin. The 1751/2 print, however, proves to be different: each fugue is entitled ‘Contrapunctus’, and to some of these, Italian subtitles have been added. In the case of the canons, two have been provided with Italian titles whilst for the third (Contrapunctus 12), Bach’s title of ‘Canon in Hypodiapasion’ has been translated to ‘Canon all Ottava’. The fourth (15) retains its Latin. Whilst it is probable that the term ‘contrapunctus’ is of Bach’s devising, since it would not only suit the scholarly intentions of the work, but also allow the movements to be seen as examples of contrapuntal settings, it is important to question the source of the Italian. A note written by Johann Christoph Friedrich Bach demonstrates that the subtitle of Contrapunctus 15, ‘per Augmentationem contrario motu’, was his father’s. But this does little to explain the other descriptions, unless they were scribed by another whom, taking Bach’s precedent, added similar subtitling to the proofs. If this is the case, then the use of Italian can be easily explained: Bach sons generally added Italian titling to works whilst he himself preferred Latin. Thus, the resulting non sequitur is inconsequential: either it was too difficult or too much trouble to translate from the Italian (the myriad inconsistencies and inaccuracies in the titles throughout the cycle is suggestive of a less than masterful command of the language), and the subtitling of Contrapunctus 18 becomes nothing other than a description of the fugue as it comes down to us. It cannot be used as a counterargument to the proposal that Bach planned a fourth subject: problems have thus far arisen over the order of the fugues and the cycle’s intended instrumentation, and since this suggests that Bach’s executors had little or no knowledge of the intended architecture of the work as a whole, there is no reason to think that they had much idea concerning its individual movements.

作曲者嘅署名手稿 also provides evidence of how much of the unfinished fugue is missing. Bach allowed the complete fugue six pages, taking up five with the surviving fragment. It is possible, therefore, 巴哈打算寫多大概四十幾小節。Whilst this might seem to suggest a somewhat terse ending, especially when considering the fugues of his Weimar and early Leipzig periods, the six-part ricercare from the Musical Offering demonstrates that, by 1747, Bach’s interests lay more in concision than expansion. Not only is there enough space for the completion of the third part, but also for the introduction of a final subject, which could easily have taken the guise of a reworking of the subject of Contrapunctus 1.

每首賦格都用簡單嘅d小調，除咗未寫完嘅對位十四。

  

### 源

1751版中，啲學章大約跟住對位技術嘅淺深來排。標題嘅阿剌伯數通常指聲部數，除咗最後一支嘅*a 3 Soggetti*指「有三項主題（"with 3 subjects"）」：

單賦格：

1. *Contrapunctus I*, and
2. *Contrapunctus II*: Simple monothematic 4-voice fugues on main theme.
3. *Contrapunctus III*, and
4. *Contrapunctus IV*: Simple monothematic 4-voice fugues on inversion of main theme, i.e. the theme is "turned upside down".
Counter-fugues, in which a variation of the main subject is used in both regular and inverted form:

5. *Contrapunctus V*: Has many stretto entries, as do *Contrapuncti VI* and *VII*.
6. *Contrapunctus VI, a 4 in Stylo Francese*: In dotted rhythm, known as "French style" in Bach's day.
7. *Contrapunctus VII, a 4 per Augmentationem et Diminutionem*: Uses augmented (doubling all note lengths) and diminished (halving all note lengths) versions of the main subject and its inversion.
Double and triple fugues, with two and three subjects respectively:

8. *Contrapunctus VIII, a 3*: Triple fugue.
9. *Contrapunctus IX, a 4 alla Duodecima*: Double fugue
10. *Contrapunctus X, a 4 alla Decima*: Double fugue.
11. *Contrapunctus XI, a 4*: Triple fugue.
Mirror fugues, in which the *complete score* can be inverted without loss of musicality:

12. *Contrapunctus XII, a 4*: The *rectus* (normal) and *inversus* (upside-down) versions are generally played back to back.
13. *Contrapunctus XIII, a 3*: The second mirror fugue in 3 voices, also a counter-fugue.
Canons, labeled by interval and technique:

14. *Canon alla Octava*: Canon at the Octave. The two imitating voices are separated by an octave.
15. *Canon alla Decima in Contrapunto alla Terza*: Canon at the tenth, counterpoint at the third.
16. *Canon alla Duodecima in Contrapunto alla Quinta*: Canon at the twelfth, counterpoint at the fifth.
17. *Canon per Augmentationem in Contrario Motu*: Augmented canon in inverted motion.
An arrangement of *Contrapunctus XIII*, see below.

18. *Fuga a 2* (*rectus*), and *Alio modo Fuga a 2* (*inversus*)
Unfinished quadruple fugue:

19. 三部賦格(*Fuga a 3 Soggetti* ,*Contrapunctus XIV*)：一支四聲部三重（甚或四重）賦格曲，其中第三主題基於BACH 動機－ B♭–A–C–B♮ 四粒音符。
啲賦格同卡農嘅次序有得拗, especially as there are differences between the manuscript and the printed editions appearing immediately after Bach's death. Also musical reasons have been invoked to propose different orders for later publications and/or the execution of the work, e.g. by Wolfgang Graeser in 1927.

The 1751 printed edition contained — apart from a high number of errors and other flaws — a four-part version of Contrapunctus XIII, arranged to be played on two keyboards (*rectus* BWV 1080/18,1 and *inversus* BWV 1080/18,2). It is however doubtful whether the printed indication "a 2 Clav.", and the fourth added voice, that is not mirrored according to Bach's usual practice, derive from him, or from his son(s) that supervised this first edition.

The engraving of the copper plates for the printed edition would however have started shortly before the composer's death, according to contemporary sources, but it is unlikely that Bach had any real supervision in that preparation of the printed edition, due to his illness at the time.

The first printed edition also includes an unrelated work as a kind of "encore", the chorale prelude *Vor deinem Thron tret Ich hiermit* (*Herewith I come before Thy Throne*), BWV 668a, which Bach is said to have dictated on his deathbed.

A 1742 fair copy manuscript contains *Contrapuncti I–III*, *V–IX*, and *XI–XIII*, plus the octave and augmented canons and an earlier version of *Contrapunctus X*.

### 未完成嘅賦格

《對位十四》(*Contrapunctus XIV*)喺第三樂段中間，第239節忽然斷咗：

署名手稿上有作者個仔Carl Philipp Emanuel Bach手寫嘅字，話："Über dieser Fuge, wo der Nahme B A C H im Contrasubject angebracht worden, ist der Verfasser gestorben." (「喺呢支賦格嘅呢度，B A C H 呢個名被引咗入對位主題，作曲者死咗。」）但係，現代學人唔同意呢種觀點，尤其係的豆豉都好明顯係巴哈自己手寫嘅，喺佢視衰手乜之前，大約係1748年同1749年之間。可參攷Christoph Wolff本 
*Johann Sebastian Bach, the Learned Musician*，ISBN 0-393-04825-X，入便嘅論。

好多學者，包括Gustav Nottebohm (1881), Wolff and Davitt Moroney, ，話have argued that the piece was intended to be a quadruple fugue, with the opening theme of *Contrapunctus I* to be introduced as the fourth subject. The title *Fuga a 3 soggetti*, in Italian rather than Latin, was not given by the composer but by CPE Bach, and Bach's Obituary actually makes mention of "a draft for a fugue that was to contain four themes in four voices". The combination of all four themes would bring the entire work to a fitting climax. Wolff also suspected that Bach may have finished the fugue on a lost page, called "fragment X" by him, on which the composer attempted to work out the counterpoint between the four subjects.

A number of musicians and musicologists have conjectured completions of *Contrapunctus XIV*, notably music theoretician Hugo Riemann, musicologist Donald Tovey, organist Helmut Walcha, and Moroney. Ferruccio Busoni's *Fantasia Contrappuntistica* is based on *Contrapunctus XIV*, but is more a work by Busoni than by Bach. Moroney's completion (a midi file can be found here) is the shortest, and regarded as the most convincing by some. Glenn Gould錄時特登喺第233節第一拍強音急停（呢度係1751年版嘅尾）； the manuscript continues until the first beat of bar 239 and the tenor voice until the end of that bar. Most performers add these bars, and execute a fade out on the last few notes.

#### 置換矩陣

In 1991 an astonishing discovery was published by Zoltán Göncz answering the question with fairly great certainty how Bach planned the appearance of the fourth subject, the main subject of the cycle:

In the course of the exposition of the first three subjects (first subject: mm. 1–21, second subject: mm. 114–141, third subject: mm. 193–207), Bach applied a serial sequence of voice entries decided in advance, by which he determined the space and time parameters of the subject entries. The superimposition of the three exposition matrices foreshadows and develops as a negative the sequence of the voice entries of the fourth subject. The copying of the four subjects onto each other displays a characteristic construction of Bach’s oeuvre occurring mainly in the vocal fugues: that of the permutation fugue. 

Commons

點矛盾都好， it follows from the logic of composing a quadruple fugue that the combinations joining all four subjects and rendered the *latest* when performing the work were already completed in the *very first stage* of composition because the possibility of overlapping the four subjects (1+2+3+4) is the *sine qua non* of writing a quadruple fugue. 

Contrapunctus XIV 入便， 巴哈用咗 *whole expositions*嘅「（en:stretto)，疊句，拉丁源，原意係夾緊」。 呈示頭三主題時， 佢編寫(programmed)咗後面嘅「置換 stretti」（ permutation stretti），然後應用啲呈示部做「程式」同埋「算法」。 個置換矩陣 (*en:permutation matrix*）, apart from originating authentically with Bach, can be proved to have been ready at the time of the genesis of the work (that is, earlier than the surviving section).

The discovery of the *permutation matrix* was one of the most essential conditions to achieve that the reconstruction of Contrapunctus XIV could come near to *the original form planned by Bach*. (Göncz, Z.: Reconstruction of the Final Contrapunctus of The Art of Fugue, in: *International Journal of Musicology* Vol. 5, pp. 25–93. 1997 ISBN 3-631-49809-8; Vol. 6, pp. 103–119. 1998 ISBN 3-631-33413-3) Score published by Carus-Verlag [CV 18.018] (see External links).

演
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See here for a more complete list.

*Harpsichord*:

* Davitt Moroney (1985)
* Menno Van Delft (1999)
* Sébastian Guillot (2006)

*Piano*:

* Charles Rosen (1967)
* Tatiana Nikolayeva (1992)
* Walter Riemer (2006), using a fortepiano of Mozart type

*Organ*:

* Helmut Walcha (1956, 1970)
* Glenn Gould (1962) incomplete
* André Isoir (1983[?])
* Marie-Claire Alain (1993)

*String quartet*:

* Italian Quartet (1985)
* Juilliard String Quartet (1989)
* Keller Quartet (1997)
* Emerson Quartet (2003)

*Orchestra* :

* Milan Munclinger with Ars Rediviva (1959, 1966, 1979)
* Hermann Scherchen with Orchestre de la RTSI (1965)
* Karl Ristenpart with Chamber Orchestra of the Saar (1965)
* Jordi Savall with Hesperion XX (1986)
* Erich Bergel with Cluj Philharmonic Orchestra (1991)

*Other*:

* Musica Antiqua Köln (director Reinhard Goebel) for string quartet/harpsichord and various such instrumental combinations (1984)
* Berliner Saxophon Quartett for saxophone (1990)
* József Eötvös for two eight-string guitars (2002)
* Amsterdam Loeki Stardust Quartet for recorder quartet (1998)
* Fretwork for Consort of Viols (2002)

註
-

1. 1 2 3  The recordings by Walcha (1970) and Moroney include both their completion of *Contrapunctus XIV* and the unfinished original, while Bergel's includes only his attempt.
2. ↑  Partial performances on organ (*Contrapuncti I–IX*) and piano (*I, II, IV, IX, XI, XIII inversus,* and *XIV*).
3. ↑  Except the canons, which are played by harpsichordist Kenneth Gilbert on the recording.

Media
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Contrapunctus I

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*播呢個檔案有問題？睇吓媒體幫手。* |

出去睇
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* Web-essay on *The Art of Fugue*
* Introduction to *The Art of Fugue*
* *Die Kunst der Fuge* (scores and MIDI files) on the Mutopia Project website
* *The Art of Fugue* as MIDI files
* image of the ending of the final fugue at external site
* Public Domain Scores of the Art of Fugue at **IMSLP**
* Contrapunctus XIV (the reconstructed quadruple fugue) – Carus-Verlag
* Contrapunctus XIV (the reconstructed quadruple fugue) – part 1 (YouTube Video)
* Contrapunctus XIV (the reconstructed quadruple fugue) – part 2 (YouTube Video)
* Glenn Gould 彈 Contrapunctus 14