Norden Hugo_Foundation Studies in Fugue

8/2/2019 Norden Hugo_Foundation Studies in Fugue

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FOUNDATION STUDIES IN FUGUEby

HUGO NORDEN

J~ ,

CRESCENDO PUBLISHINGNew York, New York


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FOREWORDThis book on one of the most important aspects of music composition

has a two-fold purpose. First, it is meant to provide the student with aworking technique in fugue so that he can approach the composing of afugue systematically and with a well-defined order of steps.

Secondly, i t is intended that through the observations presented, thestudent can formulate a method for examining fruitfully the fugues of thegreat composers of both the past and present.

Two pieces of advice may be ih order. The first is that before under-taking the study of fugue, the student should be well equipped in all aspectsof harmony and counterpoint, and especially canon. A deficiency in any ofthese techniques will make itself evident in fugal composition which callsfor the ultimate in smoothness and elegance. Secondly, the student mustnot feel that this book isthe last word in the study of fugue. Onthe contrary,i t is only the beginning. There are so many excellent fugues to be studiedand so many other treatises to be read, but-as the title implies-it ishoped that the present book, small as i t is, will provide a solid foundationupon which to build.

HUGO NORDEN

Boston, MassachusettsApril 1977

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CHAPTER IThe Subject

The theme upon which and around which a fugue is constructed iscommonly identified as the SUBJECT. A melody to be suitable for thispurpose will generally exhibit three characteristics:

1. i t is usually brief, but embodies a complete musical idea;2. i t is easily recognizable as it weaves through the contrapuntal

texture, and3. it lends itself wel l to contrapunta l and harmonic manipulationand exploitation.

From the craftsman's point of view the last at tribute is the most important.Bach, as also did other composers, employed subjects that range in

length from a few notesEx. 1 WTC I, Fugue No.9 in E major

' i f J I J J J J J lto several measures.

Ex. 2 WTC II, Fugue No. 24 in B minor4 ' a 1 ) , 1 fj4 I n m I J J ] I J J J I J J 9 J) I ) 1Rhythmical ly, a subject may be of the utmost simplic ity,Ex.3 WTC I, Fugue NO.4 in C-sharp minorr . .

or of considerable complexity.Ex. 4 WTC II, Fugue No. 17 in A-f lat major

f ~ ' V ", r r r tJ l f) I C l f r < " > r r r rau l ~In the matter of tonality, a subject may be starkly diatonic,Ex. 5 WTC II, Fugue No.9 in E majorr r I r r r

or highly chromatic.Ex. 6 WTC I, Fugue No. 14 in F-sharp minor

Ex. 7 WTC II, Fugue No.6 in D minor


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It may even be a "tone row," employing all twelve notes.Ex. 8 WTC I, Fugue No. t.4 in B minorf * a . , in!] e o Iwm_I,DJ]J.JjJJOI ),A subject may flow continuously as in those quoted above, or it may

be fragmented as in the fol lowing .Ex. 9 WTC I, Fugue No. 22 in B-flat minor

1 * r r r I eEx. 10 WTC II, Fugue No. 20 in A minor9 : ( ' l f r f I II r ! r r I T I Y

Ex. 11 WTC II, Fugue No. 11 in F major~ ~ a ~ _tD3) r gJ I Q:; ;m 1 )1A useful feature of a successful subject is a strong opening motive that

can be readily identified by the listener so that he will know when theentrance is being made. This is sometimes referred to as the 'head' of thesubject. French treatises speak of this device as "La tete du sujet." Thefollowing are typical.

Ex. 12 WTC I, Fugue No.5 in D major?:I~13 I E r i t C ' r ' c r { ' 1C J I P '

Ex. 13 ' ~ ~ # . ~ I t ,'I WTC I, Fugue No.3 in C-sharp majory r ) ; J f i f r J ; ; I n r P pI]Ex. 14 WTC II, Fugue No.7 in E-flat major

Ex. 15 WTC II, Fugue No.5 in D major? : ' a e , t r I O r ' C i t r f r r

Ex. 16 WTC I, Fugue No. 16 in G minorf ) : J ' B ;(ti~F'I' gE"" r

The foregoing general observations are introductory. I t is now necessaryto examine in detai l the specific types of subjects and their respective answers.

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CHAPTER IIThe Answer

The A~SWER, under t .he simplest condi tions, i s the subject tr ansposedto the dominant key; that IS, a perfect 5th higher or a perfect 4th lower.Ex. 17 WTC II, Fugue No.9 in E majors..bi'1

" 1 1 ' 1 e " F r I r r Fr f I f r r -

When ~ subject in a major key contains the leading-tone, this will requirean accidental for the leading-tone of the dominant key.Ex. 18 WTC I, Fugue No. 9 in E majors .bJt

~.a1J(,* ~;d f J jJ IJ J J J J 1

When a subject is in a minor key, accidentals will be required for both theII and VII notes of the dominant key.Ex. 19

Sub i f 'C1WTC I, Fugue No.4 in C-sharp minor

I , ; ( ' .. I II " f IT I * . . i f 'IAUWtrS H a 'I@ ! II

When the subject contains additional accidentals, that is, any besidesthose enumerate? above: these too must be reflected in the dominant key.~e .two following subjects and their respective answers illustrate thisprinciple.

Ex. 20 WTC I, Fugue No. 14 in F-sharp minorSuoj1 ~,.,.........._~ ) : f t ~ 1 I 1! F f ' I r c r r r r r F !r I r q ~ ~ E F E lF s P I r

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Ex. 21 WTC II, Fugue No.6 in D minorS"bj~ct . 1 J' ~ f 1 f 1 m J l 9 a a [ P F 2 f l a m J V J jJ I J \Aruw,er,~(~@wgt~r F ~ r q r I t z o n pc r 1 ) \

J" )

In the above instance, Bach ends the subject on the III note of D minorand the answer on the I note of A minor.An answer tha t is a lit eral t ransposi tion into the dominant key is ident i-fied as REAL. When, under specific conditions that are described in detailbelow, ce rta in notes in the subject tha t a re in the dominant key and answeredin the tonic key in contrast to the rest of the subject, the resulting answeris said to be TONAL. The following examples illust rate how this principleof alternating tonality operates.

Ex. 22 WTC I, Fugue No. 11 in F major; ~ i ~ e e I { U I L ; ; ; r r i I E F r r F I Qy k~ lkty ~~~~_,~ es I n J I J J jJ J I J J J J J I J, *ty Y k ey __ _ _ _ ! . q _

WTC 1, F ugue No. 22 in B-flat minorJ I !F f r I fEx. 23Y key I Juy _

Yby thy YJu j -The ent ire system of subjec ts that require real and tonal answers is c rystal-lized into 18 specific types in the i llustrations that fol low.

Subjects That Require A Real AnswerSubjec ts in this group wil l meet two genera l condit ions:( 1) they will begin and end on the I or III notes, and(2) they will not contain a leap from the I or III notes to the V or VII

notes at or near the beginning.(a) Begins on I and ends on I WTC I, Fugue No. 20 in A minor

Ex. 24~ ~ ~?~He r t1at tt&tFtth[J ,p : r r r r r r r r r r r L IPI I

~ c n : ; m ffl n 1 _ U~tj J I I " n J J J JJjAJ11)See also Ex. 17, 19 and 20.6

(b) Begins on I and ends on IIIEx. 25

wrc I, Fugue No.1 in C majorSllbject' ( ~ , n O J . J 3 f S Ift.]

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See also Ex. 18.(c) Begins on III and ends on I Two-part Invention No. 15Ex. 26 Subi""

~.~." tmJ 9 ! 9m J Imaj;J))11/

Actually, this answer is never used literally in the composition but it doesappear with the 'head' changed as follows. '

~ etc2 D I I _ ~This kind of subject is extremely rare and is as good as never found in thefugues of the master composers. The above example is found in the Dmajorsection of the Invention in B minor.. (d) Begins on III and ends on III

Ex. 27 S."jutThree-par t Invention No.3

:ill t:rlJ ~~ ~i l IE" bitt tk[j diG .. @ i J i I g/II /II, " a ( 3 ' gO rn a I P m J P ? f i P I ~

In Fugues Nos . 2 (C minor) and 16 (G minor) a curious use of subject sthat could be treated according to (c) above are answered tonally. Whentaken out of c~ntext the . fol lowing two subject s could be seen as beginningon III and ending on I in the keys of E-flat and B-flat major respectively.Instead Bach has them begin on V and end on III in the keys of C andG minor (cf. Ex. 22) in WTC II.

Ex. 284 i% { H J J J lJffll J7


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Subjects That Require A Tonal AnswerGroup I

Ex. 33 Violin Fugue in A minorS",bjt'f@fJIUUI~Subjects of this kind will begin with one or more notes in the dominantkey and complete the subject in the tonic key. Whatever is in the dominantkey willbe answered in the tonic key, and vice versa.(a) Begins o nV and ends on I WTC I, Fugue No.3 in C-sharp major

Ex. 29 Sub;1 ~'U#~I('1 , ; ~ - 1 m o I J :m E P fJ 1 1 1

v k ey 1 keyA .... .",I'W ; ~ I r o l iP I J I

The next two subject types are so rare as to be vir tually non-existent inpractical usage by the masters. The examples provided below are syntheticmodels contrived to illustrate the theoretical principle involved.(c) Begins on VII and ends on I Norden

Ex. 34Y by I k~

'jQ#I~~W ~j ~ - ; J J JD I J 7 J J P J1 fO I J I~~------------------------~ SI I b/ f


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Ex. 37 Art of the Fugue, Contrapunctus 10 (b) Begins on I and ends on VIIEx. 41

Bach: Two-part Invent ion, No. 10Subecr

,~(' 1# J J J I t J J j : JVII

F r I t r r r 1 F A . I U W ! " P " [1/\ H s c : V C j i r ,f I EII of SwbdmnilJlPtl KeyGroup II

Subjects in this group begin in the tonic key and end in the dominant key,while their answers do exac tly the opposite , beginning in the dominant keyand ending in the tonic key.(a) Begins on I and ends on V WTC I, Fugue No. 18 in G-sharp minor

Ex.38 SuN'5 .~#~ :!c Y U I rT f F r r r lip

Vlt')

The following is a longer and more involved subject of the same type.Ex.42 Klengel: Kanons und Fugen, Vol.I,Fugue No.17 in A-flat majorSubect ~ ..---....._. VII' ~ 5 h ,m : a z ti r 1 a T ("cj r? Tffl n - P t ! ! 7 1 J

V Juy

V ke) J k.eySubjects that begin on III and end on either V or VII are very rare. Theexamples tha t fol low are aga in syntheti c model s cont rived to demonstratethe principle involved.

~~ ~~Vkn I tn,A spectacular subject and answer of this type appears in Fugue No. 11

in F major of Klengel's Kanons und Fugen, Volume I.SlIliit

Ex. 39 ' ~ B f ;tJl E 1 r e U i bJ ] J d J J J ; ] J a J 1 lJ (c) Begins on III and ends on VEx. 43 NordenSub/t III V~"l J J L-J_ _J _ J _ _I_I_ I _ J1keyA"""t"r ti tdj J J ; J J I key Y kryA..." ...,., VII$ "! r J J I J r r r 1 r~~-------------hy Iluy-:;::. Vu,f ~~ 2 _ ! , J J J J J J p ? J I J J J j J J J ] J J i s J 1 ~

vu,

(d) Begins on III and ends on VIIEx. 44 t $ " " III VlI@ 2 J 1 j J I J 3 ; 1 1 pSiJ

___1kry Y key

Norden

~L- ~~~ ~ __ ~~ ---fl,:In Variation 26 of tne Goldberg Varia tions Bach employs the followingsubjec t in which the dominant key is not introduced unt il the third note fromthe end. It i s quit e unusual for the dominant key to be introduced so la te.

Ex. 40 ;i'l br r rgg E r r r ; , F r F r ! r I Pr I J r r lF " D O r

_______JV kf!Y I fuy

Ambiguous tonality comes about in a minor key when a subject of type(d) above does not contain the leading-tone of e ithe r the tonic or dominantkey. To illustrate, the following subject begins on the III of D minor andends on the III of A minor, the natural leading-tone of D minor. However,it would be entirely reasonable to view the subject as beginning on I ofF major and ending on the Iof C major, being the same as V of F major.

Vkq

, . 1 0 J 3 J J X i J j j 3 j j J J J J J J I lV kq I kq

10 11


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Ex. 45 Norden Ex.49 KlengeI,Kanons und Fugen Vol. I, Fugue No. 22 inB-flat minorSuh~CI

f ~ I V ~ ! ~ 1 fJ ' P ; IfJ ~PIG EH Blp - z n )Sub~cr,~"~n J r I J e J r [[1 I rV It.ty I key V K.t)k.ty V.kty

AlLS 'K 'u ,,-... ...f PlH U J j! I'r I f r t U I rV fty {Iuy

But, when essentially the same problem contains the raised leading-tones ofthe minor keys involved, no such ambiguity exists.Ex. 46 Norden

S.b/r ~* , 1 1 n J J I J (JiJ J J I F.. , ~(b) Begins on V and ends on VII

Ex. 50 NordenSl .IbJt~I' I J nJ J Din n J___, "-------Vby Ikq Vkeyi J~"j I f U lc e r IEttir

I~ ~ '7k:-.,-------=--

Group IIIGroup III of subjects that require a tonal answer combines the properties

ofGroups I and II. A subject in Group III consists of three sections: -1. An opening effect in the dominant key as inGroup I,2. A middle section in the tonic key, and3. A closing section in the dominant key as in Group II .

(c) Begins on VII and ends on VEx. 51 Norden

S"b;ect

f f i t c I ~ Fill] J J r o I JVkty Ike:, VktyAJ 'UWtrf f ' JI J J I ) 1 JThe following illustrations are self-explanatory.

(a) Begins on V and ends on VEx. 47 WTC I, Fugue No.7 in E-flat major

S!JIb~1 17~~Ij,j~ fjJjJJDO " , 1 0 I r; r (d) Begins on VII and ends on VIIEx. 52 Norden;0 1"


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Group IV(a) Any of the preceding sixteen types-four Real and twelve Tonal-containing a leap from the I or III notes to the V or VII notes at or nearthe beginning. The following are typical.Ex. 53 WTC I, Fugue No.8 in E-flat minor

Subjec, J* ~ ; ; 'g l f d : F - - i l i 1 J I ; - 3 F J 4JR J J____. .. I ViQ 'IkIy ::::::Ik.,

~ A I ' " v : . . . . - . . . . . : ~ ~ , - - . . . , . y& ~ i , ) I ( ' r i F r : g f r ( u V r r 1 C P r F-"-' ' Itl)' Vl., "----"v .,Except for the second note, V approached by upward leap from I, this subjectis in effect a Real (a). Thus, the second note is a dominant "island," as itwere within a subject that is otherwise in the tonic key.Ex. 54 S w' WTC I, Fugue No. 22 in B-flat minor

' ~ I p@~e r i - j -: I ! r r r 1 F7;;; : !'??- ~ Ikey

The following answer is most unusua l. The subjec t is Tonal, Group I (b),beginning on V and ending on III. The third note is V, preceded by VI. Theunusual feature here is that this V is answered tonally in spite of the tradi-tional 'rule' that a V or VII after II or VI is to be answered Real andnot tonally.

Ex. 57 S.bjw WTC I, Fugue No. 21 in B-flat major~ ~ 2 ' J J ; { ; J I a r m J ! 5 o 1 " J ! ) .

V~~k~~:/--C.


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(b) The ent ire subject answered in i ts subdominant key. Subject s answeredin this way usually begin on V or VII. The following examples show howthis answering process comes about.Ex. 61 Bach, Partita IVISubJ


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ExercisesIdent ify each of the following subjects as to type, and writ e answers to same.

1 tr6 e - . pCt f n J ~ I J J (-3j I I~ I ~ 2. I~ , h k k i U J J J J J I ~ ~l ;f a IJp 1 . 2 1 :=== - : = _ _ _ _ _

3. ~, - g P I m f 1 J I fn J~m; I r4 .'~ I t J J J I J J J 4 J) jJ J g r I F

CHAPTER IIIThe Countersubject

The COUNTERSUBJECTis in effect a second subject that accompaniesthe subject and answer throughout the course of the fugue, or as long as thecomposer wishes it to do so. Generally, in the opening section of the fugue,it appears first against the answer (which normally is the second entry)and subsequent ly against the subject in the third entry. The following Bachquotation demonstrates in the simplest possible way how a countersubjectoperates.

Ex. 69 WTC II, Fugue No.2 in C minorA,.._

5 .~~I

~ ~ : " ' : ~ 1 1 ' : , : : ; : - 1 ;S.bitttThe subject , answer and countersubject are emphasized by the larger note-heads. The remaining material, in smaller notes, is free and not relatedto the subject and countersubject.As is illustrated in the above example, a countersubject must be soconstructed that it will function equally well both above and below thesubject and answer. In order to achieve this result it is necessary to composethe countersubject so that it will operate in Double Counterpoint at eitherthe 8ve or the 15th (1. e., the double 8ve) against the subject. In thepresent instance, the inte rva l of inversion is Double Counterpoint at the15th, thus

Ex. 70

6.

~ e!r j J 1 1 Ln J p I Fj 4 . . . . . Q J,hJ I F7.f ~ e 8 I" I". .8 .f e j J J I J J I I J . 5 J I ]9.~~~I j ~ Ij. )iU In J J J P IJ . pOI J10.

COmf(-rYlh"cr1 0)

11.'VII m I Ii!]7 1 1 ralll 5 1 V J . ; Q I J12.~ . " io JJ

Compose original subjects within the various types in both major andminor keys and write answers for same.18

While Double Counterpoint at the 15th affords greater freedom ofmovement within the countersubject because of the la rge r interval of inver-19


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s ion, i t issometimes expedient for textural purposes to construct the counter-subject wi thin the narrower limita tions of Double Counte rpoint at the 8ve.By this means the subject and countersubject will be kept closer together.Such is the case in the following example where al l of the inte rvals betweenthe answer and countersubject are less than an 8ve.Ex. 71 Bach: Three-Part Invention No. 8

-Tonal Group II (a)-appear in the fugue, the operation of the counter-subject in the second, third and fourth entries is as shown below.l e : ~ j : ; OI J ' l : : : U i ; l ~ ~ ~ i P i_~ _._~ _CtxUlkr~i"r

The compositional process for writing the above countersubject would bea counte rpoint invertible at the 8ve, thus:Ex. 72 CULO 'fsuh CI

~II~ c_Jt' I J . II --, r---- -------------~ -,;;.;.------------ ---~l " ' 2:-,., '~r :.~ ';" ,) ~ , ~, ~I-J::I ~ r=

The technical it ies involved in writing countersubjects for all conditionsinvolving tonal answers follow herewith.

A somewhat unusual construction occurs in Fugue No. 23 in B majorin WTC II, wherein the countersubject crosses the answer in the secondentry. La ter, when the countersubject is placed above the subject, the voicecrossing does not come into play. The following two quotations from thefugue-measures 5 - 8 and meas. 19 - 22-show how this comes about.

Ex. 73 WTC II, Fugue No. 23 in B major

Process For Composing A CountersubjectFrom the point of view of composing a countersubject, subjects fall

into three general categories:1. All of the subject is in the tonic key.2. Most of the subject is in the tonic key.3. Most of the subject is in the dominant key.

Every subject wil l fi t into one of these categories.1. WHEN ALL OF THE SUBJECT IS IN THE TONIC KEYSubjects of this kind are those that require a Real answer. But, thesein turn subdivide into two classes of contrapuntal construct ion:(a) those without tied notes, andEx. 75

An . rw c r

",~a.s.5-ii

II

~ ISub~cr

I1

When a subject is so constructed that atonal answer is required, thesubject and answer are not alike with the result that a more complicateddouble counterpoint format is required. The subject and answer in Ex. 38 (b) those with tied notes.

20 21

,I

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EX. 76 voices such wide intervals may cause dif ficult ies in achieving a satisfactorycontrapuntal and harmonic texture.By constructing the countersubject within the limitations of DoubleCounterpoint at the 8ve instead of Double Counterpoint at the 15th such

wide ver tical intervals wil l be automatical ly avoided. Ex. 79 shows a counter-subject composed within these added limitations.Ex. 79

ASK~r6 2 ' ) 1 r z - s J jJ J I l f - b t l " ' , Hr I ~Turning first to the subject and answer given in Ex. 75, wherein no

tied notes occur, one can proceed in either Double Counterpoint at the 8veor at the 15th. At every point in the structure the longer note, whether in thesubject or in the countersubject, will serve as Cantus Firmus while the partin faster notes will act as Counterpoint. This is the same process shown inEx. 70 and Ex. 72. The following is such a treatment of Ex. 75 in DoubleCounterpoint at the 15th.

Ex. 77

COUJ1r"rsub c(

~~- - - - --

" Subject - - -., . -Cm.uJtI!r5l.1hjt'c/ !~~7r- ~!, ~!! ... . b - : : e - ; ._:. . . . .-~~ . . . tr

)I~Subjtt. - -I~ . -

Co,,"'tr$wbjtC1 .- - . . . tr( -.;; -" - - - - - -

) ! '" ' ~J A AlLJ1I.ll'r{

U _ . t. q . . .. . ~7"'"...-,;-CoufJUr,fu!Jicct . . . .~ ..- - , . tl'_.--! ' " " ~When the above is transposed to its dominant key, in this case A minor, i t

automatical ly provides the countersubjec t for the answer.Ex. 78 In t~is case the two themes are never separated by an interval greater than

a mmor 7th, thereby facilitating the weaving of a tightly knit contra-puntal texture.. When a subject features tied or dotted notes-whether these be suspen-sions, retardations or simply ties-as in the following,Ex. 80

", oJ ..- - - ~

Aruwt' .. .U - : J_ . q . . .. . . q .. . ' " _ _ _ '47 ~I C()&Intt! 'r1ub jec1 - -t r- - - = = - ' ~ ~

Subjtf IjI t _ J J J 1 1 i " i J J I m ~ n J a I JAMIJerf i' ) 1 rOi)1J'lnft [W?sJffl, J

It will be observed that in the above example the intervals between thesubject and the countersubject, and inevitably between the answer andcountersubject range from a major 3rd to a major 10th when the counter-subject i s above the subjec t and from a major 6th to a minor 13th when thecountersubject is below the subject . If the two themes are placed in adjacent

a specific order of steps in the composition of the countersubject maybe suggested.Step 1 Trea t all suspensions and re ta rdations in the subject .

22 23

it


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Ex. 81 poser's artistic requirements. The composer should decide in advancewhether he wants the countersubject to be1. as complex as the subject,2. more complex than the subject, or3. le ss complex than the subject.

~) Ie ) 2 3 .I. 6

Sub~cl Su so . SISp.

It-' . . - _ v' ;J:. -#.,; ... '--"' ., ., -#

4 37 6Coantersubiect

- -* * * * *

2. WHEN MOST OF THE SUBJECT IS IN THE TONIC KEYSubject s that fal I into thi s ca tegory wil l general ly be ei ther Rea l with theGroup IV(a) effect, as illustrated in Ex. 53,54, 55, 58 and 59, or Group I

as in Ex. 29,30 and 31. There are rare cases in Group II wherein the modu-lation to the dominant key occurs so late in the subject that most of it is inthe tonic key. This situation is illustrated in Ex. 40, 41 and 42.

A typical case would be the following subject which is in effect a Real(b) into which is inserted the Tonal Group IV(a) effect at the secondnote, being a V note approached by leap from the I.

Ex. 84 Subiea

Step 2 Treat all suspension opportunities against the subject.Ex. 82

Couruersubiect SS ' : ! ! : ; A'P,\U 43 76

1 . , . , Sub;rIt-' . . _ ' ';-:;- ~.-.,;-",. . ._ . . . .. .23

SlUp. Susp.CoullUrIJlb~ C~.fI.__,AJUWt'r r-- ---,

~ . " J . ; j. i I J . ( ~ raj n J I JI I :.. - ---_ ..To compose a countersubject that will operate successfully with both thesubject and answer, proceed as fol lows:Step 1 Set up a three-stave Double Counterpoint format with the subject

on the top line, and on the bottom line the answer transposed toits subdominant key. The middle line is left blank for the counter-subject. Now the subject and answer are so aligned that the tonicportions of the subject relate in Double Counterpoint at the 8vewhile the single note in the dominant key rela tes to the correspond-ing tonic note in the answer in Double Counterpoint at the 9th.

Ex. 85

Step _ j Fil l in the remainder of the countersubject and embellish the suspen-sions if so desired.

Ex. 83~ . . .u - - - -

SJ.lbj"1

U . . -'---V ~.*.... . . . _ _ . . . .. - - -COU&Uwb~ct ~ .f l. . . ' . +.-. - -

SJ,jbjft"1. . 1 I _ _ . . . - - - . . _ _

~ F-lu I " -8 9 8..II~.r

~. _ . t!:-~ ~fR-.~-{'- ~ .,ICOU1UvnbjK

The discretion of the composer is of utmost importance here. This part ofthe procedure can ei ther be overdone so that the counte rsubjec t becomes toocomplicated and "fussy," or be made so sparse that the countersubject isineffective. The degree of complexity must be in accordance with the com-

AIS'W'flrJTL1N~td"iu~u.bdomilWJ11lJ:q

24 25

,I


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Step 2 Fill in the countersubject on the middle line so that it will produceco rrect counterpoint aga inst both subjec t and answer.

Ex. 86The same contrapuntal problem comes about in a somewhat differentway when the subject begins with a dominant note as in Tonal Groups Iand

III. The fol lowing subject and answer can be c lassif ied as Tona l Group I(b)with a Tonal Group IV(a) effect inserted at the third note.Ex. 88l ,becf A ..t IU

Ti


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It can be noted parenthet ica lly tha t when Bach const ructed the counter-subject for the subject illustrated in Ex. 69 and 70, he avoided any involve-ment with Double Counterpoint at the 9th by starting the countersubjectafter the initial dominant note of the subject.

When the countersubjec t developed above is shown against the subjectand answer separately, with the answer in its original key, it will appear asin Ex. 87.

Ex. 91Count~,suhi1

To compose a countersubject to operate with a subject and answer of thiskind, proceed as follows:Step 1 Set up a three-stave Double Counterpoint format with the subject

on the bottom line, and on the top line the answer transposed to itsdominant key. As in Ex. 85, the middle line is reserved for thecountersubject. Here the subject and answer are so aligned thatthe dominant por tions of the subject relate in Double Counterpointat the 8ve while the note in the tonic key relates to the correspond-ing dominant note in the answer in Double Counte rpoint a t the 9th.

Ex. 93~I~; . - - . . _ _ . . . . . . . . ~ ..--.....

--Sub;ecrIt) v . .. ._____. . .. .-_ . ." . .

CDW1~J l.bj"fj U~".'-"'f:.- ~ f> " ~"f>". ~ . . . . . . . . . .IAnswertransposed to itsDomlnaru Key

-"") t) - .. "It - . . -, ; . . .

"JI8 9 8

It). .IOU trsMhrcl Subject.. - - -

It) - "---"AlU'Ml t ,"

It) . - "It < t_ _ _ _ _ _ _ _ ~ #11 ;J. "ItI COUr'r.Uf ;t~1r-: r.. ..--.t - - I

Step 2 Fill in the countersubject on the middle line so that it will producesui table counterpoint aga inst both subject and answer.Ex. 94

AnJ "Wt ' rrrrurspwed 10 iSDoM1Uht Key

.. . ,.I~ - . . .." . . . . . . . . ..A>!.

t) . .~ J"';~' ~~ . . . ~~~ . . . . . , .. 1~ :J .""-I

* * **3. WHEN MOST OF THE SUBJECT IS IN THE DOMINANT KEYThis kind of subject occurs mainly in Tona l Groups II and III, although

it is quite possible to have a subject in Group I that meets this condition, asis illustrated in Ex. 32. The following illustration is a simply conceivedsubject within the Tonal Group III(b) framework, with the modulation tothe dominant key coming between the repeated notes in the first measure(d. Ex. 65).

Subject

Ex. 92Subjecl Step 3f * i j ! I -i J ~ ! ~~ ~ I ~. J S J J I J~ . . IV VAUW'Ff~~ I r r ~ I ~~ ~ r I ~. ~II J

28

Copy out the countersubject in Double Counte rpoint against bothsubject and answer, the lat ter being retransposed back to its originalkey. As was the case in Ex. 87, the countersubject, too, must betransposed to match the key of the answer.

2 9


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.. _-------------------------'An.nl.lt" COIlI7(lrY Mt>li~ma I t f Td ; II Jt r va c aJ> ,.

CHAPTER IVThe Stretto

By STRETTO is meant the contrapuntal device wherein the subject oranswer is played against itself canonically, or when the subject and answerare made to overlap. The following quotation from Fugue No.2 in C minor(measures 23-27) from WTC IIs an interesting and fairly complex exampleof stretto technique. The subject is the one in Ex. 31, demonstrating TonalGroup I (b).Ex. 97C!fl~~!II!ff:1~ .-

~- - - - - - - - - - - - - - - - - - - - - - - - - - - -A 1UWr in contrary motion( endng a te red)

As demonstrated above, it is not necessary in a stretto formation that allentries of the subject or answer be stated with all intervals being used literally.The themes can be modified intervallically or used only partially. Often in astretto modifications are made to achieve greater harmonic interest.In measures 14-16 of the same fugue from which Ex. 97 istaken (WTCII,No.2) there is an ingenious three-part stretto consisting of the subject inits original form as wellas inaugmentation and the answer in contrary motionwith some intervallic alteration. The subject in augmentation is the middlevoice and serves as cantus firmus. The passage appears as follows:Ex. 98

Suljr inA~m~nIQtiLJlI: , t - . - - - - - - - -- ~ ;The entire stretto technique can be crystallized into twelve specific types:1. Literal2. Contrary Motion3. Retrograde4. Augmentation

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5. Diminution6. Contrary Motion + Retrograde7. Contrary Motion + Augmentation8. Contrary Motion + Diminution9. Retrograde + Augmentation10. Retrograde + Diminution11. Contrary Motion + Retrograde + Augmentation12. Contrary Motion + Retrograde + Diminution

Ex. 101l ~ ; ; n . i _ : 1 : ; : ' ; : : & t ! - JW -- -- -- .. -.. - - -.- .. --.

1. Li ter al Stre ttoIn the Fugue in F major, WTC I, No. 11, Bach makes interesting useof a s traightforward s tret to at the 8ve, both above and below. First , beginningat measure 25 it is presented between the two lower voices with a free partin the soprano.

Ex. 99 In none of these three stretto structures does the answer appear. In eachcase the harmonic context is quite different, and in the third one (Ex. 101)the last measure of the subject is embellished into a continuous 16th-note figure.

A subject that will perform as shown in the three preceding examplescan be structured as a canon at the 8ve, in Double Counterpoint at the 15th.This process will automatically produce one canon at the 8ve above, andanother at the 8ve below. In Ex. 100 and 101 the first and third entries neednot enter into the canonic format ion as they never overlap. Were the sub jec tlonger, the canonic process would have to involve an invert ible three-pa rtcanon instead of a two-part canon.

1 . . . . . ~ ~ . ..l

S"bjr

Next, beginning at measure 36 it appears as a three-part stretto in the key ofD minor, the relative minor of the key of the fugue, beginning with the 2. Stretto by Contrary Motion

A st retto in contra ry motion can be constructed in two ways:1. by exact intervallic imitation (i.e., a minor 2nd imitated by a minor2nd, a major 2nd by a major 2nd, and so on), or2. by inexact inte rval lic imitat ion (i. e., a minor 2nd imitated by a major2nd, a major 2nd by a minor 2nd, and so on).A remarkable fea ture of Bach's subjec ts i s that they provide a grea t many

stretto possibilities that he never utilized. Thus, in the remainder of thei llust rations in thi s chapter some of Bach's subject s will be shown in st ret toformations that do not actually appear in his fugues. Ex. 102 shows thesubject of Ex. 97 in three contrary motion strettos with a free part added todemonstrate a possible harmonic context. At this point, however, it mustbe stressed that this example and the ones that follow that are not by Bachhimself are in no wayan attempt to suggest that this is something that Bachshould have done. Obviously, he knew better than anyone else what hisartistic requirements were and how to develop them. What is shown hereis to be construed as nothing more than a technical demonstration of whatcan be done with a well const ructed subject if the composer wishes to do so.lt would be silly to think that a composer of Bach's calibre was unaware ofthese stretto opportunities and how to use them. The only reasonable

j " j ICW __-' I iL- _ __ _ _ _ _ __ _ __ _ _ _ _ _ ~ - - - - - - - - - - _.- - - - - _. - - - - - - -- - - _ ...Su.bjt

Then, beginning at measure 46 another th ree-pa rt st rett .o appears, thi s ti~ein G minor, which is the relative minor of the subdorninant key to that IIIwhich the fugue is written, and with the order of entries in the precedingstre tto reversed; tha t i s, st arting wi th the lowest voice.

34 35


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assumption is that he simply saw no place for these strettos in his music.And, herein lies an important lesson for every composer: one must learnto become highly selective in hi s use of contrapuntal possibili ties and rejectthose that are not art is tically suitable to the needs of a par ticular composit ion.The following example shows Bach's theme in three di ffe rent strettosin contrary motion:

(a) at the 3rd above after two Sth-notes,(b) at the 2nd above after two Sth-notes, and(c) at the 7th below after four Sth-notes,

Thematically (b) and (c) are the same, but the relation to the originalsubject both in pitch and in time are dif ferent , thereby giving r ise to differentcontextua l harmonies. In all three st rettos, however, the part that strettosis identical intervall ical ly to the original subject .

Ex. 102

In measures 39-42 the following stretto appears. The 2nd, 3rd, and 4thentries are not complete. The two middle voices enter together in a MirrorCounterpoint. It is here transcribed into open score as the original pianoformat is vi sually very compl icated due to the several part crossings.

Ex. 104SlIhul i.r-- -~------- ---_---------~-----------..------ ----------------~

--

Subjecl ~

-

-It.)

ISltbecf in Cor;tr{vy Mornnt : : ! { , ; ! ! \ : ! ' Swbj1 r3. Retrograde Stretto

A retrograde stretto is one in which the subject or answer is playedbackwards against i tself . This is what is commonly called either Crab Canonor Cancrizans Motion. It is encountered only very rarely. There are severa lreasons why a retrograde stretto may prove to be art ist ical ly impractical :

1. i t cannot be recognized audibly,2. many rhythms, especially dott ed and tied notes, become disjointed

and ineffective when heard backwards, and3. a ret rograde stretto is like ly to set up awkward harmonic situations.

Ex. 105 is not from a fugue, but from a chorale harmonization, "Wer nurden lieben Gott lasst walten," No. 112 in the 371 Chorales by Bach. It willbe observed that the first seven notes of the al to consti tute a Real (d) subject(beginning and ending on the III note) in D major while the tenor in thecorresponding notes is the answer in retrograde. From the purely theoret icalstandpoint this retrograde stretto is a Crab Canon constructed within theavailabi liti es of Double Counterpoint at the 7th.It can also be pointed out that while the subject and answer are strictlyin D major the contextual harmony consists of a modulation from F-sharpminor to B minor via D major.

Ex. 105

Subj~cr ;1 1 COnmTj Mon""

Subf)c t

(c)

. . . -___""5Jdtject inCorn,ary Motio"

A curious four-part stretto with two parts moving in each direction isfound in Fugue No.3 of Klengel's Kanons und Fugen, Vol. I. The subjectin D-flat major is Real (b) as follows:

Ex. 103

36 37


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The subject from Fugue No. 2 in WTC II that is shown in contrarymot ion stre ttos in Ex . 102 can likewise func tion in retrograde. Such a trea t-ment with a possible contextual harmony is shown herewi th. As is the case inEx. 102, the following stretto treatment of this subject does not appearin the music of Bach.

Ex. 106

Theoret ica lly any proport ion i s possible for augmentation st reUo pro-vided the subject is calculated so that it will function successfully. Thefollowing shows a subject with the answer forming a stretto in the pro-portion of 2 to 3.

Ex. 108 Norden

In the above example the crab canon that produces the retrograde strettobetween soprano and tenor is cast in Double Counterpoint at the 17th; thatis, Double Counterpoint at the 10th expanded by an octave. It is, of course,impossible to construct a crab canon within an even-numbered DoubleCounterpoint inversion.

5. Stret to by Diminut ionStruc tural ly and mechanica lly there is no difference between stre tto by

augmentation and .stretto by diminution. If the original subject is in shorte rnotes, the stret to is by augmentat ion, and if the original subject i s in longernotes the s tret to isby diminution. But, canonical ly the process of construct ionis the same. Ex. 109 quotes measures 30-32 of Fugue No.9 in WTC IIwiththe subject in its original form in the alto and the answer in diminution inthe bass. This subject is given also in Ex. 5.

Ex. 109 WTC II, Fugue No.9, in E major

4 . Stret to by Augmenta tionA stretto by augmentation by Bach himself is shown in Ex. 98, between

soprano and tenor .It is customary to think of augmentation in doubled note values; that is,in the proportion of 1 to 2. But, there is no reason why other proportions ofaugmentation should not be employed. Ex. 107 shows a stretto in tripleaugmentation; namely, in the proportion of 1 to 3.

Ex. 107 NordenSub;~t

, . . - - - - - -1 - ~~ ~__ ~ .J

A IIJ'Wf'r b, [)imi1Wlio"

6. Stretto in Contrary Motion and RetrogradeThis kind of st retto enta ils a c rab canon in contrary mot ion. The versat il e

subject from Fugue No.2 in C minor in WTC IImakes such a stret to ava il-able, as shown below. Again, this stretto does not appear in Bach's music.It is merely another in the series of adaptations made for demonstrationpurposes.

Ex. 110SW h ; f t : 1r-- --- --- - - - - - - - - - - - - - - - - ~ - - -...,

Neither the stretto nor the subject shown above are to be found in anyexisting fugue. It was cont rived simply to i llu st rate the contrapunta l possi -bility. A stretto in triple augmentation is impractical unless the subject isdesigned especially for it. Only a subject in triple rhythm will lend itself totriple augmentation.

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7. Stretto in Contrary Motion and AugmentationThe ingen iously contrived subject of Fugue No.2 in WTC II again makes

possible the demonstration of a stretto that does not appear in the fugue;namely, a stret to in cont rary motion and augmentation.

Ex. 111

10. Stretto in Retrograde and DiminutionThis stretto is made possible by the subject from the Fugue in B-flatminor, No. 22 in WTC I.As in the case of several of the preceding i llustra -

tions, this stretto does not appear in Bach's music.Ex. 114

l~:. ~ '~ fM : !~ ' ! ! r rSUbi f 'C I.!'!~t _

8. Stretto in Contrary Motion and DiminutionA stretto of this type is worked into a massive stretto involving all four

parts in measures 9-12 of Fugue No.9 in E major in WTC II. It begins onthe III note in contrast to the othe r four entri es which consist of two subject sand two answers that begin on the I and V notes in the usual way. Also,the entry in contrary motion and dimunition is slightly disguised by theaddit ion of a pass ing-note as indicated in the following example.

Ex. 112 WTC II, Fugue No.9 in E major

11. St retto in Contrary Mot ion, Ret rograde and Augmentat ionThe remarkable canonic quali ties of the strangely chromatic subjec t ofthe Fugue in F minor, No. 12 in WTC I, makes possible a stretto of this type.

Due to the chromatic structure of the subject the resulting harmonies aremost unusual . This stretto does not appear in Bach's music.

Ex. 115

c : : : ' 0 ; - ' t r - ~ ~ _ ~ r ' r f ' ~ ; = r r ~ ~ ~ : ~ " ~ I ; r -"b;rct, .--------------_. __ - - - - - - - - - - - - - - _ .._------------------,1I

9. Stretto in Retrograde and AugmentationOnce again the subject from the Fugue in C minor, No.2 in WTC II,

makes possible an example of a stret to in re trograde and augmentat ion. Thestretto is in the two outside voices with two free parts between them. Thisstretto was contrived by means of Bach's subject and does not appear inthe fugue.Ex. 113

12. Stretto in Contrary Motion, Retrograde and DiminutionThe subject used in the preceding stret to also makes possible the present

stre tto in Contrary Motion, Re trograde and Diminution, which, as in the caseof Ex. 115, does not appear in Bach's music . The harmonies resul ting fromthis st retto are even more unexpected than those in the preceding example.

Ex. 116-=~~~ Dl ~,)\~ 1- -.. -_ ...- _- - - - - - - - - _ . - - - ---- --- -- - - - - -~Sub /


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Solving the Stretto ProblemThe stretto problem confronts the student in two ways:1. constructing strettos to a given subject, as in a fugue examination; and2. constructing an original subject to yield pre-planned strettocombinations.In order to demonstrate the first situation a subject attributed to Matthe-son and used by Bach in his C major Fugue for unaccompanied violin willbe treated according to a definite and highly organized system. The subject,Tonal Group I(b), and its correct answer follow.Ex. 117

In the above example the entrance on "A" isthe subject transposed literallyto the key of D major. Quite different harmonic results would occur ifflats were placed by the "A" and "B" with the F-natural, thus:Ex. 120t r l J DJ J I J JJ I J J r J I ~ ; E r r r I F -

Subjtct

f e W I J J J J J I . I J J I J jJ J 3 1 J (b)Aj_ In\ J J 1 \ J I J . J " " J I~ I J "-J I I I.d de.

I4 1 ' r I r E J r F ir J F IF F r U I F 1

ProcedureFirst, fi t the beginning of the subject against the end of the subject inas many ways as possible. For instance, by means of a 7th chord it ispossible to achieve a stretto at the octave below as follows:Ex. 118

At the 6th below: F_: V 17Ex. 121

l J IJ D J J I J J J I J J ; n l ; u r r I F .d J J Q'i r At the 5th below:Ex. 122f IJ D J J I J J J J J ; n ~C m a i o v6 VJ6A stret to at the 8ve, such asthe above isprobably the one most l ikely to beencountered in a traditional fugue. The harmony shown above is likewisethe one that is most l ikely to be used, but this should not deter the studentfrom trying more venturesome harmonizations. Strettos at any of the follow-ing intervals are, of course, possible and probably more interest ing thanthe one at the 8ve in Ex. 118.

At the 7th below:Ex. 119

F m a ; . 16 V 1 l 1In this case the entry of the subject at a 5th below brings about the sub-dominant key, both harmonically and thematically.At the 4th below:

Ex. 123;J J JJ ~ D ; I ~ '1 r r( I t t T J3 l J I J J J 1 J J ~ fJ I ; i l r r I r _

C""'i " gC mai: IY 1116

42 43

- - -


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This brings about a unison at the last note which could weaken the harmonicand contrapuntal texture at that point.When the stretto is at a larger interval it may prove expedient not to

place it in two adjacent voices. The following shows the same subject in ast retto a t the 9th below in alto and bass, with free parts in soprano and tenor.

Ex. 124l ~ r , " . ~ ,~u r r I r rr I r r ~ ' U ] : ~t rr I rSubjtct etc.Fma;.: YJ6 YllfNext, in the same manner try a stretto at every interval above. Ex. 125shows a stretto at the 5th above, which places it in the dominant key so faras the theme i tself i s concerned. However, the contextual harmony may turnout to be in some other key, in the present instance in D minor. This does notmean that the harmony will remain in D minor throughout the stretto.

Ex. 125

1

1L 1 Sub;~cr 1 1 " " 1 I etc.._, I S.b;.." I I . . . J I I r r I r ILJ r

J----- r - - . . JD min.; J \,6

After all strettos at this entrance point have been explored, the next'step is to move back a half measure, or a full measure as desired and repeatthe process. The following examples show a few of the stret to possibil iti esa half measure ear lier .At the 4th above:f ~ : ~ f u r r l rr I r ~ ' " r l i ' r J I ~M T n J

F ml]. Y/ Y61V6 IJJ6

The stretto shown above can invert in Double Counterpoint at the 9th.In this form it could be placed in alto and tenor thus:Ex. 12 7

1" J r"1If) IsI Ib;r I ~.Jr I I r r I r I L . . J I I IStreno in Ex. 116"t If"td trw DoJbte Cpl. OJ 1N.(j,1IbJ .J n I I lJz.d_ I .i .h,1 I d bn I:

I I

44

The above strettos are shown with a possible contextual harmony in thesection where the themes overlap. Whether a st retto is a rtisti cal ly practi caldepends in large measure upon the harmony it produces. In many cases thisharmony may not fit with that of the rest of the fugue, and when such asituation comes about the stretto in question may have to be rejected,although it is entirely poss ible theoret ical ly .

At this same point a Re trograde st retto i s possible .

t ~ ~ ~ i ; 8 u r r' r r I r ' ; " ; : f : " ; ; j I d J ,J I n jJ JDmln. V I IV VThe above retrograde stretto can be inverted in Double Counterpoint

at the 8ve, 9th and 11 th, with different harmonies being produced at eachDouble Counterpoint inversion.

Ex. 12 9f J ( a l l J 8 ; J " u r J " J a t r c : ~ ~ : I ' : ~l~(bll f 8 7 r i J ' P J n ~a tr cI~:!~(c) Double Counterpoint at the 11 th

Moving back another half measure a Stretto in Cont rary Mot ion comesinto play. This stretto is invertible in Double Counterpoint at the 6th. Boththe original stretto and its inversion are shown below. In the inversion atDouble Counterpoint at the 6th the flat is inserted to avoid the crude out-l ining of the Augmented 4th melodically. It goes without saying that in anystret to formation acc identals may be added free ly in order to

(1) make the harmony more interes ting,(2) avoid awkward contrapuntal or harmonic situations, or(3) carry out a pre-determined modulation plan.

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Ex. 130 rhythm,3 measures long (i. e., ending on the 4th measure), to yielda stretto at the 8ve above after 1 measure.

Step 2 Set up the canon problem so that the first and last notes will conformto the terminal specifications stated in Step 1.

Ex. 132l ~T ) j j T F ~ t ! f i :; rr I r r r u i rC mai V f }1 I6J IV 11/6 II 16 11/6

(b) Double Counterpoint at the 6th

Step 3 Comple te the canon. The fol lowing i llu st ration shows three solu-tions to the above problem.C mai. v6 I l/ I Vl/611yf VII 7 IBeg inning at the first ful l measure , two excel lent st ret tos of the answer

against the subject are available: at the first part of the measure the answercan enter below the subject, and at the third quarter in the measure it canenter above the subject, thus:

Ex. 131(a) Subi~cl

~ J 1 2 ~ 3 H P F r fJ I n P l h t rEx. 133

(a) Streno III rhe 8 " , , . . ooove

(b) Streno al lhe 8 v, above

The strettos in Ex. 118-131 are given without any consideration ofqual ity or musical e ffect iveness. They are presented purely on a technica lbasis to provide a workable method by which a student can assess quicklyjust wha t the st retto possibil it ies of a given subjec t are . Once al l of the avai l-able stret to combinat ions of a subject a re recognized, it is up to the composerto exploit only those that meet his artistic needs.

(c)

Composing a Subject with Pre-es tablished Stret to RequirementsAll of the preceding str ettos have been developed out of a given subject .The stretto problem becomes somewhat different and under a different

kind of control when a composer constructs a subject to yield specific pre-planned stretto formations. Since every stretto is a canon, the reader isrefe rred to THE TECHNIQUE OF CANON by Hugo Norden (Branden Press,Boston), wherein every kind of canon is discussed together with the exactprocess for composing it.

To compose a subject to meet specific stretto requirements proceed asfollows:Step 1 Formulate the problem. For example:

Compose a subject of type Real (b) in the key of D major, 2/4

When two different strettos at the same time distance are required, theproblem will entail a canon in the Double Counterpoint resulting from thesum of the intervals of the two strettos. For example,

Compose a subject of Tonal Group II(a) in the key of F major, 4measures long (ending on the fifth measure) in 3/4 rhythm to yieldstrettos at the 5th above and at the 6th below after 2 measures.

Step 2 would call for a canon problem in Double Counterpoint atthe 10th as shown below.

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The strettos specified in the problem and worked out by means ofwhatever canon st ructure is appropriate wi ll usual ly not be the only stret tosthat such a subject will yield. If a subject composed in this way is thenexamined as shown in Ex. 118-131, it will undoubtedly be found that agreat many other stre ttos of the same subjec t are avail able.

Other kinds of stret to involving cont rary motion, retrograde, augmen-tation, and diminution can be pre-planned in the same way by canons incontra ry motion, retrog rade, e tc. Any composer wi th an adequa te techniquein canon writ ing should not experience any diffi cul ty in composing subjectsthat will stretto in any way he requires .

_A

It) / too.At) . . . . . . . . . . . . . .

~. .

':- - - -Sob/


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CHAPTER VThe Codetta and Episode

The terms codetta and episode both refer to connecting passages thatare generally composed of free material that may be either similar to ordifferent from motives found in the subject or countersubject. The codetta isused to lead to a new entrance of the subject or answer in the exposition,while the episode is designed to link entries of the subject or answer after theexposition is completed.The first codetta appears at the end of the initial statement of the subjectwhen there isa time lag between the end of the subject and the beginning ofthe answer. In such a case it is usually the function of the first codetta toform an unbroken line from the end of the subject to the beginning of thecountersubject which is played against the answer when it comes in as thesecond entry of the fugue. The three quotations that fol low show how thiscomes about.Ex. 138(a) WTC I..Fugue No.7 .. in E-flat majorSlIblrcr Lo*ttG LOIIflterSllbj1f , \! !nJJP~n,;J j la J j j : J r n ' i i - =(b) WTC II, Fugue No. 23, in B major

AUW'I

~ ____l_ . _._._ l '- __~_~ a _Subjrcl

( c) Art of the Fugue, Contrapunctus 8SubitCl Codt ' r lG CouItlD'nb}.

,J ~ r 2 J , ~ . J } 1 J J t ? J I J l ~ - - , - ~ ' I f ; l ~ - i - r ~'heA UW I dC.When the end ofthe subject and the beginning of the answer overlap sothat no time lag exists, as in Ex. 139, there is obviously no place for acodetta at this point in the fugue.Ex. 139 WTC II, Fugue No.2, in C minor

f i b " jW U at I r q } . ~~~ I ;Subject I__ .__ ~ . J

CQU I t lsl4b;tc-rThe codetta between the second and third entries is usually longer andmore substantive than the one between the first and second entries. InEx. 140 no codetta occurs between the first and second entries while a two-measure codetta is employed to link the answer to the second appearanceofthe subject. The lower part in the codetta reappears later in the fugue andbecomes a unifying element in the form of the composition as a whole.51

Ex. 140 WTC I, Fugue No.2, in C minor Ex. 141 (continued)


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. . . . _ - _ . __ . - _ . _ _ . _ . . _ . _ _ . _ _ . iL . . . . . . _Couruersutnect (ouerea)

Counur.1llbl'The episode differs from the codetta chiefly in its location in the fugue,

which is after the exposition; that is, after all the voices have come in withthe subject and answer for the first time. After the exposition in an intelli-gent ly planned fugue, the entri es of the subject and answer, whe ther singlyor in stretto, will occur according to a prearranged plan. For instance, inBach's Fugue in C minor, WTC I, No.2, five entries occur at the seconde ighth -note in odd-numbered measures. This plan brings about a subject inCminor in the bass at measure 7 and a subject in the relative major key ofE-fl at at measure 11, thereby leaving measures 9 and 10 open for an episode.

Ex. 142 WTC I, Fugue No.2 in C minor

A four-voice fugue has a place for a codetta between the third andfourth entries. Since there have been three entries before such a cod e tta, itwi ll have to be in three-pa rt counte rpoint , and may be of considerable com-plex ity textural ly. Ex. 141 shows such a codett a, and it is inte resting to notethat there is no codetta before this one as no time lag was built in betweenthe first and second and between the second and thi rd entri es. I n the presentinstance, the codetta develops imitatively thematic fragments from thecountersubject.

Ex. 141 WTC I, Fugue No. 12 in F minor

COlfll~.mbj~cl J~7 ~-~--~-~~---~~-~---~---------------------9~

"[tJ Subj:!


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Ex. 143 The episode proceeds as follows:Ex. 145 Couruersutnea 1~tc.

r r f etc.etc

I . _ _ _ _ _ End 0/ COIJntu3ubjt'ct I~ End of Counrersubiect 1/The third episode serves quite a different purpose than the first two,in that it leads to the sixth entry which isthe first time that the subject entersinan even-numbered measure. This entry of the subject at the second eighth-note of measure 20 is extremely important in that it defines the proportionscheme of 19:12within the 3l-measure t ime span of the fugue as a whole.The problem can be seen in the following form, with the three measuresbetween the end of the answer in the fifth entry and the beginning of thesubject in the sixth entry.Ex. 1 4 " " " ,m u b ; , 1._.-.-._-_._._._._-------_._._._._._.-._.,A well-constructed episode can be a miniature composition in itself, ademonstration of the "wheels within wheels" principle. A careful examinationof the above example willshow that structurally it consists of a canon at the5th in the two upper parts accompanied by a running bass l ine. Sti ll closerexamination reveals that the canon is made up of a sequential treatmentof the head of the subject while the running bass part is correspondinglysequential and is derived from the beginning of countersubject I. The com-

ponent elements of the episode are identified in the following diagram.Ex. l~~ntJ1,~j'1J 0"Mod oj fht' svbjr9 ,.----.--- _- ------- ---- - - .. - -------;--------------------------------------------.-~ --,

16 17

L _ _ __ _ __ ._ _. JCOl lut'nIbjecl JJ Subjl19

This episode is built around the opening motive of the subject inc~3mbi-nation with the theme in the codetta shown in Ex. 140. This codetta themeappears only in the episodic sections. Compare this episode with the codettainEx. 140.Ex. 14 c O W l, < = l! . ! ! : : ! ! _ . _ . . _ . _ . _ . ._ . _ . ~ F~~==~~==

"-.~.-.- - _._---_.-CQUJ1 l e r ndJi r IThe next scheduled entry is an answer that comes in at the secondeighth-note in measure 15 in the alto part accompanied by countersubject Iin the uppermost part. Thus it is necessary to insert an episode from thebeginning of measure 13, the last note inEx, 143 to the beginning of measure15. This episode calls for a modulation from E-flat major at the beginningof measure 13 to C minor at the beginning of measure 15. It embodies thefollowing features:

1. These are the onlytwo measures so far that have no rests.2. The running passage derived from the opening motive of counter-subject Iis placed in the uppermost part with the direction of theruns reversed, ascending instead of descending.3. The two lower voices move in parallel 3rds that are based on thesecond motive of countersubject I.No fragments of the subject itselfappear in this episode.4. This episode ends on the highest note in the entire fugue, the high 'C',which appears nowhere else in the composition. L..

C. 5.1154 55

The rhythmic difference between the upper voice of the canon in the


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second half of measure 23 and the lower voice in the first half of measure 24serves a highly practical purpose. By means of this rhythmic alterationmeasure 24 has no rests, as is the case in measures 13 and 14. These threemeasures are the only ones in the entire fugue that have no rests.Measures 13 and 24 mark the beginning and end of a 12-measure sectionthat is of formal significance within the 31-measure form, which i s dividedin at least three ways into the 12: 19 proportion. The operation of thiscompound proportion scheme is as fol lows:

1. 12 measures before measure 13 where there are no rests, and 19measures from there on.

2. 19 measures before the first entry in an even-numbered measure inmeasure 20, and 12 measures from there on.

3. 12 measures within the portion that begins and ends with measureswithout rests (measures 13-24), and 19 measures in the rest ofthe fugue.

Only one more short episode remains to be examined; namely, fromthe middle of measure 28 to the middle of measure 29, where the last entryof the subject and the tonic pedal-point in the bass begin.

Ex. 150

~ \ ~ ~ . !_ ~ _ ~ ~ _ ! . _ _ : _r- .r r~_ . _V ' 1---C. S.1

~'~-\27counurmtJ;t JJ.. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . . . . .28 _

-ubjulThis episode is constructed in much the same way that the first episodeis written (d. Ex. 144), with a canon at the 5th in the two upper partsabove the running passage in the bass. Thus, the first and fourth episodesby being almost identical structurally serve to unify the composition as awhole. The episode is diagrammed in detail in Ex. 149, but compare itcarefully with episode 1 as shown in Ex. 144.

Ex. 14922 r _c ; ~ ~! '! _ ~ _~ ~ _ ~ _~ ~ _ '! [_ ~ _~ ~ l! :. ~ - - - - -- - -- - - - -- - - - - - - - -- - - _23 ..i4. . . . .. . . . . . .. . ,

. .Sub/~CI

In the fugue presently under examination the following componentsare seen in Ex. 140-150:

8 entries-5 subjects in C minor, 1 in E-flat major, and 2 answers,1 codetta,5 episodes.

When these are "assembled" to utilize the 31-measure form plan, the com-plete fugue appears as in Ex. 151.

Sllbj~ct56 57

Ex. 151 J. S. Bach Ex. 151 (continued)


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FUGA II.

\ ~ : ! 3 , : yIF.~r::s:J~The episodes examined in Ex. 142-150 determine the form plan of a

three-part fugue, WTC I, No.2, in C minor. The thematic material thereinconsi sts o f a subject and two countersubjects tha t operate in t riple counte r-point. However, countersubject IIs treated so freely that at times it canalmost be considered as a free part.

In a four-part fugue the problem is not significantly different. Such afugue is the Fugue in G minor, WTC I, No. 16.

The subject is Tonal Group I(b), beginning on the dominant note andending on the mediant note. Thus, the head of the subject and that of theanswer are not alike at the opening interval, so that the effect of theent rie s of the subject and those of the answer are quite diffe rent .

The countersubject is derived from the closing motive of the subjectin cont rary mot ion, the reby giving uni ty and continuity o f the rhythmic andmelodic flow of the en tire composition. The three themes-c-subject , answerand countersubject- -can be isolated as follows.

Ex. 152 Sub;t'ct~~IHn J # d J 15; n J

jCountersubsect

13;]; I J J J ~J j58 59

The complete fugue appears as in Ex. 153. - Ex. 153 (continued)


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Ex. 153 J. S. BachFUGA XVI.

60 61

The complete fugue appears as in Ex. 153. Ex. 153 (continued)


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Ex. 153 1. S. Bach

60 61

The form plan of Fugue No. 16 in WTC I as determined by the entries Except for the first and the penultimate measures, there is not a measure


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of the subject, answer and the episodes can be tabulated as follows:measure 1, 2nd Sth-note, Subject in altomeasure 2, 6th 8th-note, Answer in sopranoCountersubject in al to

CODETTA in soprano and altoSubject in bassCountersubject in sopranoAnswer in tenorCountersubjec t in bassEPISODE 1Subject in B-flat major in al toCountersubject in tenorAnswer in B-flat major in bassCountersubject in tenorSubject in F major in sopranoCountersubject in bassSubject/ in B-fla t major in bass }Subject in F major in alto Stretto 1Countersubject in sopranoEPISODE 2Subject in C minor in bassCounte rsubject in a ltoSubject in C minor in sopranoCountersubject in bassAnswer in C minor in altoCountersubject in sopranoEPISODE 3

measure 4,measure 5, 2nd 8th-note,measure 6, 6th Sth-note,measures 8 - 11measure 12, 2nd Sth-note,measure 13, 6th 8th-note,measure 15, 2nd Sth-note,measure 17, 2nd Sth-note,6th Sth-note,measure 19,measure 20, 2nd Sth-note,measure 21, 6th 8th-note,measure 23, 2nd Sth-note,measures 24 - 28measure 28, 2nd Sth-note,

6th 8th-note,measure 29, 2nd 8th-note,

in the entire fugue that does not include the thematic material of the endof the subject and the beginning of the countersubject. The episodes,however, are distinguished by new and unrelated thematic material, butthis is played against tha t of the subject and counte rsubject.

This economy of material and texture is very important in achievingunity and continui ty and merit s care ful study.No pedal -points are included. But , if the bass 'D' were held throughoutmeasure 28 and into measure 29 it would make a dominant pedal; so thiscould be considered a pedal-point by implication.

The fugue with all of its component parts are diagrammed in Ex. 154.Ex. 154 FUGA XVI. 1. S. Bach

Subject in G minor in sopranoCountersubject in altoSubject in G minor in tenorSubject in G minor in bass(head only)EPISODE 4

} Stretto 2

measures 30 - 31measure 31, 6th Sth-note, Subject in G minor in altoCountersubject in bassmeasure 33, 2nd Sth-note, Subject in G minor in tenorFrom the above tabulation it will be seen that the fugue embraces 17

entries of the subject or answer (half as many as there are measures in thecomposition), two strettos, one codetta, and four episodes. Slightly overone third of the time span of the fugue is given over to episodic material.

There is not one full measure without a rest. And only a very smallport ion of the composition conta ins four-pa rt harmony. In most instancesstudents tend to overharmonize with the result that they find themselvesst ruggl ing with contrapuntal textures tha t a re far too thick and excessive lycumbersome. 62

Ex. 154 (continued) Ex. 154 (concluded)


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It may be helpful to crystallize a process for the writing of an episode.As a practical example, let us suppose that a degree candidate is writing anexamination fugue on the following subject and countersubject. The problemis a relatively simple one as the subject is a Real (b) in that it begins on thetonic note and ends on the mediant . The countersubject has been constructedin Double Counterpoint at the 8ve so that no inversion problems willcome about.

Ex. 155, r & D e @ jsJ'ii itlfC()14nr~rsubi(c'

As the fugue has been plotted out, there is a section in which an entryof the subject appears in the uppermost voice, the countersubject in thebass and a free part in the middle. After a three-measure episode the plancalls for a subject in the bass, the countersubject in the uppermost voiceand a free part in the middle. The problem is set up thus:Ex. 156S.&jectu. C nwJjo,.1J. _.,. . . . . ~ J-..J 1 ~ .. I II 1 ~pix>


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opening motive of the countersubject. This choice ispurely arbitrary. Itmighthave- been a series of harmonies to be activated contrapuntally or a line toserve as cantus firmus or any other type of imi tat ive or even a non-imita tivedevice. However, the choice has been made for the canon in contrarymotion and it i s insert ed as fol lows.

Ex. 157

The Pedal-point and CadenzaAcademic exercises and examinations in fugue generally require twopedal-points, one on the dominant note to be followed by another on the

tonic note. The main purpose of these two pedal-points is to reestablish thetonality of the composition in the event that it has become clouded by thenumerous modulations and digressions along the way within the middleentri es of the subject and answer and in the connecting episodes. However, i tis safe to say that most of the fugues in the l ite ra ture do not contain extendedpedal-points. Thus, the pedal-point, as a fugal device, can be thought of asla rge ly, though certainly not entire ly, an academic fea ture.Two unusually interes ting pedal-points by Bach occur in WTC II,FugueNo.1, in C major. These are quoted in full in Ex. 159 and 160. The measuresare numbered so that it can be readily seen where in the 83-measure fuguethese occur. The first one consists of a tonic pedal-point in the uppermostvoice that dove tai ls with a dominant pedal -point in the bass. Against thesetwo inter locking pedal-points the subject and answer are played respectively.The free running passages in 16th-notes a re derived from the second ha lf ofthe subject. As noted by the measure numbers, this pedal-point formatbegins at measure 47, five measures aft er the middle of the 83-measure form.It is ext remely rare to find a pedal -point in the first half of a fugue.

Ex. 159

1.- __1COWlt~r$Ubj'

The final step in the composing of an epi sode i s to connec t it art isticallyat both ends so that there are no "seams" or "rough edges" to disrupt thesmooth flow of the composition as a whole. This part of the process maywell be the most difficult and demanding artistically. Example 158 repre-sents an attempt to complete the episode developed in Ex. 156-157. Theconnecting material that ties in the canon in contrary motion and its accom-panying free bass part to the entries that precede and follow the episode isindicated by brackets. If such joinings are so smooth and so elegant that thel istener remains unaware of them, the composer is to be complimented.

Ex. 158

V pedal-pointAn.f)I.Ierl~=:++I~"C

___________ . JThe second of the two pedal-points in Fugue No. 1 of WTC IIsconsiderably longer and employs only the tonic pedal-point, and begins atmeasure 68, well within the last quarter of the 83-measure form. Unlikethe earlier pedal-point shown above, this one employs only the head of thesubjec t which i smade to operate in double counterpoint against a 16th-notepassage derived from the second half of the subject. These two themes areidentifi ed by a single line and a double l ine respect ively. It will be notedthat when these two themes appear in measures 72-75, they are identicalto the presentation in measures 68-71 except that they have been invertedin Double Counterpoint at the 8ve, a ll of which takes place against the tonicpedal-point which appears first between the two themes and afterwardsabove them. When the pedal-point is taken into consideration contrapuntally,thi s passage can be thought of as being in t riple counterpoint.

~_. ._. . __._. _J FrlttpGl1COUf-u3JIb~

66 67

Ex. 160 so completely different from each other within so short a time span, andyet so unified in musical effec t when performed.


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Ex. 162 Bach: Organ Fugue in D minor9J ~nro"J ~ 94 ~

.~d


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~~\("fjJ1) O J I Jcf. Ex. 161

The ExpositionThe exposition of a fugue is quite simply the opening section of the

composit ion wherein each voice is broungh t in singly, a lte rnat ing be tweensubject and answer.The exposition of Fugue No. 18 in WTC I, in G-sharp minor, is given

in Ex. 74. The fugue is in four parts; it has a two-measure subject; theentries are not separated by codettas, so the entire exposition is only 8measures long. The entrance scheme is as follows:

(b) Organ Fugue in D minorf~(Dman I U s!cf . Ex. 162

Soprano:Alto:Tenor: SubjectBass:

AnswerCountersubject

SubjectCountersubjectFree part

Counter subjectFree partFree partAnswer

The final entry of the exposition is reserved for the bass. The free parts areshown in small notes to contras t with the subject, answer and countersubject.

The exposition of a three-part fugue, No.2 in WTC I, is given in Ex. 140,this having a two-measure codetta between the second and third entries.The entrance scheme is carried out thus:

(c) Violin Fugue in G minor~~h!O F F F E J t tr I pcf. Ex. 163

Soprano: Answer Countersubject ICodetta

Subject Countersubject Ilto:Bass:

Countersubject IISubject

In this instance, Countersubject IIreplaces the free part thereby makingunnecessa ry the invention of any addi tional themat ic mate ria l.

A four-voice exposition with a codetta between the third and fourthentries is given in Ex. 141, the fugue under examination being No. 12, inF minor, in WTC 1 . The entrance plan follows:Soprano:Alto:Tenor: SubjectBass:

SubjectAnswer Counter subject Free partCountersubject Free part Codetta Free part

Subject CountersubjectThis exposition differs from that in Ex. 74 in three ways:( 1) the order of entries is so planned that the soprano takes the

final entry,(2) the final entry is a subject instead of the usual answer, so that the

exposition consists of three subjects and one answer instead ofthe usual two subjects and two answers, and(3) the third and fourth entries are separated by a three-measure

codet ta. In Ex. 74 the re i sno code tta .The entrance plan of the fugue examined in Ex. 154 requires specialconsideration. This is WTC I, Fugue No. 16 in G minor. Itwill be noted tha t

70 71

the final entry of the exposition, beginning in measure 6, comes in the tenor.General ly in academic exerci ses the student is d iscouraged f rom plac ing an

- Ex. 166 (Continued)


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exposition entry between two voices that are already in motion lest theentrance lose it s impac t by being covered over by the surrounding counter-point. However, it can be pointed out that at the time of the fourth entryin the tenor, the soprano and alto are resting so that there actually is nosurrounding counterpoint to conceal the answer. In fact, the texture is sothinned out that at the beginning of measure 7 only the answer is beingheard. The complete entrance plan follows:Soprano: Answer .......................ountersubject Free part

CodettaAlto: Subject Countersubject .......................ree part RestTenor: AnswerBass: Subject Countersubject

In a three -voice fugue there a re six possible orders of ent ry, two of whichpresent the added problem of bringing in the final entry between twovoices already in progress. A four-voice fugue makes avai lab le 24 ordersin which the exposition entrie s can be brought in , but 16 of these necessita tethe bringing in of entries between voices already in progress.

For the student about to compose a fugue or to write an examinationa systematic order of steps is available. To illustrate the process, subjectNo.3 of those given at the close of Chapter IIwill be used.Step 1 Write the correct answer and compose a sui table counte rsubjec t.Ex. 165

Subiect In~--------------- --'If~!:~,'~!t~~l~bl~fl~~,'~ ~ I4 i i ~ Inlln l f l J b 3 l;gQ rfj Ir

oun~ J~b,t'CI L ._. ._._._. ._~_. ._L ._Step 2 Decide on an entrance arrangement, and write in the subject,answer and countersubject where desired. (The exposition to bedeveloped is for a three-voice fugue.)166x Ar-" ~ ~ 1 2 J 4 ! .,! )l

lo J ~ LJ-I ~ r . : . . . . . _ _ l - - C , . . J . . . J -...!Subiect

9

L_~. ._._._.~. . .. __JCowuersubiecs

72

Coumersubiect _ _ _ --,~ o "'}-'-'-i" .~'~I _ 12 = 1 l !l J . .

-~c[The entrance scheme decided upon in the above distribution of subject,answer and countersubject i s the following:Soprano: Answer Countersubject

CodettaCountersubject... Free part

SubjectAlto:Bass:

Subject

Step 3 Complete the exposi tion by supplying the necessa ry code ttafree part, as demonstrated in Ex. 167.

Ex. 167and

A ., 1 2 3 4 } . , } j)I~ ~ l.J-.I l..l...Jr'-- r C - J - J ~ . . . . . . . . . . . . . . . -_ _______-~---------C


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A special situation comes about in the setting up of an exposition whenthe fugue is either for voices or for instruments of a limited range. Forexample , a composer about to writ e a fugue on the fo llowing subject

Ex. 168

""I~ C i " " ' " ( A J ~ r, 1 FreePQTfS~i'cl(T:;1 cJ~ COUJIIUnIb JKIFree pgrt Fru pgrI

I* 1 1 2 J I J j J I j. J s l l ; o t I J I H~ se-ndr:th Ihe springs in-to 1M ",Jl - /.q.r,

for mixed chorus could proceed as follows:Step 1 Writ~ the answer and compose a suitable countersubject, being

certain that the double counterpoint is satisfactory as this is mostimportant in choral composit ion.

Step 2 Tabulate the ranges of the subject, answer, and countersubject toboth the subject and answer in both the treble and bass clefs inorder to asce rtain which themes are suit able for the various voices.This is done in Ex. 169 under Ranges.

Ex. 169

A rough and ready rule for writing vocal fugues is to place the counter-subjec t to the answer in approximately the same range as the subjec t, whichwill automatically put the countersubject to the subject in about the samerange as the answer. In this way the vocal continuity of each voice partwill be assured.

Rang e s. . . . . .I~ H< $urd~lh th~ sprngs in to tilt ",u- Ery s ,. .. SubjectI~ . . . . H< srM~rh /lu springs ; "-10 t I lL - Yal- '.rys.

Coul1tasubjtC't 1 . . . - - 11 " - ~ . - . . . n__.-....~ 1 "-:H, smdnh rile springs in / 0 t he \'a/- leys, r

Counursubirci. . . .r e : r '-" . . . . . . . -~ . . .H< undcll! the . sp ri ng s; 11 - to duo ~'a1- l~'s.. " A1TWC I 1 _ _ _ 1loJ ~ . . . . . . . . To ~'-,fl'" o . . r - - - - - r IH, s enaem rht sprrws in-fo the ,al- leys.Couruersubiect ~:

.stl1dnh Inc .rpril1gs in to rht val- It)'_L

From the above tabulation it will be noted that the countersubject to thesubject does not fall within a convenient range for the bass. Thus, an expo-sition should be devised in which this countersubject is never assigned tothe bass. An entr ance scheme for an exposit ion that avoids such an awkwardvocal situation fo r the bass is suggested below.

74 7S

CHAPTER VIII


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HarmonyFugal harmony must be considered on three levels:

1. The key scheme of the composition as a whole.2. The underlying chordal structure.3. The vertical relationship between the several voices.

The key scheme of the fugue as a whole is generally defined by a fewdefinitive cadences. For instance, in WTC I, Fugue No.2 in C minor, asgiven in full in Ex. 151, the following overall plan comes into focus:

Beginning in C minor.measure 11: a powerful cadence in E-flat major followed by a statementof the subject accompanied by the two countersubjects inE-flat major.measure 17: an equally powerful cadence in G minor followed by anepisode.measu re 23: a cadence in B-flat major preceded by a less conspicuouscadence in C minor.measure 27: a deceptive cadence in C minor, bringing in for the firsttime the A-flat chord at the beginning of a measure.measure 29: final cadence in C minor.

Thus, the broad outl ine of the fugue as a whole could be seen as:- C minor-E-flat major-G minor-B-flat major-C minor- (A-flat major asVI in C minor)-C minor.Some such comprehensive plan is necessary to give the entire compo-sition unity and a sense of direction. Otherwise, there is the danger of afugue "losing its way" through a number of unplanned middle entries andepisodes that come and go without purpose or destination.The underlying chordal structure of a florid fugal passage is closelyakin to chorale harmonization as Bach practised this great art. Herein liesan unlimited field for analytical study in the music of Bach and all othercomposers. One brief example will suff ice to i llustrate how to proceed.Picked entirely at random is a short excerpt from Contrapunctus 7 of theArt of the Fugue. The passage chosen begins at measure 20 and extendsinto the beginning of measure 23.Ex. 171 Art of the Fugue, Contrapunctus 7

7 7

In order to see the underlying chordal structure it is necessary to removeall embellishing materials. These are as fol lows:It would be difficult to say whether or not the above passage is typical

of the chordal structure behind florid fugal harmony. Bach's technique in


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passing-notes,auxiliary-notes,suspensions,appoggiaturas,echappees,rhythmic embel lishments, such as rests, note repeti tions, etc.,other notes of the harmony that do not play a part in the progressionprocess.

this respect is as varied as it is in the chorale harmonizations. But, it can bestated as a principle that the more static the chordal progressions may be,the more is the opportunity for the embelli shing discords to func tion freelyand effectively.The verti cal relationship of the several voices can best be seen throughthe so-called "Two-Voice Principle":

Every mult i-voiced s tructure consists of a triangular number of two-partcombinations. If all of these constituent two-part combinations arecorrec t, i t follows that the structure as a whole is correct . In a four-voicetexture the two-part combinat ions are as follows:

Fmajor I VI! 116 VI 6 {{7 1/ IVO 1__Bb""'jor V---lIIf V/6

1. Bass -Tenor2. Bass- Alto3 . Bass - Soprano4. Tenor-Alto5 . Tenor - Soprano6. Alto - SopranoIII VI 16 V __

Since this underlying harmony contains no E-natural or E-flat it can beconsidered as being eithe r in the key of F major or in the key of Bvfla t major.The passage is absolute ly neutra l insofa r as the tonali ty is concerned. Thus,the feeling of a modulation from B-flat major to F major occurs only inthe pass ing-notes and auxil iary-notes , and not in the harmonies as such. Incont rapunta l harmony this i s an extremely valuable technique , that is usedextensive ly in the music of Bach. However, it should not be assumed tha t thechorda l background of fugal harmony must be so neut ral tonally. Extensiveresearch will put this aspect of what has proved to be successful harmonyin perspective.

Simplifying the harmony even further one discovers that there is acontrapuntal background behind the chordal structures. The followingdiagram shows how the chordal structure is the effect of two pairs of linesmoving in parallel lOths operating s imultaneous ly. These can be identif iedby the dotted lines in contrast to the dot-and-dash lines. In the secondmeasure the "F" in the soprano that becomes the 7th of the chord in thelast quarter of the measure is not resolved in the harmony itself, but by the"E-flat" passing-note that has been dele ted (d. Ex. 171).

Ex. 173

A rel iable, but probably a bit too conservative, method for determining thecorrectness of a passage can be formulated from the fol lowing table .

~ : ; ~ : I , ~ - : t ~ d F f S.__ ~ J

78 79

The two-part combinations:Must not conta in a 4th

Itwas mentioned previously that every multi-voiced s tructure embodiesa "tri angula r number" of two-part combinations. These a re as follows: _


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............................................................................. , .

2. Bass-Alto

3. Bass-Soprano

4. Tenor-Alto*)

except1. as a correctlytreated dissonance

2. in a correctlytreated six-fourformation

3. in a correctlyresolved inversionof a 7th or 9thchord

Must not contain5. Tenor-Soprano a 9-8 or 2-1 **)suspension

6. Alto=-Soprano)

2-part structure3-part structure4-part structure5-part structure6-part structure7-part structure8-part structure

1two-part combination3 two-part combinations6 two-part combinations10 two-part combinations15 two-part combinations21 two-part combinations28 two-part combinations

Must not contain1. The doubling ofany note tha t must beresolved in only Onedirection, such as(a) the leading-tone,(b) a suspension,( c) the 7th or 9th

of a chord,(d) chromatically

treated notesthat requireresolution suchas those in anAugmented 6thChord.

2. Consecutive Sths,8ves and unisons.

These numbers, that is, the two-part combinations within a multi-voicedstructure, can be seen in the following triangle. The first row is, of course,1; the sum of the first two rows, 3; the sum of the first three rows, 6;and so on.

3. A 7-8 suspensioneffect.

*)Should not be separated by an interval greater than an 8ve.* *) The 2-1 suspension is to be found with some frequency betweenTenor and Bass.

As suggested previously, the above table may be considered by some asbeing ~oo conservat ive for a sophisti cated fuga l t echnique but i t providestwo thmgs:

1. a norm of correctness, especially for the preparation of work to besubmi tted for academic evaluation, and2. a norm of correctness by which one can put in perspective the liber-tie s tha t a ll venturesome composers take .

80 81

Norden Hugo_Foundation Studies in Fugue

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