Johann Sebastian Bach’s The Art of theFugue forces us to become aware of

the ontological character of the relation-ship, in musical composition, betweenthe principle underlying generation ofthe Lydian mode, and broader applica-tions of the principle of inversion. Tomost readily appreciate this, it is impor-tant to grasp the term “principle” inrespect to LaRouche’s conception of rev-olutionary axiomatic progress, wherebythe development of man’s knowledge ofdiscovered and realized Classical-artisticprinciples advances, anti-entropically, asexpressed by the function (m+1)/m.

Usually, musicians only considerinversion as a “technique” of counter-point, or as an “element” of composition,and not as bearing upon principles of dis-covery. Thus, the import of Bach’s workin The Art of the Fugue has until nowbeen appreciated only by a few greatcomposers. While there are certain diffi-

culties that need to be overcome toknow this composition, it is nonethelessa transparent composition, which excel-lently illustrates LaRouche’s discussionof the generation of new, validmetaphorical principles.

The progress of hypotheses in thecomposition occurs, in first approxima-tion, as one moves from one fugue to thenext in the series, and from one set offugues to the next. The current discus-sion focusses on the discovery unveiledin Fugue IV, relative to Fugue I, withsome reference to Fugue III.

Preliminarily, it is possible to sum-marize that discovery as follows: Bachdemonstrates, in the “unfinished busi-ness” left over from Fugue I and real-ized in Fugue IV, the generative signif-icance for all keys, of the Fˇ major/minormode, which is derived from the registershift of the soprano voice. The Fˇ major-minor modality is demonstrated as an

extension of the simple Lydian modality.In other sections of this report, we showthat the simple Lydian modality, cen-tered on F˝, arises from inverting the Cmajor scale. In Fugue IV of The Art ofthe Fugue, Bach demonstrates that thereis a higher principle involved, in thedeceptively simple effort to shift the F˝Lydian modality to the locus of F ,̌ thesoprano register shift.

As W.A. Mozart clearly grasped(although he reportedly never saw TheArt of the Fugue manuscript itself),Bach’s conception of inversion, exempli-fied in this extension of the Lydian prin-ciple, allowed for a much greater densityof lawful change. Bach’s use of inversionacross voices, incorporating the signifi-cance of registral transformation andinversion as a unified, single type ofprinciple embedded within the well-tempered system, had a far-reachingimpact upon Mozart’s own ideas.

49

The Scientific Discoveries of Bach’s3 The Art of the Fugueby Renée Sigerson

extremely important, owing to the wide-spread, but totally fallacious claim, thatClassical music evolved “naturally”toward the atonal cacophony of so-called modern music. In fact, far frombeing a step toward arbitrary chromati-cism, the C-Fˇ-based Lydian mode, asunderstood by Mozart and Beethoven,achieves an enormous increase in the“Cantorian” ordering-power of tonalcomposition. Thereby it became possibleto eliminate any remnants of arbitrarychromaticism that might otherwise behiding between the toes of the earliermajor-minor system.__________

1. What is commonly referred to as“melody,” including so-called solo melody,is nothing but a derived feature of vocalpolyphony. Strictly speaking, monophonic

melody does not exist. What we call themelody of a solo voice, for example, is noth-ing but that voice’s singing of an intrinsical-ly polyphonic composition. A relevantreflection of J.S. Bach’s views on the poly-phonic principles of so-called melodic (orbetter, motivic) development, is contained inthe first biography of Bach, written by Nico-laus Forkel [“On Johann Sebastian Bach’sLife, Genius, and Works,” in The BachReader, ed. by Hans T. David and ArthurMendel (New York: W.W. Norton, 1966)].Otherwise, the cases of Gustav Mahler andRichard Wagner typify the way in which, assoon as composers depart from the rigorousprinciples of well-tempered polyphony, theirmelodies degenerate into nothing but uglygroaning.

2. In Book III of his Harmony of theWorld, Kepler polemicized against theempiricist, mechanical theory of musicalconsonance and dissonance, which had been

put forward by Vincenzo Galileo, the fatherof Galileo Galilei. Vincenzo is regarded asthe pioneer of the reductionist musical theo-ry later associated with Jean Le Rondd’Alembert (1718-1783) and Jean-PhilippeRameau (1683-1764), which became virtuallyhegemonic by the end of the NineteenthCentury, thanks to Hermann Helmholtz(1821-1894).

3. Further exploration of this pointmight usefully focus on the significance ofvibrato in the bel canto singing voice—avibrato which, in strong contrast to theRomantic’s pathetic tremolo, is defined as avariation of pitch within a well-temperedpitch-corridor. Apart from the role of vibra-to in the technique of bel canto singing, onecan demonstrate how passages sung withoutthe vibrato, i.e., at a “mathematically fixed”pitch, are correctly heard as wrong, destroy-ing the fabric of explicit and implied cross-voice relationships.

The introduction of a manifold ofkeys around Fˇ minor occurs in the criti-cal passage beginning measure 72 ofFugue IV, resolving to C major in mea-sures 86-87 (see below). The discoveryand situating of the Fˇ mode, is theproduct of a revolution of axiomaticprinciples, which begins with the para-doxical implications of a discovery inFugue I. Any ensemble of musiciansattempting to play Fugue IV necessarilyexperiences the referenced passage ashaving bearing upon Ludwig vanBeethoven’s late string quartets.

As we present the musical demon-stration of this discovery, it will be use-ful to keep the following excerpts fromLyndon LaRouche’s main essay, “TheSubstance of Morality,” in mind:

“With Plato, one begins with propo-sitions being entertained as prospectivetheorems, and then follows the approachtaken in his dialogues, as a way ofsearching out discoverable fallacies inthose underlying presumptions. . . .The challenging of such prejudices, pro-vides the user of Plato’s method withwhat appears to be, for the moment, a

refined array of mutually non-contra-dictory definitions, axioms, and postu-lates; this refined array, taken as awhole, is an hypothesis. . . .

“The method of Plato starts with therecognition that all . . . hypotheses,including what were previously themost refined ones, must include somesignificant, axiomatic fallacy of somekind. . . .

“Truth, then, does not lie in any onechoice of hypothesis. . . . Truth lies inthe always radically revolutionaryprocess, by means of which valid new

50

Comments on Bach’s Fugues by His Contemporaries ‘He who is not acquainted with

Bach’s fugues cannot evenform an idea of what a true fugue isand ought to be. In fugues of theordinary kind, there is nothing but acertain very insignificant and sloppyroutine [Schlendrian]. They take atheme, give it a companion, transposeboth gradually into the keys relatedto the original one, and make theother parts accompany them in allthese transpositions with a kind ofthorough-bass chords. This is afugue; but of what kind? . . . Bach’sfugue is of quite another kind.”(Johann Nicolaus Forkel, “Biographyof J.S. Bach,” in The Bach Reader, ed.by Hans T. David and ArthurMendel [New York: W.W. Norton,1966], p. 324.)

‘In composition, [Bach] started hispupils right in with what was

practical, and omitted all the dryspecies of counterpoint that are givenin Fux and others. His pupils had tobegin their studies by learning purefour-part thorough-bass. From thishe went to chorales; first he added thebasses to them himself, and they hadto invent the alto and tenor. Then hetaught them to devise the bassesthemselves. He particularly insistedon the writing out of the thorough-bass in parts. In teaching fugues, he

began with two part ones and so on.”(Letter from Carl Philipp EmanuelBach to Forkel, in The Bach Reader,op. cit., p. 279.)

‘The true fugue is two sorts, dis-tinguished according to their

treatment of the fugue subject:“(A) A strict fugue, fuga obligata, is

one in which no other material thanthe subject is treated throughout, i.e.,in which the subject after the exposi-tion . . . makes its appearance in oneentry after another, so to speak, andin which, consequently, all the coun-terpoints and interludes are derivedfrom the principal subject or from thecounterpoint that first appears againstthe answer, by means of division,augmentation, diminution, contrarymotion, etc.; all this however, beingbound together through imitationand a coherent and solid harmony.When such a strict fugue is workedout at length, and all kinds of otherartifices (made possible by the manykinds of imitation, double counter-point, canon, and change of key) areintroduced in it, such a piece is calledby the Italian name of Ricercare orRicercata—an art fugue, a master-fugue. Such is the nature of most ofthe fugues by the late CapellmeisterBach.

“(B) A free fugue, fuga libera, solu-

ta, sciolta, is a fugue in which theprincipal subject is not continuouslytreated; that is, in which it does notmake its appearance in one entryafter another, although often enough,and in which, when the principal sub-ject is abandoned, a brief, well-choseninterlude is worked out by imitationand transposition—which has a simi-larity to the principal subject or to thecounterpoint that first appears againstthe answer, and is related to the same,even though it is not always derivedfrom it. Such is the nature of most ofthe fugues by Handel.” (FriedrichMarpurg, 1753, quoted in The BachReader, op. cit., p. 254.)

[Note: Marpurg was no friend ofBach’s. While his distinction betweenfree and strict fugue is somewhatuseful, he, a typical musicologist,thinks in terms of form, not ideas. Infact, A Musical Offering is of the char-acter he indicates, but the fugues ofThe Art of the Fugue are much moregroundbreaking and complex. Theuseful distinction to be made, isbetween Bach’s type of thinking, andthe sort of fugues Haydn wrote,before 1782. Examine, for example,Haydn’s String Quartet in F minorOp. 20, No. 5: Every entrance is on aLydian interval, but the principleassociated with the Lydian mode isnot even referenced.]

principles are generated, new principleswhich take into account the contradic-tions inhering in the previously pro-posed hypothesis.”

Later in the same essay, LaRouchewrites:

“We have to consider the cases, inwhich a particular colligating set ofprinciples is in error only because itlacks some additional principle.”

Relationship to A Musical Offering

According to all accounts, The Art of theFugue was composed by J.S. Bach in theyear of his death. It consists of 19fugues. It was written less than twoyears after his A Musical Offering. AMusical Offering centers around a six-voice fugue; The Art of the Fugue con-tains fugues with four, three, or twovoices. Many musicians have com-plained about the form in which themanuscript was written, in “openscore,” with each part on a separate line,and its own unique clef. This makes itdifficult to play the composition, at

sight, on keyboard. However, it wasnecessary for Bach to leave the primarymanuscript in that form. It forces themusician working with the compositionto always think of the individual part asassociated with a voice species, such assoprano, alto, tenor, or bass. Usually,which voice is intended, is adequatelyindicated by the clef and line on whichthe voice appears. That, however, is notalways the case, so that sometimes thetop voice may in fact be indicating analto part, for example. The reason isthat in the actual working through ofthe composition, sometimes Bach isworking, implicitly, with the full sixvoices of the Musical Offering in mind.

The unifying thread of developmentacross the 19 fugues, is encompassed inthe seeming irreconcilability of theopening, and the concluding ideas of theentire work. That opening idea is thedeceptively simple, seemingly inconse-quential four-measure phrase that opensFugue I (Figure 3.1).

By Fugue XIX (Figure 3.2), Bach

has arrived at what was undoubtedly hisoriginally intended goal, a fugue con-taining a countersubject spelling out hisown family name.

When thinking of the great distanceone will traverse, from the openingstatement, to that conclusion, there is tobe remembered the awe-inspiringinscription engraved by Bach’s son, CarlPhilipp Emanuel, onto the originalmanuscript plate: “N.B. While workingon this fugue, where the name BACHappears in the countersubject, the com-poser died.” (In German, the letter “H”denotes B˝, and “B” denotes B .̨) Thus,when we think of The Art of the Fugue,we are virtually compelled to rememberthat Bach immersed himself in this com-position, in his final moments, giving tofuture generations the benefit of hisknowledge. In that respect, Bach exem-plifies LaRouche’s view of a world his-torical personality.

Now, we enter upon the demonstra-tion.

In all of the opening four fugues, theopening four-measure statement isalways followed by a fragment, which isessential to the unfolding of a manifoldof ideas. In the case of Fugue I, thatfragment is, with one change, lifteddirectly out of the 1748 Musical Offeringcomposition (Figure 3.3). Note the dif-ference introduced in the later fragment.In the Art of the Fugue, the f˝ at the topof the phrase descends by a Lydian inter-val downwards to b˝. Thus, the materialhas been transformed to incorporate theprincipal discovery of the Musical Offer-ing, the Lydian principle, as a point ofreference.

In Fugue I, as each voice enters, aninherent paradox emerges, showing thatthe original idea was not as “inconse-quential” as may have appeared. Thehalf-step motion into the third measureof the theme (e.g., in the first statement,d down to cˇ ) generates a sequence ofparadoxical cross voices, dominated bythe Lydian interval. The purpose of theoriginal “fragment” revised from theMusical Offering becomes clear: to pre-pare the mind for the sequence of Lydi-an intervals that will occur—for exam-ple, in the passage shown in Figure

51

B b C˙

˙ 2˙˙

3

˙#œ œ

4˙ œ œ œ

œ

5œ

FIGURE 3.1

Opening of Fugue I from J.S. Bach’s The Art of the Fugue

From The Art of the Fugue, Fugue I:

From A Musical Offering, six-part Ricercar:

?

b

b

b

C

œœ

180

œœ œ œœœ

˙n˙n

181

œœ Óœ

œ œœœ

œ

182

Ó Œœ œœ œn

B b C

6

Œœœn œœ

œ

7œ

œn

.œ

FIGURE 3.3

From A Musical Offering to The Art of the Fugue

B bC Ó

˙

B(Bb)

˙˙

CA

.œn

H(Bn)

FIGURE 3.2

Counter-subject of Bach’s final fugue

3.4a—or, when all four voices havefinally entered, on measure 15, the voic-es form a double Lydian interval of f-g -̌d -b (Figure 3.4b).

The tension between the originalidea, rooted in D minor, and the Lydianintervals, which imply motion towardany number of potential modes,requires the introduction of a new idea,to forge progress. The idea introducedby Bach is a rising fourth, which beginsto predominate and shape the directionof the earlier material. The risingfourth becomes pervasive throughoutthe entire fugue.

The passage beginning measure 36(Figure 3.5) exemplifies this approach,in the way the bass voice is organized.Do not think that this is somehow thefirst time the fourth appears in the score,for it is not. That is not the point.Rather, the emerging predominance ofthe fourth occurs in the same way that,in a drama, a character in the back-ground—perhaps a member of acrowd—suddenly steps forward andplays an important role. Bach’s determi-nation that a third idea must always beintroduced in these fugues, underlinesthe difference between his concept offugue, as well as of music overall, rela-tive to lesser composers.

Prior to Bach, many other composers“used” the Lydian interval, but only as“another” device, or “element” of com-position. The underlying principledimportance of the Lydian as discussed inother sections of this report, eludedthem. Unlike Bach, they confined them-selves to writing “strict” fugues, wherethe theme would be repeated, theninverted, or changed rhythmically; butthere was no ordering principle govern-ing the ideas of the composition.

Let us take another example, inwhich the role of the fourth becomeseven more significant. The passage inmeasures 36-40 (Figure 3.5) concludeswith a very strong resolving interval:in which e in the bass voice movesdownward by a fifth to A. This is veryimportant, for the following reason:The inversion of that interval, e mov-ing upward a fourth to a, is the high-point of the movement. In measure 49

52

&

?

b

b

C

C

36

.œ

j

œ.œ#

j

œ

œœœ˙

∑

˙ œœœ

37

.œ

j

œ .œ

j

œ

œœœ

˙

∑

˙ œœœ

38

.œ

j

œ .œ

j

œ

œœœn

˙

∑˙ œ

œnœ

39

.œ

j

œ .œn

j

œ

œœ#œ

˙

∑˙ œœœ

40˙#

œœ œ

˙

.œJ

œ

FIGURE 3.5

Fugue I, measures 36-40

With soprano register shifts:

(a)

(b)

&b C

49˙

˙

mi la

50˙˙

fa re

51˙# œ

œ

do re mi

52˙

fa

&b C

49˙

˙

mi la

50˙˙

fa re

51˙# œ

œ

do re mi

52˙

fa

FIGURE 3.6

Fugue I, theme introduced in highest voice

&

?

b

b

C

C

75

˙

‰

œœœœ

œ# œœ

˙b

˙˙

w

76œœœœb œœœ#œ

œ

œœ œ

˙#œ œ

w

77œ œ œœ

œ# ˙

œ

œ œ

œ œœ

˙ œœœ œ

w

78w

U

œ œ#œœ ˙

u

wU

w

u

FIGURE 3.7

Conclusion of Fugue I

(a)

(b)

&

?

b

b

C

C

9

œ

œ

˙ œ#

œ œ˙

˙

˙

10œ

œœn œœ œ

.œ

J

œ .œ

J

œ

˙˙

11

œ

œœ

˙#

Lydian

&

?

b

b

C

C

14

œ œœ#˙

.˙ œ

œ

˙˙

Œ

œ œnœœ

œ

15

˙nœ œ

œœœ œœ Œ

˙# œœn

œ

œn

œ œ œ œœ

doubleLydian

FIGURE 3.4

Lydian intervals introduced in Fugue I

(Figure 3.6a), the topmost voice intro-duces the theme, though this timeintroduced by the interval of thefourth, situated as e moving upwardto a . Thus, the soprano inverts the ear-lier bass voice resolution, referencedabove (the last beat of measure 39going into measure 40). This sopranoinversion of the bass voice resolution isa turning point in the movement.These are the “highest” tones on whichthe soprano voice introduces thetheme. Even more significantly, in thisposition, the uppermost voice is pre-senting the theme, for the first time, insuch a way as to cross from the secondto third register of the human sopranosinging voice (Figure 3.6b).

This coupling of inversion withregistral differentiation—includingacross two different voice species, name-ly bass to soprano—is essential to whatis meant by ontological principle in musi-cal composition. Although these fugues

are “instrumental” works, the underly-ing conception is entirely consistent withthe bel canto-trained “chest” of humanvoices. Vocal registration is an ontologi-cal characteristic of musical art (seeChapter 1). As we shall see, Bach wasintently focussed on the implications ofthe difference implied between a particu-lar interval, that interval in respect to itsinversion; and that pair of invertedintervals relative to changes of vocal reg-istration, in different voice parts, asreflecting ontological principles of musi-cal composition.

The importance Bach attributes tothe shift in soprano vocal registration, isindicated by the final four measures ofFugue I (Figure 3.7), in which thesoprano voice evokes a cadenza passage.Though there is a d pedal-point in thebass, the soprano voice is spelling out anascending C minor “scale.” This “scale”is actually composed of the identicalmaterial upon which the Musical Offer-

ing is based, namely, the paired Lydianintervals of C-Fˇ and E˛-A. (Note, fur-thermore, the downward diminishedseventh from the high b˛ to the cˇ atthe end of the phrase, again an intervalreadily identified with the Musical Offer-ing.)

Now, to have a clearer view of theprinciple indicated—and to experienceits profound implications—we turn toFugue IV.

First, a chart which simply situatesthe reader in respect to the material(Figure 3.8). In Fugue I, the theme isascending. In Fugues III and IV, thetheme is inverted to assume its descend-ing form. The inversions denote theonset of more developed hypotheses,inclusive of the emergence of new con-structive principles of composition.Bach’s recognition that inversionrequired such a development of newhypotheses, is what distinguishes hisconcept of fugue, from schoolbook ver-sions of “strict fugue.”

In private discussion, LyndonLaRouche has pointed out that The Artof the Fugue properly situates what isoften called “chromatic” motion.Throughout the composition, Bachshows that “chromatic” motion is notsome kind of sensual effect, but rather isa necessary theorem of inversion. This isparticularly evident in the canonicalduet, Fugue XV, not shown here.

Important to our investigation, is thatboth Fugues III and IV, which areinversions of the opening idea, introduceas companions to the root theme, chro-matic countersubjects, that is, phrasesbased on motion by half-step.

To make this clear, we show againthe opening measures of Fugues III andIV (Figure 3.9), accompanied by theirfragment countersubjects, which arequite different from the fragment dis-cussed in respect to Fugue I. Considerfor one moment the “chromatic” frag-ment attached to Fugue IV: Implicitlythis is a statement of inversion. Themiddle tone is a root. The half-stepabove and below the middle tone aremoving in inverted directions from oneanother (i.e., the gˇ at the end of measure5 moves back up to the a).

53

B b C

Contrapunctus I˙

˙˙˙

˙#œ œ

˙ œ œ œœ

B bC

Contrapunctus III˙

˙˙˙

˙œ œ

˙# œœœn œ

B

bC

Contrapunctus IV˙

˙˙˙

˙œ œ

˙ œ œ œœ

FIGURE 3.8

Art of the Fugue theme and elementary inversion

&

?

b

b

C

C

Contrapunctus III

∑

∑

˙

˙

∑

2∑

∑

˙˙

∑

3∑

∑

˙œ œ

∑

4∑

∑

˙# œœœn œ

∑

5∑

˙

˙

œ˙n œn œb

∑

6∑

˙˙

œ œ# œœn œ œ#

∑

7∑

˙ œœ

œœ œœœ

∑

&

?

b

b

C

C

Contrapunctus IV

˙

˙

∑

∑

∑

2

˙˙

∑

∑

∑

3

˙ œœ

∑

∑

∑

4

˙ œœœ œ

∑

∑

∑

5

˙ œ œb œœ#

˙

˙

∑

∑

6

˙ œ œb œœ#

˙˙

∑

∑

FIGURE 3.9

Fugues and fragment counter-subjects, III and IV

Immediately, the propositions beingpresented in Fugue IV are moredensely organized, per interval ofaction, than those in Fugue I. Thatshould not surprise us, since what Bachis pursuing here, is to further developthe “unfinished” question left overfrom Fugue I.

This greater density of principles isexemplified by the soprano voice inmeasure 13 (Figure 3.10). At this earlypoint in the composition, the sopranovoice moves into the third register,directly referencing the poetic high-point of Fugue I. The reference to theFugue I is explicit. The soprano movesexactly as before, upwards by a fourth,from e to the third-register a .

From this point on, there is a muchgreater density of interaction betweencolligating principles, relative to FugueI. The reason for that will becomeclear.

For example: As in Fugue I, Bachwill introduce a “new” interval, to re-sit-uate the paradox created by the fuguetheme placed against its countersubject(in this case, the step-wise [chromatic]motion). Here, the “new” interval is nota rising fourth, but rather a descendingthird, consistent with the fact thatthroughout Fugue IV, the overall direc-tion of everything (except the sopranovoice!) is downward.

Note, however, that this descendingthird is an interval of a more complextype than the fourth in Fugue I. Why?Because Bach always presents the thirdin duplicate, across two voice parts. Forexample, in measures 19 through 23(Figure 3.11), the soprano and altovoices are in such a dialogue. Implicitly,the paired dialogue of descending thirdsis spelling out an inverted fifth, or,sometimes, Lydian interval. Thus,implicitly, the paired thirds occur as aninversion of the fourths and fifths up tothis time.

There is an additional clue concern-ing the purpose of this process. In bothfugues, there is a significant occurrenceof Lydian intervals. In this fugue, how-ever, Bach meticulously postpones theintroduction of the interval Fˇ-C, untilwell into the development of the compo-

sition. Despite one early reference to C-G˛, the interval Fˇ-C only occurs inrespect to the soprano voice entering thethird register!

For example, the passage beginningthe second half of measure 34, through37 (Figure 3.12), illustrates this pairingof the Lydian interval c -fˇ with theevoking of the soprano third register. Inthe course of measure 37, for the firsttime, the double Lydian sequence a -e˛ -c -fˇ occurs, explicitly spelled out inthe bass and alto voices. The soprano

voice, meanwhile, is entirely in the thirdregister!

This occurs yet again, in measure63-64 (Figure 3.13). The soprano iscrossing back and forth between thesecond and third registers. As it doesso, the tenor voice executes a remark-able Lydian interval: e˛ down to a(obviously closely related to the ascend-ing soprano e˝ up to a ). As it enterson the a, the same double Lydiansequence occurs across the four voices:fˇ-a-c , with the e˛ from the tenor’s

54

&

?

b

b

C

C

63œœœ# œ œœ œ

˙ .œ

J

œ

˙

Œœb

˙œ œ

64.˙n œœœb œœ˙

˙ Ó˙# œ

œœ œ

FIGURE 3.13

Fugue IV, measures 63-64

&

?

b

b

C

C

34œ

œœ œn˙

˙ œ œ œœ#

∑

œœb ˙

35œ œ˙b

˙ œœbœ œ#

˙

˙

œœ œœ

œ

Œ

36Œœœ

œ

˙ œœbœ œ#

˙˙

∑

37 œ œ œ œ œœ#œ

œ

œ

œ#œ

Lydian˙b œ

œ

∑

FIGURE 3.12

Fugue IV, measures 34-37

B

B

b

b

C

C

.˙

3rdœ

19

œœœ ˙

.˙

3rdœ

20

œ

3rdœ˙

.˙

3rdœ

21

œ

3rdœ˙

.˙

3rd

œ

22

œ

3rdœ˙#

˙

23

˙

FIGURE 3.11

Pairs of descending thirds in Fugue IV

&b C

13

œ

œ œœœ œœœ

œ

œ

œ#œ

14 ˙

œ

FIGURE 3.10

Soprano and alto voices, Fugue IV, measures 13-14

previous tone implicitly included in thepaired Lydian intervals.

Now, we have arrived at the “punc-tum saliens” of this fugue. Somethingfairly remarkable is about to occur.

According to different source mate-rials, Ludwig van Beethoven copiedseveral measures from this section ofFugue IV into a notebook associatedwith his late string quartets. His entryincludes measure 61, then a doubleslash on the staff to indicate a jump,and then three and a half measuresbeginning with the second half of mea-sure 72. (See Figure 7.2 for a transcrip-tion of these passages from Beethoven’snotebook.)

The passage beginning with mea-sure 72, through to the C-major resolu-tion in measures 86-87 (Figure 3.14),records a demonstration of the processwhereby a new principle is introducedto the manifold of validated metaphori-cal discoveries [(m+1)/m], not merely as

such discoveries have bearing within aparticular musical composition, but,rather those higher order discoverieswhich bear upon the entire domain ofmusical art.

What Bach shows, is that the “solu-tion” to the unresolved crossroad of therelation of the soprano register shift tothe set of contrapuntal problems posedthus far, lies outside the domain of whatmight be called “contrapuntal” consider-ations. Beginning with the measurecopied by Beethoven, the bass voicedescends to the lowest pitch for its voicein this composition, a low D. Againstthe backdrop of two references to theinterval C-F ,̌ for the next seven mea-sures, the composition becomes, instages, increasingly “blurry.” It is almostas if each of the voices “has a mind of itsown,” typified by the soprano in mea-sure 75 sounding c , against the cˇ in thebass.

For several measures, there is a key-

less mode, until the soprano enters,asserting the opening theme, in themode of A minor. However, the out-come of this placement of the theme inthe soprano voice, is the exact oppositeof what one would expect. Rather thanthe composition becoming more simplyordered, the opposite occurs.

As the soprano moves upward, to athird-register g , coming down to a sec-ond-register f˝ , all of the other voices areemphasizing Fˇ, the precise value atwhich the third register shift occurs.The blurrying motion intensifies, and aheretofore unknown mode, F ˇmajor/minor, against B minor, takesover. After five measures of this treat-ment, everything comes together aroundC major!

This passage evokes precisely the“eerie” quality LaRouche discusses inrespect to great tragedy—where onebecomes conscious that it is the ideashovering above the individual elementsof composition which are governing thedevelopment. In the case of music, atthis precise moment, one cannot helpbut think of Beethoven’s late quartets,even if one knew nothing of the passingreference to this fugue passage found inBeethoven’s sketchbook.

What is Bach proving to us? Do notlook for the root of great musical com-position in the formalities of counter-point, or in any one of the principles.Rather, seek the root of composition inthe generative capacity to improve theprinciples which bring these elementstogether. It is actually the irony of thesoprano register shift, emphasized hereby the interplay across the passage ofF˝ against F ˇ , relative to other colligat-ing principles, which is driving Bachto focus on the underlying method ofordering these principles. Density isdemonstrated by the very compact wayBach moves from the “eerie” realm ofF ˇ major/minor to the resolution of Cmajor. In so doing, he has extendedthe notion of Lydian principle, in amost profound, and scientifically validway, by emphasizing its ontologicalroot in the soprano register shift. Thisalso has extended his conception ofinversion.

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FIGURE 3.14

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