In his article “The Substance of Moral-ity,” Lyndon LaRouche presents a

conception of the Universe as a multi-ply-connected manifold of the type (N-manifold)/(M-manifold). “M” signifiesan ever-expanding array of principles ofdevelopment of human culture, and “N”signifies a growing array of principles ofphysical action. These two sub-mani-folds, of order “N” and “M,” do not existapart from each other, but are multiply-connected by Man’s culturally-deter-mined action upon the physical Uni-verse, and the impact upon culturaldevelopment of changing physical con-ditions of human society’s existence inthe Universe. The inner developmentalcharacteristic (curvature) of physical “N-manifold” is called anti-entropy, and thecharacteristic curvature of the “M-mani-fold” of human cultural development isagape. The two are inseparable, neces-sary expressions of the Principle of Cre-ation (God).

Before turning to the musical side ofthis question, it will be useful to clarifythe meaning of “multiple-connected-ness,” and in what manner we are toconceive of a manifold that is governedby not one, but a growing multiplicity ofdevelopmental principles. To make ashort work of this, I emphasize onlysome key points, followed by an elemen-tary illustration from astronomy, whichleads us directly to music.

It is impossible to reduce the rela-tionship of events in multiply-connectedmanifold, by any deductive or similarmeans, to a single formal principle.Rather, action in the manifold is gov-erned by a multiplicity of principles,

none of which can be reduced to orderived from the others in a formal-deductive manner. Any process in themanifold is simultaneously co-shaped byeach and all the principles in any arbi-trarily small region of action. The activeprinciples, mutually irreducible andincommensurable in the just-mentionedsense, constitute true singularities—indi-vidual existences underlying the wholestructure of the manifold. We encountersuch singularities in physics in the formof creative fundamental discoveries ofprinciple, and in music as entirely analo-gous discoveries of principle of bel canto-anchored motivic thorough-composi-tion. The following sections will reviewsome of them, such as Haydn’s discov-ery of Motivführung, and Mozart’sbreakthrough on the significance of the“Lydian” major/minor mode, firstexplored in the late works of J.S. Bach.

In first approximation, one might betempted to think of each active principleas analogous to a coordinate axis in an n-dimensional space, n representing thenumber of an irreducible array of prin-ciples governing the manifold at a givenstage of development. In reality, howev-er, the principles of development, whilemutually irreducible in a formal sense,are never independent of each other inthe manner implied by the Cartesiancoordinates or the use of “independentvariables” in a formal mathematical rep-resentation. As an “n-manifold” devel-ops to an “(n+1)-manifold” and so forth,the integration of each newly discoveredprinciple modifies the entire previousarray of active principles.

Indeed, this process invariably

involves the generation of paradoxes andanomalies: events are demonstrated tooccur in the Universe, which are incom-patible with the given set “n,” point to aflaw or at least an inadequacy in thatexisting set of principles. Generallyspeaking, the newly hypothesized prin-ciple does not replace or supersede theexisting ones; rather, the latter must bereworked and redefined from the stand-point of the new discovery. Thus, theprocess of lawful generation and resolu-tion of dissonances through motiviccross-voice development in well-tem-pered polyphony, mirrors the universalfeatures of development of any multi-ply-connected manifold.

The growing array of principles issubsumed within a higher principle ofgeneration (a “One”), whose essentialcharacteristic, anti-entropy/agape, islocated in the process of change from thelower- to the higher-order manifold.Although that process involves the suc-cessive integration of singularities, eachformally incommensurable with theothers, that higher principle of creativeself-elaboration remains everywhereself-similar to itself. The proper mea-sure of the ordering of development isnot “number of dimensions” in the for-mal sense, but rather increasing cardi-nality or power in the sense developed byGeorg Cantor.

The ordering of the process of devel-opment of the manifolds by increasingCantorian cardinality, does not at allcoincide with time in the ordinarychronological (i.e., clock-time) sense. Onthe contrary, time and space are merelysubsumed physical principles, which are

38

J.S. Bach and Inversion as2 A Universal Principle of Development

In the Continuum of Musical Compositionby Jonathan Tennenbaum

ironically multiply-connected with theCantorian axis of development. Weknow this negatively, from the sad wit-ness of rise and decline of civilizations oreven human culture as a whole. We alsoknow this positively, by the fact that allacts of creative discovery involve someor another degree of apparent “timereversal.” Rigorous composition alwaysproceeds backwards from the effect to beachieved—which exists, as it were, out-side ordinary time—to the means andtemporal pathway of events required toachieve that effect. Thus, music anddrama are typified by ironic anticipa-tions and premonitions, and otherexpressions of temporal inversion.

Music and Keplerian Astronomy

The development of astronomy, fromthe most ancient times up to Kepler andGauss, provides the most direct access tothe notion of a multiply-connected man-ifold, just sketched above.

The study of the motions in theheavens leads to the discovery of moreand more astronomical cycles as principlesof motion. Thus, observing the risingand setting of the sun and the stars, weconceive the cycle of the day. Noting,however, that the path of the sun shiftsslightly from day to day, we discover thelonger cycle of the year. What at firstglance appear to be very slight discrep-ancies in the yearly cycle of the sun withrespect to the stars, reveal a much longercycle of the precession of the equinoxes.Later, additional cycles emerge, con-nected with the non-uniform (elliptical)motion of the Earth around the sun. Inaddition to these solar-terrestrial andstellar cycles, we must also take intoaccount the cycles associated with themotions of the planets. The latter revealthemselves, upon closer examination, toinvolve more complex considerations,going beyond the principle of simple cir-cular action.

Thus, as astronomy develops, we dis-cover new principles of motion not onlyas new cycles per se, but also as internalprinciples of organization of the cycles,and principles of multiple-connected-ness or “colligation” among the cycles.Thus, Kepler’s discovery of the “arealaw” of motion in conic-section orbits,

and his discovery of the harmonic prin-ciples underlying the entire array oforbits.

The observed motion of any planetor other heavenly body is the resultantof all cycles and related principles actingconjointly. So, for example, even thoughthe equinoctial cycle has a length ofsome 26,000 years, it acts efficientlywithin any arbitrarily small time inter-val, to produce a distinct, implicitlymeasurable modification of anyobserved motion. The manner in whichthe characteristics of any planetary orbitare reflected in any arbitrarily smallinterval of the observed motion, wasdemonstrated by Carl Friedrich Gaussin 1801, when he determined the orbitof the unknown planet Ceres from onlythree, very close-spaced sightings.

This concept of “curvature in theinfinitely small” of astronomicalmotions, has an unavoidable, paradoxi-cal feature: The motions we observe,

embody not only the cycles which areknown to us at any given time, but alsothose we do not yet know explicitly—cycles whose future discovery is inherentin the self-similarity of the principle ofcreation underlying the Universe as awhole. Hence, the curvature in thesmall, as reflected in the fine “articula-tion” of the heavenly motions, containsan element of creative tension, associatedwith the anti-entropy/agape of a Uni-verse constantly developing M → M+1,M+2, . . . ; N → N+1, N+2, . . . .

As Kepler demonstrated in detail forthe case of the solar system, the highercoherence of the astronomical “n-mani-fold,” is reflected in harmonic orderings,of the same type as characterize artisticbeauty in the domain of human Classicalculture. In the dialogue Timaeus, Platorefers to this common higher principleunderlying astronomy and Classical art,by declaring the Universe to be a contin-uously unfolding composition of “Godthe Composer.”

Reflecting this, Kepler’s determina-tion of the harmonic ordering of theplanetary orbits specifies certain band-like regions or corridors as the locationof the planetary orbits, and not fixedalgebraic values (Figure 2.1). The exactorbits of the planets, while remainingwithin their harmonically “quantized”corridors, are constantly changing andevolving together with the Universe as awhole, in a manner Kepler likened tothe performance of a polyphonic compo-sition.

Kepler’s Astronomical Inversions

In his New Astronomy, Johannes Keplerpresented a series of devastating anom-alies which overturned the prevailingassumption, that the planetary motionswere based on nothing but the Ptolema-ic-Aristotelean notion of uniform circu-lar motion as the basic physical princi-ple. In order to determine the actualmotions of the planets, however, Keplerhad to overcome the difficulty, that theorbital motions of the planets, includingof the Earth itself, cannot be adduced inany direct manner from the observedmotions as they appear to an observer onthe Earth. Indeed, as already remarkedabove, the apparent motion of any plan-

39

FIGURE 2.1

Kepler's determination of the harmonic ordering of the solar system, from his New Astronomy

et, is the resultant of a complex combi-nation of motions, including the Earth’srotation, the Earth’s motion around thesun, and the true orbital motion of theplanet. The true motion of the Eartharound the sun, which we can neithersee nor sense in any direct way, can onlybe determined by reference to the actualmotions of the other planets; but, to dis-entangle the real from the apparentmotions of those planets, it would seemnecessary to first know the motions ofthe Earth, from which we observe theplanets. How do we get out of this circu-lar paradox? Kepler’s ingenious solutionwas based on a method of inversion,closely akin to J.S. Bach’s method ofwell-tempered polyphony.

Kepler asked the hypothetical ques-tion: How would the Earth’s motionappear, relative to the apparent motionof the sun, if we were to observe theEarth and the sun from Mars? Anobserver would have a different solarcalendar, whose basic cycle (the Marsyear) makes a specific ratio to the Earthyear. At first glance, such a hypotheticalshift of locus of action—analogous to amodulation or more general inversion inmusic, as we shall see below—seemsonly to compound our ignorance.

Kepler, after all, had no means to actual-ly place himself on Mars! Yet it wasexactly by juxtaposing the motion ofMars as seen from the Earth, with themotion of the Earth as seen from Mars(all relative to the sun as “tonic”), thatKepler was able for the first time todetermine the orbits of both the Earthand Mars! (Figure 2.2) By thus exloit-ing the additional dimensionality pro-vided by the Mars orbital cycle, Keplerwas led to the discovery of the ellipticalform of planetary orbits, and a revolu-tion in astronomy. The key here is thetransformation between two or more setsof angular intervals (e.g., observationsreferenced to the Earth’s cycle, versusobservations referenced to the Marscycle).

Kepler was fully aware of the kin-ship of his method with the Platonicdialogue and the polyphonic principle inmusic. Just as one can only know one’sown mind and the assumptions whichshape it, in the mirror of our interactionwith other minds; so, in well-temperedpolyphony, the motivic idea emergesonly through a process of contrapuntalinversions; and so in astronomy, themotion of our Earth would never beknown—Kepler loved to say—had God

not given us Mars and the other planetsas celestial companions.

The Well-Tempered System, Briefly

Turning to musical composition,remember that the “n-manifold” ofmusical development lies entirely out-side the audible domain of musicaltones per se. One might say, that musi-cal ideas themselves are soundless. Yetthese soundless entities generate all theevents in the audible domain and ruleover it absolutely. For example, as theperformances of Wilhelm Furtwänglerand Pablo Casals demonstrate mostforcefully, a musical interval is notsomething determined by a pair oftones, like a line segment drawn to jointwo points. Rather, the interval precedesthe tones, both ontologically and in theconsciousness of the composer andgreat performer, just as the idea of thecomposition precedes the ordering andshaping of all intervals in a composi-tion. Lyndon LaRouche emphasizesthat the least “unit” coherent with theexpression of a musical idea, is a pair ofintervals in the sense of an intervalbetween intervals.

The mere acoustician will puzzleover the paradox: What could be the dif-

40

sun

M2

E1

M1

E2

(b) Earth and the sun, as observed from Mars

FIGURE 2.2

Transformation of angular intervals by change of locus of observer

sun

M2

E1

M1

E2

(a) Mars and the sun, as observed from Earth

ference between merely playing tones,playing intervals, and playing the inter-vals between intervals, in the mannerFurtwängler brought his orchestras todo? Where resides the difference, giventhat the instruments themselves producenothing but tones? The “extra” whichdistinguishes the performance of inter-vals between intervals from the meresounding of a succession of tones, isclearly heard in the mind, but is other-wise a virtual infinitesimal in acousticalterms—often nothing more than a bare-ly perceptible, specific shaping of thetones in a musical line.

That shaping of the tones by musicalintervals, and intervals by intervals ofintervals, embodies the same principleby which the well-tempered system as awhole is determined by the curvature ofthe evolving manifold of bel canto-basedmotivic thorough-composition. Thatdevelopment is bounded by the require-ment, that the creative principle embod-ied in the conception of the bel cantosinging voice, be extended in a self-simi-lar manner to an ensemble of bel cantovoices having differing registration. Inthis process, the harmonic principles ofbel canto vocalization, investigated byLeonardo da Vinci and described in partin the preceding section of this report,are “turned inside-out,” as it were, tobecome principles of well-temperedvocal polyphony.1

The result, evolving in the course ofa long, implicitly still-ongoing historicaldevelopment, is the Classical well-tem-pered system, with its various species ofharmonic intervals (octaves, fifths,fourths, thirds, etc.).

The mature chorus and orchestraensemble, as understood by Beethovenand Brahms, must sing as a single voice,even while performing the most intri-cately articulated polyphony. Con-versely, instrumental and choralpolyphony are nothing but a self-simi-lar extension of the polyphonic princi-ple inherent in the single bel cantohuman voice with its characteristicregistral differentiation.

This is exactly the conception under-lying Johannes Kepler’s famous deriva-tion of the musical intervals and scales,by harmonic division of the circle and

sphere, which were crucial to his investi-gation of the musical principles govern-ing the multiple-connectedness of theplanetary orbits (see Kepler’s Harmonyof the World, Book III).

Unfortunately, Kepler’s construc-tions are often misread to signify nearlythe opposite of what they were original-ly intended to demonstrate.2 The mod-ern reader must never forget, that thecircle and sphere of Kepler signify some-thing very different from the mere geo-metrical shapes which carry the samenames. Kepler explicitly refers to Nico-laus of Cusa, and the latter’s discovery ofthe ontological significance of the circle’srelationship, as a higher species, to itsinscribed and circumscribed polygons.Cusa and Kepler stressed two elemen-tary points in this context: First, thepolygons and the discrete whole num-bers associated with them, do not existself-evidently, apart from circularaction; and there is no valid determina-tion of the polygons which does notoriginate in the circle. Second, while thepolygons are generated and everywherebounded by circular action, it is impossi-ble to go backwards and derive the cir-cle from the polygons, even if the num-ber of their sides were increased beyondany limit.

Exactly in this sense, the generativeprinciple or curvature of the n-manifoldof bel canto-based motivic thorough-composition, bounds the process of suc-cessive discovery of principles of compo-sition, including the system of harmonicintervals, tuning, keys, modes, andeverything else. There can be no self-evident algebraic determination ofmusical intervals, nor any valid con-struction based on “empirical facts” con-cerning acoustics and the physiology ofhearing, as Helmholtz claimed. Thewell-tempered system is everywherebounded by the creative process of musi-cal development.

Thus, contrary to a nearly universalmisunderstanding, the well-temperedsystem not only does not prescribe analgebraically-fixed set of pitches andintervals, but it absolutely forbids any such“fixing”! Bel canto-based well-temperedcomposition dictates the necessity for aspecific “shaping” of each and every

tone and interval in a composition—including lawful variations of pitch with-in the harmonically-ordered “corridors”identified with the scale-steps, in such away that the infinitesimal “curvature” ofeach moment of articulation expressesthe creative tension underlying the com-position as a whole.3 Unfortunately, thecapability of distinguishing such smallbut crucial nuances, possessed by com-posers and to a large extent even theeducated musical audiences ofBeethoven’s time, has virtually died out.

By contrast, the concept of strictmathematical equal-tempering, is a fal-lacy rooted in the vain attempts to col-lapse a multiply-connected manifoldinto the “flat” space of a single (mono-phonic) formal principle.

Inversion of Intervals

As indicated, inversion is a universalprinciple of musical development. Togain some insight into this, we can startby examining the manner in whichClassical composers employ elementaryinversions of intervals as instrumentali-ties of the process of motivic-polyphonicdevelopment. As we move forward inthis series of articles, we will workupward from the simplest cases, dis-cussed here, to the higher conception ofinversion which underlies the late com-positions of Mozart and Beethoven. Inthe process, we must constantly reflecton the way our minds “hear” both theexplicitly stated intervals, and thosewhich are only implied by the composer,and which are often even more impor-tant than the stated ones. These distinc-tions, reflecting changes in assumptiongoverning any given phase of a composi-tion, must be expressed in performance,by the articulation and “shaping” oftones and intervals “in the small”(including lawful nuances in pitch into-nation).

In its very simplest formal manifesta-tions, inversion involves one of threeforms of transformation of an intervalsubsuming two tones:

(1) By sounding one or both of thetones in a different octave, voice, or reg-ister; usually in such a way, that thehigher of the two becomes the lower inpitch, and the lower becomes the higher,

41

while retaining their values within thescale (inversion of order in pitch). So, forexample, a soprano and bass singer. Inthis case, the magnitude of the interval ischanged; a fifth becomes a fourth, amajor third becomes a minor sixth, andso forth.

(2) By reversing the direction of theinterval’s motion, as taken from eitherof the tones regarded as the origin, i.e.,from upward to downward and vice-versa, while retaining the relative mag-nitude of the interval. In this case, notonly the relation of higher and lower inpitch is reversed, but also the scale-valueof one of the tones. So, the fifth frommiddle c upward to the g above it,inverts to the downward fifth from mid-dle c to the f below it. (Note: this kindof inversion is more than a simple trans-position of the interval; the directionalityis also changed.)

(3) By reversing the temporal order ofthe two consecutive tones, so that thelater now becomes the earlier, and viceversa, while maintaining their pitch val-ues.

It is important to bear several thingsin mind: First, each event of inversion,constituting a transformation of intervals,involves no less than a pair of inter-vals—the original and its inversion.Inversion can thus be considered as aspecial type of interval between intervals.In many cases (see below) the originalinterval is merely implied, but notexplicitly stated; or vice-versa, the origi-nal interval may be stated explicitly, andthe inversion only implied. Sometimesneither of the two are stated explicitly,but are unmistakably implied. Relatedto this, inversions can occur for intervalswhich span entire sections of a composi-tion, rather than merely consecutivetones, and so forth.

Now let us look at some examples ofelementary forms of inversion in com-positions of J.S. Bach. I want to empha-size that the following remarks by nomeans amount to an adequate analysisof any of these compositions. They areintended to open doors for an apprecia-tion of the role of inversion in composi-tion, starting from the very simplestsorts of cases, and working upwardtoward the more complex and pro-found. [text continues on page 43]

42

Soprano I & II

Alto

Tenor

Bass

&

&

V

?

#

#

#

#

c

c

c

c

1 (7)

œ œœ œ

Je -ach

su,wie

mei -lang,

neach

œ œœ œ# œ

Je -ach

su,wie

mei -lang,

neach

œ

œ œ

œœ

Je -ach

su,wie

mei -lang,

neach

œ œ.œ

j

œ

Je -ach

su,wie

mei -lang,

neach

2 (8)

˙ ˙

U

Freu -lan -

de,ge

œ œ#˙

U

Freu -lan -

de,ge

œ œœ ˙

U

Freu -lan -

de,ge

œœ

˙

U

Freu -lan -

de,ge

3 (9)

œ œ#œœ

mei -ist

nesdem

Her-Her -

zenszen

œ œœ œn œ

mei -ist

nesdem

Her-Her -

zenszen

œ œœ# œœ œ œ

mei -ist

nesdem

Her -Her -

zenszen

œ

œœ œ œ œ

mei -ist

nesdem

Her -Her -

zenszen

4 (10)˙ ˙#

U

Wei -ban -

de,ge

œœ œ ˙

U

Wei -ban -

de,ge

œ

œ ˙U

Wei -ban -

de,ge

œœ œœ ˙U

Wei -ban -

de,ge

S.

A.

T.

B.

&

&

V

?

#

#

#

#

.

.

.

.

.

.

.

.

5 (11)œœ œ .œ

J

œ

Je -und

su,ver -

mei -langt

nenach

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j

œ

Je -und

su,ver -

mei -langt

nenach

œ œ œ œ#

Je -und

su,ver -

mei -langt

nenach

œœ œ œ

œ

Je -und

su,ver -

mei -langt

nenach

6 (12)w

Zier,dir!

w

Zier,dir!

w

Zier,dir!

w

Zier,dir!

13

œ œ œ œ

Got - tes Lamm, mein

œ œ œ œ

Got - tes Lamm, mein

œ œ œ œ

Got - tes Lamm, mein

œœ œ œ œ

Got - tes Lamm, mein

14

.œ

j

œ ˙

U

Bräu - ti-gam,

œœ˙

U

Bräu-ti - gam,

œ œœ ˙

U

Bräu-ti - gam,

œœ

˙

U

Bräu-ti - gam,

15

œ œ#œœ

au-ßer dir soll

œ œ œ œ œ

au-ßer dir soll

œ œ œ œ

au-ßer dir soll

œ œœœ œ œ

au- ßer dir soll

S.

A.

T.

B.

&

&

V

?

#

#

#

#

16œ œ

œ# ˙

mir auf Er - -

œ œ œ œ#

mir auf Er - -

œ œ œ œœ

mir auf Er - -

œ

œ

œ œ

mir auf Er - -

17

˙

U

œ œ

den nichts sonst

˙

U

œ œ

den nichts sonst

˙#

Uœœ œ

den nichts sonst

˙

U

œœ

den nichts sonst

18

œ œœ ˙

Lie - bers wer - -

œ œ œ œ#

Lie - bers wer - -œ

œœ œ œ

œ

Lie - bers wer - -

œœ œ œ

Lie - bers wer - -

19

w

U

den.

w

U

den.

w#

U

den.

w

U

den.

octave

FIGURE 2.3a

J.S. Bach, Jesu, meine Freude, opening chorale

J.S. Bach’s Motet Jesu, meine Freude

The first two measures of the openingchorale of J.S. Bach’s motet Jesu, meineFreude (Figures 2.3a and 2.3b), presentus with an anomaly: The soprano voicedescribes in stepwise motion, thedescending fifth b -e , while the bassmoves downward from e, and then backto e. Consistent with this and the motionof the inner voices, we hear e as thebase-tone and the downward fifth b -eas a return (from where?) to the basetone. The whole motion of the voices ismore like the end of a statement, thanthe beginning.

Now glance at the intervening devel-opment. From measure 3 to the begin-ning of measure 4, the soprano voicegoes upward from b to e , spanning anupward fourth b -e , which is the inver-sion of the downward fifth b -e of theinitial measures. Thereby, in our mindwe “hear” the octave e -e as confirmingan implicit development e -b -e , inwhich the first interval has been time-inverted in the opening statement.

The movement e -b -e would haveachieved a certain closure, but that thesoprano, instead of resting at the newlygained e , falls back to the adjacent dˇ ;while the bass voice articulates theupward fifth e-b, which is an inversion ofthe soprano’s descending fifth in measure1. At this point, we reach a maximum

tension, associated with the unresolvedjuxtaposition of the intervals b -dˇ , b-fˇ (in the tenor voice), and the expectedclosure e -e . The resolution to e -e isachieved in measures 5-6, by the sopra-no—anticipated already by the tenor’smotion in measure 4—breaking into thethird register to reach e from above, viathe g and fˇ .

With the consolidation of the octavee -e , the chorale moves downward toits conclusion. Then, after some prepa-ration in measures 13-15 (themselvesexpressing an inversion), in measures16-17 the soprano moves stepwise downto b (inverting the upward fourth b -eof measures 3-4), from which it descendsthe remaining downward fifth b -e toclose the descending octave e -e andend the chorale. That downward fifth,quoting the initial statement of thechorale—albeit with a shift in meter anda significant change in the tenor voice—resolves the original paradox: Thebeginning originated from the end!

All of this is nothing more than themost elementary kind of intervalicinversion, associated with the naturalstrophic organization of the chorale.The point is to see how the counterpointdeveloped by Bach in the bass and innervoices, defines and brings out thechanges in meaning associated with theindicated inversions of what is at firstglance one and the same interval.

43

Soprano

Bass

&

?

#

#

c

c

.

.

.

.

œ œœ œ

1 (7)

œ œ.œ

j

œ

˙ ˙

U

2 (8)

œœ

˙

U

œ œ#œœ

3 (9)

œ

œœ œ œ œ

˙ ˙#

u

4 (10)

œœ œœ ˙

u

œœ œ .œ

J

œ

5 (11)

œœ œ œ

œ

w

6 (12)

w

S.

B.

&

?

#

#

œ œœœ

13

œ œœœœ

.œ

j

œ˙

u

14

œ œ

˙

U

œœ# œ

œ

15

œœ œ œ

œœ

œœ œ# ˙

16

œ

œ

œ œ

˙

U

œ œ

17

˙

U

œœ

œœ ˙

18

œœ œ œ

w

19

w

downward 5th b′ e′

upward 4th b′ e″

upward 5th e b

unison

downward 4th e″ b′ downward 5th b′ e′

octave e″ e′

&

#

w

w

w

w

w

w w

w

w

b′ e′ b′ e″ e′ e″ e″ b′ e′

FIGURE 2.3b

Schematic of J.S. Bach, Jesu, meine Freude, opening chorale

J.S. Bach’s The Art of the Fugue

We concentrate first on just a few mea-sures of the opening statement of thefugue. (See Figure 2.4a for the entirefugue, and Figure 2.4b for conceptualsketches of it.) What we are about topoint out would be immediately per-ceived by any musical audience inBeethoven’s time. Today, however, thesame things would pass unnoticed bymost listeners, on account of their lackof grounding in composition. Hence theneed for the following, relatively minuteexamination.

The fugue begins with a first state-ment of the theme (measures 1-5) withan initial contrapuntal elaborationthrough measure 8. Our initial hearingof the fugal theme is dominated by thestatement of the upward fifth d -a inmeasure 1. In measure 2, the upwardmotion is reversed; a downward third f -d is stated, closing back to what wehave already sensed to be the base-tone(tonic) D. At that moment, we “hear” inour mind two additional, impliedintervals: first, an implied unisonbetween the initial d of the first mea-sure and the final d of the second mea-sure; and second, an implied downwardfifth a -d , which is the reversal of theupward fifth d -a of the first measure.This is the first reversal/inversion.

Our initial hearing of the followingtwo measures 3 and 4, is dominated bythe downward half-step d -cˇ , from theend of measure 2 to the beginning ofmeasure 3, and the fact that the reversalof that interval (i.e., cˇ -d ), which theearlier reversal d -a , a -d makes usexpect to hear, is not really accom-plished until we reach d in measure 5.Indeed, although cˇ -d occurs nominal-ly already in measure 3, the d is sound-ed off the beat, as a quarter note—tooshort and with too much the characterof a passing note, to fully resolve thepreceding d -cˇ , which was statedstrongly in half-notes and with cˇ onthe beat. In any case, the relationshipbetween d -cˇ (measure 2 and 3), andthe cˇ -d implied between the cˇ ofmeasure 3 and the d at the beginning ofmeasure 5, constitutes a second inversion.

The intervening passage, from mea-

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FIGURE 2.4a

J.S. Bach, Fugue I from The Art of the Fugue

continued on following page

sure 3 to the beginning of measure 5, issomewhat inconclusive at first hearing;what stands out is a third reversal/inver-sion implied between the sequences cˇ -d -e -f upward, g -f -e -d downward,in measures 3 and 4 respectively. Thelatter is clearly heard as quoting thedownward third f -d of measure 2, andthe former as stating its reversal/inver-sion. However, the sense of reversal is“modulated” by the intervention ofneighboring tones cˇ in the first caseand g in the second, plus the syncopa-tion and acceleration of motion. Manythings are suggested by this articulation,which are only actualized later in thefugue, and in later fugues of the entireArt of the Fugue cycle. Finally, note thatall three reversals/inversions pivot onthe common d (as, in a sense, a pedal-point), strengthening our sense of d asthe hypothesized pivot or base-point ofthe whole composition.

All of this is preparatory to the sec-ond entrance of the theme, in measure 5.At the sounding of the a , we immedi-ately “hear” an implied unison with thea of measure 1, and recall the initialupward fifth d -a , which the initial(lower) voice now once again quotes instepwise motion from the beginning ofmeasure 5 to the beginning of measure 6.

At this point a potential conflictappears.

In the original theme, the upwardfifth d -a subsumes an implied registershift (relative to soprano registration),from first to second register, establishinga initially as the dominant tone in thatregister. This already creates the sense ofa as a second potential pivot-point orfocus of the developmental action. Thispotential focal-point function isstrengthened by the reversed pair ofintervals d -a upward, a -d down-ward, which can be heard from thestandpoint of either d or a as the pivot-point. As a result, our mental earalready “hears” as a strong implicationthe upward fifth a -e ; it is implied as theinversion of the downward fifth a -dand as the transposed quotation of theupward fifth of the fugal theme to a as anew focus.

On the other hand, other strong rea-sons point to an upward fourth a -d as

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the lawful sequel at this point. In fact, ifd remains the focal-point, then alreadythe first upward fifth d -a in the veryfirst statement of the fugue, calls for itscontinuation in the upward fourth a -d ,which would thereby complete theoctave d -d . In this way, the originalreversal, namely:

d -a (upward fifth) reversed to a -d(downward fifth) in the statement,

would be quoted via the second voice as:

d -a (upward fifth implied by plac-ing the second entrance the themeat a , against d as a pedal-point),inverted to the upward fourth a -das the first interval stated by thesecond voice.

In other words, the original motion d -a -d becomes d -a -d .

We therefore have a dissonancebetween two (as yet unheard!) invertedintervals: an upward fifth a -e , and anupward fourth a -d , both of which areinversions of the interval that com-mences the composition. Which of themwill actually occur? Bach, in fact, choos-es a -d ; but the dissonance with theimplicit a -e is still heard in the mind,and acts to drive the development for-ward.

Let us briefly note some features ofthe rest of the fugue, which confirm thisreading of the indicated passage.

First, the upward fourth a -d , whichis an inversion of the original interval ofthe theme d -a , becomes the ever-more-dominant motif throughout the subse-quent development. It is first echoed inthe counterpoint in measure 7, and istaken up as a subsumed motif in subse-quent counterpoints and especially theinterludes of measures 17-21, 36-40, and44-46. It evolves into the increasinglypowerful counterpoints of the secondhalf of the fugue, in the turning-point inmeasures 48-53 (and sequel), as well asthe final development of measures 66-70, leading to the coda, where it is com-pressed to a new figure in the sopranovoice.

Second, the turning-point beginningin measures 48 and 49. Here, the origi-nal upward fourth a -d is stated again,as if to repeat the fugal statement at a .But the listener is surprised: what fol-lows instead is the dramatic entrance ofthe soprano voice with a second upwardfourth e -a , reaching into the soprano’sthird register and initiating the state-ment of the fugal theme in the highestregister-range of the fugue. The disso-nance between the intervals a -d anda -e , now explicitly re-created by thejuxtaposition of the fourths a -d , e -a ,is effectively resolved by the f in theupper voice of measure 50, by complet-ing the upward sequence d -e -f . Notehow the alto counterpoint adds cˇ toyield cˇ -d -e -f , which is exactly theoriginal statement of the third measure

of the fugal theme, stated one octavehigher.

J.S. Bach’s Mass in B minor

One of Bach’s most condensed master-pieces of vocal counterpoint, is the six-part double fugue “Gratias agimus tibi”in the “Gloria” of Bach’s Mass in B minor(the four-part vocal chorus is expanded,in the second half of the fugue, by twotrumpet voices). The same figure recursin slightly altered form as the final sec-tion of the mass, “Dona nobis pacem.”

The initial statements of the fugue,introduced in a “canon of canons”between the two sets of voices (bass-tenor and alto-soprano), appear at firstglance to be dominated by the notion ofd as the base-tone, the upward majorthird d-fˇ and fourth d-g in the fugaltheme, and the rising fifth d-a betweenthe bass entry and tenor entry. (See Fig-ure 2.5a, and Figure 2.5b for conceptualsketches.) The inversion a-d of the lat-ter interval, is stated by the tenor (mea-sure 2) itself, and between the tenor andalto entry in the same measure, and isthen repeated in different registrationbetween the alto and soprano entries(measures 2 and 3).

However, the rhythmic and contra-puntal arrangement of the voices alsoimplies other intervals and inversions.Prominent among these are the intervalsd -b and b-d , implied, for example, inthe tenor voice line (measure 2) and var-iously in other registers between thebass, tenor, and soprano (measures 3 and4). All these intervals are heard in theinitial section essentially from the stand-point of d as a kind of pedal-point.However, beginning in measure 10, anddecisively in measure 13, the appearanceof b in the bass line, redefines the entireset of relationships, now obliging us tohear the original sequence d-e-f -g froma completely different standpoint,defined by an inversion of the originalrelationship of d and b which places b asthe pivot in the bass (see also the discus-sion below).

The moment of this redefinitioncoincides with an implied bringing-together of the two double-fugalthemes, already implied by the bassline’s reference to the second fugal

46

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!

vs.

conflict resolution

expected actualdevelopment

compare to original thememeasures 2-4

FIGURE 2.4b

Two key passages from Fugue I of J.S. Bach’sThe Art of the Fugue

theme in measures 10 and 11, and inmeasure 13. The basic movement ofthe second theme, which is first statedin measures 5-6 and recurs in variousmajor/minor variations in the course ofthe fugue, is the repeated initial note,followed by a cascade of sixteenthnotes which elaborate a descendingsequence a-g-fˇ-e. This is referenced bythe bass with its descending sequenceb-a-g-fˇ in measures 10 and 11, sound-ed against the rising major tetrachordd -e -fˇ -g in the alto. The second ref-erence is in measures 13-15, where thedescending scale steps are inverted tob-d in the elaboration by eighth notes.The second reference confirms the firstone.

The special significance of this juxta-position of the ascending major tetra-chord d-e-fˇ-g against the descendingminor b-a-g-f , is that the two sequencesare exact inversions of each other: theycontain the same sequence of steps, butin the opposite directions.

The same inversion is reaffirmed at acrucial transition, in measures 25-27 (notshown), where the bass voice’s descend-ing motion is continued to a decisivestatement of b-d, thereby reversing theoriginal inversion d -b and restoring das the base-tone.

This fugue is a good example of theabsurdity of all formal definitions of“key.” To the question, in what key thefugue is actually written, the textbookanswer would be, “in D major, ofcourse.” Yet, such an answer is incom-patible with the entire effect of thefugue. Nor could we characterize thecomposition adequately by simply call-ing it B minor. It were more accurate tospeak of a B minor seen through D, or aD major/B minor “mode,” developingthrough inversions around the intervalb-d .

Inversion and the Lydian/Major-Minor Mode

To conclude this discussion of inversion,let us look ahead toward the genesis ofthe more advanced conception which isexemplified by such later works asMozart’s C major/minor Fantasy K.475, and Beethoven’s late string quar-tets.

47

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FIGURE 2.5a

J.S. Bach, ‘Gratias’ from Mass in B minor

Those compositions embody a fun-damental discovery, which integratesthe major and minor modes of the well-tempered system into a new principle ofcomposition, sometimes called the “Lydi-an major/minor mode.” That discovery,which involves not one, but many prin-ciples of composition, will be elucidatedfrom various angles in the following sec-tions. Here, we focus on one aspect ofthe relationship of major/minor with theprinciple of inversion.

First, we should emphasize, that theentities we call “keys” and “modes” arenot formal constructs, but—to theextent they mean anything at all—signi-fy sets of assumptions or hypothesesgoverning specific phases of composi-tion. The difficulty is, that the assump-tions involved, cannot be identified withspecific scales or other literal feature insome formal, “algebraic” fashion. Thus,it is easy to demonstrate that the mostelementary and ubiquitous features ofJ.S. Bach’s music are incomprehensiblefrom the standpoint of any formalnotion of musical key. The assumptionsand hypotheses are not located in thenotes, but in the thinking process“behind the notes.”

That said, the characteristic distinc-tion of hypothesis between (for example)C major and C minor would seem to liein the different manner of formingthirds from C and its closest relations, Fand G. So, C major features the majorthirds (more appropriately termed inGerman “große Terzen” or “greatthirds”) C-E, F-A, G-B; while C minor

features the “kleine Terzen” (“smallthirds”) C-E˛, F-A˛, G-B˛. Looking atthe intervals between these intervals,note that E, A, and B are neighbors byfifths, as are E , A , and B . Moreover,the first group is related to C by a seriesof successive upward fifths:

C-G-d-a-e -b ,

while the second group is related to C bya downward or inverted series of fifths(i.e., by fourths):

c -f -b -e -A .

Related to this, the downward tetra-chord in C minor, C-B˛-A˛-G, is theexact inversion of the upward tetrachordin C major: C-D-E-F. In this sense, Cmajor and C minor appear related toeach other by a series of inversions.

The crucial missing singularity,needed to bring together C major and Cminor in a closer unity, is expressed inF , or rather the interval C-F , which isthe pivotal singularity of the wholemusical system, corresponding to thearithmetic-geometric mean of the octaveC-c and the anchor for the whole arrayof bel canto register-shifts. If we adjoinFˇ, then we obtain a notion of Cmajor/minor as a multiply-connectedmanifold which “grows” from C in bothdirections, ascending and descendingfifths, as follows:

(ascending) C → G → d → a → e → b → fˇ

(descending) c → f → b˛ → e˛ → a˛ → d˛ → G ,

the latter (Fˇ and G˛) belonging to thesame tonal corridor in the well-tem-pered system. Thus, the coherence andconnectivity of the manifold lies in theso-called Lydian interval, C-F , which isthe anchor of our new major-minormode.

The ascending set of tones forms ascale

C-D-E-F -G-A-B-c,

which coincides with the so-called Lydi-an mode in the ancient Greek musicalsystem, and is characterized by the cru-cial interval C-F . The descending set oftones forms a second scale

C-D -E -F-G -A -B -c

which is the exact inversion of the first,Lydian scale.

Now, some might shrug their shoul-ders at this, pointing out that all this isnothing more than the “circle of fifths”producing a perfectly symmetrical,chromatic, twelve-tone scale. Thisabsurd conclusion completely ignoresthe bel canto principles determining anon-algebraic, non-equal-tempered sys-tem, principles which determine theunique, pivotal role of C-F .

The result, as the use by Mozart andBeethoven of this “Lydian major/minormode” demonstrates, is not to lessen thesense of tonality in music, but actually togreatly strengthen it. This remark is

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FIGURE 2.5b

Schematic of ‘Gratias’ from J.S. Bach, Mass in B minor

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.

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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.