That the creation and re-creation of great Classical. What Mathematics Can Learn From Classical Music THE CREATIVE PRINCIPLE IN ART AND SCIENCE

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1 Clik here or Full Issue o Fidelio Volume, Number 4, Winter 1994 YMPOIUM THE CRETIVE PRINCIPLE IN RT ND CIENCE The Royal Colletion 1994 Her Maesty Queen Elizabeth II What Mathematis Can Learn From Classial Musi by Brue Diretor Heard melodies are seet, but those unheard are seeter John Keats That the reation and rereation o great Classial musial omositions is a soure o human hainess, is undeniable Yet, suh hainess annot be relegated to the domain o leasurable distrations In at, it is in the domain o great Classial musial omosition, that human beings an and do disover undamental riniles o human knoledge and o the hysial universe itsel s ill be shon belo, by ay o examles and otherise, the soure o human hainess assoiated ith reating, erorming, and listening to great Classial musial omositions, arises solely rom the mental roess o reative disovery assoiated ith suh omositions Consequently, a rigorous examination o the riniles o reative disovery, embodied in Classial musial omosition, has roound imliations ar beyond the domain o art Fundamental issues onronting mathematis and hysial sienes are dealt ith in the realm o musial omosition Muh eort has been made by the Venetianled enemies o human knoledge to obsure, degrade, and destroy the siene o musial omosition On the one side, the soalled Romantis and modernists have degraded musi to the level o ure sensuality, here Leonardo da Vini, Netune ith Four eahorses hiller Institute, In ll Rights Reserved Rerodution in hole or in art ithout ermission stritly rohibited

2 musial develoment is loered to a Benthamite leasure/ain rinile uh evil eorts redue musi to mere alternating eroti eisodes o dissonane (ain: tension) and onsonane (leasure: release) On the other side, attemts have been made to sterilize musi into an axiomati array o ixed rules and las Unortunately, most erormers and audienes today suer the eets o these deliberate orms o debasement Friedrih hiller, in Kallias, or On the Beautiul (letter to Gottried Körner, Feb 19, 179), deines this issue in terms o aesthetis: The eret, resented ith reedom is immediately transormed into the beautiul It is, hoever, resented ith reedom, hen the nature o the thing aears harmonizing ith its tehnique, hen it looks as i it ere loing orth voluntarily rom the thing itsel One an also briely exress the reeding so: n obet is eret, hen everything maniold in it aords ith the unity o its onet; it is beautiul, hen its eretion aears as nature The beauty inreases, hen the eretion beomes more omlex and the nature suers nothing thereby; or the task o reedom beomes more diiult ith the inreasing number o omounds and its ortunate resolution thereore, even more astonishing ll Classial musial omosition is inherently aradoxial The beauty o a musial omosition inreases ith the ability o the omoser to maintain a unity o eet throughout the omosition, hile inreasing the omlexity (density o hange) in the omosition Hoever, it is imossible to state exliitly the uniying idea o the omosition; it exists only in the mind o the omoser, and subsequently, in the minds o the listeners, as hat Lyndon LaRouhe has deined as a thoughtobet This thoughtobet is as real as the notes that are erormed and heard, but it annot itsel be heard in the audible domain This thoughtobet is ommuniated as a roerly erormed musial omosition that is unolded by the erormers over a inite eriod o time, rom the oening notes to the inal adene s the iee unolds, the listener is led on a ourney o disovery t the very moment ater the inal adene, the entire omosition is enolded in the mind o the listener and grased as a virtually simultaneous hole That uniying enolding ours only in the sovereign mind o the listener It is an irreduible One, yet it an ome into existene only in the mind o the listener, as a result o the unolding o the omosition over time The omosition exists in time, yet its entral idea exists in virtually no time It unolds through audible tones, yet its subet, that rom hih it is generated, is inaudible The oer o a musial omosition is assoiated ith Middle C F do re mi a sol la ti do re mi a sol la ti do re mi a sol la ti do C C C D E F G B C C FIGURE 1 The six seies o the human singing voie F Contralto orano lto (mezzosorano) Eb E n F F D C D C D G Eb E n F F BC G * b n B B Bass D F Tenor Baritone G B B C FF BC F Eb E n b C D G First register eond register Third register Fourth register * Mezzosorano Verdiana is not stritly a ourth register 8

3 the ability o the omoser to reate inreases in beauty or the listener, as identiied above by hiller But suh inreases in beauty are not ahieved by simly making omositions more omlex ny inreased omlexity a Many must result only rom the reation, irst, o a higher onetion a One rom hih the omlexity is unolded Paradoxially, this roess is heard by the mind in reverse: the omlexity o the omosition is hat is heard; the higher onetion, the One, arises in the mind o the listener, but is unheard uh inreases in oer an themselves be ordered aording to hat Georg Cantor deined in mathematial terms as ardinalities Eah higher ardinality is haraterized by a unity (One), o inreasing omlexity (Many) Eah higher ardinality subsumes all loer ones Eah suessive higher ardinality is generated as a solution to a undamental aradox existing in its redeessor uh aradoxes are ereived by our senses, in this ase hearing, as mathematial disontinuities or singularities s e ill see, the generation o rising oers (ardinalities) o reative thought, is the subet matter o great Classial musial omosition The Physial Priniles o WellTemered Polyhony Beore turning to the musial examles, let us irst examine some o the hysial roerties o the audible tones o the elltemered system o olyhony hih externally bound all musial omositions The seven otaves eah divided into telve tones o the Classial elltemered system o olyhony, are hysially determined singularities o hysial saetime Contrary to the raud eretrated by Hermann Helmholtz, hih sadly is generally aeted by most musiians and hysiists today, these tones are not arbitrary values hosen merely by him Eah tone is a oreree region o hysial saetime, hose value (requeny), like Keler s values or the orbits o the lanets in the olar ystem, is determined by the harateristi roerties o a quantized ield underlying hysial saetime The seii requenies o these tones, entered on middle C = 256 Hz, are revealed by disovering the biohysial roerties o the bel anto qualities o the human singing voie These disoveries sho that there are six searate seies o the human singing voie: sorano, alto (mezzosorano), ontralto, tenor, baritone, bass Eah seies has a harateristi set o hysiologially determined regions here the singer, in order to maintain a beautiul tone, must shit rom one register to another FIGURE 2 Intervals in the elltemered system Maor mode sending Desending Minor mode sending Desending b b b b b b For examle, hen C = 256, the sorano and tenor must shit rom their middle to high register beteen F and Fs [EE Figure 1] When to or more voies, either a aella or in ombination ith musial instruments, sing together, it is quikly disovered that a system o tuning is required hih is not determined by ratios o hole numbers uh a nonlinear system o tuning is knon as the elltemered system, and as most thoroughly elaborated by J Bah The totality o intervals in the elltemered system o olyhony an be bound together onetually, by a uniying ordering rinile (order tye) The most generalized order tye, is assoiated ith the maor/minor mode This order tye, elaborated by J Bah, is identiied ith the tentyour key, elltemered system (or, ortyeight key, i asending and desending are onsidered, as they should, to be dierent seies) [EE Figure 2] Consequently, disoveries in the musial domain are valid disoveries o obetive hysial riniles, as ell as subetive ones The audible tones o the elltemered system are externally bounded by a quantized ield hih is itsel unheard ; yet its roerties are nonetheless disoverable through the audible realm In the orksho o musial omosition, the bounding harateristis o this unheard quantized ield are revealed by the behavior o the singularities that emerge as musial aradoxes This roerty enables the omoser to ous the mind o the listener not on the notes, but on hange For examle, the listener never hears a single tone in a musial omosition The listener alays hears the interval beteen tones, that is, the hange beteen tones nd yet, in the elltemered system, an interval beteen to notes does not exress ust one tye o hange Look at the interval o a ith in the key o C maor/minor, ie, CG, shon in Figure 2 sending rom CG, the listener hears an asending ith This simle interval divides the otave into to distint intervals: 9

4 FIGURE Tones o the elltemered system, reresented as divisions o a siral The intervals orresond to degrees o rotation, here all halstes have equal rotation, although their arlengths inrease G b G G b F F E Fith G b G G b F F E Fourth G b G G b F F E Fourth B b C C Db D E b D Fith B b C C Db D E b D Minor ixth B b C C Db D E b D Maor Third B C B C B C G b G G b F F E G b G G b F F E Fourth Maor ixth B b C C Db D E b D Minor Third Fith B b C C Db D E b D D B C B C a ith (CG) and a ourth (GC) Going in the reverse diretion, ie, desending rom C, generates the interval CF, ie, a eret ith n asending ourth, CF, is dierent rom its reverse, desending rom F to C, beause F to C imlies the desending ith In addition, any artiular interval is hanged in relation to seii keys For examle, in the key o C, the ith CG, is the interval beteen the irst and the ith notes o the sale In the key o F, this same ith, CG, is the interval beteen the ith and the ninth (seond) In the key o G this same ith, is the interval beteen the ourth and the otave In the key o D, this ith is the interval beteen the ourth and the diminished seventh Thereore, a seemingly simle interval is not really so simle ater all imilar ambiguities exist or other divisions o the otave [EE Figure ] In the maor/minor mode, sequenes o intervals an be reordered, or inverted, aording to a seii rinile For examle, the irst ive notes o the maor mode an be inverted, thus generating the minor mode, and vie versa The maor mode an be inverted at the otave to generate a minor mode o a dierent key [EE Figure 4] singularity o signiiant imortane is the interval hih divides the otave exatly in hal, misnamed the devil s interval or tritone, hih ere better alled the Lydian interval or reasons hih ill beome lear later in this reort [EE Figure 5] The Lydian interval is the only interval hih annot be generated by the rinile o inversion o omlementary intervals ithin any given key This interval uniquely divides the otave exatly in hal: that is, the interval rom the toni to the Lydian tone is the same amount o hange as the interval rom the Lydian tone to the otave In the key o C maor/minor, or examle, this orresonds to the interval beteen C and Fs, hih also orresonds to the hysial singularity o the register breaks in the sorano and tenor singing voie [Figure 5(a)] Divide an otave in hal This generates a Lydian interval In the maor/minor mode, the Lydian interval is a dissonane ith reset to any given key For exam 40

5 le, in the key o C maor/minor, the interval CFs is suh a dissonane Yet this Lydian interval has the unique roerty o being a athay rom one key to the next, by ay o the leading tone o that next key (FsG in the key o G maor/g minor) [Figure 5(b)] It is a tye o singularity to be resolved through the develoment o the omosition When the otave is divided in hal again, to Lydian intervals are reated There are only three suh ombinations ossible in the elltemered system [Figure 5()] n analogy an be made ith Cardinal Niolaus o Cusa s treatment o the transendental nature o π When a irle is divided in hal by irular ation, a diameter is generated, the ratio o hose length to the irumerene o the irle, is π Cusa shoed that hile the value o π an be alulated to any deimal value, its exat value is not determinable, beause the diameter, a straight line, is o a dierent seies than the irle When to suh dierent seies are omared, a mathematial disontinuity, or singularity, results imlarly, hen an otave is divided in hal, the Lydian interval, a singularity, is generated hih is transendental ith reset to that otave Make a artial list o the audible singularities o the elltemered system o olyhony: Inversions sending/desending Intervals Comlementarity o Intervals Lydian Intervals Voal Registration/Instrumental Registration FIGURE 4 Inversion o the maor mode generates a minor mode o a dierent key Maor b C Maor Minor F Minor b 1 b 1 1 b b No ste bak and think o a given key, or examle, C maor, as a unity ee the singularities No think o C maor along ith C minor, and think o these singularities ane No think o all tentyour keys as a hole No think o all tentyour keys as asending and desending, ie, all ortyeight keys With eah ne mental at, the mind generates a suession o order tyes But the un is ust beginning What has been desribed above is simly the natural beauty o the elltemered system o olyhony, resented to us by God and disovered by that hih makes us in the image o God, reativity No e must turn to the great sientii geniuses o musial omosition, ho ontinue God s reation ith orks o artisti beauty We ill illustrate the riniles outlined above ith examles rom our string quartets For the uroses o our illustration, e ill examine here only the generative riniles out o hih eah o these omosi 1 2 FIGURE 5 (a) The Lydian interval (b) Leading tone FsG in the key G maor/g minor () Possible Lydian intervals in the elltemered system Leading Tone (a) Lydian G b B b G B G b F C C F C Db E Lydian E b D D (b) Lydian G b B b G B G b F C C F C Db E Lydian E b D D () b G B b G B G b F C C F C Db E D E b D 41

6 FIGURE 6 Jose Haydn, tring Quartet O, No in C maor, measures 16 1 Violin I llegro moderato Violin 2 B Violonello? Lydian interval n Ṡ tions develos, or the otential develoment o eah omosition is ontained entirely in the generative idea hih reates bounding onditions on it It is the omoser s genius to disover and reveal this otential s the omosition unolds, the singularities embedded ithin the generative rinile are subeted to transormations by the omoser, and the listener hears develoment in the omosition in terms o the hange assoiated ith these transormations By omaring the generative material rom omositions o three omosers enomassing the san o aroximately orty years, it is ossible to see the suessive breakthroughs in human onetual oer assoiated ith these reative disoveries The string quartet [to violins, viola, and violonello ( ello)] virtually ame into existene through the ork o Franz Joseh Haydn This medium allos or maximum musial transareny, beause o the registral harateristis o these stringed instruments hen made to sing by erormers amiliar ith the riniles o bel anto singing This ensemble o our voies atually omoses a ith voie, that o the quartet as a hole The string quartet serves both as the ore o the entire symhony orhestra and as a test bed or omositional innovation These illustrations, hile neessarily artial, nevertheless should serve as a suiient guide to the reader, to hom e leave the task o exloring, ith the ears and the mind, the ull rihness o these omositions Haydn s Ous, No Franz Joseh Haydn s Ous, the olletion o six string quartets omosed in 1781, reresented a maor breakthrough in human knoledge, ith the develoment o a ne rinile o musial omosition alled Motivührung, or motivi develoment Here Haydn takes an extremely simle idea and subets it to transormations aross the musial sae The eet is to enable the listener to hear a greater density o hange, hile maintaining a unity o eet Look at the oening seventeen measures o the irst FIGURE 7 Jose Haydn, tring Quartet O, No in C maor, measures Violin I B? Violonello b n Ṡ 42

7 FIGURE 8 Jose Haydn, tring Quartet O, No in C maor, measures Violin I B? Violonello b b b n n Ÿ ḟ movement o O, No in C maor, the Bird In measure 1, the viola lays reeated eighthnote E s ith the seond violin laying reeated eighthnote C s above [EE Figure 6] This interval divides the otave into a maor third and a minor sixth In measure 2, the irst violin enters on a G above as a hole note, adding the additional interval o the eret ith, and imliitly the minor third and maor sixth In measure, the irst violin adds a little grae note Fs to its G, introduing the interval hih divides the otave exatly in hal lso, the irst violin no divides the measure rhythmially in hal s e identiied above, this interval (CFs) is at the boundary ith the adaent key o G, and is dissonant ith reset to C Finally, on the irst beat o measure 4, the irst violin extends the grae note rom a halste belo the G, to a holeste above o, in the irst three measures, Haydn has, ith the simlest means, resented the listener ith the rimary divisions o the otave in a ondensed idea (the ith, ourth, maor third, minor third, maor sixth, minor sixth, halste, and holeste) In measures 4, 5, and 6, these divisions are simly unolded The ello lays the CEG o the viola, seond violin, and irst violin, as an asending areggio The irst violin lays a desending lourish ith the Fs hanged bak to F natural, thus undersoring the aradoxial harater o the initial Fs grae note The rest at the end o measure 6 gives the listener a hane to no hear, in his mind, the irst six measures as a One In measures 712, the quartet roets the same motivi idea to a dierent lae, hih hanges the intervalli relationshi beteen the instruments and generates a ne array o singularities [EE Figure 7] The viola is dislaed u a halste, rom E to F; the seond violin is dislaed u a holeste, rom C to D; and the irst violin is dislaed u a holeste, rom G to No the interval beteen the viola and the seond violin is a minor third/maor sixth, the omlement o the maor third/minor sixth o the irst measure The grae note no beomes a Gs, thus reserving the Lydian interval FIGURE 9 Jose Haydn, tring Quartet O, No in C maor, measures Violin I b b B? Violonello b b b b J b b b b b b b b n n n n n 4

8 ith the seond violin s these next six measures unold, a ne unity is generated in the mind o listener, hih is dierent rom the One assoiated ith the irst six measures, but, beause it is generated out o the irst One, the mind easily uniies both ideas into a ne One hih is assoiated ith the hange beteen the to setions The mind abstrats rom this hange that hih is universal, but unheard The rest at the end o measure 12 direts the mind o the listener toard this obetive In measures 117, a still neer transormation is generated out o the revious telve measures, hih, hen releted in the mind o the listener, generates again a ne One, ausing the mind to abstrat again a ne universal rom this ne degree o hange [EE Figure 8] s they unold in time, eah transormation is disontinuous rom the one reeding, yet thinking bak, one hears the transormations as a ontinuous roess o hange Haydn laes no rest at the end o measure 17, and rom measure 18 onard this roess ontinues, unolding the imliations o the initial idea, hile the mind enolds these develoments into an unheard One, hose beauty inreases as the eretion beomes more omlex and the nature suers nothing thereby In measures 667, the density o hange is greatest as the imliations o the initial motivi ideas are no themselves the subet o develoment [EE Figure 9] FIGURE 10 W Mozart, tring Quartet K 464 in maor, measures 124 Violin I B Violonello? B 9? B 17? llegro n J n J n n n n n n n n J n n n n n n J n J n n J n J n n n n J n n n n n n 44

9 The movement ends ith the restatement o the initial idea, but this time it is heard in an entirely dierent ay Mozart s tring Quartet K 464 In resonse to Haydn s Ous, Mozart, over the next e years, omosed six string quartets hih integrated Haydn s develoment o the Motivührung, ith J Bah s disovery and elaboration o the roerties o the maor/minor mode Mozart dediated these six quartets to Haydn O artiular interest is the string quartet K 464 in maor, or it is knon that Beethoven studied this quartet losely Here Mozart inreases the density o hange by generating the omosition rom a more ondensed motivi idea [EE Figure 10] Notie the ay Mozart onstruts the initial idea suh that a seemingly simle, tomeasure idea has embedded ithin it a undamental aradox In measure 1, the quartet begins ith a simultaneous division o the otave into a maor third and a eret ith (CsE) In the last hal o the seond beat o measure 1, the irst violin desends one halste to a Ds, hile the other instruments hold their original notes The listener is resented ith the singularity o the Lydian interval hen he hears this halste hange rom E to Ds in relation to the s o the ello and viola No the violin, laying alone, asends u a minor third to Fs, taking it aross the sorano s register shit beteen F and Fs Then it desends steise a ourth to Cs (laying a D natural in ontrast to the revious Ds) Then it asends again a minor third to the E on hih it started, and then it desends steise a ith to an, here it is oined by the other instruments hih, hen ombined, lay the same notes as those on hih the iee began, exet, signiiantly, in a dierent order (CsEE) This time the E is doubled instead o the From measure to measure 4, all our instruments lay simultaneous quarter notes The ello desends steise a ourth (CsBGs); the viola asends steise a ith, to B (EGsB); the seond violin, starting rom the same note (E) as the viola, reverses the viola s diretion, and desends to the B an otave loer than the viola, a steise ourth (EDCsB) To summarize, both the viola and the seond violin start on the same note (E) and end on the same note (B), but an otave aart Thus, the seond violin asends a ith, hile the viola desends the omlement, a ourth The irst violin then asends steise a ourth (B CsD) In addition, the seond violin also lays the same E ith eah note Even though this E doesn t hange its ith, it is heard dierently every time it is layed, beause everything around it hanges ne Lydian interval, GsD, is ormed on the seond beat o measure, beteen the viola and seond violin, and on the irst beat o measure 4, beteen the ello and the irst violin ll instruments rest on the seond beat o measure 4, direting the listener to bind hat has ust been heard, into a One Look at the tyes and density o audible singularities hih are resented in these irst our measures For examle, the Lydian interval in measure 1; the hanges in registration in measures 1 and 2; the dierene beteen asending and desending intervals, and the omlementarity o the ourth and the ith, in measures and 4 ll these singularities are embedded in hat Mozart has organized into a seemingly simle idea, hih the listener then rereates in his mind as a thoughtobet hih itsel is unheard From the last beat o measure 4 to the irst beat o measure 7, the quartet subets hat has ust been heard to a transormation hih rodues a signiiant dierene in the behavior o the singularities reviously generated The statement oens ith the intervals generated in measure (EGsBD) hih ontain the Lydian interval (GsD) ormed in measures and 4 The irst violin then lays the same intervals in measures 5 and 6 as it did in measures 1 and 2, but dislaed don one ste, starting on D instead o E This dislaement hanges the behavior o the singularities thus rodued The violin lays: desending halste (DCs), asending minor third (Cs E), steise desending ourth (EB), asending minor third (BD), steise desending Lydian (DGs) (instead o desending steise ith, as in measure 2) In oosition to the oening, the ne Lydian interval is this time heard in the notes layed by the hole quartet at the beginning o the statement, but it disaears hen the irst violin lays alone Note the generative quality o the halste interval D CsDCs in the irst violin in measures 5 and 6, in ontradistintion to the generative quality o the intervals E DsEDnatural in measures 1 and 2 Measures 7 and 8 imitate measures and 4, but again ith signiiant dierenes The irst violin asends steise a ourth, as in measure, but this time starting a halste loer (GsBCs) (O ourse, this hanges the internal relationshis beteen the stes rom holeste, holeste, halste, in measure, to halste, holeste, holeste here) The seond violin asends steise a ourth (EFsG), in ontrast to the steise desending ourth rom the same note in measure The 45

10 FIGURE 11 W Mozart, tring Quartet K 464 in maor, ourth movement, measures 14 B 1 Violin I llegro Violonello? n ello takes the E rom the seond violin in measure, but lays this note as one ontinuous tone, instead o our searate ones, and then desends a ith to The viola desends a minor third by halstes (EDsDCs) Notie the ormation o the to reviously heard Lydian intervals, Ds on beat to, and, DGs on beat three The seond beat o measure 8 is a rest, to allo the listener to orm hat has ust been heard into a ne thoughtobet s e have seen, measures 58, being a transormation o measures 14, seem to imitate them, but ith signiiant dierenes Those dierenes are 1 Violin I llegro B 6 8 Violonello? B? n n π π π heard as hanges in the behavior o the audible singularities resented No the listener orms in his mind an abstration o hat is the same, and hat has hanged, beteen these to thoughtobets, a ne One hih, hile unheard, is as real as the audible tones rom hih it as generated From the last beat o measure 8 to the irst beat o measure 12, all our instruments lay the same notes (ith the ello laying an otave loer) uh a unison ouses the listener intensely on this more ondensed orm o hat has been heard in the revious measures In measure 9, the quartet asends a maor third (Cs) taken rom the very irst notes o the iee; then asends and desends the halste CsD (the generative interval o measures 5 and 6); and then desends a holeste, to end on a B In measure 10, ater a rest, the quartet, draing on the generative interval o measures 1 and 2, starts on that B, and again asends a maor third (BDs), asends again a halste (DsE), but then desends a hole ste to Dnatural, and then desends a halste to Cs This is a ondensed and more generalized restatement o the singularities embedded in the irst nine measures; yet, suh a restatement ould have been imossible ithout hat has been reviously unolded ter another rest, the quartet lays the oening notes (CsE) in dierent registers, and in measure 1, the π ṗ ṗ π n π π n n π b n b n n FIGURE 12 Ludig van Beethoven, tring Quartet O 59, No 2 in E minor, measures

11 irst violin asends the maor third (Cs), then desends steise an otave and a ourth, as the quartet omletes the initial develoment on the same notes an otave loer on hih it began (CsE) [measure 16] Think bak on hat has been unolded over these oening sixteen measures The iee began ith a seemingly simle statement, rih in singularities This statement as then subeted to a transormation, resenting to the listener aradoxially hange and no hange Then, the generative rinile o the transormation as abstrated out, and ut in ontradistintion to the generative rinile o the original statement Finally, everything as brought together in a onluding statement Nothing as arbitrary Every develoment arose out o the oening idea hih Mozart had so skillully hosen Eah ste required ne oers o thought or the listener The listener s hainess inreases as he disovers the ease ith hih the oers o his on mind rise, seemingly eortlessly, on the beautiul tones o Mozart s omosition nd this is only the irst sixteen measures! The reader is enouraged to omare the kind o hange reresented here, ith hat as examined reviously in the Haydn examle ae limitations revent us rom ully exloring this omosition We ill, hoever, indiate several laes or the reader to exlore on his on Look at the last beat o measure 16 Here Mozart hanges the Cs to a Cnatural, bringing us into the minor mode The oening idea is no subeted to even more raid transormations in measures 1624 Here, Mozart ahieves a density o hange hih is not ahieved in the Haydn examle until measure 66 [EE Figure 10, measures 1624] In addition to the inreased density o hange in this Mozart examle, there is a unity aross the movements, extending the Motivührung rinile to the entire omosition For examle, the entire last movement, is generated rom the three desending halstes EDsDCs (minor third) in measures 12, stated by the viola in measure 7 o the irst movement, embodying the initial aradox stated in the irst movement, but no heard by the listener in an entirely dierent ay [EE Figure 11] Beethoven s Ous 59, No 2 ter extensive study and develoment o all revious musi, Beethoven extended human knoledge urther, reahing the greatest omositional oer in his late string quartets To more adequately gras the signiiane o those develoments, let us irst sto along the ay and see the ormation o his later ideas in their earlier stages In 1806, Beethoven omosed three string quartets dediated to the Russian Count Razumovsky, ublished under Ous 59 In these quartets, Beethoven exlored the boundary onditions o the maor/minor mode, embarking on a ourse hih ultimately led to his disovery o a higher ordering rinile o musial sae, elaborated in the late quartets look at O 59, No 2 in E minor illustrates the oint The irst movement oens ith to hords layed by all our instruments o the quartet (EBEGBEE minor; DsBEBFsBB maor) marked orte and searated by an eighthnote rest [EE Figure 12] These to hords resent the listener, imliitly, ith the boundary onditions o the maor/minor mode The ull measure rest [measure 2] direts the listener to orm in his mind a uniied onetion o these onditions In measures and 4, measure 1 is unolded by the irst violin and ello laying in otaves, a desending ourth, EB, asending minor sixth, BG, and asending maor third, GB The irst violin ontinues, asending to C, and then laying three raidly desending minor thirds (CFsDs), a desending maor third (DsB), and an asending ourth (BE: the inverse o the irst interval o measure ) Embedded ithin this desending violin line are to Lydian intervals (CFs, Ds: the otave divided in hal tie) The ull measure rest that ollos [measure 5] direts the listener to enold into a One hat has ust been unolded In measures 6 and 7, oined no by the viola, this same idea is roeted u a halste to F, ith the ello droing into a loer register This roetion hanges the omlementary intervalli relationshis o measure : desending ourth (FC), asending maor sixth (C), asending minor third (C) The order o the intervals in the desending violin line in measure 7 is hanged to, irst, a desending maor third (DB), olloed by to desending minor thirds (BGE), a desending maor third (EC), and an asending ourth (CF) Notie the invariane o the ourth hile the other intervals hange The ull measure rest that ollos [measure 8] no direts the listener to hear as a One hat this roetion has hanged, and not hanged, and to abstrat in the mind that hih is universal about this develoment In measure 9, Beethoven resses the boundaries o the maor/minor mode The viola begins alone on Fs, u another halste rom the revious setion, but instead o desending a ourth as in measures and 6, the viola desends a minor third, as in the violin line o measures 4 and 7, bringing it to the Ds o the ello in measure 1 Then, instead o asending a maor or minor sixth, the 47

12 FIGURE 1 Ludig van Beethoven, tring Quartet O 59, No 2 in E minor, seond movement, measures 18 B 1 Violin I Molto dagio Violonello? ṗ ṗ i tratta questo ezzo on molto di sentimento ṗ ṗ n n n n n viola asends a Lydian interval to Beause the viola kees the same rhythmial attern as in measures and 4, the listener hears this intervalli hange in relation to those measures The line ontinues, uming no to the irst violin, hih in a ne register begins on Ds as the viola holds its Thus, the listener hears the unique halving o the otave by the Lydian interval (DsDs) Folloing the viola, the irst violin desends a minor third (DsC), and then asends a Lydian interval to Fs o the sequene ust heard is FsDsDsCFs Instead o the ull measure rest that olloed every revious statement, the quartet no rests or the irst hal o measure 10 on the seemingly unstable Lydian intervals no layed together (DsCFs) In the seond hal o measure 10, the irst violin desends a halste to Bnatural, resolving one o the Lydian intervals into a ith (BFs) This ne tye o rest direts the listener to orm a ne tye o One In measure 11, the irst violin begins here it last ended on Fs, and lays, an otave higher, the viola art rom measure nine (FsDs) The seond violin lays dierent intervals starting on Ds, desending a maor third to B, and asending a ith to Fs This is olloed by the ello, hih reserves the intervalli relationshis o the irst violin, but starting on, desending a minor third to Fs, then asending a Lydian to C In measure 12, the quartet rests again, on the same notes as in measure 10, but in a dierent order (CDsFs) Taking advantage o the leisure aorded us by our ritten medium, let us summarize hat has ust been unolded In measure 1, the quartet resented us ith the minor mode, and ater a rest, e ere resented ith the maor mode We ere given a ull measure rest, to orm in our minds something that had not been stated in this ombination, the thoughtobet o the maor/minor mode Then this as unolded, irst rom E in measures and 4, then a halste higher in measures 6 and 7 Eah time, ne singularities imliit in measure 1 ere revealed This hange aused us to orm a ne thoughtobet in our minds The Lydian intervals embedded in measures 4 and 7, ere then unolded in measures 9 through 12, hih hange aused us to orm yet again a ne thoughtobet This roess ontinues throughout the movement For examle, in measure 19, the Lydian FIGURE 14 Ludig van Beethoven, tring Quartet O 59, No 2 in E minor, seond movement, measures 447 B 4 Violin I Violonello? n n n n n n n n 48

13 FIGURE 15 Ludig van Beethoven, tring Quartet O 59, No 2 in E minor, seond movement, measures 5255 B 52 Violin I Violonello? Ÿ~~~ J ale n intervals arrived at in measure 10, are no stated as the oening hords o measure 1, but hih no are themselves unolded in yet another ne ay hih is itsel not ossible ithout the develoment that reeeds it The reader should no omare this oening ith the Haydn and Mozart examles ee the greater omlexity (density o hange), hile the unity o eet is not lessened, indiating to us a higher unheard idea To get a sense o hat otential has been develoed in this oening, let s look briely at the seond movement, marked, Molto dagio This iee is to be treated ith muh sentiment, hih resents a urther develoment but in an entirely dierent mood rom the irst movement, here the unolding roeeds more sloly and ontemlatively [EE Figure 1] This movement has to arts, hih, ombined, orm an unheard third idea s e ill see, Beethoven ill turn bak to this orm in his late quartet, O 12 In the irst seven measures, the quartet untions as a our voie horus, resenting various singularities o the maor/minor mode This is olloed by an unolding In measures 5255, this horus returns, olloed by a ne unolding In measures 18144, this horus returns again, olloed by a inal unolding Eah time the horus returns, the unolding hih ollos is hanged In measure 1, the irst violin begins ith a desending halste (EDs), extrated rom the ello in measure 1 o the irst movement Eah voie enters, in the lo register, a ourth belo and to beats ater the revious entrane This is heard in reerene to measures and 6 o the irst movement The ello enters on Cs and, like the irst violin, desends a halste to Cnatural; this leads in measure to the Lydian intervals (CFsDs), hih are again heard in reerene to the irst movement s develoment In measure 4, the seond violin desends to halstes (EDsDnatural) When the Dnatural is heard, the rest o the quartet stos Instead o a omlete rest, as in the irst movement, to allo or the ormation o a One, here the listener hears both the Dnatural and the unheard rests When the irst violin reenters in measure 5 on a B, layed no in the middle register, this ombination o the heard and the unheard, reates an ambiguity hih allos the listener to hear the maor and minor as one uniied mode This is the inverse o the oening o the irst movement (E minor/b maor in the irst movement; E maor/b minor here) On the irst beat o measure 7, the horus resents the listener ith a ne Lydian interval (Es), hih orms the boundary ith the next key, B maor This interval is layed orte (loud) or emhasis, and in ontrast to the iano (sot) layed both beore and ater it What ollos is a beautiul unolding beginning on B, hih is best heard diretly by the reader In measures 447, the irst violin ounteroses this transition to B ith all telve tones o the ell temered system [EE Figure 14] In measures 5255, the horus returns, this time in a higher register [EE Figure 15] When the n π FIGURE 16 Ludig van Beethoven, tring Quartet O 59, No 2 in E minor, seond movement, measures Violin I B Violonello? n n n 49

14 FIGURE 17 Ludig van Beethoven, tring Quartet O 59, No 2 in E minor, seond movement, measures B 18 Violin I Violonello? n FIGURE 18 The Lydian mode J ale Lydian n unolding resumes, the s has hanged bak to an In measure 12126, the irst violin ounteroses the notes o E maor to all telve tones o the elltemered system [EE Figure 16] In measures 18144, the horus returns a inal time, olloed again by a urther unolding oda [EE Figure 17] Eah horus resents a variation o the oening seven measures, but eah is heard in relationshi to the revious unolding This uxtaosition o suh seemingly dierent ideas, generates a ne thoughtobet in the mind o the listener Beethoven s Ous 12 roximately tenty years ater omosing the Razumovsky quartets, in hat ere to be the last years o his lie, Beethoven omosed a series o ive string quartets and Große Fuge, hih omrise the highest develoment o musial siene ahieved to this day In these quartets, Beethoven launhed a revolution in musial omosition, by elaborating a ne ordering rinile or the elltemered system, hih orresonds to a higher oer o thought The singularities o the maor/minor mode are seen in an entirely ne ay; the boundaries FIGURE 19 Ludig van Beethoven, tring Quartet O 12 in maor, measures Violin I ssai sostenuto C π C π B C π Violonello? C π 9 llegro n B? π π π π n π π π π n 50

15 beteen keys are dissolved and the entire elltemered system is onetualized under a ne more generalized order tye In mathematial terms, this ne ordering rinile externally bounds the maor/minor mode o the elltemered system t a uture time e ill examine more ully all these revolutionary omositions For no, let s look at Ous 12 in minor, hih inludes the amous third movement marked, Heiliger Dankgesang eines Genesenen an die Gottheit, in der lydishen Tonart ( holy song o thanks to the Divinity, in the Lydian mode ) The Lydian mode is a division o the otave based on the Fmaor sale ith a Bnatural instead o the B This ordering has the unique roerty, that the interval hih divides the otave in hal, identiied as the Lydian interval above, is art o the mode itsel, ie, FBnatural Hoever, the otential or omositional reedom is less than in the maor/minor mode, beause the Lydian mode laks the maor/minor mode s omlementary relationshi beteen the ourth and the ith [EE Figure 18] Ironially, Beethoven generates, by reerene to this ombination o the Lydian mode and the maor/minor mode, a higher modality, subsuming the roerties o both This higher modality or ordering rinile o the elltemered system, is neither the Lydian mode, nor the maor/minor mode, nor a simle summation o the to The higher ordering rinile orresonds to a higher oer o onetualization, hih Beethoven elaborates through the omosition o these last quartets The listener hears the results o this reative roess in the greater density o hange ahieved in the omositions t all times, hoever, Beethoven maintains a unity o eet, beause he adheres rigorously to all the riniles o Classial musial omosition established by the reative disoveries o his redeessors ter Beethoven s death, the soalled Romantis and moderns made every eort to destroy Beethoven s aomlishment, by laiming that Beethoven, in the late quartets, had liberated them rom the onstraints o Classial tonality They ere enamored ith the greater otential or sensual eets resulting rom this ereived liberation Not surrisingly, hat they rodued is best desribed as organized, or in some ases disorganized, noise Beethoven s ne onetion is exressed in the oening eight measures o Ous 12 Here, the quartet imitates a ourart horus o human singing voies, announing the ne idea in song [EE Figure 19] The iee begins ith the ello laying alone, in a lo register, and very sotly the asending halste Gs Eah tone o this interval is equal in time and volume, reating an ambiguity beause the listener annot tell hih one is rimary This interval reerenes hat a hysiist ould all a boundary ondition The Gs is the leading tone to the toni, and at the same time the Lydian interval to D But there is a urther ambiguity Is this interval that o the leading tone to the toni, or is it that o the ith to the minor sixth o C minor, or that o the third to the ourth o E, et? Or is it many things at one? In measure 2, the viola, in its lo register, reverses this interval at an otave higher (Gs), hih reinores the ambiguous nature o the oening interval t the same time, the ello, in a dierent register, also lays a desending halste, but on dierent tones (FE) This also reerenes a boundary ondition: the minor sixth to the ith, and the toni to the leading tone in F ided by the transareny rovided by the hange in registration the mind o the listener goes to laes simultaneously: GsGs ( ello to viola); and Gs FE ( ello) While the mind is orming an unheard idea rom the omarison o these to heard intervals, the seond violin enters, on the last to beats o measure 2, on a B in its lo register In measure, the irst violin, in its lo register, roets the oening interval u a ith, laying the asending interval DsE This also reerenes a boundary ondition, as Ds is the Lydian interval to the toni, and the leading tone to E imultaneously, the seond violin lays exatly the same tones as the viola in measure 2, the desending halste Gs; but, signiiantly or uroses o transareny, the dierene in tone beteen the viola and the violin auses this to sound dierent The viola, at the same time, roets the ello s FE u a ith, laying another desending halste (CB), hih reerenes a dierent but similar boundary ondition Measure 4 ontinues the roetions ust begun The irst violin ontinues the roetion u a ith rom the ello line, hanging registers and laying the desending interval CB an otave higher than hat as ust layed by the viola In imitation, the viola lays hat the irst violin ust layed, but dros an otave instead The irst violin returns to the lo, adding an Fs and a Gs Beore going on, let us look at hat has been generated in these irst our measures By simle roetion, inversion, and registration o the oening asending halste, Beethoven has uxtaosed a multitude o boundary onditions or the listener, hih enable him to orm in his mind an unheard onetion o the maniold rom hih these boundary onditions arise Evidening these 51

16 FIGURE 20 Ludig van Beethoven, tring Quartet O 12 in maor, measures Violin I B? Violonello boundary onditions are the Lydian intervals (Ds) rodued beteen the to violins in measure, and the Lydian intervals generated in measure 4 (Ds beteen the viola and seond violin, and FsC beteen the to violins) The ello reenters in measure 5, reeating measure 2 (FE) The viola ums to a high register and inverts the irst violin line rom the revious measure 4, laying the asending interval BC The seond violin, still in a lo register, lays the desending holeste DC, and the irst violin ums to a higher register and reeats its irst to notes (Gs) Measure 6 ontinues this roess o inversion, roetion, and registration The violin, still in the high register, mimis the viola s inversion o its on line, laying the asending halste BC The seond violin mimis the ello s FE, and adds a C The viola mimis the seond violin rom the revious measure (D C), and the ello mimis its on oening (Gs), hih is no heard in an entirely dierent ay rom hen those same notes ere irst resented to the listener With eah ne transormation, the listener hears an inreasing density o ombinations o boundary onditions, as evidened by the generation o Lydian intervals: in measure ive, FB beteen the ello and viola, and D Gs beteen the violins; and these same intervals in dierent order in measure 6, GsD beteen the ello and viola, and FB beteen the to violins Further roetions and transormations our in measure 7 The irst violin inverts itsel (CB) The seond violin inverts the oening ello notes (Gs) The viola mimis the seond violin rom measure 4 (FsGs) The ello dros to its loest register, roeting its oening notes (DsE) don a ourth,, hih ere themselves already layed by the irst violin as a roetion u a ith in measure gain, ne orderings o these boundary onditions are rodued, evidened by the generation o more Lydian intervals: Ds beteen the ello and seond violin; FsC beteen the viola and irst violin ne ordering is generated in measure 8, leading to the raidly desending minor thirds o the irst violin in measures 9 and 10 Relet again on these amazing eight measures Beause everything in them is generated by roetion, inversion, and registration o the ello s ambiguous oening to notes, there is a total unity o eet aross all voies and all measures nd, beause these to notes themselves reerene a boundary ondition, hile at the same time are ambiguous, there is a greater density o singularities rodued s a result o these to onsiderations, the mind is led to reate a ne and higher onetion, a reletion o Beethoven s ne ordering rinile hih is o a higher ardinality Comare these irst eight measures ith the revious examles o Haydn, Mozart, and Beethoven In those examles, the singularities ere embedded in the oen FIGURE 21 Ludig van Beethoven, tring Quartet O 12 in maor, measures Violin I B? Violonello n 52

17 ing intervals, and revealed as those intervals ere unolded Here the oening interval is itsel the singularity Having reated a ne and higher onetion in the mind o the listener, Beethoven no uts the listener to ork exloring the imliations o this disovery First, in measure 11, the ello ondenses the irst eight measures into a singing line layed in its highest register ( BCBGs), ending ironially on the leading tone (Gs), the same tone on hih the iee began These tones have been abstrated rom the oening eight measures, in the manner Cantor deines or the generation o transinite numbers; that is, eah tone layed by the ello in this measure, is heard in relationshi to the tonality deined in the irst eight measures For examle, in measures 10106, the entire quartet restates ortissimo the irst violin s oening line, hih itsel is a roetion rom the ello s oening [EE Figure 20] In measures 19202, the oening notes are uxtaosed to the ello s abstration, orming a ne thoughtobet rom the ombination o the to [EE Figure 21] FIGURE 22 Ludig van Beethoven, tring Quartet O 12 in maor, third movement, measures 14 Violin I B? Violonello 1 25 B? B? Heiliger Dankgesang eines Genesenen an die Gottheit, in der lydishen Tonart Molto adagio 1 sotto voe sotto voe sotto voe sotto voe ndante Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 8 Neue Krat ühlend ten J ale ten J ale 5

18 No, look at the reviously mentioned third movement This movement is omosed o ive setions: three are marked Molto dagio and ritten in the Lydian mode, alternating ith to marked ndante and ritten in D maor [EE Figure 22] The movement begins ith one o the Molto dagio setions Here, the quartet again assumes the role o an imitation human voal hoir, singing, as a anon, this holy hymn o thanks to God These setions are ritten solely using the eight tones o the Lydian mode, ith no shars or lats, yet there is no lak o develoment The irst violin starts alone, asending a maor sixth (C) hih rosses rom a lo to a middle register The violin then desends a hole ste to G, and asends a ourth to C t the same time, the seond violin enters, reeating the asending maor sixth (C), but then desending an otave to G in the lo register, and then reeating the irst violin s asending ourth to bring it bak to its original C [measure 1] The seond violin also adds an E, making exliit the minor third imlied by the maor sixth (C) On the last to beats o this measure, the viola enters, in its lo register, reeating again the rising maor sixth (C), and then ontinues the line in the next measure, droing to G and asending to C, as the ello enters ith the asending sixth (C) [measure 2] o in these irst toandahal measures, the intervals, asending sixth, desending holeste, and asending ourth, are layed as a sequene horizontally (over time) as ell as vertially (simultaneously), exliitly direting the listener to hear the interval beteen the intervals Here the audible singularities generated by omlementary intervals asending in omarison ith desending and registration, are learly heard In measures 68, these intervals are roeted and inverted The irst violin, in a high register, starts on, desends a ith to D, asends steise a ourth to G, desends steise a minor third to E, and inally B 84 Violin I? Violonello n n n n n n n n n n J J J desends a maor sixth to G The seond violin ollos the irst, exet the desending ith (D) is in a loer register, and then it desends a ith to G ontrary to the irst violin s asending ourth to G The viola enters ith the desending ourth (D), but the ello introdues something ne, a desending minor sixth (GB) hearkening bak to the asending minor sixth (F) o the irst movement This generates the Lydian interval FB beteen the ello and the irst violin, hih is generated again to beats later beteen the viola and seond violin In measures 121, these same intervals are roeted in the ello, don and into a dierent register, to G; they are then roeted u into a dierent register in measures 1819, by the irst violin In measures 2425, e are bak to the oening, but in dierent registers s this setion unolds, the listener seeks to orm a One, but beause o the ambiguous ay the intervals are resented, that One annot be assoiated ith any artiular key or suession o keys Instead, the listener s mind must orm a One that is above the searation beteen keys Comare this to the horal oening measures o the irst movement, hih ahieve a similar eet, but rom a dierent diretion, so to seak The thoughtobet thus deined is a ne and higher onetion o the ordering o the elltemered system The entire Molto dagio is brought into a One ith the abrut shit to the ndante at measure 1 uddenly, the quartet sithes to D maor, ormed by hanging the F to Fs (the Lydian o C) and the C to Cs (the Lydian o G) Beethoven subtitles this setion Neue Krat ühlend ( Feeling ne oer ), something exeriened immediately by the listener With the unolding o this setion, the listener orms to distint obets in the mind: the One assoiated ith the Molto dagio and the One assoiated ith the olloing ndante These no orm a ne unheard One in the mind The Molto dagio returns, but this time hanged FIGURE 2 Ludig van Beethoven, tring Quartet O 12 in maor, third movement, measures