Bartok - Bela Bartok an Analysis of His Music 1971

FwtguMmW 1971 bSWwMLwte aud ipt: A&Avc CgghtQ&6Lvæ1 9 71 Rcdrdt 1979 Mw cpc Ø qmmþHpmW& Hwk Lld ad A A Xw Lu (Umw&m). T pub wu M w d At Oga wr m æmæ in thc pri o 0mt oµæ IS8N0 900707 ÛÔ Print ad bud In ¼w¡Brh þ RmW BUW L Tmwbm¿c Contcnts ¡amwm V To Pple ÄA5)le Fo Pp OmSlion '7 Fibc&= ² ] 3ÎMo CæIntm CwS)t Ã0 nu.=Sye '7 Appd I Û App n '0 Ap III ito Ïntroductìon Te publton of Û.tudy of t muc ofBa mkban imprtt evet. Many deiptve analye of partcula work ofh have app, but here for the fit tme mthe Englih langage M a. autorittve and convincing epsiton of the theretcl prncple which the comper worked out for mbut ree, W far W i kw,fom epunding to anyone durng m lifetme, ether m wrtng or by word of mouth. Tu we owe bth t author a t publiher a æ debt. Æ. Er6 Ldv"; Þ0te fct tat WB6k m mearly te, evolved for ha metho ofintegratn, æ the dement ofmuic; te ÞMthe chOrdalltuctW wth the meloc mot apprprate to tem, together with thC"pf prtiOD of length a between movement i • whole work, main dvision witin a movement such W eposition, develoP" ment and repitulaton w even baancng pbru withn at01 of movemet, according to ODC sigle bac prncple, that of the Golde Secton. Sme luch matematcl prprn wu 6ntpmpW M"cthetc prnciple by Calde ..in the W 3rd miUennium Ð.C.g taken up by the Greeks two thousand years later and rediscovered during the Renaissance, but never systematically applied to music at any time. (There exists onc single string quartet movement by Haydn, composed in Iength according to Golden Section proportions, but this is morc of an intellectual quirk of the compwer's than a principled pro­ cedure.) Bart6k discovered a way of deriving the basic pentatonic intervals A-G-E and the frst inversion of the major common chord E-G-C from the Golden Section in its prac­ ticable fonn of Fibonacci's series of whole numbers. From there Bart6k proceeded to the establishment of two fundamental scales, dCKribed by Lendvai a "diatonic" and "chromatic", containing respectively seven and eight notes inside the octave. Within thiJ framework Bart6k applied Îm theory of "tonal axes" as the basis of tunality. It is an implied thesis of the bok that the pentatonic scales of the earliest folk music, the modes of oiental iDd medieval art and folk mwic and lastly, the major and minor scale idiom of European art music of the 17th, 18th and IDth centuries, are stage. on the road towards Bart6k's complcte integration of the deepest fundamentals of tonality with perfect formal proportion. During the pat ñµ years there have been various scienti­ fcally orientated attempts within musical theory to show the way forward to the composer and to help him to fnd a fnn fothold in the period of chaos which followed the dis­ integration orthe major and minor scale period at the beginning of Ucentur. The most important in order oflheir appearance have been Aaviev's uMusikalnaya Forma kak Prouess" and Ulntonatsia" (1930), Hindemilh's "Craft of Musical Composi­ tion" Vol. I (English Ed. 1937), Derck Cooke's "The Language of Mwic" (19 5 9) and Emest Ansermct's "Ï Fondements de la Musque dans la Cnscience Humainc" (lg61). To these major worD should now be added Lendvai's exposition ofBart6k's mwical theories. Though thee fve work V propagate theories which are mutuaUy contradictory in onc respect or another, they arc all in agreement on onc fundamental proposition. namely, that tonality, that tonal relatons of,ome kind or another are an Cential framework for any COll$truction of tone' which can be rightly considered B a work of musical art. Aaviev' , concept of "intonation" , Hindemith' , "SerieJ I ; _ Co kc' , "pinpointing of the inherent emotional characterlation of the major, minor and chromatic scale",- Ansermet' , exposition of the space between the notes makng up the octave B a "structured IpaCC, divided unequally at the perfect ffth and perfect fourth". and now Bart6k'I tonal axe, operatng within his particular "diatonic" and "chromalic" scale (the latter not the chromatic scale of twelve semitone) arc all based upon the admision that there exist a h.ierarchy of intervals, proceeding from the essential nature of musical tones themselves, which may nOI be diregarded if music is to result from composing or the putting together of tone. Some readers may wonder why Ï have not included among the important theoretical writings of this century Arnold Schoenberg' s essay entitled "Composition with Twelve Tones" (1941), the argumentation of which in support ofr method of composing with twelve tones which arc related only with one another (now known as serial dodecaphony) advances it, in the author' , opinion. "to the rank and imporLance of a .cicnttc theory".·· A study of the theoretical paragraphs of this cssay dispels any such illusion. The whole justifcation orlhe method of comping with twelve tone depends upon the roHowing two sentence: "The tenn emancipation of thc dissonance refen to its comprehensibility. which is considered equivalent to the consonance's comprehensibility. A style based on U • Lkc: T�rgmwu,pagczii q •• Shonb: S! mIwa,Q@c ¡Q pUNmææMOMÞMOßadtÐ0ua0æ ß WOU • LO mMæM=mVMt t0 Om0c W mc M mßt 0m M gM0y gbÎc mpmu0l mæ0M æ Üvt 0M b0t WW W OM0] Ít b æ 00Ucat æwwwdgby æHm.Tmm aßa0t0vgbt a0tæmmwWÍlÅtwctcHmmwmObæÞßa0c.ÅdÍa ßayOtbctcnvaOß0 0b0lßt0næccattcdwb0tl0Îl0wû0m wy gNW0w y tuw gt0gÌtÎ0n ßnd u mctcÎy ß d0@ßtÌ0 æ0 0l W0 Omg8 bdAtucb Ìt U t0tßÎly wm0ut tbc æcnmc~MdÍty wbÎcb bc C l0t Î� ßad tbctN0tc m æy bædly mcnU mdmeaßm0ag mcÅmgMat tbÞtc00ßl wtÍbn@wm0bM mcn00acd ß0~c. M fa æ I ßm ßwßtc a0 äuggttct 0f ßt0nßbty, 8cnßl 0t 0thcwc, bæ gt0wdcd ßny gtml 0lÌUtbÞtctÍO ~ßÎÍdÌtyW ß g blcûcw0tkÎ0t mæ æ ¯cWmdßbÎc cbægí0a 0l tbc NM b0 0l U ct, Ter WÎwa@nd AOnO, mW "WgbÍcdct ncvæNw¨ (19) WUD tm�ßgßNb0mÜMókMdb0Vky, 0my bbbtg,Üt æd Wcm æ Ü l0M0wO w wttby M b W = m g<y W W M . ¯ ¯ � ¯ W Æ @W =»=>�=#" v = W ¯W Im¯¿ W mmW Ü m @ w ccam wm d Nm Oæ OW Þc m it aot­ m ½ 0b[M0~cÎy ßad dæctÌW mc Îßtgc wrb m bÞt mtvtc g æ "Wctkc dæ p �gm°, •• wbÎcbOvÎdbÎÎMyUNæ"w0tæm mß@ t Ww¯- M ÂW æ wÞÎc to gtum ßw btwit æmæt0ßddmßtÜmcgat0l~Îcw0lw0æ mwæ W, d gttlN00w cbÍt<bßt 0l Æ.§0ba OP a0twn ß m0mcat`8c0mdctß00a e • &Þ·8pmMPI9¶ ••Æ: mÆm,gg & In concuson, te publicton of à Lndva'• bo k C ony 0welcomed. It ahould be lIudied, not only by ¢Bat6k'. admrer. tlether wt the other UD abve-mentone, but by æ IlUden W o cmpoiton who W ! to 6ght ter way out o the prent ðWMmcnfuson i te mu world, and Iw to build a famework for their ceatve work, which U not a chace mixture of te latet .tyle 8l preent in vogue amonl onc or other M cique, bUI a logically inlegte develpment fm te mu a of pat peo. In thil6ght the ,uggle of Ba6k a eunde by Lvai, Ma iupira­ lon, even Ümsluton may not b the one whch proVQ to b the mot widely accepte. Aan Bush, Ra|eu,191'_ æ JonaI !rtnctplcs The As Sytem "Every &has the rgbt to Itrke it rot in the &of a previow @c¦ it Dot only Þ the right to but it mu.t .tem from it", Bart6k once delarcd. Hi tonall)ltem @m out of functonal music. B urunter· rupted Wmevoluton can 0followed from the bgng of functional concepts, trough the harmone of Viennee O O and the tone-world of romantcam to B m.sl. By an analysis of W compitons, Ú m system can pry b shown to gthe es ental proprie of claieal harmony, 1c [aj the functonal aftie of the fourth and  degree (6) the relatonship ordativc major and minor keys (c) the overtone relatons (4) the role of leading Dote Cl) the oppite tenson of te dominant and ,ubominant U te duait of tonal and ditance principle I [ej 1o bwm¿ Wl M tQ to utvAtc B`z waM Qtcm mmcccoͱ.LtMt C Wmcwæc[1j.¯ca F, tbc Íowm U tbc 8vboæt [ôj¦ Ü, tbc bm dc@cc¿ u tbcdoæt[Üj¦A, tbcÞXmdc@ccMdMÎAbvcoÍUctonì0¿ ÍwW & tom0¡ Ü, Ucæadm æd æabvc wUc 8vbomæt,ñW • wbomat¡ E, tbc 0d dc@ aad ÞAbvc oÍ Uc domt¿ Å0m W Å dMt. T ææ o Â, E-F 0oÞÿam to Uc Ívn0bonaÎ M W ÜT¬ 1¬ W0. f ¾ð wM UAt mc MvcaÞ Ü1¬ wgaU it. We Ü gdquee OTWæmmcmMædMcmc oÍ tbc ælt my b ð@$ • • M • W O.M z M Îl M "para.e W Û W W æm 0IOnic, 'Ub dominant and domnant &g nptvc1y. 60800MlßkßI kXl6 G "f : • 6P I0ßl0 kXl6 W0.@ 00MIßkßI kXl6 Chon m on the fundamenta C, E� (=D#), F# (-G�) and A have a Amfuncton. Cnn bu on the fudamet E, G, B� (=At), Cl (=D�) have a ..functon. Con mon the fndamental D, F, A� (=G#). B have ÅWWfuncton. It UW0that the puar &HmDt b cDldu . chordl of the de aeveDthJ but W te functona rationhip o four m W l tnt, wh may bt b cmpa t te rjomr ratioD of Uæ muc (e.g. C major ad A nioor, E� major ad C mj. å It should b noted, þwM, that a much more aeNitive rdatoDp cu betwen the 'f ple o an athe "count:u, e.g. C a F�t thos atuated net to each other, e.g. C and A. A ple Ualways interchangeable with its cuntle without any change in iu functon.· The ple-counterple rdationhip M the mot fundamental lUctUral principle in Bartk', music, in repet to bth 8m æ large form. Aready 'he WM form of BIwhø4 å Cat& wu conceive in ple.counterple tenions. It starts at the dark F# ple. rle to the bright C major chord (the realm m Bluebean) and deends agan to the glomy F#. The cour o Û Soll fo, TU Pi w P"nn O from the deptu to the heights: from F# '0 C, the beginning and end of the work. In Fig, '5 the Fl and C entrie (bs. 0-5) repreent the tonic, ÜG and Dl ente (from the end orb. 8) Ü dominant, the Ab w D entric (b. 12-17) the .ub. dominant counterple. The B major tonic mte røt=Czmno i replaced in the developmen, by it coun'erpo1e F (b. ll5). Sily the F major tonc of the Di/i Ureplaced by B in te develop­ men. (b. &}. Tmovemenu of Mw/or Stgs, FnmawÛtmlahave the foUowing structure: IOWK I 11 III 1 XÅOUO A C F, A MUL E� (b. 56) F# (b. 063 ) C (b. 46) E� (b. 83 ) W M A C F, A • A mmþWµ o c, B-A-DF m m §M of . mWbvu,! i t mmway: E-A-A�DI-C-F. m =¤D wUmrqþA� .. D�, Wà W Þ- Wm«¬mMµ]a8ms,PmmwOu(Me. It.176 181), w m mµ o FI'-B'-E'-A'-D'-'-C'-F @ @� F,'-B'-E'-A'-GI'-C$'-(C')-F"· å ±table teache yet another les n. ÅWwmovemcnU MÎ on the tonic m8, A-EIF#. Thw the ÜMl and Íuw movemeDt are suppned by the "prncipa branch". A and E�; the middle movement. however, by the "seondar branch", C and F#. Thw each 8 ha a two-fold afnity depending on whether we opp the ple with the counter­ ple, or the principa branch with the secondary branch . FB ¡Þ í W • � f¤v Fe!e WO.g CDIquently the compnents of the Aetem are as folows: ple branch æ - ple+counterpole M principal +sccondar branch æsY'tem º T+D+S U e (no dimenion) (1 dimenion) (2 dimenn) (3 dimenion) The Slow Movement of the 8:mw[:rTw ÎætæPttmd i baed on the lubdominant a¡ BDF-A�, complying with the tradition of clas ical compitioD. The modal amgWt o i1 principal theme M symmetrical: the bginning and end luppned by the Band F countcrples (i.e. the ;rinipal branch 8.8.-2 5 oÍ mm),whc dMM a fu mc lia ral eadD and A� cunte (i.e. te m brch o te =jwit te aner o te tanging-fI(E) W te mddl.· UW ¦ 8 9¬| ¾ �*¼ 0 g� ß~F � � �� ª ·Õ • 3 mw M m « W æ m w m my MmwæÞ U • Uw Mdo-t Ha (C.et.) CI wt. dom m&wmµ =W)ê· 6 The meloy contitutng the core o the movement U $ centred arund the subdominant æÆ The 0# opning and dæare replaced Vthe middle of the teme by the counterle D. Ever main mctric and motivic pint revolve around the subominant æ ?WN mo.ê Thee two melodie truly rdet the Itrcture o the movement. one o them being attached to the principa BF. te other to the lCodar Gfn branch of the lubdomnant æ The seond teme of the ¥Øh B C:æø, the famoUl  æto be smewhat more intricate. Although the twelvetone meloy touche ever degre o te chromatc scale. there UW doubt a to iu tonalit. In it a we M0 the A and D# counter­ ple (bginning. middle, end) and the broken-up F# major and C major-minor counterlC. �� • • 3 W • W0. ] 7 For ÛHMdetaib, MApp. I, p. 99. (|} A 8wQof te eoluton of haronie tg lead to the conclUn that the birh of te 8 l)tem ¾ • BWDO nccaty, repreag te logica contnuatol (and in a ce WDthe completon) of European functonal music. It c b demontated that the æI)Item, with it characteritic feature 0¿ in efet, ben u by te Vienne "Greau". Indeed, it m ben rcd by Bach, in 1 chrmtcism. The MD uÎ j1 crtion in muc wa introuced in praetce by te rca ton of the I-IV-V-I afty (in meieval mo muic, at fnt in cadence form only) In the cue o the C tonc: SUBDOMA F 3NIC C DOMA G The clacal ther of hæny already .paks of prima and aeondary U Ww m the C may b my by i . relatve A, the F by it relatve D and the G by ill relatve E. A 2NfU C Z Romantc harony @ 80 fun, making freuent w of the uppr relatve. (Natl y ony mjor and mnor key. of wmkcyolignature may b pmmrelatvc. c.g. C major and A mor, Q C Wæ El major): 8 3NO C Z` A Ej One more step complete the system. The & extend the application of relative t o the wWÌt system. The a aystem implie the reognition of the fact that the common relative for A and E� is not only C, but MF# (=G�); that D and A� not only have F a a common relative, but mÛ] and that E and B� not only have G, but alao C# (=Db} a common :e|.u.µ, 8UBOOMA F ·` D A� `· B 2NfU C ·` A E� `· F# MOMÆ G ·` E B� `· C' ¶ i well known, Bart6k showed a p:e|e:eaee for the M of a: aUed major-minor chords (sce Fig. 32b). For instance, iu ronn in C tonality i: The function remains unchange even úthe C major $c a shown in the abve chord-is replaced by the relative A mnor. or when the E[ major tonalty replace the relative C minor. ¯ technique ocun :ep|a:|,in Bart6k'l mwic: C major-minor · ` A minor Eb major The .ubtitute chords may a b employed in major-minor for, which bring! the system w a clsI. sn t relatve of A major (FI minor) and that of E� minor (G� major) met at a pint of enharmonic ccincidence, FI=G�. 9 A minor A major C majo�minor × ` ` E� major E�minor ` . F# minor =G� major Tec relatvC, applied w dominant and lubdominant hamony, agai reult W te MOmthe æstem.· (6) The ther oCthe mItem i a lubtantatcd by the WWoC awt. Autcally, arrving fom the WMWthe k¤u¿ M to reach the rot from & overtone-all cadental re­ latoD rCt on the prnciple o interconnecton between rots aDd their ovcrtonC. ThUl, the domnant of C i not ony G but æthe next overtODC E and B�. Tercfore te c of domno •OmqybmþWDmmWO t I de ��F ÆM Æ dt N m mm m. H,mR'sqmubWmpwwWm m M&T æqu mw W W M mm • æm mmwæumO m 8it o • d mi.e.±8u gW wm UmmB wmwt 6t o t tn Fo m&w'rC�W ¼,W Ï#Æ |m==WpW(b)}=;MWÞÑ Þ , L•U&tmmw3dmwmæ mk æ m mwqæoæwWmmM M bO W W M m m w (F¬-0-ß-8) w • b b I B a P. No I b m mm ww @æ¡tbÞtvyMmgmmw¬|.mm mmæµmmC�æAmmmp bUþW mmtmtmQmM 8uµmy o. IbtO mt I m W.Tæag W M c>n�mp ¡t U æ b Ræ m m NwwÞ0 ætmmp æÆ r cor wWcæø,ltbmt•co b oa n�,bw Æ Nw fm &(mm k´ß mm±en[ mtmQæ. wM<,wmWM am- m }. Æ Wm me&mmææmw• æanb æmææmæ.MWhm�æ mÞ@,mw,±mmw "mwm'*ww±w V% IO at·tonic rclationship w expanded to include E. C and BIO. Since the DT relatonship correpnds· rdalvely w the T- and the S-D relatiObip, ovcrtonerot attracton et btween the T - and the SD, W well. ROÚl OVRTONB MU&TÆ tonic C E and B� � dominant dominant E G#and D � subdominant dominant B� D andA� W .ubdomnant ,ubominant A� L and G� & wm0 lubdomnat D F#ad L W lonic Ç F # Ä E-B� Ñ 0 y •3 ðamt o d dm t (d dwt) mæ d M o te rboc whe t dmto mtvW æmmo t æ 11 If we add mrole o t Movenone, i.e. t fth, then we Cdeduce the cmplete mI)tem frm thcc relaton. (d) In the aplCt cadence, that of V-I, te m role U playe by Üsl ed stve note whch prouce the pull o the domnant towa the tonc. Te le Dote pulb to t rot M the seveth towan the third degee of the tonic, i.e. the leading note B relve OD C and the Kventh F on E or E�. ËÛø 10 Te impnant atve note W&ÞWrelatonp to each othe. The ultonehalf the otave interal-iI Çæ- terled by the interhangeability o iu WM without changing the interval. ThU. dthe BF relatioPhip U converted into an F-B one (a U freuenty the C wit Bak), the. the F ( _E#) aume the rle o the leadng note, puling towad the FI intead of E, while the .. enth B pulh toward A# or A itead o C. So, wtcad of the Qpted tonc C mjor, the tcwþck, the equaly tonic F. major (or mnor) emergC. Ho. t i I2 ¯ relution M reered by BartOk for a ludden c of scene. The crcumstance of an epcted G'-C cdece emerging a G'-F# gve w a llBart6keao p:udo-cadenc". (e) Starting from the tonic centre C we reach the dominant in one diretion and the lubdominant in the other, in iitd latitude. At a distance 0 a fth we fnd the dominant G upwards and the subdominant F downwards. Regarding :moæ relatons we m get the dominant G. E, B� in the upper and the subdominant F, Atp, D in the lower directiona • 80MWlM gz 0WlW 0l8tCl08 0lM6B F • Æ E • W W0. 12 But what happens if the pendulum coven the latitude 0 a tntone? In U cae the deviations made upwards and down. wards meet, both ending at FI (-G�), and iwe were to take one a the dominant, then the other would have to asume the subdominant function. By Úcoincdence, however, a neutral· isation of their functons take place. domnant and subdomnant merging are rendered inefective in the interaction of their oppsite rorce�. Cnuently .e balance is saved, and the function is jnvuriably that of the tonic. The counterpole is br: Similarly the distance between the tonic C and F# is bisected by E� ("nl) in the one and by A in the other directon; % lying in tensionJC. neutral eection point, they æ have to be interreted a tonic. No more than four tonic pole can b surmised, since the intervab CE�, EltF" FIA. A-C provide no funher pints of biction. ' 3 F·m|y,whatg thouIdb ztuM a wo z cmomucdcgmc,o�B æiuwunmCC§[=Dg Y u ÛM W WM thc dmt znd which thc rub domt|malRcIztæcB, C§domzdcgræof Ucvztion o t f. wmch might corrqnd % thc SD intcr- dcgadcncc,botnotteiuopptc.Anywzy,t rubomnzut foncuoa o B æ mc domaznt |uncdon 0 C§ B un- quuuoazbIcwhcathcywcrc|ztcdtethctenicF§countcrpIc. []) Thut,omthc |ogco macuoazI intcrconnccuon o| Ü MN 0¿ æÜ iatcruuag pìat m- Tc IUb doæzut æd domazat & rcprucotcd D8Î cñ±tivcIy m¡ by mc dcg ¡V zad V bot, ia thc C o| C Int. thc rubomnznt byA� [zadìucouatcrp|c), thc domin æ t by E [zndiucomtcrpIc). ¯M¿ dtcrm¿ nothingacwmrc thcrcU¿ |ormW, thc dmt t=adzry thcmc il E m &cthovcu't WGUi, $.(0 mz[or}orthctubmtöIowMovcmcntin Aþo| Ü ÎNh@ [C mm). Thc movcmcut: o| Br' Fis' 5yMghzvc t |o|Iowiag kcy-mucncc: OßA� in mc m o tooioomnzat-robdomnzat-tomc,cR Howwcr, mc zbc mzuoa o thc=ryttcmlzJt to czp|aa whyBk ÿtgmmuc zogmcntcd D rcIztio W W dcuadiuoæJ-lV-V-¡.[ForCpIu,W App.11. p. 103.) Tamiutu zncw zpprozch tothcryttcm. lt U gcm y zocptcd dzt twcIvc-toac mu dom z tuocgtcadcucytoiad crcct toaUrcIzuom. Ateaú rcIzuo W C b mmt tuitzb|ycß`mtcd bymct¡æ diwono thcotzvc, oro| thccirc|co 6(t. Bydìvidngthc octzvcmtwcIvcquUpzruwcgctt chromaucvzIc;int C oImcgoa|pzruwchzvcmcwho|c-tencvz|c;|ourcguzI pzrugivour thccltord oImcmcdwvcnm; m c thc zogmcatcd m, zadmy by diwdìng thcotavc into two goúpzmwczmvcztt Uwm ' 4 For the PWt we NM cclude the whole-tne M Ww of il WN pbilte: T whole-tone lae produce the chromatc ae by interlocking. Ever tonal Item praupplC & centre W weD W ,ub ordinate relaton dependent OD the centre. Takng agan Ç W the towc centre. the three functon W repreented moat ptenty by thoe degreC dividng the crcle of W into three equal pau, i.e. in the augented triad CEA�. Propertie inheret in casa hWny are repnible for the E auumDg a d�mt functon and A� a lubomoant functon in relation to the towc C. Each o tee man note prmt their lubttuton by teir cunterplc, i.e. their trtomc equvalenu. Thus, C my be replaced by FI. E by B� and A� by D. If we divide the twelve-tone chrmatc se proprtionaly between the three functona, each functon whave four plt, and thcc-imfar W we keep M the m.tance prnciple-arc @ ged U dimnihed-seventh rtlations, dvding dle cle into four equa par. Accordingly. CEtF"A belong to the range of the C tonc. E--B�CI to that of the domnant E mun note, and A�B-DF to that of the subdominant A� man Dote. S, the tonal syltC reulting from a division oCthe chromatc sale into equal parts agree completely with the æsystem: 80800M¡ßkßI T0ß¡0 00M!ßkßI � O O §... ÆÆÆ¿ #� == ***W ~~= ®^^ ©M d fÞ b� rto. ig '5 ¥l cooy, given Û twevetne Item and ttac three fuoctoD ÜUte .n.l)t tat Cbræbÿmean of dwdiviion. Viewed WwDæÿ¿ the æ system re8ccl the ag�ld stggle betwee te princple of Illi! aud tgtmMt¿ with the @adm æcudæ@ of te laller which W ÿ reulted in the fre a eua teaWt of Û chromatc twelve nOle!.· Here we have to draw.liDe btween Bk', lwelve-tone sy1.em w te Zw6lftoDu m SDbug. Sch6nbrg .n ale ædWlvC tnity whereu Batk incorpratea te priocpJa o mmµgµ a pet aynthc. To pnetrate into Bak', creatve genus i W diIover the Ww m0æ aud intc psibilite. inhereDt in the musical materal • •3mm�o d wpwcwabut tbc mq ç mm .6 Iorm !rtnctpIcs Golden Secton Golden Section (".cetio aurta", and henceforh CS) means the division of a distance in IUch a way that the proprton of the whole length to the larger pat correponclgeometrically t the proporton o the larger t the smaller part, i.e. the larger part i the geom Æ of te whole length and the .maUu part. A simple calculation shOWl that if the whole length i taken B unity, the value o the larger secton i 0.618 • • • | 1 Ü*¥Ì W0+ tq T` x=x: (I-x) (ICe upper formula on page 78), and hece te smaller part i 0.38 •• • • Tus, the larger part of any length divided into CS is equal 10 the whole length multiplied by 0-6,8 • , • '7 Bak'. m, i h cntt ol W m bmmoay, 18 O y mæ wwth c lw o c Üâ. ¯ u a lormæ cÎWcat wh i alÎcæt æN@0aali Bak'. mm0 æ lbc 9+1,g+g¿ 0+0 bmgNomor tbcovcrloacbarmoæaboai c NìcmæOæ&QÎc. �MCgÎc¿ Îct M W lbc mt mvcmæloltbc &æ ]aJWÏæFn�a.¯cmvWtcOmgrm@gban, ÆíbÜö&gdaÞvcformula-i @gxO·010¿ì.c.1ÿ], wh it t ct o gvt i t movcmcat: lbc rccagìlwadoa8W gmyW t a)]tb0. movcmcal ¡ o C1.u cn.t o gg ban. aad íu Üâ [ggxo·010j agaìaÆWmcb gof UE gìluÎaboa ìa mcæddÎco bmgÿ. movcmæl Î o tbc DW� eonmt o g0g triplet urt (the numbr of ban Uu Ocv&t owing to their varable time­ signatures). Åc Üâ ol g0g [g0g>O·010=g]0] again coincides with the recapitulation. In Vot. VI of MikrDMst the Gâ of "Free Variations" can be seen to touch the "Molto piu calmo"-82) O·010¬g1. Åc Üâ of "From tbc Diar ol a Fly" come al the climax: the double WMMd (if the gj]i taken B a 11 bar, calculating in a]] ban). In "Broken Chords" wc fnd the recapitulation at the Üâ [0O¤O·010=]g}, etc. Åc 10 introductor ban of the SDMajDr TW PilRDS m P,mmt repreent a model example of GS construction­ or more precisely, b. a-1],because it is here that the organic life ol the work begins. • • IR. t@ ' 9 Its fnt pari is in the sphere of the tonic (b. 2-5) . the second within the dominant (bs. Bg) and the third part in that of the subdominant (b. 12 on). This third part is thematically the t¤Ntt0¤ of the fnt two. So, to summarise: Teme in r06l positon-tonic: FtC Theme in t00lpsition-ominant: O-DU Theme iltrttd-subdominant: Afr-D entrie .t ». Considering the change of time-signature, it is more practical to calculate in units of 3/8 time. The whole form consists of 46 unit. Its OS is 46 Æ 0·618 -28, and this covers that part up 10 the inDn o the theme (see the main section of Fig. 16). It can be observed that OS always coincides with the m0tl signifcant turning point of the form. Let us now separate from the whole the parts in root position, i.e. the frst 28 unt. Now 28 7 0·618 ..17'3. At this ver point the tonic part ends-at the frst third of the 18th unit (see the dminant entr in Fig. 16). ßûûI Fû5ITIûN T BBk C • l l iìH +×t¶t im W = u w • it •• 9wímt C •• 1 J ¯ W Mt§e 8e¶eti ve ?0IH¥f FlG. ¡Û INVLß5IûN Tw Re »t•• OS division may be seen to follow one of two possible courses, 2O depending on whether the longer or the shorter section comes tlrst. Let us call one of the possibilities ptitiw: long section followed by the short one-and the other negativl: short section followed by the long one. In the structure of both tonic and dominant parts the cymbal­ stroke creates a sharp duality. The position of the cymbal-stroke is in both cases determined by the GS, but whereas the tonic unit (at the sign "cym" in Fig. 16) is divided so as to make it P03itiw (I 7 '3 xo·6lB ¾11), the dominant pan. on the contrary, becomes a rgaJivt division (it consi3ts of 10 units and is divided 4+6). The positive and negative sections complement each other as something with its own mirror-image. But the meeting-point of the two (the dominant entry) has a positive sign. In other words, condensation and dispersal of the nodes cause a longitudinal undulation, the wave-crests meeting in a positiw section. Its rgative counterpart u found at the entry of the tar-tar (in the inversion) so that the positive section of the root and the negative section of the inversion arc again joined symmetrically. Not only the entire formal arc but even the form-cells submit entirely to the strictest geometric analysis. For instance, in the dominam part. we fnd up to the cymbal.stroke, eleven eighth­ note. ÏI1 positive CS point (7+4) determnes the position of the only musical stress in the unit-by means of elongating the E� note. This is soon counter-balanced by the negative section­ point, at the side.drum beat, in ban 10-11. Similarly, the positive section of the tonic part up to the cymbal·stroke is marked by the most important turning­ point, by the third (C�n timpani entry-counted in eighths: 33 X0·61B ¾20. Precisely here, the thematic condensation begins: also, with the 21St eighth. On the other hand, the complementar, negative section of the part following the cymbal-stroke is indicated again by the side-drum (see Fig. 16). B.8·-3 X Summarising the abve, bth Uthe smaller and larger form­ detaiu, there is a symmetric joining of the psitit and Igatitt sections. From thee concatenations a single great "ptentia'" fonn arise, wherein the smaller parts are fnally summarised in a pnli" main secton. ¯proces is therefore coupled with a powenul dynamic increase, from pianisimo to forte-fortissimo. Analytical studies permit the conclusion that the poitive section i accompanied by intensifcation, dynamic re or concentration of the material, while the negative section by a falling and subsiding. The sections always follow the contents and form-conception mthe music. By way millustration let us subject Movement III of the Sonata for Tr PitS m4 Pncssin to a detailed analysis, Exemplary, is the unity of proprtions of the exposition: the principal theme ha a psitive and the closing theme a negative Iction, while the Icondary theme developed between the two i symmetrically arranged, Thus, the principal theme· (43'S bn long) is divided a follows: Al + J+ B, The psiton of B i determined by: 43'sxo·618-27'S. while the two A's are rdated to each other according to: 27'S Æ0,6,8 =tQ. `� . . ¹ ¦ . ~.=.-- ¿y·_ =.~.. ..Wæ.•• æw æ••• æ gg·g æ,ææ........ no . " • T incmpIcteh·b atte µm age|mmovemtat1tte1takcn inmcemwat|eawaeomwmmam, The symmetrical division of the secondary theme can be expresed as following: 12+17'S+17'S+U! (bs. 44-102). The geometrical cenlre (b. 73) accords with the lonal construction of the theme al. The negative main secton of the closing theme (bs. 103-133) is given in D. liS (see Fig. 18). Within this, b. IIS-133 have a positive section in D. 127 because of the pwerful dynamic ascent, and the static construction in 4+4+4 units of bs. 103-114 produces a solid bae for U rise. WÎÊÅ stetk 4141ÅW 0Þ lÏ w0W8+pB8ìtÍw Þ¡D. tÛ ¡3 Ukewise the proportions of the development are symmetreal (bo. ' 3 4-0 47) , Its negative main section-countecbalancing the poitive main section of the development of Movement I-is determined precisely by the point of climax in b. 177 (F, tonic counter­ ple). Te pe:ítíxsection of the part preceding the climax and the iitgclíw section after the climax indicate the most imprtant turning point: U.160 the fugato of the principal theme, while b. 20S the return of the frst theme of the development (xylo­ phone entr): Þtf M4 låð r0$I¡l¥£ ttlNAx Ú 4ÞÞ 8f64II¥£ wo. t y Ï4â The build.up (owards the climax is always marked by a positive section: from b. 140-159 it falls on b. 152 (psitive) ., b. 16-176 .. _ h. 170 .. ,, D· 160-I6g .. .. b. 166 .. .. b.1 70-176 .. .. b.1 74 .. From the point of climax on, however, the negative sections show inverted proportions: from b. 177-204 it falls on b. 18g (negative) " b. 189-204 .. " b. 195 . . .. b. J95�0 4 .. " b. 199 .. The climax itself i divided statically into 6 + 6 bars (m. 177-188). The negative main section of the recapitulation (bs. 248-350) coincides with the watenhed, a it were, of the thematic material, i.e. with U. 287. Bs. 287-350 form one single broad wave, and it structural view usimilar to that of uebeginning of Movement Ï (cf. Fig. 16): ctIw •• I ?ðI ðI5 å0å Æ j · , | ¹ ¡ ͯ JH . w< U POIllf( • IlGATlYI 4 . . ?0lTME • IIGATllr • F051T¡¥I FlC. 9U The negative main section of the coda· (bs. 351-�20) coincide with the thematic centre of gravity of the whole coda: at the same time the return of the C tonic, in b. 379. Ugiven a greater emphasis by 0 lengthy preparation. Corresponding to its static structural character this thematic centre has an 8 + 8 bar division (m. 37!-394). The frst par. of the coda (b. 351-378) combines a positive tat W • 1ä l1t� M 49ã l¯ I ¦ · , F�Þ Î ¡ • ät It I • IrIlc Ì 8u IB Î E¯ l . MH¥I • BfWM VE PsmY( + 8tW M¥I • N.D. T Wt ine o m .-.sWÏÆ {qi t] i erroneou. '5 and a negative section in units of 9 + 5 and 5 + 9 bars. The second part {m. 319-420). a shown in Fig. VI¿contains at the same time a pitve (b. 405) an a negative (b. 395) section. Finally, the poitive section of 00. 395-404 (in b. 401) and the negative section of b. 405-420 (in D. 411) are again sym­ metrically related to each other. At frst glance it may appear contradictor that the points of lection determined by the laws of as can remain unfultd by the changing tempi. ¯phenomenon i easy to understand if we consider that music breathe in metric pulsation and not in the absolute meaurement of time. In music. pasing time is made: realisable by beats or bar whose role is more emphatic than the duration of performance. Subjectively we feel time elapse more feverishly in a movement with a quick time-bcat and more sluggihly in a slow puuaton. Finally, let me give an eample to those who reproach Bart6k for not having efected the "total and radical reorganisa­ tion of the materal". The complete fonn of the 5enaloyerTw PitS I Ptesin is divided into " slow-fast +.Jow-fUl t" movementl. Te as may therefore be expected to aQQc.\t 0l the beginning of the second slow movement. Our eXI.ectatiouJ are wholly fulfled; the time value of the complete wurk is 6.432 eighth notes, and the as is at the 3,975th cÌghIh nu!c. which is precisely where the movement begins . • 6 Fibonacci Series All of us who have played Al Barbaro, have been troubled by the FI minor throbbing, extending over 8 or 5 or 3 or even '3 bars. The proportion of 3 : 5 : 8: ' 3 contains a Lb sequence, approximately expressed in natural numbcn: the Fibonaui numbers. A characteristic feature of tm sequence i that ever member i equal to the sum of the two preceding members: 2, 3. 5. 8, 13, 21, 34, 55. 89 . . . and further, it approximates more and more to the irrational key.numbcr of the Lb" (the Lb of 55 is 34. and that of 89 i 55). Let U compare Um aequence with the proportions of the fugue (fit movement) of Mw/or Slrings, P"nusion mCel,sla. b¡ar¡1ng pianissimo Ρ gradually rises to fortc·fortissimo. then again recede to piano-pianissimo. The 89 bars orlhe movement WC divided into sections of 55 and 34 bars by the peak of this pyramid-like movement. From the point of view of colour and dynamc archtecture the form sub-divide into further units: • The aquare or mQnur Mequa to the prout oflhe preng æ foBowmQDumbc, plu WÆWWOBc. by the removal of the mute in the 34th bar, and its use again in the 69th bar. The section leading up to the climax (b. SS) shows a division of 34 + 21 , and that from the climax onwards, 1 3 + 21 . Thus, the longer part comes fnt in the rising section, while in the falling section it is the shorter part that precedes the longer, so the section.points tend towards the climax. Positive ;lDd negative sections ft together like the rise and fall uI R single wave.- J J Jð ²Î ²Î H¬, Po. 22 The proportions follow the Fibonacci serie. lt is no accident that the exposition ends with the 21St bar and that the 21 bars concluding the movement are divided into 1 3+8. The proportions of Movement ÎÎÎ of Mwu ycr Slríagt, Immtíon e¤d LrÍOla also reflect the Fibonacci series (if wc calculate throughout in 4/4 bars and consider the occasional 3/2 B I i ban). Its formal and corrCponding geometrical structure is shown in Fig. 23 • •The Ü mn Dthe Å mut D completed by a whole-bar ral, tn accordane with the BUlow analy o BlhoYfl • •8 @¶==WWW = ¶§ æææÆæææ. g~~µ ~~ �... þ¶ ~...~ & ØAPf#&W I1 • ¥nt¤t¤a ! W T W • lM W × W¡ MÞ¤ Ì sl ZæWW 1M ,..,, ... , µ..:.. MC. 23 The Fibonacci series refects, in fact, the law of natural growth. To take a simple example. If ever branch of a tree, in onc year shoots a new branch, and these new branches are doubled after two yean, the number of the branches shows the following yearly increase: 2, 3, §, 8, 13, 21 , 34 æ æ • "Wt folow naturt in composition," wrote Bart6k, and was indeed directed by natural phenomena to his discovery of thee regularities. He was constantly augmenting his collection of plants, insects and mineral specimens. He called the sunfower his favourite plant, and wa extremely happy whenever he found fr-cones placed on his desk. According to Bart6k "also folk mwic is a phenomenon of nature. Its formatioru developed as spontaneously as other living natural organisms: the fowen, animals, etc." ("At the Sources of Folk Music": 1925). This is why the form-world of Bart6k's mwic reminds w most direcdy of natural pictures and formations. 2 9 The C5of a circle, having 360°, subtends an anRle m 2�2·g o on onc hand, and 137"5° on the othcr. It can wobered in a large number of plant. c.g. pm¸pplars. catkns, etc., that each bud, twig or leasubtends an anglc of 137'5° with the next onc. 1 � - ' • MO. 2g Al, each new branch divide the jortf feld of section according to te rule of as: % twig 3 divide the right-hand feld between I and 2; twig 4 the left-hand feld between I and 2; twig 5 does the same with the feld between 2 and 3, ad inf- •Te Fibi MWa@andOÆWw¡l:ñUbwen 2 mgív divide by¿,WW gamS b Û, WW @æÛb I,. etc. 30 ÎÏwe r.onsidcl" lhc Qrocc ol the luguc oÏ Music(«r Sì iu¿s, Í¢rw 1¤n od CL/uta (analysed on pages 27-6) as a circum� volution,· it structure wiU surpriingly correspond to Fig. 24- Or let us examine the diagrammatic sketch of the chambered shell of the cepalophod nautilus-J ule Vere was %intereted in this sea shell that he named his famous WcultÍut aer it. The diagon.lIs drawn in any directon through the centre provide a patl.!rn in which the centre always remains in the positive or negative GS section of the felds marked A-B, BC. C-D. D-E. E-F. F-G. A B mo.$_ ¯ scheme is strikingly similar U the musical structures illustrated in Figs. 16 and 22 • •Æ te te ÆW m tm c D 6frI A ctre bk to the A ctre. 3 ' But Ihe most revealing example is presented by Ihe structure of IheJr-colle. Proceeding from the centre of iu disc. logarithmic spirals are seen to move clockwise and anti·clockwisc in a closed system where the numbers of the spirals 4/w�s represent values of the Fibonacci sre. \'11 VIII I'XII! • tafà Î ¬ Ñ H ·jntaÎÞ A~I´ �· !I·¹¹²'¹ 'f""h (If we turn the cone upside down, we can als sce the system of two spirals along the junction line of the scale). Each of the spiral Iystenu contain all the scale of the cone. There are cone in which the numben of the spirals present still higher serie values: 3. 5. 8, 13, RI. 9 7 w¯ Õ I�( PO, 26b �t r ×` • � • |' ll • tt _ Ó ^ ¬ Ô XY XlY ¯e 1 'vii 33 Simlar anangemenu C be o1ered in sunfowen, daiie, ananu, etc., æ in the convolutions of the stem of IcOlve on numerous plant. Frequently the serial numben 21 , 34, 55. 8g and even 14 and 233 are encountered in thee spiral sy�lcms. For exampl�, the sunfower has 34 pdals and it spirals have the values of 21 , 3 4 , 55, 8 9 . 144. It is interetng to nole that the as is alwayt usociated only with O'IIU matter and i quite foreign to the inorganic world.- • Te irralional numbr in U6fmulaoG5proIudæiuocurreace in cruaJ·form 3 4 \sc oI Jhc and !ntcrvaIs Chromatic System Chords The study of thee proprtion leads us immediately to the queton of Ban6k's ue of chords and interal. Hi chromatic system i based on the laws of GS and epecially, Fibonaci's numerical serie. Calculated in semi-tones: 2 stands for a major second, 3 " JJ minor third, 5 1J g_ perfet ÍuuM¿ ¸ 8 .. 9$ minor sxth, 1 3 J an augmente otave, etc. For the preent, the musical tisue may be imagined a buih up exclusively of cells 2, 3, 5, 8, and 13 in sile, with sub-diviioD following the proportion provided by the abve serie. Thus, me 8 may D broken up only into 5 + 3. (Te posibility of a diviion int 4 +4 or 7 +1 i preluded by the s)tcm.) This cell division can b well obsered in the fnale of the DiDtI;mu. The principal theme appears in the course of the movement in fve variations: in Fig. 27 we have grouped them according to size, and indicated with each variation the characteristic division. The initial form oflhe theme is 3 ¬¶Ã 5. 1 l å ø g «¯t ³ Î ¥¡O. $§ Since the ffth line (in Fig. 27) continues on the previous one, in its fourth bar the melody rises not by a minor third (3), Win the previous line, but by a perfect fourth (5), thus conforming to a CS augmentation. Fig. 28 gives the successive themes in the frst movement or the Sonal for Tw Pianos m4 Ptrtwsion. The range of the leitmotif is 8 semi-tones, divided by the fundamental note C into 5 + 3 semi-tones. The principal theme compriso ' 3 semi-tones divided by the fundamental note C into 5 + 8. (Sec also Fig. 6 .. . ) The frst phrae of the secondary theme extend! 36 1 3 semi-tone, from C down to F#i while the second phrase, 21 semi-tone from 8 down to D. The melodie follow each other in CS order: Leitmotif Principal theme Secondary theme 3+5=8 5+8- 13 13. 21 rto• ¸ zð From the point of view of haronic architecture. this expition also bean witness to a systematic arrangement. The principal theme gcu its magical tone-colour from a pmlalonic harmony I.B·- 4 3 7 (see Fig. lga),· the formula of which i Q+3 + Q. In the middle of the principal theme there come & otinato built 3 + 5 + 3, A� major-minor (se. Fig_ 2gb): C-EIJAIJB, the fourth, EIAb, i further divided by an FI into 3 ¬2. Parallel fourths (5) and minor sixths (8) join the sccondar theme (%Fig. lgc). Ti i scen clearly also in the recapitulation from b. l92. Finally (see Fig. lgd) the closing theme u accompanied throughout by parallel minor sixth (8) • • ) (|t) Ë I B# Å +B 110. 29 Tu each new harmony Donc Itep higher in the GS order, i.e. principal theme mddle part secondar theme closing theme Asimilar correlaton of mot Mencountered Uthe MirdU/OUS M .. .,: • H ¤pp a i te my. Þ �n-9: AtFIEIDt ad FI�B. 58 F¡G.¿o It U interestng to note that in Bartok's music, in spite of the frequency of paraUet, major thrd and maor lixth parallels seldom occur, beeau�e such parallels cannot be ftted into the GS system, being quite incongruous t it. We could even speak of the pohibition of these parallels in the same sense that parallel ffths and otave are forbidden in clasical harmony. On the other hand we meet at every step with minor third (3), perfect fourth (5), minor sixth (8) , and even major second (2) parallels. The major third ha no noteworthy mIe4ít function either, the more natural, almost selfevident i the motivic role of the minor third: 39 $'Ï1. ¿i This is the reason why, whenever llart6k uscd a triau in Q ChrQ1tic movement, he placed the minor third evrr thc fundamental note and the major third lr|eu it, the churd thus acquiring the proportion 8:5:3. ½O. ¿za From the synthesis of U¢ two emerged the most typical Bartok chord, the well·known "major·minor" form, consisting of B minor third-perfect founh-minor third (3 + 5 +3). TIlis major·minor chord is often completed by the seventh of the root, e.g. an E-G-C-Eb chord with a Bb {see al Fig. 2gb}. 4" 4' ¯ major-minor chord ha a number of synonym for, to whch we shall give (for want of a better term) the coUective designation: type mýÂ0 {¾]_ and we shall call the different sections of it by the letters 0rl0 [þ¡¸ gamm (y·¡, dÍ¡ø (I> and rý:ílzn [t}.¯ type occun as frequently in Bartok's music as do the seventh-chords in nineteenth-centur muic: rto. ¿¿ These chords are exclusively bult up ofGS intervals (2, 3, 5, 8), a follows: Ho.@ and do not contan the characteristic intervals of the overtone system-fth, major third and the minor seventh.· •From here arile the eblracteriltic "Slow " of the alpha harmoniC. Pcthap the ¾ÛÎ teNt chord in Baroue mwic wu the dimiliish«f Kenth. 3 Ie,on it increaed in &nk', 1p/ chorda throuah m meig o tw dmmædcM 9ª Type ¾ can readily D reduced t the relations oC the m sst. In order to Cecl the tonality oC a chord. we need at leat two notcs! in the simplet case the rot. say C. and ÎH ffth G. or jts major third E. when G or E respectively supprts the C.· Let M put Ü relation in GS form: Ñ%o gg According to the axis system. tbe tone G (or E) may b replaced by any other of the correponding axis (CE-BI:#) without Changing the tonal character oC C. We can therefore subtitute E. Bb or even Cl for G. The four interals sounding together reult in the chord èHø (�). It should be noted that the combination of the fnt three intervals is no novelty to us, since it i identical with the chord of a major seventh: CE-G-B� . • Tonæity W oy be æUmm tgh t æmHdm d¡vbiono Utolte; mÅD eua divuion¾Ñwbubl t dcc ucæ t A similar æsubstitution may be carritd out with the note C without changing its function. We can thus replace C by E�. F# or A, all belonging to Ihe same æ. T " * æ 4 • = ò Ilo· 36b I n the form ofila the frst thrce intervals are summarised. Chord alp"a is therefore practically an axis-like application of the simple CG, or CEG relation, the only stipulation being that the chord should be composed of tw IDYn("axes") : that of the tonic and the corresponding dominant.· c ÞW- ¿ 7 • The two iayen (T and D)corepond to the Oand overtone rclilion of mmLharmony. It æpertinent that æin U"aditional muic, funclional aUr;ctiolU were D on thnc two layen. T authentic (e;delltiil) conn«te chords require th . t the rot of the lnt chord b«ome an ew� of the chord following. (Chsical harmony can. these £UÞt1Iut1 nota.) Tw, in the prU£rcs ion T to S, the rot of I (C) become a lifth in IV. or Wxventh in 11. Connecting Sand Dthe rot of 1I (U) or I V (1-') bc OUC ffth or KVenth in V. Cnnecting D and T the root of V lU) become ffth in I. m wq1h.¡ N#q e.I., ¶ 45 V Å ÆW W *wæ4#Æ••••••••••• 4� • •• ••q#.·, = •••_ $Ð ®I L# M 4¤9w¢ J¢ M •• ¬l Type rytílea [t} u sddom used since its tonal character u unstable, due to (he abence of G without which the root does not receive sufcient suppon. Certain sectons of the tp/ chord have been familiar to us from cI:icil harmony: E-G-BtC is the C major seventh, G-Bt-C-EI i the C minor seventh, Bb-C-EtFI (Gb) is the C iCventh chord baed on a diminished triad. Novelty is produced by the introduction of the relative A. and primarily by the Cl. In fact the chord 0rlo is an inversion of the ninth chord: C-FG-B�D� (Cll to C"E-CB�C. 46 Essentially, type 0l]h0 uan axis harmony. A an example ¡c us take the simplest case. If the L major and its relative A minor arc replaced by L mtn0r and A mjer, P1u,gg and thae two chords are combined. then btta, ¿mm and æll0 will be equally readable in the resulting harmonie!. This chord bears a high counterpole tension due to the diverse tonal character of its component, expressed by the diflcrence of six accidenta-the three fat signs of the L minor and the three sharp signs of the A major. In accordance with the stratifcation of the 0l]¤0 type it u possible to build up a still more extended 0l]À0 pile: 9t0.q0 4 7 From a succession of diminished triads a "closed" sequence i derived since, by the periodic repetition of the intervals we arc taken back to the starting point: rto. qi And now we come to the very gist! That CS is not an external retriction but one of the most intrinsic laws of music i demonstrated by ptntatD1perhaps the most ancient human sound system-which may be regarded as a pure musical expression of the CS principle. In the l0-1õ-mí fgures of the oldest children songs the notes of the mc10dy are tuned afer the geomdric mean, i,e. afer OS. Pentalony, particularly lile most ancient Cornu of minor pentalony (la and re), rets on a patter refected by the melody steps of major second (2), minor third <3) and fourth (_) •• • In tbe old.typ pntalonic meloiC witb a changing.firh .tructure (wuaJly ¼-&Ø m+wÆ Ø l« ot ¼-¼-mÌ-w+M¬-H .caIe) the major third plM y a ae ndary pri. To quote Koily: "It æ clmt that the pcntalul lY w Úfinh conttrction are indepndent. WTW¢ 'lyliically "I' I _ite," According to Koly the railing minor third, æM, rather thall 1.:..~iur any other simple confguration Mwbat the child ÅW fnt to fed W M U·U1L mwical relationhip-repreenting the mrlíct muical exptniulI uf +f human being. rto. qz ! " À ¬ º � This aspect of CS architecture is markedly evident in the DtJt Suit which appropriately has been called the "Eastern European Symphony". The make-up of the CS system can here be (allowed step by step, for this work-a rich and complex musical universe based on the primordial elements of pentatony -reveals the evolution of this technique. The frst movement arises from major seconds (2); the second is built on minor thirds (3); lhe third summarises these former elements (2 + 3 + 2 + 3 + 2), presenting a pure pentatonic scale. The harmonies of this movement arc based on 5 + 5. Finally, the melody of the fourth movement follows the patte 8=5 +3. where 5=3+2. rto.qg 4 9 Type 01# can æ D derived from pt. Ti i how Bart6k transfrms a pentatonic sale into trm and gæmo structure: 8 ? C w 8 M0.qq This type of harmonies originating from folk song was suggeted by Bart6k himself in "The Folk Songs of Hungary" (Pro Mwica: 1928):- ±tw î« Æwl 4 4R1 1l 4¥4¡: «¤ ]wge F ¯ Î �0 ¾¹¹ � no. g_ • AI 6ntiI may seem uloniJhinl thatin B6k'. muc pntatony iJ 2O dO!y aie to chromaticim. But th rdation u ntura, u wuh 8rtok te primordia attraction Dpntatony O Q¡ æ]mt tH teme:· fncb auate rorm or cpraion in hi G5sytw We now mention a frequently recurring group of CS.type chords which structurally n'prl'sl'nl intrrv"ls oÏ t:�. 1 :3 anU I :2. The CS relation between thee three formulae reults from the proportion g¯g2. Each of these 3rue from the periodic repetition of intervals 1 :5. I :3. or 1 :2 respectively. Their strcture i, consequently, B follows: Motl ':5 alterating minor seconds and perfect fOllnh e.g. C-C�-F.-G-C # æ • Moc' 1:_ alterating minor seconds and minor thirds e.g. C-C .. E-F-G .. A-C æ • • Mot' J´X alterating minor and major seconds e.g. C-C .. EI-E-F .. C-A-BI-C . # . and hereby, they form clearly dz:rdsystems.· I,. MODFL l:' MOOEL MD. qö t: "'ODE • The iuccc o folk mu:i. pI ibly a mpmiMcÎ0Mod t:5. e.l. Movement III of 5uiu@. rq wæ iNpired by Arab folk muc. Perfect example of t !2 an 1 ¦g moel have beround ill compitioN of Lit and Rinty.Konakov. � �Ld.,£ II[LS I·J C_ WWJ wÜh@l PlO. 4Gb ¤.a. ._ å3 MM6M Î* G·•HqHÖM g-lg �¹* ' � MHa_ �·•,l Ml We attribute the greatest importance to Model 1 :2 since it actually represents a scalc-group of the mushown in Fig. qQ¸ i.e. C-Cj-EIE-Fj-A-Bp. ,,os P1O. gg It can also he called the "basic scale" of Bartok', chromatic system, with whose help the tonality of even his most cam· plicated chromatic melodies and chords can be determined. And here we arrive at an important discovery. There exists an organic correlation between the mu system, the alpha chords and Models 1 :2 and 1 :5- If we detach the upper C-A-FjEp and the lower G-E-CjBp lay ... of the axis (see the centre part of Fig. 48) and pile up one on the other. we obtain the aJpl chord (sec top le of Fig. 4). lwe separate the pole.counterpole �e1ations (C-F# and A-Eb. repectively) of the axis, we have Moel 1 :5 (see right bottom of Fig. 48). Ifwe combine the notes of the axs we get a Model 1 :2 (see top right of Fig. '8). 5 5 ¥* ¯f¯" .0.48 P &BÃL t2 Õ In respect to tonality these formulae are inseparable. The fundamental role of the I.9 model is only emphasised by Ihc inclusion of all the potentialities of the tonic (�J:I1F#A) major, minor, seventh, and algha chords, as well as Models 1 :5. • These formulae merge into each other % that it is sometime rather difcult to define where one of them ends and the other begins .. � !t-_|. ¢«,leI, p· í ÅÁ¼- V¬L^Î¥¬A •• v:u. go • The tonal reting pint in Mod 1:2 always r uIb on the Ånote of the millor Kcond. which U the upper note Wthe major .ecnd. In the KWT or Moel 1:$ on 1 lonie, it Þ 1. or Eþ, M }', or A. Thu the bu nOle m Ihe minor accond, major third, ffth and millor ICvcnth Is alw:ys the Îuwc lIote, while that of the major second, fourth, minor sixlh and major iC\'cllth, is the upper nOte. In the LW of the minor third, ttitenc or m;jor aiJth, .ny of the notes may Krvc a b:, at Ihey alI lie on tbe lame aw. åJ And this uthe reasn why the most characteritic æmeloie in Dart6k are cxchuivcly ruled by CS principJ-s (sec bottom ler, or Fig. 4). AXH Ms¿l • • Ho.§: Within the range of the twelve-tone scale three different .:2 models can bconstrcted: a l •• u c. lEI-E-FlG-A-B�; a '.mi Æ " Cl-D-E-F-G-AI-BI-B; and a "bd,mi.,." D-EI-F-Fl-AI-A-S-C. Everother form agree with one or other of the abve rormulae. 5 8 I would like to illustrate the interelations outlined abe. by three bref Cple. Te Notllm in Mi,oo follows the tonic-tonic-ominant-tonic structure of the new-type Hun­ garian folk songs. So its ñnt,second, and fourth line ful tonic functons. accentuated by the tune which corutitutc a ",nU Mol l:2. mo.¸a 1µtonal character is determined by the A-fourth step (E-A), completed by the harmonie into a complete tonic æ! w · Y 59 The piece called FrDm ll Islad o B(/i (Mikrokomos No. tog) rests on the G#-B-D-F axis. hs scale provides a full Moel I ·× (G"'A-B-C-D-EIF-G�) which, as apparenl from Ihe fnal chords can be considered B a B-flh· {b-Cy=h-¡§] and GI-fth (G"'E�=Ab-FO), and as a F1,",lh (C-F) and D-(urtl (A-D), covering the complete axis. W o.fV P F � MC. r _ Both right and left hands play separate I ._models (C#-A-D-Eb and B-C-F-Gb)·· and these are characterised by ltlt'ir counlt�r· pole relations : left hand, GI·firth ·!D-fourth, right hand, U.ffh+F-fourth. Abo the formal construction of the piece is adjuslt:d ÎO the • lie,, we men1ion Iht problem o( Àd dD� aher;uion. "It i. Ifi.hly desir:lJh: Ih,., we have M 5yt.h:m U\ttu¡a¡Íun of Iwdve equiv .. klll lymbuls." writc1 D¤¡!Úk, addiHIo thal lll.· ¼W :lways guid c Uby ¡ l uL¼t¡owof 1c:\+!u tJility wllt'lI wrili"! his W¾f¼ = ¯hat is the n::n why we (requelltly ÜuU thr (lIharmollic varillllS in Ihe I»ano rWU(liolll mhi. orchetral worb. Ollr methOt of lIul u tillll "ol;l;ill:11 in the Jiatonic system and tben:(ure it t» O¼ utterly ules tol when it comn to recording twdvc-Ione music"­ U;rll,k: 7mPræm«}ÂwWn mu:ít(t920) • •• Thu by ahe merging WIW Moels 1 :5 we obtain moH I :z. b F-B-GI-D axs. The frst section closes in Ï, ending at the double·bar. The middle frst move around B, then G#, with an extended D pedal·point at the second double·bar. nle fnal chord i a synthesis of D major and F minor. and may be considered at the same time as type alpha (F#-A-C-D-F-At). Þ1U. __ Our third eXaDgIe is the recapitulation theme of the Violin Conc,rto, representing axis E-G-A#-C#. Its scale is of Model f ·� (E-F-G-G#-A#-B-C#-D). Bars f and � arc based on thc C l. E (melody) and G (harmon y ) pole5 of the axis. Rar 6 circumscribes the £.gamma chord (E maor-minor, G#-�E-G). and the melody or bars 5-13. the I ._ model (I&-F-Afl· ¯ | r~ r¯Î�î 6, We have to mention a a third typ of chromatc chord­ namely the chorW of tt mm. Its mot fruent fornu U the GS system are the whole-tone Kale. chord of diminished seventh, chord in founhs and the augmented triad. The last has its justifcation in Bart6k's chromaticism only in so far a it is built of minor sixths (8+8+8). Whole-lone scale Diminished aventh Chord in fourths Augmented triad 2+2+2+2+2+2 3+3+3+3 �+�+�+5 · . . 8+8+8 In our tone systcm two whole-tone scale C be distinguished: they are "geometrical dominant". complementar patter of each o.her: C-D-E-F.-Gl-A. and Cl-E�-F-C-A-B. 6. Aflwl=�+ 0Jk,\8Þ¤� ¯ ¯¯¯ l T Ho.§) Bart6k liked to use whole·lone chords hiort climtuu, since it has the ef ct. a it were, of "melting" the sounds (ICe Fig. 5 7 : Bluehtlrd's Castle No. 136, TI Woodn Pinct No. 123. Muc Mov. I b. 48, Mov. 11 b. 5 6, Mav. III b. 14). Harmonisation and theme construction in fourth chords are strikingly frequent, due to the influence of Hungarian peasant music. Ho.§ Chords 1n founhs generally aUow two combinations: un¡� according to the 2:3 p�ntatonic grìnc:gÌc, the other after the I !_ model. (a) LÍ the two fourth chords in the 2:3 scale wc can treat the one, which ¡ìcs a major second (o] higher ora minor tb|rd (j) ìVwcr Ihan thc ut hcr, us tuatr, unu Iìus (;In ltt· rcuut:t·u U the du-so_la cadence olthe older luÌk songs: FlG. _g (b) A good cxamgÌc of I .§ association M the closing theme in Movement ÎÎ of the klwit for Strings, Percussion tJd Celesta. The 1 :5 models are based on two fourth chords: D-G-C-F and A�-D�-GI-Cl-F�. 64 1 :5 models { AI-DID-G DI-G!-G-C GIq-C-F ?tG. Û The GS chords and chords or equal intervals onen combine together, in pracliee. Fig. bt shows an ostinato from Mov. l of thc Scoa|ajor ! wePiaoatar:dPcra.ioi:.TII<: twt·lvc tot:cs of the ostinalo contain tbc cntìrc chromatic scale. MC. Ût The upper part U baed on the A-B- Df-EI-F-G whole-tone scale, and the lower on the complementar F#-G#-B�-C ·O-E whole-tone scale. Each part is composed or minor sixthsj the upper of A-F-Ob and B- G-E� augmented triads, and the lower of F#-D-Bf and GI-E-C augmented triads (8 + 8 + 8). The twO parts move in parallel minor thirds [ 3 }. The ostinato U characterised by the 1 :3 models and the ¿amma harmonies l3 + 5 + 3)· t f g~§~§~§~§~@ • The beginning and fnal notes asume a ple-counterpole relationship: in the upper part, A and E�. and in the lower, F, and C. When viewed together they fonn an axial arrangement, FIA--Eb. Ever compnent of the structure u of GS fonula. M Diatonic System Bartdk'. matony M simply an exact and systematic ilsDr of the laws of A chromatic tehnique, i.e. the CS rule. I. The mt charactetic form of BartOk', "diatonic" system i the ttwli (overtone) scale, C-D-E-FI-G-A-Bp-C, and te IDdi cd (major tiad with minor sevent and augmente (ourth, e.g. C major with B� and F#). It is called acoustic becaue 1U tones derive fom the natural ovetone .eres. ·C • 4¢ ~ bô rIo. 6] In the fnale of the Sonata for Two Pianos and Pmws;on, for example, the acoustic scale C-D-E-F#-G-A-Bb enfolds itself above the C-E-G (C major) chord: see Fig. 64. This scale is dominated by the major third, perfect fifth, "natural seventh", and further by the augmented (acoustic) fourth and the major sixth (with D;rtok, the "pastoral sixth"). All chis in contrat to the minor third, perfect fourth, minor sixth (3:5 :8, C-Ei-F-Ab) milieu of the CS system. Let us place the principal themes of the chromatic Firt Movement and the diatonic Third Movement, side by side. The "chromatic" theme is composed of GS cells, the melodic line hinges on minor third, perfect fourth, minor sixth intervals (3-5-8). The "diatonic" theme is D perfect acoustic scale. FlO. 14 11.11.--6 Te two sphere of harmony complement each other to such meaure that the chromatic scale can be separated into a OS sequence and an acoustic scale.· Ho. 6g In the acoustic scale the major third replaces the minor tird Í3). the augmented fourth replace the perfecl fourth (�), and the major sixth replaces the mnor sixth (8). Incidentally. let me refer here to the la-sfmi fgures in the oldet childrens' æu@ and primitve folk music, whch, by no stretch of imagination can be regarded as products of sme delberate planning, though the notes accord with the "geo­ metric mean". i.e. GS. Likewise, when ltenng to tradtional muic, it seldom occun to us that the consnance of a simple Wjor mæ mght result from the coincidence of the nearest natural overtone: our ean limply register the fundamental number relation in the vibrations of the perct fh and major third. In Movement I of the S01tafr TU P;aMs mdpssion, the melodic and haronic devise are derived from the most prmtive "taton;, elements, whle the principal theme in the Finale limply evolve the natural overtone tcale over the C tjor chord (ICe Fig. 6). Yet this major triad CUIDW as a revelation. How can a simple major chord produce such BD explove effect? Lo ke at from another angle, may a comper with a •TÇæB. æOmticinlcr, ruir a chtc intcration, 1Ý pretence of being up-to-date avail himself at all of the major triad, whose vital signifcance has long so worn ofT and became an empty husk? Actually, the eitmtal efect of Bart6k'. music is due, for the most part, to his method of reducing expression to simple and primary symbols. The major triad may in itself be a hollow cliche, but when brought into a polar-dual relation­ ship with another system-as done by Bart6k-it may regain its original and potent signifcance. The explanation is that the GS between two points always cuts into the most tenst point, whereas symmetry create balantt: the overtone series is devoid of tension because its notes are integer multiples of the fundamental note's vibrations.· The thrilling efect of the major triad in the Finale of the Sonata is a direct r • .ult of it being completely released from the constraints of the GS system. So the la-so-mi (pentatony) and the m)er triad are not only symbols of the purest music but also elements of structre and formation, which, in Bart6k's interrelation regain the fre only they may once have possesed. This is what I would like U denote as the elemental rebirth oCmusic through the reconstruc­ tion of its means. Let us set up the formula of the work: ONNAM¡C proportion " GS-forms " pentatony -opening movement 5A3IC proportion -symmetry=overtones -closing movement •The cs epres tbe law or &he ,lOW mean, &he overtone reRct tbe Law or &he mWUmean. � we know, harmonic overtone @ prouce by tbe vibration or atrinl, air in tub, etc.; thCe not only vibrate to ther Îul¡ lenJth but nW in h¤ÎVm. &hird., ÎÞuMW. etc. of the ImBth-proudnJ 1_Uu noc on the .trinJ or in the tube. The overtonc combine with the Uc note, ad the mm or the tone u determined by the etet to which thce overtone moify the 80un0. We ,he�ro� mB the bDrmoniC of the acoustic system "colour chordt". It U no accidcnt that the w0UC effect in Bart6k', compitioR origi nate primarily i &he colour chordJ m French imprcionism. Bart6k himself use to allu.de to uu inpiration. J¹ This implies that the 5ymmetrical perioisation of the Viennese classical school and its harmonic system of overtone rdations are phenomena not independent of each other; they only represent diferent (horizontal-vertical) projections of the same basic concept. V- The two systems reOect each other in an inverse rdation. ship. Through the inversion of CS intervals. acoustic intervals are obtained-from a major second {×] a natural seventh (e.g . from Bb-C, C-Bb). from a minor third (3 ) a major sixth. from a perfect fourth (5) a ffth. from a minor sixth (8) a major third-the most characteristic acoultic interals. Therefore nol only do they complement. hut also rtttl each other organically . The opening and closing of the Call1ata Profana oilers a beautiful illustration. two Kales mirroring each other note for note-a GS scale (interals 2, 3, 5, 8 with a diminished lifth) and a pure acoustic scale: ¤¢¢ �tÎtt 4 ••v eto. Û It is worth clarifying Ihis interrelation from another point of view. The harmony which appears beneath the atowlu melody of Fig. 64 produces perhaps the greatest surprise of the work, obtained by means of a simple major chord: C-E··C. 72 This consists or the closest overtonl relations. Le. a pcrrcct frth and major third. In the chromalic First Movement the major triad always emergcs in the 3 + _* 8 di\ision of thc OS: The characteristic perfect rourth (5) ,,"d minor sixth (0) of this CS chord have been transformed oy inVlrsi(ln into the Plift" fift h and major t h irJ of the acoustic chord rcspcctively. Let us show these chords in their seventh forms too: 6 ¹ '10. 68 What is valid. relative to the C rot. in the OS system from Obve downward is equally vOlid in the acoustic system in the opposilt direction. It is thcrefore an "overtonc" chord. The cir­ cumSlance that ollr ancient melodie hOve a desunding character may perhaps be rdated to the fact Ihat pcntatony is &OS tone­ sequence. _. Although thcse features seem to appertain ÎO the outward form. this no longer applies when it is considered that only (unsunant inlclV;lls exist in the acoustic system (owing ÌOovertone 73 consnance) whereas the GS avails itself precisely of those intervals which have been considered dis onant by musical theor from the time of Palestrina. Incidentally, this diversity accounu for the tendency of Wetern music to be acoustic and of Easter to be pentatonic. ¯impJies that the relation of consonance and disonance is thus inverted in the two harmony.worlds; the purity of " diatonic consonance is in direct proportion to the overtones, while the chromatic technique attains iu highet degree of consonance when all the twelve semi.tones in the tempered scale are made to sound together-" like the roar of the sea" . to quote Bart6k. However incredible it may sound, in pentatonic melodies baed on mi & a key·note (mi·so.l-doscae: interals 3,5, 8) belonging to the most ancient layer of folk music, the geatest dissonance is represented by the pe:¡ect¡:):t (cf. Fig. 7 6) . .. . A secnt of Bart6k's music, and perhaps the most profound, is that the "closed" world of the OS is counterbalanced by the "open" sphere of the acoustic system. The former always pre· supposes the presence of the ,ompleu system-it is not accidental Ihal we have always depicted chromatic lormations in the Uv:�J circle offlfths. (See Figs. 2, q _ 46, 48.) In the lat, all relations are dependent on one tone since the natural sequcncc uÍuvcr- tone emerge from one single root: therefore it is Uptll. 5. Thus. the diatonic system has a fundamental, rool note and the chromatic system a o»t·unote. In the chromatic system all relations can be inverted without changing the signifcance of the central note. The principal theme of the recapitulation in the Vi"lin C"nttrlo has a B tonality, in spite of the fact that the B major tonic "stands on its head" (owing to the inversion ol the theme) and our ears, accustomed only to overtone relations, perceive it as having an "E minor" tonality. (The B centre is aceentuated by a shimmering pedal-point too) : 7 4 WO. Q It is this "mirror" (see Fig. 6g) which shows that the chromatic technique leaves the requirements othe overtone system out o consideration, and ideas like "up" and "down" become quite meaningltSin it. The hannony which in the preeding Cple sounds below the B centre, producc, by the negation of the overtone system, an effect a if the objeu of the physical world have suddenly become weightlCa sphere where the laws of gravity are no longer valid (sce Mov. I, b. 1 9 4). 6. And this is why Bart6k's CS system always involve the concentric txpmíon or tonlrali(1 of intervals which is as consistent as to be virtually inseparable from the chromatic technique. For example, the quoted themes from the First Movement of the S01t Jor Tw Fímt ad Fotm:íen are constructed in ever-widening orbiu (sce Fig. 28: leitmotif-principal theme­ secondary theme 8-13-21). The principal theme is augmented from bar t bar, from mnor third to fourth, sixth .d seventh intervals. And the scope of the secondary theme expands similarly step by step, frst with pentatonic turns, then with fuurth and ffth intervals, fnally reolving in a broad sixth (se, Fig. ,8). Wc frequently fnd a "funne�-shaped" (sec Fig. go) and "scisr-like" movement of notes, ( t.¢O ¥tU. }0 and sequence proceeding by wider and wider steps: 7 5 Even these processes follow a planned course, every detail showing augmentation up to the geometric centre ul the movement (b. 21 7), after which they gradually contract .. gain. On the other hand, in the diatonic Third Movcm('nt, such progressions arc quite unima,rina6Ie. The diatonic harmolljt�! nn� characteriscd by a 1Ialrr frmncss (e.g. the chord of rig. u) u radiates its energy for a long period of time with a motiuult·ss. unwavering constancy) i n contrast to the CS system, which is always of a dynamic character. 7. Bartok's closed (chromatic) world may well be symbolised by the circle, while his open (diatonic) systcm, by the straight line. Like in Dante's Di\'ine Comedy the symbol of the Inferu is thc circle, the fIng, while that of the Paradiso, the �traight line, the arrow, thc ray. ¯hc rings of the Inferno unucrg0 a concentric diminution till they :lrri\'c at the "Cucitus", whereas those of the Paradiso widen into the infnite "Empyreum". În Bartok's "cosmos" the (hrmes fullow a similar pattern; chromaticism is most naturally associated with the "circular" while diatony with Ihc "straight line" of melody: sce Fig. )z. 8. The idea of "open" and "closed" is also expressed by t¡u: organisation of the themes in relation to linu. The basis of 1º zn4 rIo. )z a classical melody is the ptriod. As a rule, the themes i n the music of Haydn, Mozart and Beethoven arc divided into 8 + 8, 4 + 4 and 2 ¶2 bars: the frst two· bar "question" being followed by a two-bar "answer"; these four bars may then be considered as a single question, the answer to which being given in bars 5-8. Thus the form dcvelopes: this simple O-har sentence ends in :, half-close and corrtponds to the tonic, full.close cnding of the 16-bar period. Thi s principle of symmetrical pcriodisation is readily discerned ill Bart6k's diatonic mode of writing as well. In contrast to this, in hjs OS technique. positive and negative sections constitute quite U different system of "questions" and "answers" (see Figs « • 6 and 22). Here the law of h:llancc and symmetrical periodisation is replaced by regularities of Itme «:¸mmtt¡). Positive and negative sections embrace each other like the ascending and descending parts of a wave. The conditions of organisation in the CS system are inversely related to those in the symmetrical pcriodisation: one providu Íor a proces of merging, the other for dividing its constituents; the former emphasises the orgallic Ulli� in time, the l:ler snrveys the material in space. The CS forms assume the character of an uninterrupted time proces, revolving in the ;lrc of a wave, while the symmetrical periodisation breaks the material into metrical components of lines, rhyme and strophes, W in the construction of a verse. J1 9. And what do the two sy.tem lok like when examined in number relations? The key�numben of the overtone system are wlwlt numben: those of the octave-2. 4. 8 j of the fh-3, 6, J2j of the major tird-5, 1o. etc. While in the OS !ystcm the key number U inatiana/: ' - -=0.618034° ¤ . . 2 The irrational character become still more explicit if the formula is written B follows (which again conceals the Fibonacci series): The acoustic system rats on aritlntti.al, the O� system on gtOmlmal proprtons. (Sce App. 111.) The characteristic 3-5-8 proportion w only approximately correct and w expres� sible only in ilational numben (e.g. 5:8'0906J • # # ). The minor third in pcntatony can be proved to be somewhat larger than it W in the tempered system.· 10. It may be symbolic that in the diatonic system the partial. toncs range abD while in the chromatic Iystem btlow the fWdamental note (see Fig. 68). It is of sme interet that Remann derives the minor triads from "under" -tones and the major um from "over" -lone. In the case of CS alp44 chords U relaton is also valid, inversely; the minor third falll abve and the major third below the key-nole. Although Riemann's concept may be • The order·nubr Wthe minor third in Ihe lempre .yatem it I " 9 æ in the pnlatonic .)tc:, 1 $t. (3 t o t .k Trans· m ubuptatony coe clo to te mr third,) contetable it u still worth considering the fact that the lwo mosl inltnse GS intervals produceable within the compass of an octave are identical with the chord Riemann produced by inverting and projecting the major triad in the lower range (C-AI-F) : 8 ¯C-A�. 5 ¯C-F. The leitmotif of the Àír0mÎ0w À0nd0rín" receives ÍU intensity fom just these CS intervals: Ap-F-GI-F. (See Fig. 31.) • In the Kore mthe ÂÍÍfmM muudunu each person urepretental by a tone.symbol with whoe aid ¾may weU "read" the plol mthe pantomime. The Mandarin may U rec08ni,aI (rom the note G'-I: (Ap-F); tÌw Girl Ítuu the nula J:;--Op ¾T l·.þ-bþ 1 {Dg·Ag M¥ O'-A'· I�)--W W the very beginning of the work. The (act that the complimena mthe Old Gallant are intended (or the Girl i. shown also by the music: Ihe basic chord 0the Old Gallant, more than thirty time, leads to the Girl', symbl: Wc mentio only three brief ampJe, After the entry o( the Mandarin the pntatonic ostinto undulate from the Mandarin', G'-I note to the Girl's tone.,ymb.: or later when he Wtrying to reach the Girl: ûiv1 We Æ the mimof the two .ymblt at Ihe W© m Mdwok: A detaile analya o the ptomime ¾ publbe by d author in 5twu mwl¢¿u (Vo!. ¡ No. R¡ pp. )¤)-{)a.mGa). 7 9 A particularly efective application of these inverted relations can be observed at tbe climax, in the C.major sccne or BIut- ',ard's C4lt, when the nage i plungcd into darknes: WO. )¿ ÌÌ.It i eay to sec that the symmetry centre of the penta· tonic scale U the re: do-,o-re=la-mi Similarly, de!ree re constitutes the symmetr centre of the major and minor scaes - ad that of the acoustic scae, too. The atithetic relationship of the two systems becomes evi· dent it wc rcaile that the basic step of pentatony. the plagal IIl�mi cadence and the baic step of clasica hanony: the dominant·tonic sO-do cadence ate precise mi"or images of each other, related to the f' symmetr centre: symmetry centre so �do re mi �/a Bartok's chromatic system results in pu¿et,while his diatonic sYltem in authentic, hamonic interconnectionl: the baic ste l ' of the fonner being T-S-T and that of the latter, T-D-T (sec App. 11). In traditional music the motif·line usualy attains itl most tcnsc gruìnt un Uu s|xtb ]·tpt·rft···t f·otth]::ftht·t·n|t·.!|u dimax of the sixtLil baicaly a pruperty olclassical music where it functions W the subdomimmt. & FIO. 74 Actualy, the most intense hannonic function in classical music is represented by the subdominat, but it comes as B surprise that the minor subdom£nant, being the in tensest of m subdominat chords, is essentially a characteristic Conn of "CS tension" (intervals 2,3,5,8): ric. ]g and conversely, the dominant harmonies arc built on deJ'ccs of the nearest overtones IC, E, 8 I. ÄU means nothing less than that the technique of tension-relaxation in classica music is closely related to the dual principle of CS and acoustic cor­ relations: subdominant tension is, in fact, a CS tensio", while dominant-tonic presems an overtolle relationship. The S-T tur in the CS system can be reduced tothe la·mi or rc-Iu close so frequently found in ancient pentatonic melodies. Compare with the " changing fifth" and six-four types of old Hungarian peasant songs: �J J =J¶ _ & #8M+ }Û 8, J 2. Baf6k's diatonic mwic is always inspired by an optimism and sCeniQ, h chomatc muic by a dark, moreover , irrational and demoniac passion. This involuntarily bring! to mind te chromatic experiments of Liszt and Moussorgsky, probing the gloomy depths of life. Let U recall the lÆte piano piece of LisZt¯ Gre Cl"uds, UnJue¸Stas, Ptludi" Funebre, the death�music R. WQgntr, Vene<ia, the ghostly Lgubre Gondola, all these are written in a tone-system of distance modcs. Or the scene of Ö0rU Godnov's ÛnZy. where Moussorgsky avails himself of a perfect "axis system". All in all, the Chromatic and diatonic systems form a coherent wmN¿ representing two sides of the same coin, one of which negates and at the same time complements, the other. They constitute contrast in unity: afrm and deny, presuppose and exclude each other.· •The ame duality appear in the frequent M Wthe ttl ltl� keys. Two triads which merge Stita9 diMlve m other bc aUe the eui­ distance creata • foating tonality, W anihilAte U. ¯ prograne W the ?Ut! SlriI C/-"iIInc" and "recovery"-i mW the duality ofF minor ad A major. TheJe two mmcomplement each other, meeting in a Umdistance moel (I: ] moeJ): F-Ap-C + A-Cf-E ÆF-Ap-A-C­ q E. ._ · ø A N ' + ² ¹° ]y º fÆw w # A =g-· In the piano piece "Se saw, dckor)-aw" the foating Í8 elprecd by the combination of E minoÏ æ Ab major (E-G-B+Ab-C-J: b givC 8. In some ofhil works Bak @so far in the plaritation and reduction of hit material, that form and content, means and meaning, W to constitute an almot inseparable entity. From tbe preceding principle ever further dj(J njc formaton become self-evident. The main diatonic intervals. i.e. the 1/l and majM tird, & mphæwbyÜ major chord, built up m succesive thirds, every second degree of which rhymes in perfect mU. • • � Ho. ]) W $1@moel). Slly & nut Ka E mmæAp m�or pan" the hS W the 6BwlfW: "A bit drnk" (I FiS. 4b ). ¹P ßIm&m4'tUürÚcÊm¤rC0aCthe fth dor it dCwyed by cminor .t & çq0Úc bm -m. A,dmplan or the opra i built up 0Ncte.rti. ¯nght Wrete by FI minor. Mdayli,hl by its cwlerpc cm]m. 1 major try b nuti by Al minor. thUl the lattu ban a "dcth"-ymblm in t wgrk. while t c­ plementar key of F# minor-BP major-u &I tc with the "'ovc" scenc. 'hae rour t includc ever dqec 0t twdve-note .le: f-A-C B�-IF Here we have the origin of the well. known Bart6k "signi1lurt�".· +¶4 ¯¯ wo. )ß (Cf. Two Portraits, Swiss Violin Concerto, Bagatelles Nos. 1 3 and Iq¡ Ten easy piano pieces: "Dedication" and "Dawn", Mikrokosmos No. 10, etc.) This type, combined with Uic acoustic fourth (e.g. C major chord with B and F#), appears at the most splendid moments, as in thcflowcr·gardcn of Bluebeard') Castle, or as thc symbol of the "faming. golden-haired noon" : Fig. 79 . • Tu chord has a counterpart: the minor chord with nujor JeveluJI. e.g. D-F-A-C" which æ ;uw�ys asi:ued with pain ;md Jloion ill Hart6k', dramatic work and songs ("Your leitmotif" wrote Bart6k ¡u Slefi Ceyer). We giyc three brief examples from MÍw0Of4´! Íw¼¦ wJ8=«ð æ18 M 41@ M +I1 4¢ $f4< +t.w+t1t¿ +. t g.... ,",,', ,i,. ,,", I . " his 1ot<11l" At the <ud mMlw0rmd'1 4ø1tÌ« ¾M Þ<«t Qt¢ t +otl 1¶ I: ^ L¸ the ín½wu= uÎ Ihe tonic leitmotif C-E;-O--U, Py .he iuven;'JI Il,e ••· .. . C-Ep-G minor chord M uæ m into the C-A-F m:aj(r cUutU, U�Î on the augmente triad C.-F-A: (rom here die dg�lIjlinJ ml´ o: ariv�. 110 · 79 Since the acoustic system u merely an inversion of the CS, we can obtain diatonic harmonies by rrwzín_ the layers of the dlpld chord: 10. 80 Te diatonic effect is due to the alpha-inversion being govered by perCect firths, major thirds ad minor sevenths (i.e. the nearest overtones -which were exclude4 oyth�¢lj/¢rh¤r4tI· Ho. &b u.u.¬Q However paradoxical it may seem. the chord which has a major third above the key-note and a minor third below it. makes the most "diatonic". most opened impresion in BartOk's music: Y H ¡ M l Ñ «ºÎ¤ { lL&� 1 And to complete the concatenation, it should be poillll'u out that the inlersion of alpha contain the very kernel of the muuxlÍc chord: FO. ßz Ït happens frequently that an ambiguous bass i sometimes represented by C and sometimes by F,: M Th is me cae in the "axis melody" ofthe theme in Movement III of Muic/or Strings. PtCS;OI ant Ctleta: � �� wo. þ We may summarise these analyses ð follows: os TPES (chromatic system) Pentatony Alpha chord . ì:9¡ 1:3. 1 :5 models Forms of equ;J interals: . whole-lone scale diminished seventh chord in fourths augmented tiad, consisting of minor sith Acoumc TPES (diatonic system) Overtone chord and scale Invcrsion or alpha Succesion or thirds and frts with major char­ acteristics Fomu of equal intervals: chord in frths augmented triad. con· sisting of major thirds Particular signifcance may be attributed to the fact that J"1tatrJlY is most characteristic of Bart6k's chromatic (CS) system while altrtone chords prevail in his diatonic system. This duality, in our opinion, would seem to express the two most ancient endeavours of music. The physiological apparatus of our ears (with the logarithmic structure of the cochlea) enables us most readily to perceive the sl-la-so-mi (2:3:5) relations at the earliest stage, of which both primitive folk music and our simplest children's songs provide unequivocal evidence. In primitve musie-cultures the sense for major tonality and functional attractions arc quite unknown.· The devl"lupmt · nt of Iron;c thinking derives from a quite different sourCe, namdy the overtone series. This could only have COlRe into its own with instrumental music, and it is no accidt'nt that functional musical thinking is hardly more than a few cellturirs old. Pentatony may be deduced from the Pythagorc;m tonal system-grouping the nearest fifths and fourths-harmonic music from the overtone serie. Incidentally, pentatony is of mtlodi" linear origin, being of "horizontal" exlent (in time) while the overtone system is of harmonic origin and has a "vertical" (spatial) dimension. Would it be too daring to suppose that the rots of pentatonic and acoustic thinking were the two points of origin of all music.·. (If so, then Bart6k hpenetrated to its inmot cor(.) •"Pentatony doe not .uler the dominant-tonic cadence:' (Bart6k: Hungarian Folk Music. 1933). "In this scale the ffth ha no prevailing role" (BMI6k: Hunguian .·olk Music and WC Hungarian Mu�ic, 19:0). On 1Þc uIÞm hand "the frequent use 0Ihe fourth interab in our meloie sU88n1cd to w the ¾ of fuufth-chords" (Bart6k: ¯Þc influence of peaant muic on moern music, 1920) . ••BatI6k himlr IlOnily believtd that "il .. iII be pble to trace baCk mÎ tÞc(olk mwlc on Ihe flce of the Slobc enli.Uy ÍO • few parent-fotm., archetyp, ancient Ityle" (Bart6k: FoIk.song rearch and nationalislll, 1937) · ö The frst is justifed by "inner" hearing, based on the ,,, riD [olicm structure of the ear; the second by "external" hearing, controlled by the physical Ia.ws of consonance. The former is, therefore (ense, expressive and emotionally charged, the latter colourful, impressive and :tmuem. The above claim u supported by the scientifc observation that GS is to be met with in organ;c matter only. Pcntatony, with æit tension, could nddler have come into being wiloUl the aid of human emotion. The acoustic harmony on the other hand, may develop independently of the phenomenon of human life or of human intervention-a vibrating column of air in a pipe (or a string) is enough to bring it about. Pentatonic and acoustic trends follow contradictor coune, Physiological eforts tend to organist and create tmsion, while physical efforts disorganise by striving to aboJish tension. Here the thesis may b advanced that the OS creates a dostd world and carries an inner tension, white the acoustic system is e]ttt and strives to release tension through its overtone consonance. It may be added that this closedness is an organic feature of GS (see Figs. 24, 25 and 26 for examples independent of Bart6k's tone-system) and this quality is responsible for the capacity of GS t organise. Æ an illustration: CS can be easily brought about if we bind a simple "knot" with a paper ribbon; without exception, every proportion of this knot will display geometric golden secdon.· Fig. 85 . • It m no accident Ihat penl_S0n.. 1 common in living nature, arc roreign 10 mc inorganic world. � � 4:rw¢:Þm b:e * ,:o · 6,3 no. 85 It i t property of the pentagram that Ûw¡M aUudes to in Faust, Part I: MPH.: Let me admit; a tiny obstacle Forbids my walking out of here: It i the druid's foot upon your threhold. F AU: The }m¿rmdistreses you? But tell me, then, you son of hell, If this impedes you, how did you come in? How can your kind of spirit be deceived? MZPH.: Observe! The lines are po rly drawn; That one, te angle pointng outward, Is, you see, a little e] 4 • • Although the important question ofthythm and metre cannot be dCultvíthhcre utut|ylcngth, «Å few OuutuudÍnglc.tturcswill be pointed out. Bartok's rhythm Î governed by Bstrict laws as has been shown to rule his form and harmony. The drcfar character of Movement 1 of the Sonala jor T U'0 Pianl and Ptreion U in no small degree due to the "absolute" odd metre, 3 times 3 eighths, while the Finale owes its static character to its "absolute" even metre, 2 times 2 eighths. In Movement M¸ even and odd bars are intentionally alternated. (Bartok was very much interested in the potentialities of "even" and "odd" metres. In the SrCru Pano Conctrlo, Mov. 11 of Music, Violin Crlo, Diverlimenlo, Mikrokosmos Je. t_y¿ themes presented in even�metred bars return in odd rhythms, or vice versa.) The rhythms with a "strong" ending in Movement I have counterparts with "weak" endings in the Finale (see Fig. 87). MU. ß6 Consequently the theme of Movement I constitute a dosrd, and those of the Finale an oJn form. But the polar principle prevails also within the even and odd metres: ¨+¬+ and '¬+¬¨ unit are periodically alter­ nating in the odd-metred theme, while an alternation of "+-+-" and ´`¬+¬+¨ umU provide the rhythmic pattern of the even-metred tune. 9' 9' N0VíMfNT 1 � � � + ¬" ^ , ¯¬ M0VENfhT | ¯ + ¯ ~+�� � '¯¯�� ¿�·�`\ -% = + + ~ ± ~ H ~ ® $ ~ q ~ ¶ ~ ¶ ~ ¯ ¯¬ ¯ w ~ � > V * ¬¬ + + .\ ._ + ± c1 w w PlO. ô] æ ¬* That is why we feel the following idioms to be 6 revealing of Barl6k. 93 rto. M As a fnal example. let W compare the opening and closing bars of the Sonata for JæÏma1 and Pemus;on. åìÌt_ « ål� 0 ••twìl Àm tiÿI m¡W» a1rk8 Æ4�q+ Æ+ . PIO. 8 The opening bar give the impresion of decent, as it were , into &well "which is immensely deep, or should we say. has no bottom at all" (Thomas Mann). The low shivering sound of the timpani really seem to emanate from the negative pole of life, from a phase of precontiousnes-the key of which is FI , the lowet point in the circle of ffths. Towards the close of the work, the "flliped" cymbal sounded with the nail and the light stick dancing on the n`m of the side.drum, produce ostinatos which gambol joyfully over the work, with "slender anklc" on the paths of |ï¿kl: in C major, the highest point in the circle of m¿ and counterple of F#. In this way the extreme points of the comption may b regarded a negative and positive pote so that the analogy of a magnetic field offers itself, a current being develope betwen two opposite pole. The Lento-with its utterly iI ral now-is represented by the lowet, the Allegro-with ÌU V3 articulated rhylhm-by Ihe rughel drum effect. On the onc side a linear, on Ihe olher, a rhythmic-spatial clement. Nevcrlheless, the mosl interesting circumstance is thal Ihe dimensions of the comple" work were not accidental: it reRccts the unilY of Ihe correlated principles of the closed circle and open snutr. 11,e symbl oflhe circle is ¼_ while the laller can be expresed by Ihe powers of2 (�'=4, 4'= 16, 16'=256; Ihe next power is already too large). Te time-value of the whole work (the above-mentioned 6432 eighths) is 804 whole notes, and this is precisdy the proucl of: It can be deduced from Ihe foregoing Ihat onc and the same regularity is established throughout many different dimensions of the work, through form, key, harmony, proportions, rhythm, dynamics, colour, etc. Considering the date (1937) and olher particulars, one may risk the supposition thal Banok probably intended the ðvnuld for Tw Pianos and PnCion to be a crowning piece: the Makrokosmos of the Mikrokomos (192�37). What role did Bartok's art play in the music of our centur? His chromatic system has its roots in Eastern folk music and in pentatony; m acoustic Ifstem he owed to Western harmonic thinking, He himlelf admitted his indebtednes to folk music and Ihe French impressionists as the IWO most descisivc inRuences on his an.· • "The two W Uor our ut ogin . ta in folk muic and the new French mutic," wrote &.16k ("Zoh£n Ko£ly": 19:U). Thi decluatton dccr attention fo it u well known th . t he rarely, ÍÎ ever, eomrlle hinuclr W hi, own eompitions-Ihouah he liked to emphuile their relation to rolklore, mainly .. ith the Intention of pro­ pal&tingfo mlDic. "Lt my music lpak ror itlf, I lay no claim to any n p lanation or my worb'" Should his posltlon in music be summed up in a single sentence it might run as follows: Bart6k achieved somelhing that no.one had before his time, the symbolic handshake between East and Wet: B synthesis of the music of Orient and Occident. • This esay is the introductory part of the author's book, "Bart6k's Style" published in Hungarian in 1953. A following chapter tackles the "dramatic" principles of Uart6k's music, especially of his instrumental works. We must not forget that Barl6k is, in fact, a dramatic temperament, as all creative genius in whose character the bents for logic and heroism, arc united. V1 Appendix ¡ Referring to the &1ta JM Tw PialS aM Pert;oT, it is of intCrtst to point out a few particulan of Movement Ï In hs. �35-247 wc again notícc thc two-Ío¡tÌ aOillity oÍthc axis; on the one hand the G# otinato running through the gætand the D counterpole. and on the other hand the constricting funnel-shaped motivic progressions, wedged in the D-GI principal branch and later in the F-B secondary branch. �:¯ = æbqt ºÎ � • • • If � WO. @ VV The four sections of the seondary theme in the recapitulation (bs. 292-33°) are based on the four poles of the IoniC axis M that their Outer and inner parts, repectively, correspnd with each other in their pole-countcrpole relations. FIC. 91 The axis construction of thC ca is equally unambiguous (bs. 41 7-431 ). In accordance with the polar ãchcmc the augmented QtÎncígßI theme appears in E\, then in A, and fnally in Eb ¶A, over the disjointed Eb-Gb-A-C major-minor (gamma) accompaniment chord5. Lì *Ül ¬Å" Û AÄl 5 FIO. 92 lO 3Lcdevelopment in Movement 11 uÎMu for Slrings. Percsion and Celesta preents an exemplary axis cutuUuctÎun! rIc. g¿ The appearance of the Et-F#-A-C is always ;ccenluated by the bass drum, The Iccomplnimcnt, which rcmainJ unclumged throughout, stresses the A and E� coulllcrpolc (blla chords): MD. @ ß.a.¬ lOl It U evideot bth from the accompaniment and the dynamics that the ElA polarity fornu the backbne of the structure. And fally Fig. 95 gives some strongly marked æmelodies. l I f ½ to9 £B � 1' rto· 9§ Appendix ¡ ¡ A few examples are given to iUwtrate the interrelations expunded in connection with Fig. 13. The order of keys of the Fi,sl ROMc i W follows: C tonic, E dominant, A� subdominant and C lonic. Movement I of the ctt i subdivided by the fve·fold recurrence of the principal theme: F tonic (expoition) h. 76 D� subdominant (fnt part of development) h. 231 A dominant (second part of development) h. 31 3 F tonic (recapitulation) h. 386 F tonic (cod.) h. V A similar arangement i to be seen in Movement Ï of Sonl IQr Tw Pims mPecssion: C tonic (expsition) h. 32 E dominant (fnt part of development) h. 161, 195 ܶ subominant (second part of development) b. 232 C tonic (recapitulation) b. 27. In the tlUrd example of page 45 thee relations are: D-AJ-F#. The principal theme in Movement 11 of Mwit jor Stngs, Ï ¤æc:l::teUparticularly cogent, because here we fnd the cupla-structure and tonic-tonic�ominant-tonic con­ struction of the new-type Hungarian folk songs. The tonic i represented by te C-F# counterpole, the dominant þ&Bb counterle (not by G). The second ent of the tonic provide the exact "tonal answer" of the C-FI æ. G-C tone change into Cand C�F� into F#-C#. ImJ 3 FIC. @ A similar & ialion of dominant and Ionic is evident in bs. 1,1-1,8 of Movement 1 or the Diu" ilf/oj the E(-A dominant counterpole correspond to the F-8 tonic counterples: mo. p) or at the recapitulation: • ª� f The lower majur second degree (e.g. Bt in C tonality) might justifably be ( alled the "bart6kcan dominant" owing ÍO its rrequent occurrence in his music: Ho. @ whch can again be explained by the regularities dealt wilh above. The opening bars of the Mirtu/ol Mandarn illustrate how the tonic and subdominant are linked; the D# tonic swings towards the subdominant F and B counterpoles: MD. ID t0 It is interting to note that, in Bart6k's music, the three functions play a symbolic role too, particularly in his stage works. In Bluebeard's Castle this sign-language always goes hand in hand with the plot and contents of the drama. The sub­ dominant has a negative meaning being reserved for the expre­ sion of fever and passion. All positive movements start with the dominant. The static pillan of the opera and the pointJ of rest are based on the tonic. The downward pull of the action in the MirDtulow Mllwin is also expressed by the functional relations. The dominant start of the pantomime (in C) reflects the throbbing excitement oflife. whereas the subdominant end of the work (in F) depicts the death of the Mandarin. The intermediate scenes-nearly half of the music-especially where the Mandarin satisfes his desire, are written in the tonic C. Beginning G QOMINAN3 Climax C 3ON¡C End F 5UBOOM¡NAN3 The succesion of scenes follows the same descending order _ a triple descent from the dominant heights to the subdominant depths, as if expressing the idea that the work moves towards a "fateful" abyss: DDM1HA 7DH1C 8UmtHA I. Viaitor (Old Callant)--I. Visilor (Youth)--3. Vililor (Man) Î+ WalZ_2. W.ltz¯- Ýuu¡ å+ Murder _2. Murder -3. Murder It is for this reason that the scenes, situated one below the other on the above plan, are varianls-devcloped from the same material; e.g. the music oflhe First Murder originate in the basic chord of the Fint Visitor (Old Gallant): IO) ¡-- ·-¿ W * f Pt0. 1Q$ In thl" plot of Ihe Hædrn Îrímr all this is inversely IM1C. `Ïh r scenes follow an u1rm ín ¿ line, incessandy going up thc T-D-8-T grades: EXPOSITION MIDDLE PART Prelude TONIC Princes DOMINANT Prince SUBDOMINANT Forest TONIC Stream DOMINANT Making of the doll SUBDOMINANT Dance of the Wooden Prince TONIC I. Scene (No. 120) 2. Scene (No. 128) 3. Scene (No. 1 32) Conclusion TONIC DOMINANT SUBDOMINANT TONIC RECAPITULATION" end of exposition TONIC DOMINANT SUBDOMINANT TONIC Wooden Prince (No. 1 4 9 ) Princess and the Stream Denouement For example, here follows the aiS structure of lÎ:c frst scene-Dance of the Princes:· • A detaile analysu or the wrk ¾Æpublished in the author's " Drllk's Dramaturgy": Stage wrks and Can.ata prorana (Editio Muic: lJlltlapol. rp¤¶)• •08 L¬ ų ~ Lr¬6 AX| 5 åßsg¬R • mx B¿ lh p Y ¡ir×Ì Þ r« " &Ì> Codrøm t-4 * wø no. ioz 1Q Appendi ¡¡¡ Bexact golden section can only he constructed gemctrically­ it cannot be obtained matheDatiClly_ i.e. by means of rational numben. The key-number of the CS i irrational (similar to T). Here i & example m the uEudoxus" construction, with square and semi-circle. ¥IU. 103 ` MÓÌÑ.. ` and another based on the Pythagorean proporton. MU. tog Io The hypotenuse <Il of the Kepler triangle subtends a "golden angle" to the shorter perpendicular (0.618 . . . } : 51° 49' 38- • • # Å� oõts W 0 I:v¥I8~0ðW'04W ?lO. ID�¿ and a chain of golden sections can Dbrought about B follows: zo. to6 t I The members of the OS chain can be obtained by subtraction Thus the OS of I is 0·618, that of the latter: 1 -0.616=0'382, dc: 0·6,8 "0'382 " 0'2,36; 0'382 -0'236 ² ('146, C1C. But thc sanuOS chain can also be obtained by involution .0.6,81 = 0·618; 0 . 6182-0'382; o·618's0·236; 0.6,8-=0' 146, etc. The sine ad cotagent cures meet in one definite point. and precisely in the g0Ìdrn0ngÌr·The vaue pertaining to it is: v0·618 . . . no. 10} With the aid of compasses it can be shown that the radius goes 6 times, while its CS 10 cimes into the circumference oÍucircle. The Cheops Pyramid in Egypt reveals the following structure:· 4¯ A ¤ + N no. 10b and so ìU sloping sides subtend a golden angle . • ÆcrK.Kcppiscb an E. Bindl. "' The dynamic quality of the Parthenon in Athens owes much to its OS dimensions, and that is why we feel the building soaring upwards, as it were.· P1U. 1U@ While Gothic architecture favoured angle of 45°, Renaissance art, following the Greek models, showed a predilection for the golden angle. The circle had been given a "heavenly" symbol­ ism, while the square R "earthly" onc. Because the Gothic concept subjected earthly affairs to heavenly concerns it forced the square to fnd a place witin the circumference of the circle. Ïncontrast to U matters of heaven :md earth had an equilibrium in Greek and Renaissance art, therefore, in •ÆcrZising. geometric terms, the circle was coupled with & isoperimetric square . Hence the triangle no more subtended B angle of 4�o, but rather the golden angle: U:r:| m d Ktm:u RÅ ÉÅ ÝÎM+ Î ÎÛ ²4 and the ratios between the three circles in the above Jetter� symbols show OS relations.- Zeising derived OS from the proportions of the human body, and 1",ld the Ilt:Ivl'dere AI)(lto «ÂÑ t l l(' l )Cr[i:cl l"mbfldimt:nt uÍìt. Eisltcin planned his flm, "Potemkin", in such a way that he placed point of uperfect inactivity" in the negative goldC section of each act and thoe of "highest activity" in the positve sections. According to Einstein, OS pTvidC a ratio which opposes the bad and fadHtates the development of what is good. "No element can be properly joined without the aid of a third one, for the two can only be united by the mediation of & link ]but of all the links that one is most beautiful which makes a complete whole oritself and of the elements united by it." Plato: Timau . • L. Otto Shubrt: Getz d BUkUNt (Scn -Lipig iy§). " 5