Fin, publUbcd iD 1971 by SWlDIorc Prcu Ltd wuIcr their auodaud imprint: KabA" Avcri1l
Copyri,bt to £m6 Lctadvai 1971
Rcvilcdrcpdnt 1979 Mw.lc:al cumplca ate quoted by k1nd pcrmiIaloD of &oc.cy "
Hawka Lld and AlCtcd A. KalmUl Lui (Univena1 EditioD). The publiahen would Iikc to &hank AteI O'1a Cor bit "'&ance in tbe prcparatioa of the mtWc c:xamplca.
ISBN 0 900707 (K 6 Printed and
bound In CIUI Brhain UDWOOO .UIlH uwrn.D Trowbridlc
by
Contents
IDtroduetion Tonal
..
Principles
The Ad 5)'1lem
Form Principlcl Golden Seclion Fibonacci Series The Vae of Chorda and Interva4 Chromatic S)'Item
Diatonic Syatem
'7 '7
..
'7
App.endUt I
..
Appendlx n
'03
Appendlx III
110
Introduction
The publication of this .tudy of the music ofB6a Bart61r. u an
important event. Many descriptive analyses of particular works ofhia have appeared, but here for the fint time
in the
English
language it an. authoritative and convincing exposition of the theoretical principles which the compoaer worked out for himaelf but refrained,
at
far
u
is known, from expounding to
anyone during hiJ lifetime, either
in
writing or by word of
mouth. Thus we owe both the author
and the publisher a
ainccre debt.
Mr. Ern6 Lendv"; has diJcloacd the fact that Baa Bart6k, iD
biI early thirtiel, evolved for himIelfa method ofintegratin, all
the dementi ofmuaic; the scaIea, the chOrdalltructW'tl with the melodic motifs appropriate to them, together with thC"'pfOa portiODl of length as between movements in
•
whole work,
main divisions within a movement such at exposition, develoP"" ment and recapitulation and even balancing pbrucs within
aecti01lJ of movements, according to ODC single basic principle,
that of the Golden Section. Some luch mathematical proportion wu fin. proposed .. an "cathetic principle by Chalde.... in the
vii
3rd miUe:nnium D.C., taken up by the Gree:ks 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 Ie:ngth according to Golde:n Se:ction proportions, but this is morc of an intellectual quirk of the compwer's than a principled pro cedure.) Bart6k discovere:d a way of deriving the basic pentatonic intervals A-G-E and the first 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 proceede:d to the e:stablishment of two fundamental scales, dCKribed by Lendvai as "diatonic" and "chromatic", containing respectively seven and eight notes inside the octave. Within thiJ framework Bart6k applied his theory of "tonal axes" as the basis of tunality. It is an implied thesis of the book that the pentatonic scales of the earliest folk music, the modes of oriental ilDd 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 complc:te integration of the deepest fundamentals of tonality with perfect formal proportion. During the past finy years there have been various scienti fically orientated attempts within musical theory to show the way forward to the composer and to help him to find a finn foothold in the period of chaos which followed the dis integration orthe major and minor scale period at the beginning of thU century. The most important in order oflheir appearance have been Asaviev's uMusikalnaya Forma kak Prouess" and Ulntonatsia" (1930), Hindemilh's "Craft of Musical Composi tion" Vol. I (English Ed. 1937), Deryck Cooke's "The Language of Mwic" (1959) and Emest Ansermct's "Lea Fondements de la Mus.i.que dans la Conscience Humainc" (lg61). To these major worD should now be added Lendvai's exposition ofBart6k's mwical theories. Though these five work! YlU
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 relations of,ome kind or another are an CSlIential framework for any COll$truction of tone' which can be rightly considered
all
a work of musical
art. Asaviev', Cookc', of the major, minor and chromatic scales",- Ansermet', exposition of the space between the notes making up the octave at
a "structured Ipa-CC, divided unequally at the perfect fifth
and perfect fourth". and now Bart6k'I tonal axes, operating within his particular "diatonic" and "chromalic" scales (the latter not the chromatic scale of twelve semitones) arc all based upon the admission that there exists a h.ierarchy of intervals, proceeding from the essential nature of musical tones themselves, which may nOI be disregarded if music is to result from composing or the putting together of tones. Some readers may wonder why I have not included among the important theoretical writings of this century Arnold Schoenberg' (1941), the argumenta.tion of which in support ofru. method of composing with twelve tones which arc related only with one another (now known as serial dodecaphony) advances it, in the author, ' theory".·· A study of the theoretical paragraphs of this cssay dispels any such illusion. The whole justification orlhe method of composing with twelve tones depends upon the roHowing two sentences: "The tenn emancipation of thc dissonance refen to its comprehensibility. which is considered equivalent to the consonance's comprehensibility. • Cooke: Tht �, f1.! Mwi&, page xii
.
•• Schoenberg: S!fo GIIIJ Idt4, page log.
A
style based on this
premise treata cIiuoDaDceI like CODlO"IOcel and rCQounca a conal ceIltre...•
Of coune diIIoDaace it equivalent to CONOlWlce iD the wue
that both at< perfecdy pemWaible But cWeonaac:e it DOt the aame
U
insreclienll of muoicaJ arL
CODlOunce ; it b.. different
&COUltical""d ph)'lio1ogicaJ eft'-. Therefore diooonancc ought not to be treated u ifit were identical with coDlOoance. And in
any CIIe the renunciati oo of a tonal centre does Dot follow from
any pmiolll1y .lated proposition and ia merely a dogmatic aaertioo of the compoteTI
belief. AA IUch it ia totally without
the acicntilic validity which he c1aima for i� and therefore hia aaay hardly menta
inclwioo amoDg the important theoretical
writings which are mentioned above. M far .. I am aware no lupporter o( atonality, aerial or
otherwise. h.. provided any proof of ita thcoretic:a1validity AI a
possible framework (or muaica1 art. The formidable champion
of the Vienna School of thia century, Tbeodor Wicxngrund AciOnlO, iD Ilia uPhiloeopbie der neum Muaiku (1948) aauma tha� apart fiom BartOk and Stravinaky, oaly Schocaberg, Bc" and Webcra and their followcn are worthy
10
be taken
."" .. ....P'*'" .. _-
. ,·",.pF,_"';'-:'�2L_"'("; 1 ._ ...."1.. NOlCl-. bill iD c:ampariDg 1he Yam'.... School or dte .am.
century with the ViCDDa claIIia, CVCD be UIeIICI ita abort cominga very objectively and dcacribca the large worb of Schocnbcr(. mature period .. "Werk< dca
�8eD1u,••
grouartigca
which could be literally tranalated .. "worb of mapi.6eeDt Callure". If AdoI'l1O ia unable to juacify aconal a:nuaic, it ia occd1eu to add that from the point ofview ofaerioUl mlll1caJ theory, the prclCDdoUI cbit-chat of Mr. John Case la not wonb a moment', consideration. • Scboc:Dbq: Str" _ u.. ... 10)•• Adomo: PAiIo,,_., ... Mw. pap g6.
"
In conclusion, the publication of Eru6 Lendvai'• book
can
only be welcomed. It ahould be lIudied, not only by all Bart6k'. admirers. tolether with the other treatisca above-mentioned, but by all IlUden.. of composition who
way out of the present
an:
.tace of confusion
!tying to 6ght their
in the muaical world,
and Ihw to build a framework for their creative work, which is not a chance mixture of the latest .tyles 1.1 present in vogue amonl onc or other amall clique, bUI a logically inlegrated development from the musical art of past periodJ. In thil6ght the ,uuggle of Bart6k as expounded by Laulvai, iJ an iuJpira lion, even ifhia solution ma.y not be the one which proVQ to be the moIt widely accepted.
Alan Bush,
Radleu, 191'_
xi
Tonal Principles
The Axis System "Every art has the rigbt to Itrike its roots in the art of a previow agCj it Dot only has the right to but it mu.t .tem from it", Bart6k once declarcd. His tonall)'ltem grew out of functional music. AD urunter·
rupted line of evolution can be followed from the
beginnings of
functional concepts, through the harmonies of Viennese claaaiciJm and the tone-world of romanticiam. to his
axis .f1sllm.
By an analysis of his compositions, this axis system can primarily be shown to _ the essential properties of claaieal harmony, Le.
(411) the functional affinities of the fourth and fifth degrees (6) the relationship ofrdativc major and minor keys
(c) the overtone relations (4) the role of leading Dotes Cl) the opposite tension of the dominant and ,ubdominant Ul the duality of tonal and distance principles
I
(.) To bqiD with, le. go try to lituatc BartOk', Iou! 'litem iD the circle offillho. Le. UI tab C u the ioDic: (T). Then F, the fourth degtee. io the ,ubdomiDao. (8); G, the fifIh degree, io the domiDao. (D); A, the obtth degree and relative of the tonic, fUDCtioDa
ua
tonic; DJ the ltCOod degree, aDd relative ollhc
subdominant. functioaa u • mbdominant; E, the third degree and rdativc of the dominant, £unctiODl u a dominant. The
aeriCl of tiftluI, E--A-D-G-C-F conaponda to the functional
oeri.. D-T...s-D-T-S:
no. I
W. oote that the "'Iuencc D-T-S "'peall itsclf. When thio
periodicity is ezteaded over the catUc circle offiftha the acme of the uio l)'ltem may be dearly _: •
,
D D
•
n... •
us "'para.e the three 1\110:0.. 0 aDd c:aU them IOnic, 'Ub dominant and dominant axes, n:apec.tivc1y.
Le.
---+.....
SUBDOMINANT AXIS 1:
,
DOMINANT AXIS
+.1--+--.... +. •
..
TONIC AXIS no. S
Chonh baled on the fundamental C, E� (=D#), F# (-G�) and A have a tu"" function. Chnnh bued on the fundamental E, G, B� (=At), Cl (=D�) have a ......., function. Chonh baled on the fundamental D, F, A� (=G#). B have a sdUmiMnl function. It is euential that the particular axes Ihould DOt be coDliducd. .. chordl of the diminilhed aeveDthJ but AI the functional rdationJhipa of four diff....... tonalitieo, which may beat be compared to the rMjor-minor rdatioDl of ctauica1 muaic (e.g. C major and A ntioor, E� major and C ntioor). 3
It should be noted, however, that a much more aeNitive rdatioDShip c:xiau between the 'IJIDsilf poles of an axia-the "countc:rpolc:su, e.g. C and F�t.ha.n those aituated next to each other, e.g. C and A. A pole is always interchangeable with its countc:rpole without any change in iu function.· The pole-counterpOle rdationship is the most fundamental lUUctUral principle in BartOk', music, in respect to both .mall and large forma. Already 'he inner form of Blwb
III IV
BEOIHHDfO
MlDDLIt
...D
A C F, A
E� (b. 56) F# (b. 063) C (b. 46) E� (b. 83)
A C F, A
• A cadCDCC-like aeqUCllCC of chonb, B-A-D-G-C-F in the l)'Iecm of .Bat\6k can aJ..o be: vu,,,!!"'" iD the lo1lowiaa way: E-A-A�DIt-C-F. the
""Foal D ... a ..... ,
This table teaches yet another lesson . All four movemcnU
real
on the tonic am, A-C-EI>-F#. Thw the firs. and COuM movemeDtJ are supponed by the "principal branch". A and E�; the middle movements. however, by the "secondary branch",
C
and F#. Thw each axis has a two-fold affinity
depending on whether we oppose the pole with the counter pole, or the principal branch with the secondary branch.
..,.
f •
.... • • •
� ... .... r.l. PlO. "
CoDICquently the components of the axis eystem are as follows: (no dimension)
pole branch
- pole+counterpole
am
_
principal +sccondary branch
axis sY'tem
III
T+D+S
..ea
(1 dimension) (2 dimenai.ons)
(3
dimensions)
$OMI4for Two PiIllUlS tmtl PtrtlUmd is based on the lubdominant am, B-D-F-A�, complying with
The Slow Movement of the
the traditions of classical compoaitioD. The modal ammgcment
of i1l principal theme is symmetrical: the beginning and end lupponed by the Band F countcrpoles (i.e. the ;rineipal branch 8.8.-2
5
of the axis), whcteU the accond and fourth melodic lina ral
on the D and A� countcrpoles (i.e. the mon4oty branch of the axil) with the answer of the tbanging-fiIIh (E) iD the middle.·
I
PRIIlClPAL BRANCH
"J.
B
D-F
....�." 11.
foII�
._,..",.
I
A\-f
�L--"S£""C,"NO "'�n)Y.-BnllHt1lnm.-----'�� ..
.... ,
• The IUIIW'CriD& lower finh ia wIUch &be .a::ood aecdoa repeaD the melody or tbc lint teeUoa. iI • chanY:tcriltic of the o1cl-type Hwtpriaa. (Cbcn:mia. etc.) ColkIODl with. dClC'mdioa mdodic lilac. Sce 6n& example
air". 76. 6
The melody constituting the core of the movement is also centred around the subdominant axis The 0# opening and .
close are replaced in the middle ofthe theme by the counterpole
D. Every main mctric and motivic point revolves around the subdominant axis ,.,
1'10.6 These two melodies truly rdlect the Itructure of the movement. one of them being attached to the principal
B-F. the other to
the lCCOodary Gf-n branch of the lubdominant axis. The second theme of the YiDll il seems to be somewhat more intricate. Although the twelve-tone
melody touches every degree of the chromatic scale. there it no doubt as to iu tonality. In its axis we sec the A and D# counter poles (beginning. middle, end) and the broken-up and
F# major
C major-minor counterpolC!.
�I� • •
• ..
f
•
PlO. 7
7
For funher detaib, ICC App. I, p. 99.
(b)
A .urvcy of the evolution of harmonie thinltiog leada to the conclUlion that the birth of the axis l)'ltem was • historical nccaaity, representiag the logical continuatioll (and in a certain SCDJe the completion) of European functional music. It can be demonstrated that the axil I)'Item, with its characteristic featurel bad, in effect, been used by the Viennese "Greau". Indeed, it had been rccogniscd by Bach, in 1W chromaticism. The aense DCj'tm&1ionaJ correlation in music was introduced in praetice by the rcaliaation of the I-IV-V-I affinity (in medieval modal music, at fint in cadence form only) In the cue of the C tonic: SUBDOMINANT F
DOMINANT G
TONIC
C
The clauical theory of harmony already .peaks of primary and aeeondary triada inaamueh .. the C may be repla
C
A
/
Romantic harmony goes ,cill funhcr, making frequent use of the upper relative.. (Naturally only major and minor key. of limilar kcyolignature may be regarded .. relativcs. c.g. C major and A minor, Of C minor and Elt major): TOMe
C
/"
A 8
Ej)
One more step completes the system. The axes extend the application of relatives to the wlwlt system. The axis aystem implies the recognition of the fact that the common relative for A and E� is not only C, but also F# (=G�); that D and A� not only have F as a common relative, but also Dj and that E and B� not only have G, but alao C# (=Db) as common relatives. C
F
/'
/'-
D
DOMINANT
TONIC
SUBDOMINANT
A
A�
G
/'-
E�
E
'-/
C'
F#
B
B�
'-/
'- /
well known, Bart6k showed a preference for the wc of ao-c::aUed major-minor chords (sce Fig. 32b). For instance, iu ronn in C tonality is: N is
The function remains unchanged even if the C major mode as shown in the above chord-is replaced by the relative A minor. or when the Rp major tonality replaces the relative C minor. This technique occun regularly in Bart6k'l mwic: C major-minor
/
A minor
'-
Eb major
These .ubstitute chords may abo be employed in major-minor form, which bring! the system to a closI. since the relative of A major (FII minor) and that of E� minor (G� major) meet at a point of enharmonic cc>incidence, FI=G�. 9
C majo�minor A minor A major
""
/
E�minor
"\.
F# minor =G�major
Theac
relativCI,
applied
E�major
to
/
dominant
and
lubdominant
harmony, again relult iD the IChcme of the axil system.· (,) The theory oCthe axis I)'Item iI abo lubstantiatcd by the laWl oC acowtia. Acoustically, arriving from the a.miMnlto the IDnieJ is to reach the root from
an
overtone-all cadential re
latioDl rClt on the principle of interconnection between roots aDd their ovcrtonCl. ThUl, the dominant of
C iI not only G
but
aho the next overtODCI E and B�.Thercfore • ODe c:ouJd euily be baSed by &be face tbat &he cbord OD the ICvaub dqree ��F) UlWDII:I a dominant f'uActioA iD. tradidoaal harmoay. Howew:r. i.D. Rimvnn'. opiniM• tbiI it hue an iacompJclc IC¥CDlb cbord OD \he 6Itb dqrce. 'ThiI amhipicy it raolved u IOOD .. cilbu • ..", or ",,,,or chon1 it bucd OD \be B inatead of • diminid",d uiad, i.e. tilt B 11 puwd an indcpawIaal role. ID sbit cue B will have the 6an£tioD. of the tubclom1n.n' Foz .... . mplc" i.D. BcctbovCD', G � Pia.ao CoDccno, Ihe FI ...jorchonlol .... priocipoI ...... (b.7)...nY .... .... . ..bdomInaaI. le. • ch-nsiQC domin.nt uuc:rpetatMm. Tbc dift'uau:e bdwcm tbc acvo DOte aad tbe twdve-eote aystema1l awpicuoua alto iD that tbc drde ot fi1II>o builI OD ... ...... or ... ....... ..... (F-c-G-D-A-I!-B) Mdfen • break betweal Ibo B and P. No IUCb break occun iD the twd\'C'oaOle ayKCID ..ll .. tNUtup at,."."..., tbca nca the aimplal rdaUvet-dlc C � .. It. miDoI' rdatioDab1p-woWcl be cft'cdeCl by the ClODcr.dicciotl that &be duxd. bucd cm B iI "ndoutMedJy of. IUbdomlD'Dt cbaracta' i.D. the It. minor key. 1be MIDC appUa to lhc cLc.-. le " bowa &oIIl Rulaau. tbat Chc Neapolitan llalb c:aDDOl be rcprdcd... areal cbord OD dle accoad dcpec; it iI DOt • cbord bucd oa bill &Il altered founb depec (in a.no.... muk me damm-D! UIUoIIIy appcan .. . UWftlA-dMJrd. wkb lcadiaa DOte atcnc ....). . Abo .... ....... oIatb dqpoc ID obe ...... ..... couId "'- • UNlk: aipificance by auiaUladon to tbe IIl.;ot 1Calc. ID aD. bQQl�UI twdvc-Docc .ys&aD. hoWCYCI', mete dia\01lic "uua.po.idouo·lolt mdt powul.
imcrval rdatioaa. Wen It otbcnotiIe
Ct-D�
D�.
10
Db
ant·tonic rclationship. it expanded to include E-..C and BI>+O. Since the D-T relationship corresponds· rdalively to the T-S and the S-D relatiObSbip, ovcrtone-root attraction exists between the T-S and the S-D, 31 well.
ROOT
OVERTONI!.
tonic C dominant E dominant B� ,ubdominant A� lubdominant D
E and B� G#and D D andA� o and G� F#and 0
auULTANT � � = = -
dominant subdominant .ubdominant tome
lonic
G-F# T
E-B� n..
9
domjnant of &.be dominallt (changins dom.iDant) acquifta the aipjWanrc of the rubdomiAaDc while the dominaDt of the lubdcwninant . • The
..uma the role or the conic.
11
If we add the role of the nearest ovenone, i.e. the fifth, then we can
deduce the complete axia I)'Item from thcac relations.
(d) In the aimplClt cadence, that of V'-I, the
main role is
played. by the so-calle d sensitive notes which produce the pull
of the dominant towards the tonic. The leading Dote pulb to the root aad the seventh towanh the third degree of the tonic, i.e. the leading note B resolvel OD
C
and the Kventh F on E
or E�.
no.
10
These imponant aensi.tive notea bear a UUoIIi& relationdUp to each other. The ultone-half the octave interval-iI chatac. teriled by the interchangeability of iu the interval. ThUJ. it the
B-F
notes
without changing
relatioPlhip i.J converted into an
F-B one (as iJ frequently the
case
with Barulk), the. the F
( _E#) asaumea the role of the leading note, pulling towards the FI instead of E, while the ...enth B pulh towards A# or
A
instead of C. So, wtcad of the Qpected tonic C major, the tOunlfrlOlt, the equally tonic F. major (or minor) emergCl.
"0. 1 1 ..
11Us resolution is reserved by BartOk for a ludden change of
scene. The circumstances of an expected G'-C cadence emerging as G'-F# gives w a llBart6keao psc:udo-cadencc:". (e) Starting from the tonic centre C we reach the dominant in one direction and the lubdominant in the other, in iimtUtd latitudes. At a distance or a fifth we find the dominant G upwards and the subdominant F downwards. Regarding
OlllJ'101ll
relations we also get the dominant G. E, B� in the
upper and the subdominant F, Atp, 'UIIIIIII"" DllllctlG'
D in the lower directiona
..,C
•
DO.IIWIT DlllctUll •
f
l
,.
..
• 1'10. III
But what happens if the pendulum coven the latitude or a tntone? In this case the deviations made upwards and down. wards meet, both ending at FI (-G�), and ifwe were to take one as the dominant, then the other would have to assume the subdominant function. By this coincidence, however, a neutral· isation of their functions takes place. dominant and subdominant merging are rendered ineffective in the interaction of their opposite rorce�. Consequently .the balance is saved, and the function is jnvuriably that of the tonic. The counterpole is born: Similarly the distance between the tonic C and F# is bisected by E� ("",nl) in the one and by A in the other direction; 10 lying in tensionJCI3. neutral eection pointl, they also have to be interpreted
as tonics. No more than four tonic
poles can be surmised, since the intervab C-E�, Elt-F" FI-A. A-C provide no funher points of bisection.
'3
FuWly. what Ji&nificancc: ohould be attached 10 a l>ling of a chromatic dep
For the Pf'CICDt we ahall czclude the whole-tone acale of ill limited poaibilities: two whole-tone lCalel produce the chromatic acaIe by interlocking. Every tonal I)'Jtem prauppolCl a centre AI weD as ,ub ordinate relationa dependent OD the centre. Taking again C u the towc centre. the three functions are represented moat potently by those degreCl dividing the circle of fifths into three equal paru, i.e. in the augmented triad C-E-A�. Properties inherent in classical harmony are responsible for the E auumiDg a d�m.inant function and A� a lubdomioant function in relation to the towc C. Each of these main notes permit their lubstitution by their counterpolcs, i.e. their tritomc equivalenu. Thus, C may be replaced by FI. E by B� and A� by D. If we divide the twelve-tone chromatic scale proportionally between the three functiona, each function will have four polt:t, and thcsc-imofar u we keep to the m.tance principle-arc arranged in diminished-seventh rtlations, dividing dle circle into four equal paru. Accordingly. C-Ett-F",A belong to the range of the C tonic. E-G-B�CI to that of the dominant E mun note, and A�B-D-F to that of the subdominant A� main Dote. So, the tonal syltCJD resulting from a division oCthe chromatic scale into equal parts agrees completely with the axis system:
because
SUBDOMINANT
TONIC
DOMINANT
c
e
�
b--d
E
- --
e�--
- .. -a.
Et>
.10. 13
g---
0.
---
b� '5
Put coocilcly, given the twelve-tone I)'Item and ttac three fuoctioDl dUI is the .ni.1l)'1teDl that can be realiJed by means of
disWlCC divilion.
Viewed historically, the axil system re8.ccll the ag�ld
struggle betweea the principlea of Ioll4li!1 and lpi-disUwl, with the cradual aac:endancy of the laller which finally reaulted in the free
and equal
tteatment of the chromatic twelve nOle!!.·
Here we have to draw.liDe between Bart6k', lwelve-tone sy1.em and the Zw6lftoDDlusik of Sc:htiDbug. Sch6nbcrg
.nnihilalel and dWolvCl tonality whereu BartOk incorporatea
the priocipJa of harmonic
thinking iD a perfect is tQ
penetrate into BartOk', creative genius
aynthcsis. To diIcover
the
natural affinities and intrinaic possibilities. inhereDt in the musical material
•
• The iatrod� of the tbIa .-
.6
tempered Kale mark.cd. about the middle or
Form Principles
Golden Section Golden Sec:tion (".cetio aurta", and henceforth CS) means the division of a distance in IUch a way that the proportion of the whole length to the larger part corresponcll geometrically to the proportion of the larger to the smaller part, i.e. the larger part is the geomdrie mun of the whole length and the .maUu part. A simple calculation shOWl that if the whole length is taken
aI
unity, the value of the larger section is 0.618
(
, A •
(1-") Plo.
t:
• • •
1
'4
x=x: (I-x)
(ICe upper formula on page 78), and hence the smaller part is 0.38•• • • Thus, the larger part of any length divided into CS is equal 10 the whole length multiplied by 0-6,8
• . •
'7
Bart6k'. method, in hie construction of form and harmony. ;, c:IooeIy COlUlCCted with cbe law of cbe GS. TbiJ ;, a formal element which ia at least as significant in Ban6k'. muaic as the 2 + 2, 4 + 4, 8 + 8 bar periocb or the overtone harmonisation in
cbe Vieaneoe dasai
� an example, let
us
take the fint movement of the
So_
for Two PiaDs tm4 P"�,,. The movement compriKs 443 b an . 10 ita GS-rollowing
the above formula-is 443 )( 0'618, i.e. 274.
which indicates the centre or gravity in the movement: the recapitulation .tartI prcciJely at the 274th bar.
Movement I of C.1dr.,u conai.tI of 93 ban. and ita GS (93 xo·6,8) again marb the beginning of cbe ....pitulation in the ntiddle of bar 57. Movement J of the DWn� eonmta of 563 triplet urnta (the number of ban it irrdevant owing to their variable time signatures). The GS of 563 (563)( 0·618 - 348) again coincides
with the recapitulation. In Vot. VI of MikrDMsmtu the CS of "Free Variations" can be seen to touch the "Molto piu calmo"-82)( 0.618=51. The GS of "From the Diary of a Fly" comes at the climax: the double sfouando (if the 3/4 is taken as a 11 bar, calculating in 2/4 ban). In "Broken Chords" wc find the recapitulation at the GS (80 )(0·618=49), etc. The 16 introductory ban of the SDMlajDr TWf PillRDS lUUI P,mumtl represent a model example of GS construction or more precisely, bJ. 2-17, because it is here that the organic life of the work begins.
•
•
FIO. 15
'9
Its fint pari is in the sphere of the tonic (bs. 2-5). the second within the dominant (bs. B-g) and the third part in that of the subdominant (b. 12 on). This third part is thematically the inDtTsion of the fint two. So, to summarise: Theme in rool position-tonic: Ftr-C Theme in foot position-dominant: O-DU Theme ilWtrttd-subdominant: Afr-D
entries .. ..
Considering the changes of time-signature, it is more practical to calculate in units of 3/8 time. The whole form consists of 46 units. Its OS is 46 x 0·618 - 28, and this covers that part up 10 the inDtTSlon of the theme (see the main section of Fig. 16). It can be observed that OS always coincides with the: most significant turning point of the form. Let us now separate from the whole the parts in root position, i.e. the first 28 uni.ts. Now 28 x 0·618 .... 17'3. At this very point the tonic part ends-at the first third of the 18th unit (see the: dominant entry in Fig. 16).
INVERSION
ROOT POSITION T..1e
..ia.ul c••
c••
I
I
I
'nilin
I
.2.- .....Ii..
.....Im
"'..U..
,_
...
•
.,,,,1..
,.,ili••
.
....,.i••
POSITIVI .'10, 16
OS division may be seen to follow one of two possible courses, ,.
depending on whether the longer or the shorter section comes tlrst. Let us call one of the possibilities followed by the short one-and the
ptnitiw: long section 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-strokes 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 PllJitiw (I 7'3 xo·6lB = 11), the dominant pan. on the contrary,
becomes a
rugaJivt
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
rugative
counterpart is found at the entry of the
tarn-tarn (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 find up to the cymbal.stroke, eleven eighth notes. Its positive CS point (7+4) determines 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
33 )( 0·61B = 20.
(C�n timpani entry-counted in eighths:
Precisely here, the thematic condensation
begins: also, with the 21St eighth. On the other hand, the complementary, negative section of the part following the cymbal-stroke is indicated again by the side-drum (see Fig. 16). B.8·-3
"
Summarising the above, both in the smaller and larger form detaiu, there is a symmetric joining of the posititlt and
IUgatitlt
sections. From these concatenations a single great "potentia'" fonn arises, wherein the smaller parts are finally summarised in a ponli" main section. This process is therefore coupled with a powe:nul dynamic increase, from pianissimo to forte-fortissimo. Analytical studies permit the conclusion that the positive section is accompanied by intensification, dynamic rue: or concentration of the material, while the negative section by a falling and subsiding. The sections always follow the contents and form-conception of the music.
By way of illustration let us subject Movement
Sonata for
Trw
PitlMS
III of the tlIId PncussiDn to a detailed analysis,
Exemplary, is the unity of proportions of the exposition: the principal theme has a positive and the closing theme a negative ICction, while the ICcondary theme developed between the two is symmetrically arranged, Thus, the principal theme· (43'S ban long) is divided as
follows: Al +As+B, The position of B is determined by:
43'sxo·618-27'S. while the two A's are rdated to each other according to: 27'S x 0,6,8 -17.
t�----
-.----
1
. .
17"S .---.
-.---••-- -•••- -4)-S .7--- . -----
no. "
+
J
-------,---
-
-
-
• The incomplctc half·bar &1 the l?csinningotthe movcmcnl.illobc: takcn into cosuidct.lion wbcn makia& lhae c:alculations.
The symmetrical division of the secondary theme can be expressed as following: 12+17'S+17'S+U! (bs. 44-102). The geometrical cenlre (b. 73) accords with the lonal construction of the theme also. The negative main section of the closing theme (bs. 103-133) is given in h. liS (see Fig. 18). Within this, bs. IIS-133 have a positive section in h. 127 because of the powerful dynamic ascent, and the static construction in 4+4+4 units of bs. 103-114 produces a solid base for this rise.
'"
'" '.3 "'
'"
.lltle 4t,,4 ..
lacrIU.' ..lit," rlo. 18
Ukewise the proportions of the development are symmetrieal (bo. '34-047), Its negative main section-countecbalancing the positive main section ofthe development of Movement I-is determined precisely by the point of climax in b. 177
(F,
tonic counter
pole).
The posi'iOl section of the part preceding the climax and the
tltgatilll section after the climax indicate the most important turning points: h. 160 the fugato of the principal theme, while
b. 20S the return of the first theme of the development (xylo phone entry):
CLIMAX
bar U4 10'
POSITIVE
on
'41 ,,,
NEGATIVE
fl(l. 19
The build.up (owards the climax is always marked by a positive section: from b. 140-159 it falls on h. 152 (positive) ., b. 160-- 176 .. " h. 170 .. .. .. .. b. 166 .. h. 160-I6g .. b. 170-- 1 76 .. .. 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�04 .. " b. 199 .. The climax itself is divided statically into 6 + 6 bars (b!. 177-188). The negative main section ofthe recapitulation (bs. 248-350) coincides with the watenhed, as it were, of the thematic material, i.e. with h. 287. Bs. 287-350 form one single broad wave, and its structural view is similar to that of the beginning of Movement J (cf. Fig. 16):
. .
CliMAX
I'"
••, ZlJ
lOt
" 1'
•
.
IlGATlYI
!.
PDSITh'(
•
'"
I'"
'" ..
POIllfV(
...
•
,
IIGATlllr
,
,aSIfIVI FIG. 2U
The negative main section of the coda· (bs. 351-.�20) coincides with the thematic centre of gravity of the whole coda: at the same time the return of the C tonic, in b. 379. is given a greater emphasis by Do lengthy preparation. Corresponding to its static structural character this thematic centre has an 8 + 8 bar division (m. 37!l-394). The first par. of the coda (bs. 351-378) combines a positive
... '"
•..
" I'" I' I'"� I
m .�.
.oo
I I
•
•,.lIe ... . , .,.
I
rosmvE
• N.D. The
I I I
•
" ". POsmY( t
1("fII'l
Wl index of ban
. ..
in the
score
.10 ,
.''''nVl
(41 1) is erroneous.
'5
and a negative section in units of
9+5
and
5+9
bars. The
(b!. 319-420). as shown in Fig. 'l l . contains at the 405) and a negative (b. 395) section. Finally, the positive section of 00. 395-404 (in b. 401) and the negative section of bs. 405-420 (in h. 411) are again sym
second part
same time a positive (b.
metrically related to each other. At first glance it may appear contradictory that the points of lection determined by the laws of as can remain
unaffultd by
the changing tempi. This phenomenon is easy to understand if we consider that music breathes in metric pulsation and not in the absolute measurement of time. In music. passing time is made:: realisable by beats or bars 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 sluggishly in a slow puuation. Finally, let me give an example to those who reproach Bart6k for not having effected the "total and radical reorganisa
tion of the material". The complete fonn of the $oniJlafor
TwO
PitmDS IJ1IJI Ptrewsion is divided into " slow-fast +.Jow-fUllt" movementl. The as may therefore be expected to appc;\r at the beginning of the second slow movement. Our eXI.ectatiouJ are wholly fulfilled ; the time value of the complete wurk is
6.432
eighth notes, and the as is at the
3,975th
which is precisely where the movement begins.
•6
eighth note:
Fibonacci Series All of us who have played Allegro 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 CS sequence, approximately expressed in natural numbcn: the Fibonaui numbers. A characteristic feature of dtis sequence is that every member is 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 CS· (the CS of 55 is 34. and that of 89 is 55). Let w compare tills aequence with the proportions of the fugue (fint movement) of Mw/or Slrings, P"nusion mul Cel,sla. Starting pianissimo it gradually rises to fortc·fortissimo. then again recedes to piano-pianissimo. The 89 bars orlhe movement arc divided into sections of 55 and 34 bars by the peak of this pyramid-like movement. From the point of view of colour and dynamic architecture the form sub-divides into further units: • The aquare orevery number it equal to the product oflhe preceding and (OUOWUl, Dumbcn, plus or mihw ODt:.
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 of34 + 2 1 , and that from the climax onwards, 1 3 + 2 1 . Thus, the longer part comes fint 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 fit together like the rise and fall or a single wave.-
"
l4
'l
H
'l
" --======�fff =:====Plo. 22
The proportions follow the Fibonacci series. lt is no accident that the exposition ends with the 21St bar and that the 2 1 bars concluding the movement are divided into 1 3 + 8. The proportions of Movement 111 of Mwu fQr Strings, PercWJiDn tmJ Ct/uta also reflect the Fibonacci series (if wc calculate throughout in 4/4 bars and consider the occasional 3/2 as I i: ban). Its formal and corrCJponding geometrical structure is shown in Fig. 23 •
• The 88 ban of the ICOI"C must be completed by a whole-bar ral, accordanee with the BUlow analywt of Bc:clhoYffl •
•
8
in
+___
+-- - "
--
It
...� -- - -
_ _ _
_
-
is -
-+
- --
- - - - - -
"",
..,u.,...
..., .... 11
•
,. ....
hd l.....
"
, I
..
"
•
cliII ..
Z"' .....
,. Ih..
... --,, - - ....10- - -.,_ . PlC.
23
The Fibonacci series reflects, in fact, the law of
growth. To
natural
take a simple example. If every 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, 5, 8, 13, 2 1 , 34
.
.
•
"Wt follow naturt in composition," wrote Bart6k, and was natural phenomena to his discovery of these
indeed directed by
regularities. He was constantly augmenting his collection of plants, insects and mineral specimens. He called the sunflower his favourite plant, and was extremely happy whenever he found fir-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 flowen, 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. 29
having 360°, subtends an anRle of 2�2"5° on onc hand, and 137"5° on the othcr. It can be observed in a large number of plants. c.g. palms. poplars. catkins, etc., that each bud, twig or leafsubtends an anglc of 137'5° with the next The GS of a circle,
onc.
1
�--7l_---r:3
•
flO. 24
Also, each new branch divides the jortnlf fields of section
according to the rules of as: field between
I
and
so
twig 3 divides the right-hand
2; twig 4 the left-hand field between I and the field between 2 and 3, ad inf.-
2 ; twig 5 does the same with
• Tbe Fibonaoci lCriel appean lhiI tinu: u well: rtdd between 2 and 3 j, divided by $. between 3 and S by 8, between S and 8 by I,. etc. 30
r.onsidcl" the pl"oces.s uf the fugue nf Mu!it fur S" iIl1:1, Ptr&llSsu,1I ond CL/uta (analysed on pages 27-6) as a circum� volution,· its structure wiU surprisingly correspond to Fig. 24Or let us examine the diagrammatic sketch of the chambered shell of the cepalophod nautilus-Jules Verne was so interested in this sea shell that he named his famous }(outiius after it. The diagon.lIs drawn in any direction through the centre provide a patl.!rn in which the centre always remains in the positive or negative GS section of the fields marked A-B, B-C. C-D. D-E. E-F. F-G. If we
A
B
PlO. 25
This
scheme is strikingly similar illustrated in Figs. 16 and 22
to
the musical structures
•
• Abo the theme
moves
back to the A centre.
nnmd lhe circle
of
6f1bs-from Ihc A centre
3'
But Ihe most revealing example is presented by Ihe structure of IheJr-colle. Proceeding from the centre ofiu 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 series.
I'XII!
1-8
•
... . I'.
tIl.-r
\'11
VIII
-
U >pi .... 11 '1"..1> �, '1''''1. l
'f""h
(If we turn the cone upside down, we can also sce the system of two spirals along the junction lines of the scales). Each of the spiral Iystenu contain all the scales of the cone. There are cones in which the numben of the spirals present still higher series values: 3. 5. 8, 13, Il l .
I - XXI • I - I .l • \- 11 • c(-l -
�x
'/,
/'
�I U 11
.piul
'fM" "
_p'." !i >pi,,"
"
"
\
,
,
s
G 'XV
----4-_ 6 F xlII
XIV
9
-'
I�(
,
,
6
PlO, 26b
33
Similar anangemenu
can
be o1»erved in sunftowen, daisies,
ananu, etc., also in the convolutions of the stems of IcOlves on numerous plants. Frequently the serial numben 2 1 , 34, 55. 8g and even 144 and 233 are encountered in these spiral sy�lcms. For exampl�, the sunflower has 34 pdals and its spirals have the values of 2 1 ,
34, 55, 89.
144.
It is interesting to nole that the as is alwayt usociated only with O'IIUIU matter and is quite foreign to the inorganic world.-
• The irralional number in me formula of CS precludes its occurreace in cryuaJ·forms.
34
The Use of Chords and Intervals
Chromatic System The study of these proportions leads us immediately to the question of Ban6k's use of chords and intervals. His chromatic system
is based on the laws of GS and especially, Fibonac:ci's
numerical series. Calculated in semi-tones: 2 stands for a major second, "
11
minor third,
5
3
11
IJ
perfect rOUM, .
8
..
11
minor sixth,
13
n
an augmented octave, etc.
For the present, the musical tissue may be imagined as buih up exclusively of cells 2, 3, 5,
8, and 13 in sile, with sub-divisioDJ
following the proportions provided by the above series. Thus,
8 may be broken up only into 5 + 3. (The possibility of a division into 4 +4 or 7 + I is precluded by the s)'Jtcm.) me
This cell division can be well observed in the: finale of the: DiDtfI;mmu. The principal theme appears in the course of the movement in five 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 + 'l 5. Cl
1 " i ' r ·-; '
.!
I, . . . ..
FlO. '27
Since the fifth line (in Fig. 27) continues on the previous one, in its fourth bar the melody rises not by a minor third (3), as in the previous line, but by a perfect fourth (5), thus conforming to a CS augmentation. Fig. 28 gives the successive themes in the first movement or the SonallJ for Two Pianos tJnd 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 first phrase of the secondary theme extend! 36
1 3 semi-tones, from C down to F#i while the second phrase, 2 1 semi-tones from 8 down to D. The melodies follow each other in CS order: Leitmotif 3+5=8 Principal theme 5 + 8 - 13 Secondary theme 13. 21
rlo 28 • .
From the point of view of harmonic architecture. this exposition a.lso bean witness to a systematic arrangement. The principal theme gcu its magical tone-colour from a pmlalonic harmony II.B·-4
37
(see Fig. litga),· the formula of which is lit + 3 + lit. In the middle of the principal theme there comes an ostinato built 3 + 5 + 3, A� major-minor (se. Fig_ 2gb): C-EIJ-AIJ-B, the fourth, EI:J-Ab, is further divided by an FI into 3 + 2. Parallel fourths (5) and minor sixths (8) join the sc:condary theme (see Fig. litgc). This is sc:en clearly also in the recapitulation from b. lit92. Finally (see Fig. litgd) the closing theme is accompanied throughout by parallel minor sixths (8) •
) (U')
•
:: ..
:
J .., ..
1'10. 29
Thus
each new harmony rises onc Itep higher in the GS order,
i.e. principal theme middle part secondary theme closing theme Asimilar correlation of motifs is encountered in the MirdlU/OUS M....,;,, :
• h .ppean abo in the melody, bI.. �n-39: AtJ-FI-EIJ-DtJ and
FI-�B.
58
PlO. 30
It is interesting to note that in Bartok's music, in spite of the frequency of paraUet., major third and major lixth parallels seldom occur, beeau�e such parallels cannot be fitted into the
GS system, being quite incongruous to it. We could even speak of the prohibition of these parallels in the same sense that parallel fifths and octaves are forbidden in classical harmony. On the
(3), perfect (8) , and even major second (2) parallels. The major third has no noteworthy melodic function either,
other hand we meet at every step with minor third fourth (5), minor sixth
the more natural, almost self.evident is the motivic role of the minor third :
39
1'10.
31
This is the reason why, whenever llart6k uscd a triau in a ChrQ1TUltic movement, he placed the minor third oIJer thc fundamental note and the major third below it, the churd thus acquiring the proportion 8:5:3.
PlO.
32.
From the synthesis of these two emerged the most typical Bartok chord, the well·known "major·minor" form, consisting of a 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 also Fig. 2gb}. 4"
4'
1lUs major-minor chord has a number of synonym fornu, to which we shall give (for want of a better term) the coUective
designation: type oJplla
(a) , and we shall call the different sections of it by the letters luta (tI). gam"", ()I), "114 (I> and 'psi/Oil (t). This type occun as frequently in Bartok's music as
do the seventh-chords in nineteenth-century music :
'10· 33
These chords are exclusively bu.ilt up ofGS intervals (2,
3, 5, 8),
as follows :
no. ,..
and do not contain the characteristic intervals of the overtone system-fifth, major third and the minor seventh.·
•
From here: ariles the e:blloracte:riltic "Slow " of the alpha harmoniCl.
Pcthap. the
mon
te:Nt chord in Baroque: mwic wu the dimiliish«f
Thia Ie:,won it increased in &n6k', 1I1p/14 chorda throuah merzing of two diminilhed KVCnth cho«ia.
KYenth.
4'
the
Type
Q
can readily be reduced to the relations oC the
axis
system. In order to Cecl the tonality oC a chord. we need at least two notcs! in the simplest case the root. say C. and ils fifth G. or jts major third E. when G or E respectively supports the C.· Let
us
put this relation in
GS form:
no.
35
According to the axis system. tbe tone
G (or E) may be
replaced by any other of the corresponding axis (C-E-BI:J-C#) without Changing the tonal character oC C. We can therefore substitute E. Bb or even Cl for G.
The four intervals sounding together result in the chord 6dll (�). It should be noted that the combination of the fint three intervals is no novelty to us, since it is identical with the chord of a major seventh: C-E-G-B�.
• Tonality un only be atabfuhed through the uymmetrical divuion of
the tonaIl}'ltem; in cue ofequal division wc woukl be unable to determine
the lOOt.
A similar axis substitution 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 axis.
r;
:
':
';
+ •
I'lo· 36b
In
the form
-
f]
S S
ofila the first thrce intervals are summarised.
Chord alp"a is therefore practically an axis-like application of the simple
C-G, or C-E-G
relation, the only stipulation being
that the chord should be composed of two IDYns ("axes") : that of the tonic and the corresponding dominant.·
e rlQ.
37
• The two iayen (T and D) correspond to the root and overtone rclillion of claaica1 harmony. It it pertinent that also in U"aditional music, funclional
aUr;actiolU were based on thnc two layen. The authentic (e;t.delltiill)
conn«ted chords require th..t the root of the lint chord b«omes an
.wr� of the chord following. (Chusical harmony can. these CORlIIIOII
nota.)
Thw, in the prU£rcssion T to S, the root of I (C) becomes a lifth D the root of 1I (U) or IV (1-') bcc;OUlCS fifth or KVenth in V. Connecting D and T the root of V (Cl in IV. or a. xventh in 11. Connecting S and becomes fifth in I.
M..... . Sir., t-... e.I., J[
45
•• V..&o.
C__fo
. ... __._._•••••••••••M�• • •
...M••• '
••_ .
1$ .gg:mWl'arall t,
...·......
I"
"
...... r
Type tpsilon (c) is sddom used since its tonal character iJ unstable, due to (he absence of G without which the root does not receive sufficient suppon. Certain sections of the tUp/uJ chord have been familiar to us from cI:wicitll harmony: E-G-BtJ-C is the C major seventh, G-BtJ-C-EI, is the C minor seventh, Bb-C-EtJ-FI (Gb) is the C iCventh chord based on a dimi.nished triad. Novelty is produced by the introduction of the relative A. and primarily by the Cl. In fact the chord IUIII is an inversion of the ninth chord: C-Fr-G-B�D� (Cll to C"'E-C-B�C. 46
Essentially, type alpha is an axis harmony. As an example Id us take the simplest case. If the C major and its relative A minor arc replaced by C minDr and A mojDr.
F10·39
and thae two chords are combined. then btta, gamma and thlte 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 accidental.s-the three flat signs of the C minor and the three sharp signs of the A major. In accordance with the stratification of the elphe type it it possible to build up a still more extended elplza pile:
'10.40
47
From a succession of diminished triads a "closed" sequence is derived since, by the periodic repetition of the intervals we arc taken back to the starting point:
fll';, 41
And now we come to the very gist ! That CS is not an external restriction but one of the most intrinsic laws of music
is demonstrated by ptntatD1pI-perhaps the most ancient human sound system-which may be regarded as a pure musical expression of the CS principle. In the la-SD-mi figures of the oldest children songs the notes of the mc10dy are tuned afler the geomdric mean, i,e. after
OS.
Pentalony, particularly lile
most ancient Cornu of minor pentalony (la and re), rests on a pattern reflected by the melody steps of major second (2), minor third <3) and fourth
(,;)••
• In tbe old.type pentalonic melodiCl witb a changing.firth .tructure (wuaJly "'J..JIIi +"od..11J or "·,..m;·,,+ ,,-4-'/I.u .caIe) the major third pl..Y' a aeeondary pari. To quote Kodily: "It b clear that the pcntalul lY and the finh conttruction are independent. more 10. 'lyliMically "I'I_ite:," According to Kodily the railing minor third, so-mi, rather thall J.,.,,·m; ur any other simple configuration is wbat the child Kerns fint to fed all a Lasic mwical relationship-representing the earliest musical exptnloiulI uf la human being.
L..2 I,d .....J . �
FIO· 42
This aspect of CS architecture is markedly evident in the DtJnCt Suite 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 ofpentatony -reveals the evolution of this technique. The first movement arises from major seconds (2); the second is built on minor thi.rds (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 pattern 8 = 5 +3. where 5 = 3 + 2.
FIO·43 49
Type 01#14 can also be derived from pt:nltWIft.1. This is how Bart6k transforms a pentatonic scale into be14 and glUtlm4 structures:
.. ..7 e.....
no. 44
This type ofharmonies originating from folk song was suggested by Bart6k himself in "The Folk Songs of Hungary" (Pro Mwica: 1928):-
� air 'rlJ J !llwrrmrlfunll�,W
':lY,. ,t.... ."I...
r
.. J
5/...1,·.... ...,
J.l! _,f
I
,
no. 4$
finl iI may seem uloniJhinl that in But6k'. muaic pentatony iJ 10 Batl6k primordial attractions of pentatony cany 1tIUi.wt, and jWll ltu. leMOI!
• AI
dOle!y aUied to chromaticism. But thiJ rdation iI natural, u whh the
fincb
adequate rorma or capratlion in hiJ CS 1}'101CQ\.
We now mention a frequently recurring group of CS.type chords which structurally n'prl'sl'nl intrrv"ls uf I :�. 1 : 3 and I :2. The CS relation between these three formulae results from the proportion S:Pi. Each of these 3rue from the periodic repetition of intervals 1 :5. J :3. or 1 :2 respectively. Their structure is, consequently, 3$ follows: MotUl ':5 alternating minor seconds and perfect fOllnhs e.g. C-C�-F.-G-C .
.
•
Mock' ':3 alternating minor seconds and minor thirds e.g. C-C....E-F-G....A-C .
•
•
MotU' J:2 alternating minor and major seconds e.g. C-C....EI>-E-F....C-A-BI>-C . . .
and hereby, they form clearly dOled systems.·
I,.; MODF.L
l:' MOOEL no.
46
l:! "'ODEl.
• The inftucncc of folk music:: i. poIIibly abo raponlible (or Modd I :5. e.l. Movement III of Sui" op. '4 wu iNpired by Arab folk mualc. Perfect examples of I :2 and 1 :3 models have been round ill compolitioN of Liat and Rinulty.Konakov.
,....... .� t .� Ir.u .. td.,lC
ItOI[LS I·J
C_.... I.. On\,..lIl
PlO. 4Gb 0.11·-5
53
"HUC I" c.-.• " . rr a-...
st,l., �.td
. "' _ .-' ""' � _
· ·, ·· f i i fili1'ii Slr4.,
�d•.t .....
We attribute the greatest importance to Model 1 :2 since it actually represents a scalc-group of the axis shown in Fig. 4-7. i.e. C-Cjl-EI>-E-Fjf-{l-A-Bp.
,",os
.IC· 47
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 axis system, the alpha chords and Models 1 :2 and 1 :5- If we detach the upper C-A-Fjl-Ep and the lower G-E-Cjl-Bp lay... of the axis (see the centre part of Fig. 48) and pile up one on the other. we obtain the aJpluz chord (sec top len. of Fig. 48). lfwe separate the pole.counterpole �e1ations (C-F# and A-Eb. respectively) of the axis, we have Model 1 :5 (see right bottom of Fig. 48). Ifwe combine the notes of the axis we get a Model 1 :2 (see top right of Fig. 'l8).
55
,.
,....110.
Air"·
10.1
:
.10.48
In respect to tonality these formulae are inseparable. The fundamental role of the 1:2 model is only emphasised by the inclusion of all the potentialities of the tonic (�J:: I1-F#-A) major, minor, seventh, and alpha chords, as well as Models 1 :5.
•
These formulae merge into each other so that it is sometimes rather difficult to define where one of them ends and the other begins...
..r1"'�Ci St'I'e.., Cel. "I . ,.0 1 Axa , C - U -Ff-A
••
flU, 50
•
The: tonal resting point in
Modd 1:2 always r..lb on the
'-r note of
the millor Kcond. which i.J the upper note of the major .econd. In the cue or Model 1:2 on C lonie, it ia C, or Bu. CN" }o', or A. Thus the bue nOle of Ihe minor accond, major third, fifth and millor ICvcnth Is alw:t.ys the lown lIote, while that of the major second, fourth, minor sixlh and major
iC\'cllth,
is
the upper nOte. In the
case
of the minor third, ttitenc or m;ajor
aiJ[th, .:Iny of the notes may Krvc as b:bC:, at Ihey
all lie on tbe lame aw. 57
And this is the reason why the most characteristic axis melodies in Dart6k are cxchuivcly ruled by CS principJt.-s (sec bottom ler, or Fig.
48).
AIU (d".ll
•
•
pto. 51
Within the range of the twelve-tone scale three different .:2 models can be constructed : a I••"" c..clf-EI>-E-Flf-G-A-B�; a '.mi..." Clf-D-E-F-G-AI>-BI>-B; and a "bd,mi.,." D-EI>-F-Flf-AI>-A-S-C. Everyother form agrees with one or other ofthe above rormulae. 58
I would like to illustrate the interrelations outlined. above. by three brief examples. The NotlllmD in MiJc,okomw follows the tonic-tonic-dominant-tonic structure of the new-type Hun garian folk songs. So its first. second, and fourth lines fulfil tonic functions. accentuated by the tune which corutitutcs a ",nU MoJ.l l:2.
Plo. 52
lu tonal
character is determined by the A-fourth step (E-A), completed by the harmonies into a complete tonic a.xU :
PlO. 53
59
The piece called FrDm llu Island of B(J/i (Mikrokosmos No. log) rests on the G#-B-D-F axis. hs scale provides a full Model I :"l (G"'A-B-C-D-EI>-F-G�) which, as apparenl from Ihe final chords can be considered as a B-fiflh· {B-Gb = B-V#} and GI-fifth (G"'E�=Ab-F.O), and as a F1,",lh (C-F) and D-f()urtll (A-D), covering the complete axis.
, �
"".. f" , �
nO· H !
Both right and left hands play separate 1 :5 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.fifth+F-fourth. Abo the formal construction of the piece is adjuslt:d 10 the
lr trf rt ..� aher;uion. "It i. IfiJ.:hly • lie,., we menlion Iht problem o( tnJ desir:llJh: Ih,., we have
3
5yt.h:m (,If 1101:\1;0» of Iwdve equiv..klll lymbuls." WaJ :llways guid...J by tlUI.'ltions of rr:ltl..tJility That is the n::uon why we (requelltly find thr.
writet Uartvk. addiHIo: thal lll.· wllt'lI wrili"!; his
KOrt"ll.
(:lIharmollic;: varilllllS in Ihe I»ano rWU(:liolll of' hi. orchestral worb. Ollr methOt.! of lIul..tillll "ol;l;ill:l11"l 3n
in
the Jiatonic system and tben:(ure it if
utterly useless tool when it comn to recording twdvc-Ione music"
U;lrll,k: T/v Prohltm. ./ Mwn" MI4';c (t920)•
•• Thus by ahe merging of IWO Models 1 :5 we obtain Modd I :2.
60
F-B-GI-D axis. The first section closes in Jo', ending at the double·bar. The middle first moves around B, then G#, with an extended D pedal·point at the second double·bar. nle final chord is a synthesis of D major and F minor. and may be considered at the same time as type alpha (F#-A-C-D-F-AtJ).
1'10· 55
Our third example is the recapitulation theme of the Violin Conc,rto, representing axis E-G-A#-C#. Its scale is of Model I :� (E-F-G-G#-A#-B-C#-D). Bars I and � arc based on thc Cl. E (melody) and G (harmony) poles of the axis. Rar 6 circumscribes the £.gamma chord (E major-minor, G#-�E-G). and the melody or bars 5-13. the 1 :5 model (Il-&-F-Afl·
F---r--+-- '--r =
-----
r
r
�
6,
We have to mention alto a third type of chromatic chord namely the chorW of ttpuU inUmJIs. Its most frequent fornu in the GS system are the whole-tone Kale. chord of diminished seventh, chord in founhs and the augmented triad. The last has its justification in Bart6k's chromaticism only in so far as it is built of minor sixths (8+8+8). Whole-lone scale Diminished ac:venth Chord in fourths Augmented triad
2 + 2 +2+2+2+2 3 + 3 + 3 +3 �+�+�+5 · . . 8+8+8
In our tone systc:m two whole-tone scales tan be distinguished : they are "geometrical dominants". complementary patterns of each o.her: C-D-E-F.-Glf-A. and Clf-E�-F-C-A-B.
6.
Mo Blwlu�l C.dk, o,.n (11. u� ... ..... _
. ....
..
_
-.-
....._ ..
--;]
r
1
no. 57
Bart6k liked to use whole·lone chords hifort climtuu, since it has the effect. as it were, of "melting" the sounds (ICe Fig. 57 : BluehtlJrd's Castle No. 136, TIu Wooden Princt No. 123. Music Mov. I b. 48, Mov. 11 b. 56, Mav. III b. 14). Harmonisation and theme construction in fourth chords are strikingly frequent, due to the influence of Hungarian peasant music.
FlO. ss
Chords in founhs generally aUow two combinations: Ont� according to the 2:3 p�ntatonic principle, the other after the 1 :5 model. (a) or the two fourth chords in the 2:3 scale wc can treat the one, which lies a major second (2) higher or a minor third (3) lower than the uther, as lunit, and Ilu$ (:;In ill: rcdut:(:d Iu the du-so_la cadence of the older folk songs :
f10· 59
(b) A good example of 1 :5 association s i the closing theme in Movement 11 of the klwit for Strings, Percussion tJnd 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>-DI>-D-G DI>-G!>-G-C GI>-q-C-F
PlO. 60
The GS chords and chords or equal intervals onen combine together, in pracliee. Fig. 61 shows an ostinato from Mov. l of the Sonatajor Jwo Piafl(JJ alld Peuun;oll. TII<: twt·lvc tOIlC:; of the ostinalo contain the entire chromatic scale.
1'10. 61
The upper part is based on the A-B--Df,-EI1-F-G whole-tone scale, and the lower on the complementary 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 is characterised by the 1 :3 models and the gamma harmonies 13 + 5 + 3)·
I
".-I-r-I-C-b
I
.
.
•
The beginning and final notes assume a pole-counterpole relationship: in the upper part, A and E�. and in the lower, F, and C. When viewed together they fonn an axial arrangement, FI-A-C-Eb. Every component of the structure ill of GS fonnula.
66
Diatonic System Bartdk'. matony is simply an exact and systematic ilUltrs1DrI of the laws of his chromatic technique, i.e. the CS rules. I.
The most characteristic: form of BartOk', "diatonic"
system is the tUtlwlic (overtone) scale, C-D-E-FI-G-A-Bp-C, and the
IJ&DUdie chord (major triad with minor seventh and
augmented (ourth, e.g. C major with B� and F#). It is called acoustic because ita tones derive from the natural overtone .eries.
-c•.:.r..
68
Pr.{-
flO. 6]
In the finale 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;artok, the "pastoral sixth"). All chis in contralilt to the minor third, perfect fourth, minor sixth (3:5 :8, C-Ei:J-F-Ab) milieu of the C;S system. Let us place the principal themes of the chromatic First 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 a perfect acoustic scale.
FlO. 14 11.11.--6
These two spheres of harmony complement each other to such measure that the chromatic scale can be separated into a
OS sequence and an acoustic scale.·
no. 65 In the acoustic scale the major third replaces the minor third
Cs), the augmented fourth replaces the perfecl fourth (�), and
the major sixth replaces the minor sixth (8).
Incidentally. let me refer here to the la-sfI'omi figures in the oldest childrens' IODgs and primitive folk music, which, by no stretch of imagination can be regarded as products of some deliberate planning, though the notes accord with the "geo metric: mean". i.e. GS. Lik.ewise, when listening to traditional music, it seldom occun to us that the consonance of a simple
WUJjor
In'ad might result from the coincidence of the nearest
natural overtones: our ean limply register the fundamental number relations in the vibrations of the perfect fifth and major third. In Movement I of the S01l4tafor TUHI P;aMs tulti ppcussion, the melodic: and harmonic devises are derived from the most primitive ",.taton;, elements, while the principal theme in the Finale limply evolves the natural overtone tcale over the
tIUIjor
chord (ICe Fig. 64). Yet this major triad
COlDeI
C
as a
revelation. How can a simple major chord produce such
an
explosive effect?
Looked at from another angle, may a composer with a • 1lIc Cl and B. uduomatic inlcrvab, require a chromatic intcrprnation, ,.
pretence of being up--to-date avail himself at
all of the major
triad, whose vital significance has long so worn ofT and became an empty husk? Actually, the
eitmlfltal effect 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 significance. The explanation is that the GS between two points always cuts into the most tenst point, whereas symmetry creates
balantt:
the overtone series is devoid of tension because its notes are integer multiples of the fundamental note's vibrations.· The thrilling effect of the major triad in the Finale of the Sonata is a direct r•.:sult of it being completely released from the constraints of the GS system. So the
la-so-mi (pentatony) and the "",jor triad are not only
symbols of the purest music but also elements of structure and formation, which, in Bart6k's interrelation regain the fire only they may once have possessed. This is what I would like
to
denote as the elemental rebirth oCmusic through the reconstruc tion of its means. Let us set up the formula of the work: DYNAMIC
proportion "" GS-forms "" pentatony - opening
movement STATIC
proportion
-=symmetry = overtones -closing
movement
• The cs expresses tbe law or &he ,lOfJUW mean, &he overtones reRcet tbe Law or &he drillundi& mean. � we know, harmonic overtones arc produced by tbe vibration or atrinlP, air in tube., etc.; thCle not only vibrate to their rull lenJth but abo in halves, &hird., rouum. etc. of the ImBth-produdnJ J,YfPUNlriud nodc:s on the .trinJ or in the tube. The overtonc:s combine with
the batic: note, and the ml_ or the tone it determined by the extent to which thc:ae overtones modify the lIOund. We ,he�ro� eaU the bD.rmoniCl of the acoustic system "colour chordt". It it no accidcnt that the acOWltiC effects in Bart6k', compositioRl originate primarily
in &he colour chordJ of
French imprcaionism. Bart6k himself used to allu.de to
this inspiration.
7'
This implies that the 5ymmetrical periodisation of the Viennese classical school and its harmonic system of overtone rdations are phenomena not independent of each other; they only represent different (horizontal-vertical) projections of the same basic concept. :z.
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 (:z) a natural seventh (e.g. from Bb-C, C-Bb). from a minor third
(3)
a major sixth.
from a perfect fourth (5) a fifth. from a minor sixth
(8) a major
third-the most characteristic acoultic intervals. Therefore nol only do they complement. hut also rtjlttl 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 (intervals
2, 3,
5,
8 with a diminished lifth)
and a pure acoustic scale:
...�.Ii.
Ie
,••
PlO.
66
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 firth 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" fifth and major thirJ of the acoustic chord rcspcctively. Let us show these chords in their seventh forms too:
i ; '10. 68
What is valid. relative to the C root. in the OS system from Olbove downwards is equally vOllid in the acoustic system in the opposilt direction. It is thcrefore an "overtonc" chord. The cir cumSlance that ollr ancient melodies hOlve a desunding character may perhaps be rdated to the fact Ihat pcntatony is a OS tone sequence. j. Although thcse features seem to appertain
the outward form. this no longer applies when it is considered that only (unsunant inlclV;.lls exist in the acoustic system (owing 10 overtone 10
73
consonance) whereas the GS avails itself precisely of those intervals which have been considered dissonant by musical theory from the time of Palestrina. Incidentally, this diversity accounu for the tendency of Western music to be acoustic and of Eastern to be pentatonic. This impJies that the relation of consonance and dissonance 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 highest 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 based on mi as a key·note (mi·so.lIl-doscale: intervals 3, 5, 8) belonging to the most ancient layer of folk music, the greatest dissonance is represented by the perfect fifth (cf. Fig. 76). .... A secn=t ofBart6k'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 tlos�J circle offlfths. (See Figs. 2, .... 46, 48.) In the last, all relations are dependent on one tone since the natural sequcncc of over· tones emerges from one single root: therefore it is Uptll. 5. Thus. the diatonic system has a fundamental, rool note and the chromatic system a ttlltrlU note. In the chromatic: system all relations can be inverted without changing the significance 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 uf' 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) : 74
PlO.
59
It is this "mirror" (see Fig. 6g) which shows that the chromatic technique leaves the requirements of the overtone system out of consideration, and ideas like "up" and "down" become quite meaninglt:SS in it. The hannony which in the preceding example sounds below the B centre, produccs, by the negation of the overtone system, an effect as if the objecu of the physical world have suddenly become weightleu-a sphere where the laws of gravity are no longer valid (sce Mov.
I,
b. 1 94).
6. And this is why Bart6k's CS system always involves the concentric
tXpamiDn
or
tonlracli(1n
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 S01l4ta. Jor Two
PiaMS and PnrwsirJn
are constructed in
ever-widening orbiu (sce Fig. 28: leitmotif-principal theme secondary theme 8-13-21). The principal theme is augmented from bar to bar, from minor third to fourth, sixth .md seventh intervals. And the scope of the secondary theme expands similarly step by step, first with pentatonic turns, then with fuurth and fifth intervals, finally resolving in a broad sixth (se, Fig. ,8). Wc frequently find a "funne�-shaped" (sec Fig. go) and "scis50r-like" movement of notes,
(... ud
rlu. 70
and sequences proceeding by wider and wider steps:
75
Even these processes follow a plan ned course, every detail
showing augmentation up to the geometric centre uf the
movement (b. 2 1 7), after which they gradually contract ..gain. On the other hand, in the diatonic Third Movcm('nt, such progressions arc quite
unima,r:ina6Ie. The diatonic harmolljt�!I nn�
characteriscd by a stalic firmncss (e.g. the chord of rig. (i7;1 radiates its energy for a long period of time with a motiuult·ss. unwavering constancy) i n contrast to the CS system, wh ich is always of a
dynamic character.
7. Bartok's closed (chromatic) world may well be symbolised by the circle, while his line. Like in
Dante's
open.
(diatonic) systcm, by the straight
Di\'ine Comedy the symbol of the Infernu
is the circle, the ring, while that of the Paradiso, the �traight line, the arrow, the ray. The rings of the Inferno undergo a concentric diminution whereas
those
of the
till
they :lrri\'c at the
Paradiso
widen
into
"Cucitus", the
infinite
"Empyreum".
I n Bartok's "cosmos" the (hrmes fullow a similar pattern; chromaticism is most naturally associated with the "circular" while diatony with the "straight line" of melody: sce Fig. 72.
8.
The idea of "open" and "closed" is also expressed by lIu'
organisation of the themes in relation to
16
linu.
The basis of
..I
"10. 7'.l
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 first 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 corrt.'Sponds to the tonic, full.close cnding of the 16-bar period. This 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 it 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
IIsymmttry.
ltnst
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
for a process of merging, the other for dividing its constituents ; the former emphasises the
orgallic Ulli� in time, the l:lI(er
snrveys the material in space. The CS forms assume the character of an uninterrupted time process, revolving in the ;lrc of a wave, while the symmetrical periodisation breaks the material
into metrical components of lines,
rhymes and
strophes, as in the construction of a verse.
77
9.
And what do the two sy.tems look 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 fifth-3, 6, J 2 j of the major third-5, 1o. etc. While in the OS !lystcm the key number is inatiana/:
v5-1 ' -=0.618034° -: 2
.
-'-'
.
.
The irrational character becomes still more explicit if the formula is written a!I follows
(which again conceals the
Fibonacci series):
The acoustic system rats on aritluntti.cal, the
gtOmelmal proportions. (Sce App.
111.)
O�
system on
The characteristic
3-5-8 proportion u only approximately correct and u expres� sible only in ilTational numben (e.g. 5:8'0906J
•
.
.
). The:
minor third in pcntatony can be proved to be somewhat larger than it is in the tempered system.· 10. It may be symbolic that in the diatonic system the partial. toncs range
aboDl while in the chromatic Iystem btlow the
fWldamental note (see Fig. 68). It is of some interest that R.iemann derives the minor triads from "under"-tones: and the major triadJ from "over" -lonea.
In the case of CS alp44 chords this relation is also valid,
inversely; the minor third falll above and the major third below the key-nole. Although Riemann's concept may be
• The order·number of the minor third in Ihe lempered .yatem it I " 9 and in the pc:nlatonic .)'Stc:nl, " 21 . (The thirds of the .o-c.aIkd Trans· daaubiaD. pentatony come close to the ma.ior third,)
contestable it is 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 Miraculous Mandarin· receives tU intensity from just these CS intervals: Ap-F- GI-F. (See Fig. 31.) •
In the Kore of the J.li,wow
Mandarin each person iJ repretental by a
tone.symbol with whose aid we may weU "read" the plol of the pantomime.
The Mandarin may be rec08ni,aI (rom the notel G'-I: from the nula
J::;--Op
ur
1':;-0;-
II
(0, -AI
ur
(Ap-F) ; the Girl O'-A' · I�)--...:c the: ve:ry
beginning of the work. The (act that the complimena.
of the
Old Gallant
are intended (or the Girl i. shown also by the music: Ihe basic chord of the Old Gallant, more than thirty timel, leads to the Girl', symbol:
Wc mention only three brief aampJeI, After the entry o( the Mandarin the pentatonic ostinato undulates from the Mandarin', G'-I-' notes to the Girl's tone.,ymb..,):
or later when he i.t trying to reach the Girl: Gi,1
We
ace
the
.iOlt of the two .ymbolt at
Ihe very end
0( the work:
A detailed analysia of the pantomime wu publisbed by the $lw/ia Mruicol'pa (Vo!. I No. 2, pp. 363-432, in German).
author in
79
A particularly effective application of these inverted relations can be observed at tbe climax, in the C.major sccne or ',ard's C4Sllt, when the nage
is plungcd into darkness:
FlO.
Dlue
73
1 1 . It is easy to sec that the symmetry centre o f the penta· tonic scale is the re: do-,o-re
+la-mi
Similarly, de!,rree re constitutes the symmetry centre of the major and minor scales - and that of the acoustic scale, too. The antithetic relationship of the two systems becomes evi· dent it wc rcalil.e that the basic step of pentatony. the plagal IIl�mi cadence
and the basic step of classical hannony: the dominant·tonic sO-.+do cadence:
ate precise mi"or images o f each other, related to the
n'
symmetry centre: symmetry centre so � do
re
Bartok's chromatic system results in
mi � /a
plagal, while his diatonic
sYltem in authentic, harmonic interconnectionl: the basic stel' of the fonner (sec App.
being T-S-T and that of the latter, T-D-T
11).
In traditional music the motif·line usually attains itl most
sixth lur pl'rf,'c'l funrthl Cif th,' honic'. TIll' dimax uf the sixth il basically a pruperty uf classical music
tense I'uinl nn du,'
where it functions as the subdomimmt.
80
FIO. 74 Actually, the most intense hannonic function in classical music is represented by the subdominant, but it comes as a surprise that the: minor subdom£nant, being the in tensest of
all subdominant chords, is essentially a characteristic Conn of "CS tension" (intervals
2, 3, 5, 8):
FIC. 75 and conversely, the dominant harmonies arc built on deJ,'Tccs of the nearest overtones IC, E, 8 I. This means nothing less than that the technique of tension-relaxation in classical 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 turn in the CS system can be reduced or
re-la
tu
the la·mi
close so frequently found in ancient pentatonic
melodies. Compare with the " changing fifth" and six-four types of old Hungarian peasant songs:
r�f!rl[fffr-J I1D1IJJJj1DJ!fh+n; ,
1'10.
76
8,
J 2. Bart6k's diatonic mwic is always inspired by an optimism and serenity, his chromatic music by a dark, moreover, irrational and demoniac passion. This involuntarily bring! to mind the chromatic experiments of Liszt and Moussorgsky, probing the gloomy depths of life. Let us recall the late piano pieces of Liszt: Grey Cl"uds, UnJueky Stars, Prtludi" Funebre, the death�music R. WQgntr, Vene<:ia, the ghostly Lugubre Gondola, all these are written in a tone-system of distance modc:ls. Or the scene of B01U Godunov's frenzy. where Moussorgsky avails
himself of a perfect "axis system".
All in all, coherent whok, which negates
the
chromatic and diatonic systems form a representing two sides of the same coin, one of and at the same time complements, the other.
They constitute contrast in unity: affirm and deny, presuppose and exclude each other.·
• The aame duality appears in the frequent use of the ttmtpll"lItlU " � keys. Two triads which merge SJIft_trital9 diMOlve each other bccaUle the equi distance creata • floating tonality, 10 annihilAte. it. The progranune of
the Firs' SlriIIt Cbw/d-"iIIncu" and "recovery"-iI
bucd on the duality meeting
ofF minor and A major. TheJe two triads complement each other, in a
closed distance model ( I : ] modeJ): F-Ap-C + A-Cf-E = F-Ap-A-C
qrE.
a® ':: ' '
,, ""
. C,
f .......
A .
' [
fl 8 .
A -j.r
In the piano piece "See-saw, dickor),-
by the combination of E
8.
minor and Ab major (E-G-B + Ab-C-J::b givCl
In some ofhil works BartOk goes so far in the polaritation and reduction of hit material, that form and content, means and meaning, ICCtn to constitute an almost inseparable entity. From tbe preceding principles every further dj(JJQnjc formation becomes self-evident. The main diatonic intervals. i.e. the fifth and majM third, are emphasised by the major chord, built up in successive thirds, every second degree of which rhymes in perfect fifths:
•
..•.�
rlO. 77
1:5 model). Slmilarly &he neutral ra.., aC E minor and Ap m�or pain" the hoverinS in the S««td BwlufW: "A bit drunk" (ICe FiS. 46b). (Q BlwHllrd', Cut/, the E major climax aCthe fifth door it dClwyed by minor .t &he colt')' of the blooc:l.motif. Also. the tonal plan or the opera iI built up of these cnter.rcbtiou. The night is represented by FI minor. the dayli,hl majeN'. C major truly be neutralized by Alt minor. by its cowllerpo1c thUl the lattu bean a "dcath"-symboliam in the wgrk. while the com plementary key of F# minor-BP major-u &IIOciatcd with the "'ovc" scencs. 'Thae rour triads includc every dqrec of the twdve-note .ale:
a
C
C
f#-A-Cf B�-I>-F
Here we have the origin ofthe well.known Bart6k "signi1lurt�".·
It
.
fA
A "-
FlO. 78
(Cf. Two Portraits, Swiss Violin Concerto, Bagatelles Nos. 1 3 and 14. Ten easy piano pieces : "Dedication" and "Dawn", Mikrokosmos No. 10, etc.) This type, combined with the acoustic fourth (e.g. C major chord with B and F#), appears at the most splendid moments, as in
the flowcr·gardcn of Bluebeard')i Castle, or as the symbol 79.
of the "flaming. golden-haired noon" : Fig.
• Thu chord has a counterpart: the minor chord with nu.jor JeveluJI. e.g. D-F-A-C" which it ;uw�ys associ:ued with pain ;md J.lousion ill Hart6k', dramatic works and songs ("Your leitmotif" wrote Bart6k 10 Slefi Ceyer). We giyc three brief examples from IJfwbumJ'J Gwlh ;
.11....' ....., .,.."....,. 11."
......... ... ..i
,.... ,",,', , i , . , , " ,
I ."
his lem:IoI" At the cud ..r IItUlIJtfuJ', (,"01,,,# wc he... d..;: ....'u,,1 Cl
I: A C,
tonic leitmotif C-E;-O--U, Py .he iuven;'JII Il,e ••'...., C-Ep-G minor chord it uarufonncd into the C-A-F m:aj()r rhunJ. h�1 on the augmented triad C.-F-A: (rom here die di.Jorg�lIjlinJ; dli..""Cl l:1riM�,
the j"vcni.on or the
1'10· 79
Since the acoustic system is merely an inversion of the CS, we
can obtain diatonic harmonies by rtllnsjng the layers of the dlplld
chord:
1'10. 8001
The diatonic effect is due to the alpha-inversion being governed by perCect firths, major thirds and minor sevenths (i.e. the nearest overtones - which were exclude4
Plo. Oob U.B·-7
by lh�qlph4 chQr4$):
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 impression in BartOk's music:
14 \I 11 L&!�UI,I V..I... c..".t.
And to complete the concatenation,
it should be poillll'u out
that the inllersion of alpha contains the very kernel of the .u;uu:Hic chord :
FlO. 82
It happens frequently that an ambiguous bass is sometimes represented by C and sometimes by F,:
86
This is me case in the "axis melody" ofthe theme in Movement III of Music/or Strings.
PtrClUS;OIl anti Ctlesta:
f� I:,z : ��
f?'
,- ft'
. . . F.'
Plo. 8.t
We may summarise these analyses
as
follows:
os TYPES
Acoumc TYPES
(chromatic system)
(diatonic system)
Pentatony
Overtone chord and scale
Alpha chord .
Invc:rsion or alpha
I :2.
Succession or thirds and
I :3.
1 :5 models
firths with major char acteristics Forms of equ;:J intervals : .
Fomu of equal intervals :
whole-lone scale
chord in firths
diminished seventh
augmented triad. con·
chord in fourths
sisting of major thirds
augmented
triad, consisting
of minor sixths
Particular significance may be attributed to the fact that J",1tatrJllY is most characteristic of Bart6k's chromatic (CS) system while alJtrtone 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 sl1-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 primitive musie-cultures the sense for major tonality and functional attractions arc quite unknown.· The devl"lupmt·nt of Iwrmon;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 series. 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 roots of pentatonic and acoustic thinking were the two points of origin of all music.·. (If so, then Bart6k has penetrated to its inmost cor('.)
• "Pentatony does not .ulfer the dominant-tonic cadence:' (Bart6k: Hungarian Folk Music. 1933). "In this scale the fifth hat no prevailing role" (BMI6k: Hunguian .·olk Music and New Hungarian Mu�ic, 19:10). On the olhu hand "the frequent use of Ihe fourth intervab in our melodies sU88n1cd to w the use. of fuufth-chords" (Bart6k: The influence of peaaant music on modern music, 1920) . . be poa.sible to trace baCk •• BatI6k himKlr IllOnily believtd that "il ....iII
on Ihe fllce of the Slobc euoenli.Uy 10 • few parent-fotm., archetypes, ancient Ityles" (Bart6k: FoIk.song rexarch and nationalislll,
a1l lhc (olk mwlc
1937) · 88
The first is justified 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 Stnsuow.
The above claim is supported by the scientific observation
that GS is to be met with in organ;c matter only. Pcntatony, with all its tension, could nddler have come into being wilhoUl 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 contradictory counes, Physiological efforts tend to organist and create tmsion, while physical efforts disorganise by striving to aboJish tension. Here the thesis may be advanced that the OS creates a dostd world
and carries an inner tension, white the acoustic system is Optft
and strives to release tension through its overtone consonances.
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 to organise. & 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 is no accident Ihat penl_S0n.. 10 common in living n:ature, arc roreign 10 the: inorganic world.
J,r_r:,_ ,:,.
..
'�=::::;= , :':;---:-�'
, : o·6,3
no. 85
It is th.i! property of the pentagram that GIJtIM aUudes to in Faust, Part I : MltPH.: Let me admit; a tiny obstacle Forbids my walking out of here: It is the druid's foot upon your threshold. FAUST: The penJagram distresses 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 poorly drawn; That one, the: angle pointing outward, Is, you see, a little open .
•
•
Although the important question ofthythm and metre cannot be dt"alt with hcre at allY Icngth, il few oU15tanuing feOlturcs will be pointed out. Bartok's rhythm is governed by as strict laws as has been shown to rule his form and harmony. The drcufar character of Movement 1 of the Sonala jor T1.£'0 PianllS and Ptrewsion is 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 n, even and odd bars are intentionally alternated. (Bartok was very much interested in the potentialities of "even" and "odd" metres. In the SrC()ru/ Piano Conctrlo, Mov. 11 of Music, Violin ClInurlo, Diverlimenlo, Mikrokosmos }lu. 137, 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).
PlO.
86
Consequently the themes of Movement I constitute a dosrd, and those of the Finale an oJun form. But the polar principle prevails also within the even and odd metres: u + _ + .. and u_+_" units are periodically alter nating in the odd-metred themes, while an alternation of " + - + -" and 11_ + _ + " units provide the rhythmic pattern of the even-metred tunes. 9'
MovEMENT
1
,....+-
r>'
��ftlgftf�<1
.
. ' :- : "' , + --=-+ _ ....
,
_
.
't"---=--
MOVEM£NT m
r:i;:r;l �� -#hl 'ii:�.,�ffinefjt!,f (l"(hrIQ.Tj - ... : �+ .. +... - ... � �... ..-
+ -
+ - +
-...-
- ... - + --
+ -1
---
....--. ...--... � -+ - +- + + - + - - +
cl...... .. ..
9'
PlO. 87
- + - + -
---. r -+- +
That is why we feel the following idioms to be so revealing of Barl6k.
93
rlo. BB
As a final example. let us compare the opening and closing
bars of the Sonatafor Two PitJ1l0S and Pemus;on.
All." . Sl� 0..•• , ",ItIt. ...., 1iV-l ul ru. ,h.I.. ... .� .f ...."'.
\JlJTIIJmJI.rm:JI... PIO. 8g
The opening bar gives the impression of descent, as it were , into a well "which is immensely deep, or should we say. has no bottom at all" (Thomas Mann). The low shivering sounds of the timpani really seem to emanate from the negative pole of life, from a phase of precontiousness-the key of which is FI, the lowest point in the circle of fifths. Towards the close of the work, the "filliped" cymbal sounded with the nail and the light sticks dancing on the
nOm
of the side.drum, produce ostinatos which gambol joyfully over the work, with "slender anklcS" on the paths of
ligh': in C
major, the highest point i.n the circle of fifths, and counterpole of F#.
In this way the extreme points of the composition may be regarded as negative and positive potes so that the analogy of a magnetic field offers itself, a current being developed between t,wo opposite poles. The Lento-with its utterly iRIIrliadald now-is represented by the lowest, the Allegro-with its
95
articulated rhylhm-by Ihe rughesl drum effect. On the onc side a linear, on Ihe olher, a rhythmic-spatial clement. Nevc:rlheless, 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 symnutry. 11,e symbol oflhe circle is w, while the laller can be expressed by Ihe powers of2 (�'=4, 4'= 16, 16'= 256; Ihe next power is already too large). The time-value of the whole work (the above-mentioned 6432 eighths) is 804 whole notes, and this is precisdy the producl 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 SunG/G
for Two Pianos and PnCWJion
to be a crowning piece: the
Makrokosmos of the Mikrokosmos ( 1 92�37). What role did Bartok's art play in the music of our century? His chromatic system has its roots in Eastern folk music and in pentatony; his acoustic Ifstem he owed to Western harmonic thinking, He himlelf admitted his indebtedness to folk music and Ihe French impressionists as the IWO most descisivc inRuences on his an.· • "The two TOOlI orour ut oOgin..ta in folk music and the: new French mutic," wrote &..16k ("Zoh£n Kod£ly": 19:U). ThiJ decluatton dcscrvcs attention fo.- it is well known th..t he rarely, ir ever, eomrrUlled hinuclr on hi, own eompositions-Ihouah he liked to
emphuile their relation to rolklore, mainly ....ith . the Intention of pro
pal&tingfoU: mlDic. "Let my music lpeak ror itIClf, I lay no claim to any nplanation 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 West:
it
synthesis of the music of Orient
and Occident. •
This essay 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.
91
Appendix I Referring to the &1Ulta JM Two PialU)S aM PertuSJ;oTl, it is of intC'rtst to point out a few particulan of Movement L In hs. �35-247 wc again notice the two-folt! aOillity of the axis; on the one hand the G# ostinato running through the part and 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.
�D
-----�
---
If,J
-----
_h'!'lo'
tJ, r H d'rEilI'JJJ4l'J1" ;! If ."[!".cr�f·-
< ."
1ft
J.,
...
•
·r
•
•
• .�
nc. go
99
The four sections of the secondary theme in the recapitulation (bs. 292-33°) are based on the four poles of the IoniC axis $0 that their Outer and inner parts, respectively, correspond with each other in their pole-countcrpole relations.
FIC. 91
The axis construction of the coda is equally unambiguous (bs. 417-431). In accordance with the polar scheme the augmented principal theme appears in E\), then in A, and finally in Eb + A, over the disjointed Eb-Gb-A-C major-minor (gamma) accompaniment chord5. E I - CI -A - C
AXIS
FIO. 92 lOO
The development in Movement 11 ofMusicfor Slrings. Percussion and Celesta presents an exemplary axis eonstruclion:
F10, 93
The appearance of the EtJ-F#-A-C is always ;l.ccenluated by the bass drum, The Ilccompll.nimcnt, which rcmainJ unclumged throughout, stresses the A and E� coulllcrpolcs (blla chords):
no. 94
B.8,--I)
'0'
It is evideot both from the accompaniment and the dynamics that the Elt-A polarity fornu the backbone of the structure. And finally Fig. 95 gives some strongly marked axis melodies.
!
�
I@f It;r r IT 1@\trT'� .
I'ltl. 95
.0.
Appendix I I A few examples are given to iUwtrate the interrelations expounded in connection with Fig. 13. The order of keys of the
Fi,sl ROMc is as follows:
C tonic, E
dominant, A� subdominant and C lonic. Movement I of the
c,metrt# is
subdivided by the five·fold
recurrence of the principal theme: h. 76
F tonic (exposition)
D� subdominant (fint part of development) h. 231 A dominant (second part of development)
h. 3 1 3
F tonic (recapitulation)
h. 386
F tonic (cod.)
h.
488
A similar arrangement is to be seen in Movement I of Sonald
IQr Two PimrDs tUUI Percussion: C tonic (exposition)
h. 32
E dominant (fint part of development) h. 161, 195 subdominant (second part of development) b. 232 Cl b. 27.
C tonic (recapitulation)
In the tlUrd example of page 45 these relations are: D-AJ-F#. The principal theme in Movement PttclLSsilft J tmtI
11 of
Mwit jor
Strings,
CtltsllJ is particularly cogent, because here we find
the cupola-structure and tonic-tonic�ominant-tonic con
is &-Bb
struction of the new-type Hungarian folk songs. The tonic represented by the C-F# counterpoles, the dominant by
counterpoles (not by G). The second entry of the tonic provides the exact "tonal answer" of the C-FI axis: G-C tones change into
C-G and C�F� into F#-C#.
(M)
T
PlO.
!)6
A similar associalion of dominant and Ionic is evident in bs. 1,1-1,8 of Movement 1 or the Diue" ilMft/oj the E(,-A dominant counterpoles correspond to the F-8 tonic counterpoles:
•
A CS9 ' .
.� F
hO· 97
or at the recapitulation:
The lower majur second degree (e.g. BtJ in C tonality) might justifiably be ( alled the "bart6kcan dominant" owing rrequent occurrence in his music :
(0
its
PlO. 99
which can again be explained by the regularities dealt wilh above. The opening bars of the
MirtUu/olLf Mandarin
illustrate how
the tonic and subdominant are linked; the D# tonic swings towards the subdominant F and B counterpoles:
no. 100
006
It is interesting 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 expres 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 MlUlwin 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 satisfies his desire, are written in the tonic C. Beginning G
Climax. C
End F
DOMINANT
TONIC
SUBDOMINANT
The succession of scenes follows the same descending orderI a triple descent from the dominant heights to the subdominant depths, as if expressing the idea that the work moves towards a "fateful" abyss: DOMINANT
IUBDOWlNANT
TONIC
1.
Viaitor (Old Callant)--+It. Visilor (Youth)--+3. Vililor (Mandarin)
I.
Wa!lZ Murder
I.
------
_+2. W.ltz
-+
_+2. Murder
-+3. Murder
----
Punuit
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 originates in the basic chord of the Fint Visitor (Old Gallant): 10'
(_. "C) r# : ,
PlO. 101
In thl" plot of Ihe: Wooden Prime all this is inversely (nu", 'rh..
scenes follow an ascntt/ing line, incessandy going up the: T-D-8-T grades : EXPOSITION
MIDDLE PART
Prelude
TONIC
Princess
DOMINANT
Prince
SUBDOMINANT
Forest
TONIC
Stream
DOMINANT
Making of the doll
SUBDOMINANT
Dance of the Wooden Prince
TONIC
I . Scene (No. 120)
TONIC
2. Scene (No. 128)
DOMINANT
3. Scene (No. 1 32)
SUBDOMINANT
Conclusion
TONIC
RECAPITULATION "" end of exposition
TONIC
Wooden Prince (No. 1 49)
DOMINANT
Princess and the Stream
SUBDOMINANT
Denouement
TONIC
For example, here follows the axiS structure of lilt: first scene-Dance of the Princess :· • A detailed analysu or the work was published in the author's " Darlltk's Dramaturgy": Stage works and Can.ata prorana (Editio Music:. lJlltlapol.
1964) •
08
•
E - Af - C' - G AXIS
An.,nU• .kr&...oI.. Cr,..,... . l,OIll 0...,...
No.l$
c..du.u
E.I
_h.
no. 102
log
Appendix I I I An exact golden section can only he constructed geomc:trically it cannot be obtained mathematically. i.e. by means of rational numben. The key-number of the CS is irrational (similar to Tt). Here is an example of the uEudoxus" construction, with square and semi-circle.
,-
,
J
,
\
\\
06111..
\
PlO. 103
and another based on the Pythagorean proportion.
PlO.
"0
104
The hypotenuse
•
•
.
.�
0'618 -
"'"
l'U'6l1. a61.t:alI2
PlO, In�1
and a chain of golden sections can be brought about as follows:
PlO. 106
'"
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, 0·6,8 ""0'382 "" 0'2,36; 0'382 -0'236 "" (,1'146, etc. But the sanuOS chain can also be obtained by involution : 0.6,81 = 0·618; 0.6182- 0'382; o·618's: 0·236; 0.6,8-=0'146, etc.
dc:
The sine and cotangent curves meet in one definite point. and precisely in the golden angle. The value pertaining to it is:
v'0·618 . . .
no.
107
With the aid of compasses it can be shown that the radius goes 6 times, while its CS 10 cimes into the circumference oCa circle. The Cheops Pyramid in Egypt reveals the following structure:·
-.' ,0 ..'
no.
108
and so iu sloping sides subtend a golden angle. • After K. Klcppiscb and E. Bindcl.
"'
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.·
,
"
PlO. IO!)
While Gothic architecture favoured angles 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 an "earthly" onc. Because the Gothic concept subjected earthly affairs to heavenly concerns it forced the square to find a place within the circumference of the circle. In contrast to this matters of heaven :md earth had an equilibrium in Greek. and Renaissance art, therefore, in
• After Zeising.
geometric terms, the circle was coupled with square. Hence the triangle no more subtended
an an
isoperimetric angle of 4�o,
but rather the golden angle:
Crtck IlIId Rtuiu...ct
1'10.
"4
1 10
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 ;L'I tll(' l)Cr[i:cl l"mbfldimt:nt uf it. Eisell$tcin planned his film, "Potemkin", in such a way that he placed points of uperfect inactivity" in the negative golden section of each act and those of "highest activity" in the positive sections. According to Einstein, OS providn a ratio which opposes the bad and fadHtates the development of what is good. "No elements can be properly joined without the aid of a third one, for the two can only be united by the mediation of a link i but ofall the links that one is most beautiful which makes a complete whole oritself and of the elements united by it." Plato: Timaeus.
• Cf. Otto Schubert: Gesctz der BaUkUNt (Scemann-Lciptig 19M).
"5
