View of Rational Order in Tone Scales and Cone Scales

R ational Order in Tone Scales and Cone Scales

T h e b e lie f th a t n a tu re m ust be co n sid ered as a s ta n d a rd fro m w hich a rt can derive its guidelines (natura artis magistra) was firm ly estab lish ed d u rin g m any centuries. N o t so firm w ere th e reasons why a rt sh o u ld a p p re n tic e itself to n atu re . T h e e ig h te e n th cen tu ry saw th e transition from a neoclassical co n ­ ce p tio n o f n a tu re as b ein g regularly o rd e re d , a n d th e re fo re an ex a m p le to m a n k in d (as in P o p e ’s Nature methodized) to th e R om antic id e a o f m a n b ein g overw helm ed by n a tu re (following B urke’s delightful sublimity). By show ing two controversies in very d iffe ren t fields I in te n d to show how, in a m o re subtle way, also in o th e r periods the idea o f an intrinsically rational order in nature com es in to conflict with a m o re practical, em pirical attitud e.

In his Istituzioni Armoniche o f 1558, Italian m usical th e o ris t G iu sep p e Z arlino p ro p o se d to co n sid er n o t only octave, fifth a n d fo u rth , b u t also th ird a n d sixth as c o n s o n a n t intervals. H istorically speakin g, this c o rre c tio n o n P y th a g o rea n th in k in g was lo n g o v erd u e. T h ird s a n d sixths h a d g rad u ally co m e to b e a c c e p te d as h a rm o n ic sh elters since th e earliest form s o f p o ly p h ­ ony cam e in to existence. B ut n o t b efo re Z arlin o d id th e m a jo r th ird ac q u ire th e p restig io u s p o sitio n o f b e in g o n e o f th e c o rn e rsto n e s o f th e h a rm o n ic fram ew ork.

Z a r li n o ’s c o r r e c tio n m a rk s th e e n d o f th e p r e d o m i n a n c e o f th e P y th a g o rea n tetraktys as a th e o re tic a l basis fo r harm on y: th e tetraktys allows only tho se interv als as c o n so n a n t w hose ratios can b e ex p ressed by th e first fo u r n u m b e rs.1 Z arlino in troduces a new co n c e p t in m usic theory: th e senario, im plyin g th a t six r a th e r th a n fo u r is th e lim it fo r th e ra tio s th a t b u ild u p c o n s o n a n t intervals. E n te r th e m ajo r th ird ( 5 : 4 ) , th e m in o r th ird (6 : 5) a n d th e ir c o u n te rp a rts , th e m in o r sixth (8 : 5, w h e re 8 is c o n s id e re d th e tw ofold o f 4) a n d th e m ajo r sixth ( 5 :3 ) . T h e V enetian m aestro b elieved th a t ju s t intonation co u ld be achieved by b asing all intervals in a to n e scale o n th e fifth a n d th e m a jo r th ird . T h a t leads to th e only type o f in to n a tio n w hich Z arlin o is w illing to c o n sid e r as natural.2 In o th e r words: Z arlin o d id n o t so 1 That is: the octave (2 : 1), the fifth (3 : 2) and the fourth (4 : 3) - and, trivially, the

prime (1 : 1).

2 Just&nà natural are still in use as synonyms for this particular intonation (in German:

reine or natürliche Stimmung).

m u c h overthroiu th e P y th a g o rea n way o f th in k in g in term s o f ra tio n a l o rd e r b a se d o n n u m e ric a l ontology, b u t r a th e r saved it by e x te n d in g th e ra n g e o f fu n d a m e n ta l n u m b e rs to six.

T h e attac k o n th e o n to lo g ic al basis o f this type o f th in k in g was le ft to V in cen zo G alilei, fa th e r o f th e fam ous a s tro n o m e r b u t also a p u p il o f Zarli- n o ’s. G alilei d oes n o t a c c e p t his m a s te r’s g u id e lin e o f th e senario. In p a rtic ­ ular, h e attacks th e status o f fifth a n d th ird as »natural« intervals. No such th in g- s a y s Galilei: all intervals, all to n e scales have co m e to be estab lish ed by h u m a n convention. E xact rational p ro p o rtio n s (in th e m ath em ad cal sense o f b e in g ex pressib le as a ra tio o f integ ers) have n o special m e a n in g h e re . T h e r e is n o p rin c ip a l d iffe re n c e , in this re sp ect, b etw een th e in terv als o f m usic a n d th e w ords o f a »natural« langu age.

G alilei’s critical a ttitu d e towards his m a s te r’s au th o rity is fu n d a m e n tal.

T h e id e a th a t a c o n s o n a n t interv al sh o u ld b e a n y th in g else b u t a ra tio n a l n u m b e r w ould have b e e n co n sid ered ab su rd d u rin g

th e m ajo r p a rt o f E u ro p e a n history. T h e fo u n d a tio n o f th a t th o u g h t goes back a t least as far as P la to ’s Ti- maeus, w h ere th e very ratios o f th e tetraktys are consti­

tutive fo r th e c re a te d o rd e r o f the cosm os. G alilei’s criticism clearly reflects m o re th a n ju s t a m usicologi- cal co m m en t; it heralds the paradigm shift with which th e n a m e o f his son will forever be linked. B ut before g o in g d e e p e r in to th e h e a te d d eb a te betw een m aster a n d p u p il, we shall first take a look a t a conflict in a com pletely different setting an d time - n o t ab o u t a hu ­ m a n p ro d u c t, b u t c o n c e rn in g th e p ro d u c tio n o f n a­

tu re herself.

Towards th e m id d le o f th e n in e te e n th century, a b o tan ical d eb a te flared u p a b o u t th e way in w hich n a tu re accom odates certain p rim o rd ia a ro u n d a ce n ­ tre - like leaves a ro u n d a stem , scales o n a p in e cone, sunflow er seeds o n a flow er h ead , etc. T h o u g h we use to w o n d e r a b o u t th e am azing spiral stru ctu res w hich th ese plants show, we o ften d o n o t realize th a t these spirals w ere n o t th e re in the first place. T hey com e into existence step by step; in fact, the birth certificates o f all th e sunflow er seeds are issued o n e by o n e, in a strict o rd e r th a t can even be traced subsequently. T h e spirals we see are n o m o re th a n an e p ip h e n o m e n o n o f a spiral we d o n « t see, b u t w hich we can o b tain by

c o n n e c tin g th e scales in th e o rd e r in w hich they p o p p e d u p . We shall call this the fundam ental spiral (in the picture: 1-2-3-4 etc.), w hereas th e contigu ou s par­

allels as they b ec o m e visible are called parastichies (in the p ictu re: 6-14-22-30, o r 19-27-35-43 etc.).

By 1830, G e rm a n b o ta n ist A le x a n d e r B rau n h a d th e b rillia n t id e a to use th e p re c ise o r d e r o f th ese scales fo r th e classificatio n o f c o n ife ro u s p la n ts.3 C lassification b e in g a favourite p astim e fo r b o tan ists, th e su b tle dif­

fe re n c e s b etw een th e im p la n ta tio n o f th e scales in the d iffe re n t species o f co n ife ro u s p lan ts see m e d to offer an id eal h a n d le to com e to g rip s w ith th e d iffe ren c es b etw een th em , a n d to label th ese d ifferences. In o r d e r to w ork o u t th ese labels, B rau n in tro d u c e d th e n o tio n o f divergence in b o ta n ic a l p a r­

lance. By n o ta tin g su ch a div erg en ce as, say, 8J21 (as in th e case o f th e p in e c o n e o n th e p ic tu r e ) , B rau n m e a n t th a t 21 scales w ere fo u n d w h e n th e fu n ­ d a m e n ta l spiral h a d ro u n d e d th e c o n e exactly 8 tim es.4

T h e p re s u p p o sitio n o f this p ro je c t is th a t th e p o sitio n o f (in this case) th e 2 2 n d scale is exactly above th e first. B ra u n ’s c o n c e p tio n im p lies th at, a p a rt from th e p arastichies, each co n e also shows parallel orthostichies (in th e p ictu re : 1-22-43-64, o r 9-30-51-72 etc.). B rau n do es in d e e d believe th a t af­

te r a n a tu ra l n u m b e r o f scales th e fu n d a m e n ta l spiral has co m e full circle, so th a t the ra tio o f th e n u m b e r o f scales a n d th e n u m b e r o f ro ta tio n s can b e ex p ressed as an ex a ct ra tio n a l n u m b er.

No such th in g - say two F re n c h scientists w ho sta rte d in v estig atin g co­

n ife ro u s p lan ts a r o u n d th e sam e tim e as B rau n did. A ug uste a n d L ouis B ra­

vais observ e th e sam e cones as B rau n , b u t see s o m e th in g e n tire ly d iffe re n t.

In particu lar, they d o n o t see a series o f distinctly d iffe re n t ratio s in th e d i­

v ergences o f th e plants. B ra u n ’s d iffe re n tia tio n is b u t an illu sio n, o r so they claim . N a tu re has fo u n d th e o p tim u m an g le fo r th e im p la n ta tio n o f every n e x t see d o r scale; th a t an g le e n su res th a t all th e p rim o rd ia have an o p ti­

m u m space to grow, a n d it rem ains th e sam e a t every turn: 137° 30' 28".r> T h a t am o u n ts to a re p e a te d division o f th e circle ac co rd in g to th e g o ld e n section, w hich is an irra tio n a l m easu re a n d can , fo r th a t re aso n , n e v e r le a d to th e ra tio n a l classification th a t B rau n p u rsu e d . It is, however, a constant m e a su re - th e only o n e th a t g ra n ts e q u a l rig h ts to all p rim o rd ia . T h e w ho le o rg a n ­ 3 A. Braun, »Vergleichende Untersuchung über die Ordnung der Schuppen an den Tannenzapfen als Einleitung zur Untersuchung der Blattstellung überhaupt«, in Nova Acta Academicae Caesareae Germanicae Leopoldinae, Nr. 15, 1830, pp. 199-401; reprinted in book form in Bonn, 1831. Page numbers in this article refer to the book edition.

4 Numerator and denominator of the divergence will generally relate as the numbers ( n - 1) : (n + 1) from the Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34 ....

5 L. & A. Bravais, »Essai sur la disposition des feuilles curvisériés«, in Annales des Sciences Naturelles, Seconde Série, t. 8ème, 1837, pp. 70/1.

ism b en e fits fro m this eq u a l division. R e c e n t re s e a rc h 1’ has show n th a t this is, in fact, th e way n a tu re behaves; o n e do es n o t n e e d to involve g e n e tic a l o r teleo lo g ical p rin c ip le s to fin d th a t the flow er h e a d o f a su nflo w er is d i­

v ided ag ain a n d ag ain , by ea ch new p rim o rd iu m , ac c o rd in g to th e g o ld e n section.

B oth contro v ersies, th e o n e in th e R enaissance a b o u t th e alleg e d ra­

tionality o f to n e scales a n d th e o n e in th e n in e te e n th c e n tu ry a b o u t th e al­

le g e d ra tio n a lity o f c o n e scales, fin d th e ir o rig in in o p p o sin g c o n c e p tio n s o f th e value o f ra tio n a l o rd e r in n a tu re . O f co u rse, b o th pairs o f o p p o n e n ts have a lo t in co m m o n , d u e to the p re co n ce p tio n s th a t even o p p o n e n ts w ould sh a re in a c e rta in age. B oth Z arlin o a n d G alilei fre q u e n tly call o n » th e a n ­ cients« to sub stan tiate th e ir own p o in t o f view; b o th believe th a t th e a n c ie n ts h a d set a n ex a m p le , n o t so m u c h by th e ir h ig h s ta n d a rd o f c u ltu ra l devel­

o p m e n t, b u t by th e ir b e in g clo ser to nature, th a t is, by th e ir b e tte r u n d e r ­ sta n d in g o f natural order.

Z arlin o believes th a t M o th e r N a tu re restricts h e rse lf to a w ell-consid­

e r e d d o se o f p e rfe c tio n by d iffe re n tia tin g b etw e en th e individuals th a t b e ­ lo n g to th e sam e species r a th e r th a n j u s t c lo n in g th e id eal a rch ety p e again a n d again. H e praises th e an c ie n ts fo r tra n sp o sin g th a t p rin c ip le to m usic, w h e re re p e titio n o f id e n tic a l c o n s o n a n t intervals is to b e avoided:

»T hus they h e ld it as tru e th a t w h e n ev er o n e h a d arriv ed a t p e rfe c t c o n s o n a n c e o n e h a d a tta in e d th e e n d a n d th e p e rfe c tio n tow ard w hich m usic te n d s, a n d in o r d e r n o t to give th e e a r to o m u c h o f this p e rfe c tio n th ey d id n o t wish it re p e a te d over a n d over again.

T h e tru th a n d e x c ellen ce o f this a d m ira b le a n d useful a d m o n itio n are c o n firm e d by th e o p e ra tio n s o f N a tu re, fo r in b rin g in g in to b e in g th e in d i­

viduals o f ea ch species sh e m akes th em sim ilar to o n e a n o th e r in g e n e ra l, yet d iffe re n t in som e particu lar, a d ifferen ce o r variety affo rd in g m u c h p lea­

su re to o u r senses. T his ad m ira b le o rd e r th e c o m p o se r o u g h t to im itate, fo r th e m o re his o p e ra tio n s re sem b le those o f o u r g re a t m o th er, th e m o re h e will be esteem ed. A n d to this course the n u m b ers a n d p ro p o rd o n s invite him , fo r in th e ir n a tu ra l o r d e r o n e will n o t fin d two sim ilar p ro p o rtio n s follow­

in g o n e a n o th e r im m ed iately

V incen zo Galilei is involved in a d iffe re n t battle. H e is a m e m b e r o f th e F lo re n tin e Camerata, th e think-tank o f h u m a n ist scholars an d n o b le m e n w ho paved th e way fo r an en tire ly new form o f art, a sp ectacle th a t w o uld con- 6 S. D ouady & Y. Couder, »Phyllotaxis as a Physical Self-O rganized Growth P attern«, in

Physical Review Letters, Vo\. 68, Nr. 13, 1992, pp. 2098-2101.

7 G. Z arlino, Istituzioni Armoniche, in O. S trunk (ed .), Source readings in Music History, Vol. I I - The Renaissance, New Y o rk /L o n d o n 1965, pp. 4 4 /5 .

q u e r E u ro p e a n stages in the s e v e n te e n th century: o p era. O p e ra is typically an a rt fo rm th a t d id n o t re su lt directly fro m any d e v e lo p m e n t in m usical p ra ctice, b u t was p re p a re d o n th e draw in g b o ard . T h e m a in im p u lse cam e fro m th e F lo re n tin e resistan ce ag a in st c o n te m p o ra ry (» m o d e rn « ) p o ly p h ­ ony. G alilei’s Dialogo della mušica antica e della modernd1 is a n a r d e n t p le a fo r a new type o f m usic (» p o stm o d ern « , so to sp ea k ), th a t w o u ld d o ju s tic e to th e n a tu ra l ex p ressio n o f h u m a n affections - a task w hich p o ly p h o n ic m u ­ sic, w ith its in tric a te s tru c tu re o f sim u lta n e o u s m elo dies giving voice to sev­

eral texts a t th e sam e m o m e n t, co u ld n o t possibly fulfd. T h e p o ly p h o n ic m usic o f G a lilei’s c o n te m p o ra rie s is an in su lt to h u m a n n a tu re (so h e b e ­ lieves) , a n d th e m usic o f an tiq u ity is p u t forw ard in his w ritings as a n in sp ir­

in g gu id elin e.

Intrinsically, differences o f o p in io n betw een Z arlino a n d G alilei a re n o t as g re a t as th e ir p e rso n a l fe u d m ig h t suggest. G alilei w ould have n o tro u b le w ith th e q u o ta tio n given above, re g a rd in g th e d esire d variety in intervals, a n d Z arlino w ould w h o leh earted ly a g ree w ith the C a m e ra ta ’s p re fe re n c e fo r w ords above m elo d y w hen p u ttin g tex t to m usic. T h o se w ere in fact th e c e n ­ tral issues o f th e tim e, a n d b o th a u th o rs w ere well aw are o f th em . B u t u n ­ fortunately, b o th m e n w ere driv en by ».... th e d e sp e ra te wish to c o n tra d ic t ea ch o th e r« .<J T h e adv an tag e o f this fo r la te r scho lars is th a t th e ir d iffe re n t a ttitu d e s tow ards th e im p o rta n c e o f ra tio n a l o r d e r received m u c h e m p h a ­ sis, a n d thus clearly ex pose th e d iffe re n c e b etw een Z a rlin o ’s n e o p la to n ism a n d G alilei’s m o re e m p irica l a p p ro a c h .

E m p irical re se a rc h , as it b ec am e to be p ra c tise d by th e investigative R enaissance m inds, d id n o t au tom atically im ply a re p u d ia tio n o f ra tio n a l p ro p o rtio n . G alilei m ad e a n a m e fo r h im se lf in th e history o f m usic th eo ry by c o r re c tin g w h a t th e M id d le Ages h a d b eliev ed was a n o b s e rv a tio n by P ythagoras him self: th e discovery o f th e p ro p o rtio n a l re la tio n sh ip s b etw een th e w eig ht o f th e h a m m e rs used by th e blacksm ith, a n d th e p itc h e s o f th e so u n d s they p ro d u c e d . Every m edieval m usic th e o ris t knew th a t if a c e rta in p itc h was p ro d u c e d by tying a w eigh t to a string , th e octave o f th a t p itch w ould be p ro d u c e d by tying the d o u b le w eight to th e sam e strin g, a n d a fifth w ith th e h e lp o f a w eig h t o n e a n d a h a lf tim es th e o rig in al, etc. In o th e r words: th ese ra tio s w ere su p p o sed to b e th e sim ple inv ersion o f th e (m o re easily m easurable) ratios fo r string lengths p ro d u c in g th e sam e intervals. N ot so, says Galilei: to p ro d u c e those intervals by tensio n, th e w eights w o uld have to b e in squared inverse p ro p o rtio n to th e len g th s o f th e strings. T h e ir re la ­ tio n sh ip s to th e p e rfe c t c o n s o n a n t intervals a re still p erfec tly ex p ressib le as 8 F lorence, 1581.

‘J D.P. Walker, Studies in Musical Science in the Late Renaissance, L eiden 1978, p. 16.

ra tio s o f w hole n u m b e rs , b u t n o t an y m o re in th e trad itio n ally con stitu tiv e n u m b e rs o f th e P y th a g o rea n tetraktys.

H ow d id G alilei fin d this out? G oing by his re p e a te d re fe re n c e to ex ­ p e rim e n ta l m e th o d (con il mezzo dell«esperienza), we m ay safely assum e: by try in g out.

Z arlino, as we saw, did n o t stick e ith e r to th e tetraktys to express th e ratios o f th e im p e rfe c t co n so n a n c e s, b u t his a rg u m e n ta tio n a l back-up is o f a to­

tally d iffe re n t o rd e r. W hy sh o u ld th e senario r a th e r th a n th e tetraktys be c o n ­ sid ere d as the basis fo r o u r h a rm o n ic u n d e rsta n d in g ? As if we cou ld n o t have guessed:

- G od c re a te d th e w orld in six days

- six signs o f th e zodiac are always above th e e a rth , th e o th e r six are invisible

- th e re are six »planets« (to Z arlin o ’s know ledge: S atu rn j u p i t e r , Mars, V enus, M ercury, a n d th e m o o n )

- th e re are six d ire c tio n s (up , dow n, a h e a d , b e h in d , left, a n d rig h t;

Z arlin o calls o n P lato to testify to this spatial insig ht)

- th e n u m b e r 6 is traditionally h ailed as th e first » perfect n u m b er« ; th a t is, it eq u als th e sum o f its d ividends 1, 2 a n d 3; m oreov er, it is th e ir p ro d u c t

- in m usic, th e re are six » a u th e n tic « a n d six

»plagal« m odes.

Z arlino gives q u ite a few m o re re a s o n s ,10 b u t th e s e six w ill s u ffic e to sh o w th e g a p t h a t e x ­ ten d s b etw e en th e m e n ­ tal w orld o f Z arlin o a n d th a t o f his p u p il. G alilei, w ho was an early p io n e e r o f equal temperament, d id n o t feel a n y th in g was lost by giving u p th e p e rfe c t­

ly ra tio n a lly o r d e r e d in ­ terv als. Z a rlin o , o n th e o t h e r h a n d , c o u ld n o t im a g in e j u s t in to n a tio n

10 See C.V. Palisca, Humanism in Italian Renaissance Musical Thought, New H a v e n /L o n d o n 1985, p. 248.

in any o th e r way th a n by th e numeri sonori o f th e senario, as this illu stra tio n from his b o o k shows: a w ell-ordered w orld o f m usical intervals, w ith th e se­

nario in th e ce n tre .

T h e contro versy b etw een A le x a n d e r B rau n a n d th e Bravais b ro th e rs is situ ated in a d iffe re n t age, against th e b a c k g ro u n d o f d iffe ren t scientific strat­

egies. E x p e rim e n ta l v erification h a d b e c o m e p a rt a n d p a rc e l o f re g u la r sci­

entific b e h a v io u r by th e tim e B rau n d ev e lo p e d his theory, a n d h e h im se lf was n o ex c ep tio n : th o u san d s o f p in e co n e s w ere c o llec te d by h im a n d his co lleag u e, C arl S chim per, a n d m eticulously so rte d o u t a n d classified. A n d yet, B rau n is ste e re d by a n o th e r drive th an co llec to r’s m an ia o r labellin g n e u ­ rosis: h e w ants to u n rav el th e h id d e n p rin c ip le b e h in d n a tu ra l o r d e r as this b ec o m e s visible in th e a rra n g e m e n t o f leaves, seeds, petals a n d scales alo n g a stem .

W h at B rau n finds is fascinating, b u t m u c h m o re fa scin atin g is to know w h at h e is lo o k in g for. B rau n was, in his ow n w ords, ch asin g th e »joyful p re ­ su m p tio n o f a law fo u n d e d deep ly in th e life o f th e plants« (freudige A h n u n g eines tief im Leben der Pflanze gegründeten Gesetzes).11 To this e n d , th e e x a c t d e ­ sc rip tio n a n d classification o f th e o u te r a p p e a ra n c e o f th e co n e s was n o t e n o u g h . In lo o k in g fo r his h id d e n law, B rau n believed h e was follow ing n a ­ tu re herself. A n d w h e n h e fo u n d th e constitutive spiral, th e row th a t d ic ta t­

e d th e p o sitio n o f all th e scales, h e w e lco m ed this » m iracu lo u s re g u la rity o f ord er« (wunderbare Gesetzmässigkeit der

Anordnung) w ith an alm ost religious respect: »In this last, O n e Row, daw n­

in g u p o n o u r e x p ectatio n , we b eh o ld th e tru e goal o f o u r h o p e , th e O n e G ro u n d o f phyllotaxis, on w hich all m u ltitu d e a n d variety o f rows m u st ^ rest.« 12

B raun ’s draw ing, within a circle, 6 o f a b o tto m view o f th e p in e c o n e shows o n e layer o f this rational order.

It is alm ost re m in isce n t o f the picture in Z a rlin o ’s book: a ro u n d e d way o f th in k in g th a t always com es back to its p o in t o f d e p a rtu re .

11 Vergleichende Untersuchung, p. 3.

12 »In dieser uns in d e r Erw artung vorschw ebenden letzten, Einen Reihe erblicken wir das w ahre Ziel u n serer H offnung, den Einen G rund d e r Blattstellung, a u f dem alle V ielheit u n d V ielartigkeit d er R eihen b e ru h e n muss.« Vergleichende Untersuchung, p. 22.

T h e r e is a n in trig u in g ten sio n b etw e en u n ity a n d variety in B r a u n ’s c o n c e p tio n o f n a tu ra l o rd e r, c o m p a ra b le to th e way Z arlin o deals w ith th e p e rfe c tio n o f co n so n a n ts a n d th e ir necessary d iffe ren tiatio n in m usical co m ­ p o sitio n . T h e u n ity th a t is firm ly estab lish ed in th e overall ru lin g o f th e fu n ­ d a m e n ta l spiral serves as a c o n d itio n to b rin g o u t a m u ltitu d e o f d iffe re n c ­ es - d iffe ren c es by w hich th e several species o f co n es can be d istin g u ish e d a n d lab elled . B ra u n ’s aim is a classification in th e lin e o f L in n ae u s, arriv ed a t by m ean s o f em p irica l o b serv atio n , b u t his reg ulativ e c o n c e p tio n is th a t o f an overall ra tio n a l o rd e r. In o th e r words: B rau n treats d iv erg en ces as if they w ere m usical intervals acco rd in g to a trad itio n al system o f tem p e ra m e n t, a n d h e d o es so o n th e basis o f a d ee p ly ro o te d in n e r conv iction th a t this is how n a tu re behaves. B ra u n ’s phyllotaxis reflects an o r d e r o f ju s t intonation.

It is to this p re c o n c e p tio n th a t th e Bravais b ro th e rs o p p o se. T h e r e is n o d iscre te classification o f d iffe re n t divergences; w hen trying to a ttrib u te o n e o f B ra u n ’s ra tio n a l labels to a specific p la n t, th e ch o ice b etw e en , say, 8j21 o r 13 j3 4 o ften seem s q u ite arbitrary. N o n e o f B ra u n ’s alleg e d o b se r­

vations is as precise as th e ex a c titu d e o f th e ra tio n a l m e a su re suggests. T h e b ro th e rs carefully ju stify this s ta te m e n t with a n u m b e r o f illu stratio ns. W h a t th ey o b je c t to is in fact n o t so m u ch th e validity o f B ra u n ’s equ ally ca re fu l observ atio n s, b u t th e very status o f th e sta rtin g p o in t w hich led th ese o b se r­

vations to re su lt in th e conclusions th a t B raun p re sen ted . T h a t startin g p o in t is th e c o n c e p t o f orthostichy, w hich, to c o n tin u e th e m e ta p h o r I have j u s t in ­ tro d u c e d , in B ra u n ’s system o f ju s t b o tan ica l in to n a tio n fulfils th e ro le o f th e octave, th e p o in t o f re fe re n c e fo r all th e o th e r intervals. T h e s tro n g im ­ p a c t o f B ra u n ’s c o n c e p tio n b eco m es clear w h en we re a d th a t C arl F rie d ric h N a u m a n n c o n sid e re d th e o rth o stich y as »the re al essence« (das eigentliche Wesen) a n d p arastich ies as »a m e re p h e n o m e n o n o f phyllotaxis« (ein blosses Phänomen der Blattstellung) . 1S T h e a lte rn a tiv e w h ich th e Bravais b r o th e r s p re s e n t com es dow n to g ra n tin g id en tical rig h ts to p rim o rd ia in th e sam e way to n e s have id e n tic a l rig h ts in e q u a l te m p e ra tu re - w ith th e proviso th a t in th e case o f th e plan ts, this eq uality is g ra n te d by n a tu re .

A p a rt fro m carefully e x p la in in g th e ir own theory, th e Bravais b ro th e rs m ake a stan d against B ra u n ’s po sitio n in a sep a rate artic le .14 T h e to n e o f this artic le is (as o p p o se d to G alilei’s to n e tow ards Z arlino ) m ild a n d resp ectfu l;

B ra u n a n d S c h im p e r a re given a m p le c r e d it fo r th e ir re s e a rc h , a n d th e o p p o sitio n again st th e n o tio n o f orthostichy is very carefully p re se n te d . B rau n 13 C.F. N au m a n n , Uber den Q uincunx als Grundgesetz der Blattstellung vieler Pflanzen,

D re sd en /L eip z ig 1845, Vorwort.

14 A ttached to th e G erm an translation o f th e ir work: L. & A. Bravais, Uber die geometrische Anordnung der Blätter und der Blüthenstände, B reslau 1839.

is less attentive in his reply to th e b ro th e rs in a la te r b o o k .15 In o r d e r to cou n- te rd ic t th e F re n c h criticism , B rau n tries to fin d a th e o re tic a l p e g fro m an a re a w h ere ra tio n a l o rd e r h a d co m e to b e u n d e rs to o d a n d g en e rally a c cep t­

ed: crystallography. As this h a p p e n e d to b e A ugu ste B ravais’s field o f e x p e r­

tise, a n d as h e h a d even b ee n o n e o f th e p io n e e rs in estab lishing w hich class­

es o f crystals w ere m o rp h o lo g ically possible, B rau n seem s to b e a t his o p p o ­ n e n t a t his own gam e w hen h e claims th a t w iping o u t th e d ifferen ces betw een th e several ratio n al divergences w ould a m o u n t to saying th a t all crystal form s a re n o t really d iffe re n t b ec au se they all have th e s p h e re as th e ir lim it.16

T h is a rg u m e n t so u n d s stro n g e r th a n it is. Crystal fo rm s a re d iffe re n t fo r co n structive reasons; as o p p o se d to phyllotaxis, each specific fo rm is th e re su lt o f a d iffe re n t chem ical bu ild -u p th a t is discretely estab lish ed fro m th e b eg in n in g . W hatev er possibilities th e re are, th e s p h e re is n o t a m o n g th em . B ut it is an ex c e lle n t illu stra tio n o f B ra u n ’s way o f th in k in g . H e w ants to see his co v erin g law as a regulative p rin c ip le , n o t as a g e n e ra lisa tio n o f e m p iri­

cal data. T ra n s c e n d e n t u n ity m u st a p p e a r to th e senses as p h e n o m e n a l vari­

ety. T h e re alm o f tru th is n o t to be fo u n d in ex p e rie n c e , b u t in th e m in d:

»All tru th is m en tal« , says B raun ; »all facts b e c o m e re c o g n iz e d tru th s only w h en we can m en tally c o n stru c t th e m « .17

It is alm ost to u c h in g to re a d how N ees von E senbeck, th e a u th o r o f the in tro d u c tio n to th e G e rm an tran slatio n o f th e Bravais w ritings, tries to u n ite th e c o n trib u tio n o f b o th p artie s in o n e e n c o m p a ssin g re c o n c ilia tio n : hav­

in g m a d e clea r th a t it was his co m p a trio ts B rau n a n d S c h im p e r w ho led th e way a n d w ho to o k ca re o f th e essential discoveries, h e c o m p lim e n ts th e B ra­

vais b ro th e rs fo r th e ir m a th e m a tic a l fin e -tu n in g o f th e issue. T h e discovery o f th e »essentially irra tio n a l p ro p o rtio n « ( das wesentlich irrationale Verhältniss) involved in th e div ergences, leads in his eye to th e »ideal infin ity o f th e fu n ­ d a m e n ta l spiral« (die ideale Unendlichkeit der Grundwendet). A n d h e c o n tin ­ ues: »bo th th ese sig n ifican t results a re re d e e m in g fe atu res n o t o n ly fo r th e m e ta m o rp h o sis o f plan ts, b u t in d e e d fo r th e p h ilo so p h ic a l c o n te m p la tio n o f th e o rg a n iz e d w orld. It confirm s th e conviction th a t even th e originally ra tio n a l a rra n g e m e n ts o f leaves are su b je c te d to th e fu n d a m e n ta l law o f ir­

ra tio n a l [phyllotaxis], a n d a re re c o g n iz e d as m e re m u ltip les o f th e m « .18 15 A. B raun, B etrachtungen über die E rscheinung der Verjüngung in der N a tu r; Leipzig 1851.

16 Betrachtungen, p. 126.

17 A. Braun, »Dr. Carl Schim per’s »Vorträge ü b e r die Möglichkeit eines wissenschaftlichen Verständnisses d e r Blattstellung«, in Flora, Jg. 1 8 ,1. Band, 1835, p. 146.

18 T h ere seem s to be a word lacking in the G erm an text; maybe the dash after irrationalen in th e m anuscript was m ean t to re p e a t B lattstellungen: »diese b e id e n b ed eu tsam en R esultate sind L ich tp u n cte nich t allein fü r die P flanzenm etam orphose, so n d e rn fü r die p h ilosophische B e trach tu n g d e r o rganisirten Welt ü b e rh a u p t. M an sie h t m it

T his is a su rp risin g p o in t o f view. It co m b in es th e m a th e m a tic a l c o n ­ clu sio n o f th e Bravais b ro th e rs c o n c e rn in g phyllotaxis w ith B ra u n ’s p h ilo ­ so p h ical idealism c o n c e rn in g th e fu n d a m e n ta l o rd e r th a t prevails in n a tu re

— a n d yet m an ag es to sq u eeze in th e id ea th a t these a rra n g e m e n ts a re »orig­

inally ratio n al« .

It is n o t very difficult to r o u n d o ff em p irica l d a ta c o n c e rn in g m u sical in tervals o r b o tan ica l p rim o rd ia in su ch a way th a t th e ratio n a lity h y p o th e ­ sis is c o n firm e d . B oth to n e scales a n d co n e scales c o m e very close in d e e d . B u t this ra tio n a lity com es a b o u t as a re su lt o f h u m a n ev alu atio n. W h e th e r, in th e e n d , n a tu re do es o r d o e sn « t show ra tio n a l o rd e r, d e p e n d s - n o t o n th e n a tu re o f n a tu re , b u t o n th e n a tu re o f o u r c o n c e p tio n o f n a tu ra l o rd e r.

verstärkter U eberzeugung, wie selbst die u rsprünglich rationalen B lattstellungen d e r P flanzen sich dem G rundgesetze d e r irrationalen - u n te ro rd n e n , u n d als blosse V ielfache d erse lb en e rk a n n t w erden (....).« L. 8c A. Bravais, Über die geometrische Anordnung der Blätter und der Blüthenstände, Breslau 1839, pp. V /V I.