Albert van der Schoot
R ational Order in Tone Scales and Cone Scales
T h e b elief th at n a tu re m ust be co n sid ered as a stan d ard from w hich a rt
can derive its guidelines (natura artis magistra) was firmly established d u rin g
m any centuries. N o t so firm w ere the reasons why a rt should ap p re n tic e itself
to n atu re. T h e e ig h tee n th century saw the transition from a neoclassical con­
cep tio n o f n a tu re as bein g regularly o rd e red , an d th ere fo re an exam ple to
m an k in d (as in P o p e ’s Nature methodized) to the R om antic id ea o f m an being
overw helm ed by n atu re (following B urke’s delightful sublimity). By show ing two
controversies in very d ifferent 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 ore practical, em pirical attitude.
In his Istituzioni Armoniche o f 1558, Italian m usical th eo rist G iuseppe
Z arlino p ro p o se d to consider 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 speaking, this c o rre c tio n o n
P y th ag o rean th in k in g was lo n g overdue. T h ird s a n d sixths h a d gradually
com e to be ac cep ted as h arm o n ic shelters since the earliest form s o f polyph­
ony cam e in to existence. B ut n o t b efo re Z arlino did the m ajo r th ird acquire
th e p restigious p o sitio n o f b ein g o n e o f th e co rn ersto n es o f th e h a rm o n ic
fram ew ork.
Z a r lin o ’s c o r re c tio n m ark s th e e n d o f th e p r e d o m in a n c e o f th e
P y th ag o rean tetraktys as a th eo re tic al basis for harm ony: th e tetraktys allows
only those intervals as c o n so n a n t w hose ratios can b e expressed by th e first
fo u r n u m b ers.1Z arlino introduces a new co n cep t in music theory: th e senario,
im plying th a t six ra th e r th an fo u r is th e lim it fo r the ratio s th a t b u ild u p
c o n so n a n t intervals. E n te r the m ajor th ird ( 5 : 4 ) , the m in o r th ird (6 : 5)
a n d th e ir co u n te rp a rts, th e m in o r sixth (8 : 5, w here 8 is c o n sid e re d the
twofold o f 4) a n d th e m ajor sixth ( 5 :3 ) . T h e V enetian m aestro believed th at
ju st intonation co u ld be achieved by basing all intervals in a to n e scale o n the
fifth a n d th e m ajo r th ird . T h a t leads to th e only type o f in to n a tio n w hich
Z arlino is w illing to c o n sid er as natural .2 In o th e r words: Z arlino 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).
Filozofski vestnik, X X (2/1999 - X IV ICA Supplement), pp. 91-100
91
Albert van der Schoot
m u ch overthroiu th e P ythagorean way o f th in k in g in term s o f ra tio n a l o rd e r
b ased o n n u m eric al ontology, b u t ra th e r saved it by e x te n d in g the 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 attack o n the ontological basis o f this type o f th in k in g was left to
V incenzo Galilei, fa th e r o f the fam ous a stro n o m e r b u t also a p u p il o f Zarlin o ’s. G alilei does n o t ac cep t his m a ste r’s g u id elin e o f th e senario. In p artic­
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 com e to be established
by h u m a n convention. Exact rational p ro p o rtio n s (in the m athem adcal sense
o f b ein g expressible as a ratio o f integers) have n o special m e a n in g h ere.
T h e re is n o p rin cip al differen ce, in this respect, betw een the intervals o f
m usic a n d th e w ords o f a »natural« language.
G alilei’s critical attitu d e towards his m aster’s authority is fundam ental.
T h e id ea th at a c o n so n a n t interval sh o u ld be an ything else b u t a ra tio n al
n u m b e r w ould have b ee n considered absurd d u rin g
th e m ajor p a rt o f E u ro p e an history. T h e foun d atio n
o f th at th o u g h t goes back at least as far as P lato’s Timaeus, w here th e very ratios o f the tetraktys are consti­
tutive fo r th e created o rd e r o f the cosmos. G alilei’s
criticism clearly reflects m o re th an ju s t a m usicological com m ent; it heralds the paradigm shift with which
th e n am e o f his son will forever be linked. B ut before
g o in g d e e p e r in to the h ea te d debate betw een m aster
a n d p u pil, we shall first take a look at a conflict in a
com pletely different setting and time - no t ab o u t a hu­
m an p ro d u c t, b u t co n c ern in g the p ro d u c tio n o f n a­
tu re herself.
Towards th e m iddle o f the n in e te e n th century,
a b otanical d eb ate flared u p ab o u t the way in which
n a tu re accom odates certain p rim o rd ia a ro u n d a cen­
tre - like leaves a ro u n d a stem , scales o n a p in e cone,
sunflow er seeds o n a flower head, etc. T hough we use
to w o n d er ab o u t th e am azing spiral structures w hich
these plants show, we o ften do n o t realize th at these
spirals w ere n o t th ere 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 at can even be traced subsequently. T he
spirals we see are n o m o re th an an ep ip h e n o m e n o n
o f a spiral we d o n « t see, b u t which we can obtain by
92
Rational Order in Tone Scales and Cone Scales
co n n ectin g th e scales in the o rd e r in w hich they p o p p e d up. We shall call this
the fundamental spiral (in the picture: 1-2-3-4 etc.), whereas the contiguous par­
allels as they b eco m e visible are called parastichies (in the picture: 6-14-22-30,
o r 19-27-35-43 etc.).
By 1830, G erm an b o tan ist A lex an d er B raun h a d th e b rillia n t id ea to
use th e p re cise o rd e r o f th ese scales fo r th e classification o f c o n ife ro u s
p lan ts.3 C lassification b ein g a favourite pastim e fo r botanists, th e subtle dif­
feren ces betw een the im p la n ta tio n o f the scales in the d iffe re n t species o f
co n ifero u s p lan ts seem ed to offer an ideal h a n d le to com e to grips w ith th e
d ifferences betw een them , a n d to label these differences. In o rd e r to w ork
o u t these labels, B raun in tro d u c e d th e n o tio n o f divergence in b o tan ica l p ar­
lance. By n o ta tin g such a divergence as, say, 8J21 (as in th e case o f th e p in e
co n e o n th e p ic tu re ), B raun m e a n t th a t 21 scales w ere fo u n d w hen th e fu n ­
d am en ta l spiral h a d ro u n d e d the c o n e exactly 8 tim es.4
T h e p resu p p o sitio n o f this p ro jec t is th a t th e po sitio n o f (in this case)
th e 2 2 n d scale is exactly above the first. B ra u n ’s co n c e p tio n im plies that,
a p a rt from th e parastichies, each co n e also shows parallel orthostichies (in the
p ictu re: 1-22-43-64, o r 9-30-51-72 etc.). B raun does 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 com e full circle,
so th at the ra tio o f the n u m b e r o f scales a n d the n u m b e r o f ro ta tio n s can
b e exp ressed as an exact ra tio n al num ber.
No such th in g - say two F ren c h scientists who started investigating co­
n ifero u s p lan ts a ro u n d the sam e tim e as B raun did. A uguste a n d Louis Bra­
vais observe th e sam e cones as B raun, b u t see so m eth in g e n tire ly d ifferen t.
In particular, they d o n o t see a series o f distinctly d iffe re n t ratios in th e di­
v ergences o f th e plants. B ra u n ’s d iffe ren tiatio n is b u t an illusion, o r so they
claim . N atu re has fo u n d the o p tim u m angle for th e im p la n ta tio n o f every
n e x t seed o r scale; th at angle ensures th at all the p rim o rd ia have an o p ti­
m um space to grow, a n d it rem ains the sam e at every turn: 137° 30' 28".r>T h at
am o u n ts to a re p e a te d division o f the circle acco rd in g to th e g o ld en section,
w hich is an irra tio n a l m easu re a n d can, fo r th at reason, n ev er lead to the
ra tio n al classification th at B raun p u rsu e d . It is, however, a constant m easu re
- th e only o n e th a t grants eq u al rights to all p rim o rd ia. T h e w hole 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.
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Albert van der Schoot
ism b en efits from this equal division. R ecen t re se 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 does n o t n e e d to involve gen etical
o r teleological p rin cip les to fin d th a t the flow er h e a d o f a sunflow er is di­
v ided again a n d ag ain, by each new p rim o rd iu m , acco rd in g to th e g o ld en
section.
B oth controversies, th e o n e in the R enaissance ab o u t the alleged ra­
tionality o f to n e scales a n d the o n e in the n in e te e n th cen tu ry a b o u t th e al­
leg ed ratio n ality o f co n e scales, fin d th e ir o rig in in o p p o sin g co n c ep tio n s
o f th e value o f ra tio n a l o rd e r in n atu re . O f course, b o th pairs o f o p p o n e n ts
have a lo t in com m on, d u e to the p reconceptions th at even o p p o n en ts w ould
sh are in a c e rtain age. B oth Z arlino a n d G alilei freq u en tly call o n »the a n ­
cients« to substantiate th eir own p o in t o f view; b o th believe th at the ancients
h a d set a n ex am p le, n o t so m u ch by th eir hig h stan d ard o f cu ltu ral devel­
o p m e n t, b u t by th e ir b ein g closer to nature, th a t is, by th e ir b e tte r u n d e r­
stan d in g o f natural order.
Z arlino believes th a t M o th er N atu re restricts h e rse lf to a w ell-consid­
e re d d o se o f p e rfe c tio n by d iffe ren tiatin g betw een th e individuals th a t b e ­
lo n g to th e sam e species ra th e r th an ju s t c lo n in g th e ideal archetype again
a n d again. H e praises th e an cien ts fo r tran sp o sin g th a t p rin cip le to m usic,
w h ere re p e titio n o f id en tical co n so n a n t intervals is to b e avoided:
»T hus they h e ld it as tru e th a t w henever o n e h ad arrived at 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 ten d s, a n d in o rd e r n o t to give th e e a r to o m u ch o f this p e rfe c tio n
they 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 ex cellence o f this ad m irab le a n d useful a d m o n itio n are
c o n firm e d by th e o p eratio n s o f N atu re, fo r in b rin g in g in to b ein g th e in d i­
viduals o f each species she m akes them sim ilar to o n e a n o th e r in g en e ral,
yet d iffe re n t in som e particular, a difference o r variety affording m u ch plea­
su re to o u r senses. T his ad m irab le o rd e r the co m p o ser o u g h t to im itate, for
th e m o re his o p eratio n s resem b le those o f o u r g re at m other, th e m o re h e
will be esteem ed. A nd to this course the n u m bers an d p ro p o rd o n s invite him ,
fo r in th e ir n a tu ra l o rd 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 ediately
V incenzo Galilei is involved in a d ifferen t battle. H e is a m em b er o f the
F lo ren tin e Camerata, the think-tank o f h u m an ist scholars an d n o b lem en w ho
paved th e way fo r an en tirely new form o f art, a spectacle th at w ould con6 S. D ouady & Y. Couder, »Phyllotaxis as a Physical Self-Organized Growth Pattern«, 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. II - The Renaissance, New Y ork/L ondon 1965, pp. 4 4 /5 .
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Rational Order in Tone Scales and Cone Scales
q u e r E u ro p e a n stages in the sev en teen th century: opera. O p e ra is typically
an a rt form th a t d id n o t re su lt directly from any d ev e lo p m e n t in m usical
p ractice, b u t was p re p a re d o n the draw ing bo ard . T h e m ain im p u lse cam e
from th e F lo re n tin e resistance against co n tem p o rary (» m o d ern « ) polyph­
ony. G alilei’s Dialogo della mušica antica e della modernd1is a n a rd e n t p lea fo r
a new type o f m usic (»postm odern«, so to speak), th at w ould d o ju stic e to
th e n a tu ra l exp ression o f h u m a n affections - a task w hich p o ly p h o n ic m u ­
sic, with its in tricate stru c tu re o f sim u ltan eo u s m elodies giving voice to sev­
eral texts at th e sam e m o m en 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 alilei’s c o n tem p o raries is an insult 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 antiquity is p u t forw ard in his w ritings as a n in sp ir­
in g guideline.
Intrinsically, differences o f o p in io n betw een Zarlino a n d Galilei a re n o t
as g re a t as th e ir p erso n al feu d m ig h t suggest. Galilei w ould have n o tro u b le
w ith th e q u o tatio n given above, re g a rd in g th e d esired variety in intervals,
a n d Z arlino w ould w holeheartedly agree with the C am erata’s p re fe re n c e for
w ords above m elody w hen p u ttin g text to m usic. T hose w ere in fact th e ce 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 them . B ut u n ­
fortunately, b o th m en w ere driven by ».... the d esp erate wish to c o n tra d ic t
each o th er« .<JT h e advantage o f this fo r later scholars is th a t th e ir d iffe re n t
attitu d es tow ards th e im p o rtan ce o f ra tio n a l o rd e r received m u ch e m p h a ­
sis, a n d thus clearly expose th e differen ce betw een Z arlin o ’s n eo p lato n ism
a n d G alilei’s m o re em pirical ap p ro ach .
E m pirical research , as it becam e to be p ractised by th e investigative
R enaissance m inds, d id n o t autom 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 ade a n am e for h im self in th e history o f m usic th eo ry
by c o rre c tin g w h a t th e M iddle Ages h a d believed was a n o b serv atio n by
Pythagoras him self: the discovery o f the p ro p o rtio n a l relatio n sh ip s betw een
th e w eight o f th e h am m ers used by th e blacksm ith, a n d th e p itch es o f the
so u n d s they p ro d u c e d . Every m edieval m usic th eo rist knew th a t if a certain
p itch was p ro d u c e d by tying a w eight to a string, th e octave o f th a t pitch
w ould be p ro d u c e d by tying the do u b le w eight to th e sam e string, a n d a fifth
with th e h e lp o f a w eight o n e a n d a h a lf tim es th e original, etc. In o th e r
words: th ese ratios w ere supposed to b e th e sim ple inversion o f th e (m o re
easily m easurable) ratios for string lengths p ro d u c in g the sam e intervals. N ot
so, says Galilei: to p ro d u c e those intervals by tension, the w eights w ould have
to b e in squared inverse p ro p o rtio n to th e lengths o f th e strings. T h e ir re la­
tio n sh ip s to th e p e rfe c t co n so n a n t intervals are still perfectly expressible as
8 Florence, 1581.
‘J D.P. Walker, Studies in Musical Science in the Late Renaissance, L eiden 1978, p. 16.
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Albert van der Schoot
ratio s o f w hole n u m b ers, b u t n o t anym ore in th e traditionally constitutive
n u m b e rs o f th e P y thagorean 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
trying out.
Z arlino, as we saw, did n o t stick eith er to the tetraktys to express th e ratios
o f th e im p e rfe c t co n so n an ces, 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 er. Why sh o u ld the senario ra th e r th a n th e tetraktys be c o n ­
sid ered as the basis for o u r h arm o n ic u nd erstan d in g ? As if we could n o t have
guessed:
- G od c reated the w orld in six days
- six signs o f th e zodiac are always above the e a rth , th e o th e r six are
invisible
- th ere are six »planets« (to Z arlino’s know ledge: S aturn ju p i te r , Mars,
Venus, M ercury, a n d th e m o o n )
- th e re are six d irec tio n s (up, dow n, ah e a d , b e h in d , left, a n d right;
Z arlino calls o n P lato to testify to this spatial insight)
- th e n u m b e r 6 is traditionally hailed as th e first »perfect num ber«; th at
is, it equals th e sum o f its dividends 1, 2 a n d 3; m oreover, it is th e ir p ro d u c t
in m usic, th ere 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 se six will su ffic e to
sh o w th e g a p t h a t e x ­
ten d s b etw een th e m e n ­
tal w orld o f Z arlino a n d
th a t o f his pupil. G alilei,
w ho was an early p io n e e r
o f equal temperament, did
n o t feel an ything was lost
by giving u p th e p e rfe c t­
ly ra tio n ally o rd e re d in ­
tervals. 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 ag in e ju s t in to n a tio n
10 See C.V. Palisca, Humanism in Italian Renaissance Musical Thought, New H av en /L o n d o n
1985, p. 248.
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Rational Order in Tone Scales and Cone Scales
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 cen tre.
T h e controversy betw een A lex a n d er B raun a n d the Bravais b ro th e rs is
situated in a d ifferen t age, against the b ackground o f different scientific strat­
egies. E x p erim en tal verification h a d b eco m e p a rt a n d parcel o f re g u la r sci­
entific b eh a v io u r by the tim e B raun developed his theory, a n d h e h im self
was n o excep tio n : th o u san d s o f p in e cones w ere collected by h im a n d his
colleague, C arl Schim per, a n d m eticulously so rted o u t a n d classified. A nd
yet, B raun is steered by a n o th e r drive than collector’s m ania o r labelling n e u ­
rosis: h e w ants to unravel the h id d e n p rin cip le b e h in d n a tu ra l o rd e r as this
beco m es 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 hat B rau n finds is fascinating, b u t m u ch m o re fascinating is to know
w hat h e is lo o k in g for. B raun was, in his own words, chasing th e »joyful p re ­
su m p tio n o f a law fo u n d e d deeply in the 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 ex a ct d e ­
scrip tio n a n d classification o f the o u te r a p p e a ra n c e o f th e cones 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 raun believed h e was follow ing n a ­
tu re herself. A n d w h en h e fo u n d th e constitutive spiral, th e row th a t d ictat­
ed th e po sitio n o f all the scales, h e w elcom ed this »m iraculous re g u larity o f
order« (wunderbare Gesetzmässigkeit der
Anordnung) w ith an alm ost religious
respect: »In this last, O n e Row, dawn­
ing u p o n o u r expectation, we beh o ld
th e tru e goal o f o u r h o p e, the 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 drawing, 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 rem in iscent 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 at 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 er Erw artung vorschwebenden letzten, Einen Reihe erblicken wir
das wahre Ziel unserer Hoffnung, den Einen G rund d er Blattstellung, a u f dem alle
Vielheit u nd Vielartigkeit der Reihen b eru h en muss.« Vergleichende Untersuchung, p. 22.
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Albert van der Schoot
T h e re is a n in trig u in g ten sio n betw een unity a n d variety in B ra u n ’s
c o n c e p tio n o f n a tu ra l order, co m p arab le to the way Z arlino deals w ith the
p erfec tio n o f co n so nants a n d th eir necessary differentiation in m usical com ­
po sition. T h e u nity th a t is firm ly established in the 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 ren c­
es - differen ces by w hich th e several species o f cones can be d istin g u ish ed
a n d labelled. B ra u n ’s aim is a classification in th e lin e o f L innaeus, arrived
a t by m ean s o f em p irical observation, b u t his regulative co n c e p tio n is th a t
o f an overall ra tio n al order. In o th e r words: B raun treats divergences as if
they w ere m usical intervals according to a traditional system o f tem p eram en t,
a n d h e d o es so o n th e basis o f a deeply ro o te d in n e r conviction th a t this is
how n a tu re behaves. B ra u n ’s phyllotaxis reflects an o rd e r o f ju st intonation.
It is to this p re c o n c e p tio n th at the Bravais b ro th e rs oppose. T h e re is
n o d iscrete classification o f d iffe ren 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 lan t, th e choice betw een, say,
8j21 o r 13j34 o ften seem s quite arbitrary. N o n e o f B ra u n ’s alleged o b ser­
vations is as precise as the ex a ctitu d e o f the ra tio n a l m easu re suggests. T h e
b ro th e rs carefully justify this sta te m e n t with a n u m b e r o f illustrations. W hat
they o b ject to is in fact n o t so m uch the validity o f B ra u n ’s equally carefu l
observations, b u t th e very status o f th e startin g p o in t w hich led these o b ser­
vations to result in the conclusions th at B raun presen ted . T h a t starting p o in t
is th e c o n c e p t o f orthostichy, w hich, to co n tin u e th e m e ta p h o r I have ju s t in­
tro d u c e d , in B ra u n ’s system o f ju s t botanical 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 fere n ce for all the o th e r intervals. T h e stro n g im ­
p ac t o f B ra u n ’s co n c ep tio n becom es clear w hen we re a d th a t C arl F ried rich
N a u m a n n c o n sid ered th e orthostichy as »the real essence« ( das eigentliche
Wesen) a n d p arastichies as »a m ere p h e n o m e n o n o f phyllotaxis« (ein blosses
Phänomen der Blattstellung) . 1S T h e altern ativ e w hich th e Bravais b ro th e rs
p re s e n t com es dow n to g ra n tin g identical rights to p rim o rd ia in th e sam e
way to n es have id en tical rights in eq u al te m p e ra tu re - with th e proviso th a t
in th e case o f th e plants, this equality is g ra n te d by n atu re .
A p a rt from carefully ex p lain in g th e ir own theory, the Bravais b ro th e rs
m ake a stan d against B ra u n ’s position in a separate article.14T h e to n e o f this
article is (as o p p o sed to G alilei’s tone towards Z arlino) m ild a n d respectful;
B rau n a n d S c h im p e r are given am p le c re d it fo r th e ir re se a rc h , a n d th e
o p p o sitio n against th e n o tio n o f orthostichy is very carefully p re sen ted . B raun
13 C.F. N aum ann, Uber den Quincunx als Grundgesetz der Blattstellung vieler Pflanzen,
D resden/L eipzig 1845, Vorwort.
14 A ttached to the G erm an translation o f their work: L. & A. Bravais, Uber die geometrische
Anordnung der Blätter und der Blüthenstände, Breslau 1839.
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Rational Order in Tone Scales and Cone Scales
is less attentive in his reply to the b ro th ers in a later book.15 In o rd e r to counte rd ic t th e F ren ch criticism , B raun tries to find a th eo re tic al p e g fro m an
a rea w here ratio n al o rd e r h ad com e to be u n d e rsto o d a n d generally accep t­
ed: crystallography. As this h a p p e n e d to b e A uguste B ravais’s field o f ex p e r­
tise, a n d as h e h a d even b een o n e o f th e p io n eers in establishing w hich class­
es o f crystals w ere m orphologically possible, B raun seem s to b e a t his o p p o ­
n e n t at his own gam e w hen h e claims th at wiping o u t the differences betw een
th e several rational divergences w ould a m o u n t to saying th a t all crystal form s
are n o t really d iffe re n t because they all have the sp h e re as th e ir lim it.16
T his a rg u m e n t sounds stro n g e r th a n it is. Crystal form s a re d iffe re n t
fo r constructive reasons; as o p p o sed to phyllotaxis, each specific fo rm is th e
re su lt o f a d iffe ren t chem ical build-up th a t is discretely established from the
b eg in n in g . W hatever possibilities th e re are, the sp h e re is n o t a m o n g them .
B ut it is an ex cellen t illustration o f B ra u n ’s way o f thinking. H e w ants to see
his covering law as a regulative p rin cip le, n o t as a g en e ralisatio n o f em p iri­
cal data. T ra n sc e n d e n t unity m ust a p p e a r to the senses as p h e n o m e n a l vari­
ety. T h e realm o f tru th is n o t to be fo u n d in ex p e rien ce , b u t in th e m ind:
»All tru th is m ental«, says B raun; »all facts b eco m e reco g n ized tru th s only
w hen we can m entally co n stru c t th em « .17
It is alm ost to u ch in g to re a d how N ees von Esenbeck, th e a u th o r o f the
in tro d u ctio n to th e G erm an translation o f th e Bravais writings, tries to u n ite
th e co n trib u tio n o f b o th parties in o n e en com passing reco n ciliatio n : hav­
in g m ad e clear th a t it was his co m p atrio ts B raun a n d S ch im p er w ho led the
way a n d w ho took care o f the essential discoveries, h e co m p lim en ts th e B ra­
vais b ro th e rs fo r th e ir m ath em atical fin e-tu n in g o f the issue. T h e discovery
o f th e »essentially irrational p ro p o rtio n « ( das wesentlich irrationale Verhältniss)
involved in th e divergences, leads in his eye to th e »ideal infinity o f th e fu n ­
d a m e n ta l spiral« (die ideale Unendlichkeit der Grundwendet). A nd h e c o n tin ­
ues: »both th ese significant results are re d e e m in g featu res n o t only fo r the
m etam o rp h o sis o f plants, b u t in d e e d fo r th e p h ilo so p h ical co n te m p la tio n
o f th e o rg an ized 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 jected 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 are reco g n ized as m ere m ultiples o f th em « .18
15 A. Braun, Betrachtungen über die Erscheinung der Verjüngung in der N a tu r ; Leipzig 1851.
Betrachtungen, p. 126.
17 A. Braun, »Dr. Carl Schim per’s »Vorträge üb er die Möglichkeit eines wissenschaftlichen
Verständnisses d er Blattstellung«, in Flora, Jg. 1 8 ,1. Band, 1835, p. 146.
18 T h ere seems to be a word lacking in the G erm an text; maybe the dash after irrationalen
in th e m anuscript was m eant to rep eat Blattstellungen : »diese b eid en b edeutsam en
Resultate sind L ichtpuncte nicht allein fü r die P flanzenm etam orphose, so n d ern fü r
die philosophische B etrachtung d e r organisirten Welt ü b erh au p t. M an sieh t mit
16
99
Albert van der School
T his is a su rp risin g p o in t o f view. It co m b in es th e m ath em atical c o n ­
clusion o f th e Bravais b ro th e rs c o n c e rn in g phyllotaxis with B ra u n ’s p h ilo ­
sop h ical idealism co n c e rn in g the fu n d a m e n tal o rd e r th at prevails in n a tu re
—a n d yet m anages to squeeze in the idea th a t these a rra n g e m e n ts a re »orig­
inally rational«.
It is n o t very difficult to ro u n d o ff em pirical d ata c o n c e rn in g m usical
intervals o r b o tan ical p rim o rd ia in such a way th at th e rationality h y p o th e ­
sis is co n firm ed . B oth to n e scales a n d co n e scales com e very close in d eed .
B u t this ratio n ality com es a b o u t as a re su lt o f h u m a n evaluation. W h eth er,
in th e e n d , n a tu re does o r do esn « t show ra tio n a l ord er, 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 the n a tu re o f o u r c o n c ep tio n o f n a tu ra l order.
verstärkter U eberzeugung, wie selbst die ursprünglich rationalen B lattstellungen d er
Pflanzen sich dem G rundgesetze d e r irrationalen - u n te ro rd n en , u n d als blosse
V ielfache derselben erk 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.