Albert van der Schoot 
Rational Order in Tone Scales and Cone Scales 
The belief that nature must be considered as a standard from which art 
can derive its guidelines (natura artis magistra) was firmly established during 
many centuries. Not so firm were the reasons why art should apprentice itself 
to nature. The eighteenth century saw the transition from a neoclassical con-
ception of nature as being regularly ordered, and therefore an example to 
mankind (as in Pope's Nature methodized) to the Romantic idea of man being 
overwhelmed by nature (following Burke's delightful sublimity). By showing two 
controversies in very different fields I intend to show how, in a more subtle 
way, also in other periods the idea of an intrinsically rational order in nature comes 
into conflict with a more practical, empirical attitude. 
In his Istituzioni Armoniche of 1558, Italian musical theorist Giuseppe 
Zarlino proposed to consider not only octave, fifth and fourth, but also third 
and sixth as consonant intervals. Historically speaking, this correction on 
Pythagorean thinking was long overdue. Thirds and sixths had gradually 
come to be accepted as harmonic shelters since the earliest forms of polyph-
ony came into existence. But not before Zarlino did the major third acquire 
the prestigious position of being one of the cornerstones of the harmonic 
framework. 
Zarl ino 's cor rec t ion marks the e n d of the p r e d o m i n a n c e of the 
Pythagorean tetraktys as a theoretical basis for harmony: the tetraktys allows 
only those intervals as consonant whose ratios can be expressed by the first 
four numbers.1 Zarlino introduces a new concept in music theory: the senario, 
implying that six ra ther than four is the limit for the ratios that build up 
consonant intervals. Enter the major third ( 5 : 4 ) , the minor third (6 : 5) 
and their counterparts, the minor sixth (8 : 5, where 8 is considered the 
twofold of 4) and the major sixth (5 :3 ) . The Venetian maestro believed that 
just intonation could be achieved by basing all intervals in a tone scale on the 
fifth and the major third. That leads to the only type of intonation which 
Zarlino is willing to consider as natural.2 In other words: Zarlino did not so 
1 That is: the octave (2 : 1), the fifth (3 : 2) and the fourth ( 4 : 3 ) - and, trivially, the 
prime (1 : 1). 
2 fust and natural are still in use as synonyms for this particular intonation (in German: 
reine or natürliche Stimmung). 
Filozofski vestnik, XX (2/1999 - XIVI CA Supplement), pp. 91-130 121 
Albert van der Schoot 
much overthrow the Pythagorean way of thinking in terms of rational order 
based on numerical ontology, but rather saved it by extending the range of 
fundamenta l numbers to six. 
The attack on the ontological basis of this type of thinking was left to 
Vincenzo Galilei, father of the famous astronomer but also a pupil of Zarli-
no's. Galilei does not accept his master's guideline of the senario. In partic-
ular, he attacks the status of fifth and third as »natural« intervals. No such 
thing- says Galilei: all intervals, all tone scales have come to be established 
by human convention. Exact rational proportions (in the mathematical sense 
of being expressible as a ratio of integers) have no special meaning here. 
There is no principal difference, in this respect, between the intervals of 
music and the words of a »natural« language. 
Galilei's critical attitude towards his master's authority is fundamental . 
The idea that a consonant interval should be anything else but a rational 
number would have been considered absurd during 
the major part of European history. The foundation 
of that thought goes back at least as far as Plato's Ti-
maeus, where the very ratios of the tetraktys are consti-
tutive for the created order of the cosmos. Galilei's 
criticism clearly reflects more than just a musicologi-
cal comment; it heralds the paradigm shift with which 
the name of his son will forever be linked. But before 
going deeper into the heated debate between master 
and pupil, we shall first take a look at a conflict in a 
completely different setting and time - not about a hu-
man product, but concerning the production of na-
ture herself. 
Towards the middle of the nineteenth century, 
a botanical debate flared up about the way in which 
nature accomodates certain primordia around a cen-
tre - like leaves around a stem, scales on a pine cone, 
sunflower seeds on a flower head, etc. Though we use 
to wonder about the amazing spiral structures which 
these plants show, we often do not realize that these 
spirals were not there in the first place. They come 
into existence step by step; in fact, the birth certificates 
of all the sunflower seeds are issued one by one, in a 
strict order that can even be traced subsequently. The 
spirals we see are no more than an epiphenomenon 
of a spiral we don«t see, but which we can obtain by 
92 
Rational Order in Tone Scales and Cone Scales 
connecting the scales in the order in which they popped up. We shall call this 
the fundamental spiral (in the picture: 1-2-3-4 etc.), whereas the contiguous par-
allels as they become visible are called parastichies (in the picture: 6-14-22-30, 
or 19-27-35-43 etc.). 
By 1830, German botanist Alexander Braun had the brilliant idea to 
use the precise order of these scales for the classification of coniferous 
plants.3 Classification being a favourite pastime for botanists, the subtle dif-
ferences between the implantation of the scales in the different species of 
coniferous plants seemed to offer an ideal handle to come to grips with the 
differences between them, and to label these differences. In order to work 
out these labels, Braun introduced the notion of divergence in botanical par-
lance. By notating such a divergence as, say, 8j"21 (as in the case of the pine 
cone on the picture), Braun meant that 21 scales were found when the fun-
damental spiral had rounded the cone exactly 8 times.4 
The presupposition of this project is that the position of (in this case) 
the 22nd scale is exactly above the first. B rauns conception implies that, 
apart from the parastichies, each cone also shows parallel orthostichies (in the 
picture: 1-22-43-64, or 9-30-51-72 etc.). Braun does indeed believe that af-
ter a natural number of scales the fundamenta l spiral has come full circle, 
so that the ratio of the number of scales and the number of rotations can 
be expressed as an exact rational number. 
No such thing- say two French scientists who started investigating co-
niferous plants a round the same time as Braun did. Auguste and Louis Bra-
vais observe the same cones as Braun, but see something entirely different. 
In particular, they do not see a series of distinctly different ratios in the di-
vergences of the plants. B rauns differentiation is but an illusion, or so they 
claim. Nature has found the opt imum angle for the implantation of every 
next seed or scale; that angle ensures that all the primordia have an opti-
mum space to grow, and it remains the same at every turn: 137° 30' 28".r> That 
amounts to a repeated division of the circle according to the golden section, 
which is an irrational measure and can, for that reason, never lead to the 
rational classification that Braun pursued. It is, however, a constant measure 
- the only one that grants equal rights to all primordia. The whole organ-
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. 
93 
Albert van der Schoot 
ism benefits f rom this equal division. Recent research0 has shown that this 
is, in fact, the way nature behaves; one does not need to involve genetical 
or teleological principles to find that the flower head of a sunflower is di-
vided again and again, by each new primordium, according to the golden 
section. 
Both controversies, the one in the Renaissance about the alleged ra-
tionality of tone scales and the one in the nineteenth century about the al-
leged rationality of cone scales, find their origin in opposing conceptions 
of the value of rational order in nature. Of course, both pairs of opponents 
have a lot in common, due to the preconceptions that even opponents would 
share in a certain age. Both Zarlino and Galilei frequently call on »the an-
cients« to substantiate their own point of view; both believe that the ancients 
had set an example, no t so much by their high standard of cultural devel-
opment , but by their being closer to nature, that is, by their better under-
standing of natural order. 
Zarlino believes that Mother Nature restricts herself to a well-consid-
ered dose of perfection by differentiating between the individuals that be-
long to the same species rather than just cloning the ideal archetype again 
and again. He praises the ancients for transposing that principle to music, 
where repetition of identical consonant intervals is to be avoided: 
»Thus they held it as true that whenever one had arrived at perfect 
consonance one had attained the end and the perfection toward which 
music tends, and in order not to give the ear too much of this perfect ion 
they did not wish it repeated over and over again. 
The truth and excellence of this admirable and useful admonit ion are 
confirmed by the operations of Nature, for in bringing into being the indi-
viduals of each species she makes them similar to one another in general, 
yet different in some particular, a difference or variety affording much plea-
sure to our senses. This admirable order the composer ought to imitate, for 
the more his operations resemble those of our great mother, the more he 
will be esteemed. And to this course the numbers and proportions invite him, 
for in their natural order one will not find two similar proportions follow-
ing one another immediately ....«' 
Vincenzo Galilei is involved in a different battle. He is a member of the 
Florentine Camerata, the think-tank of humanist scholars and noblemen who 
paved the way for an entirely new form of art, a spectacle that would con-
6 S. Douady &Y. Couder, »Phyllotaxis as a Physical Self-Organized Growth Pattern«, in 
Physical Review Letters, Vol 68, Nr. 13, 1992, pp. 2098-2101. 
7 G. Zarlino, Istituzioni Armoniche, in O. Strunk (ed.), Source readings in Music History, 
Vol. II-The Renaissance, New York/London 1965, pp. 44/5. 
94 
Rational Order in Tone Scales and Cone Scales 
quer European stages in the seventeenth century: opera. Opera is typically 
an art form that did not result directly f rom any development in musical 
practice, but was prepared on the drawing board. The main impulse came 
from the Florentine resistance against contemporary (»modern«) polyph-
ony. Galilei's Dialogo della musica antica e della moderna8 is an ardent plea for 
a new type of music (»postmodern«, so to speak), that would do justice to 
the natural expression of human affections - a task which polyphonic mu-
sic, with its intricate structure of simultaneous melodies giving voice to sev-
eral texts at the same moment , could not possibly fulfil. T h e polyphonic 
music of Galilei's contemporaries is an insult to human nature (so he be-
lieves) , and the music of antiquity is put forward in his writings as an inspir-
ing guideline. 
Intrinsically, differences of opinion between Zarlino and Galilei are not 
as great as their personal feud might suggest. Galilei would have no trouble 
with the quotation given above, regarding the desired variety in intervals, 
and Zarlino would wholeheartedly agree with the Camerata's preference for 
words above melody when putting text to music. Those were in fact the cen-
tral issues of the time, and both authors were well aware of them. But un-
fortunately, both men were driven by ».... the desperate wish to contradict 
each other«.'1 The advantage of this for later scholars is that their different 
attitudes towards the importance of rational order received much empha-
sis, and thus clearly expose the difference between Zarlino's neoplatonism 
and Galilei's more empirical approach. 
Empirical research, as it became to be practised by the investigative 
Renaissance minds, did not automatically imply a repudiation of rational 
proport ion. Galilei made a name for himself in the history of music theory 
by correcting what the Middle Ages had believed was an observation by 
Pythagoras himself: the discovery of the proportional relationships between 
the weight of the hammers used by the blacksmith, and the pitches of the 
sounds they produced. Every medieval music theorist knew that if a certain 
pitch was produced by tying a weight to a string, the octave of that pitch 
would be produced by tying the double weight to the same string, and a fifth 
with the help of a weight one and a half times the original, etc. In other 
words: these ratios were supposed to be the simple inversion of the (more 
easily measurable) ratios for string lengths producing the same intervals. Not 
so, says Galilei: to produce those intervals by tension, the weights would have 
to be in squared inverse proportion to the lengths of the strings. Their rela-
tionships to the perfect consonant intervals are still perfectly expressible as 
8 Florence, 1581. 
9 D.P. Walker, Studies in Musical Science in the Late Renaissance, Leiden 1978, p. 16. 
95 
Albert van der Schoot 
ratios of whole numbers, but not anymore in the traditionally constitutive 
numbers of the Pythagorean tetraktys. 
How did Galilei find this out? Going by his repeated reference to ex-
perimental method (con il mezzo dell«esperienza), we may safely assume: by 
trying out. 
Zarlino, as we saw, did not stick either to the tetraktys to express the ratios 
of the imperfect consonances, but his argumentational back-up is of a to-
tally different order. Why should the senario rather than the tetraktys be con-
sidered as the basis for our harmonic understanding? As if we could not have 
guessed: 
- God created the world in six days 
- six signs of the zodiac are always above the earth, the other six are 
invisible 
- there are six »planets« (to Zarlino's knowledge: Saturn,Jupiter, Mars, 
Venus, Mercury, and the moon) 
- there are six directions (up, down, ahead, behind, left, and right; 
Zarlino calls on Plato to testify to this spatial insight) 
- the number 6 is traditionally hailed as the first »perfect number«; that 
is, it equals the sum of its dividends 1, 2 and 3; moreover, it is their product 
10 See C.V. Palisca, Humanism in Italian Renaissance Musical Thought, New Haven/London 
1985, p. 248. 
- in music, there are 
six »authent ic« a n d six 
»plagal« modes.  
Zarlino gives quite a 
few more reasons,10 bu t 
these six will suff ice to 
show the gap t h a t ex-
tends between the men-
tal world of Zarlino and 
that of his pupil. Galilei, 
who was an early pioneer 
of equal temperament, did 
not feel anything was lost 
by giving up the perfect-
ly rationally o rdered in-
tervals. Zarlino, on the 
o t h e r h a n d , cou ld n o t 
imagine jus t in tonat ion 
96 
Rational Order in Tone Scales and Cone Scales 
in any other way than by the numeri sonori of the senario, as this illustration 
from his book shows: a well-ordered world of musical intervals, with the se-
nario in the centre. 
The controversy between Alexander Braun and the Bravais brothers is 
situated in a different age, against the background of different scientific strat-
egies. Experimental verification had become part and parcel of regular sci-
entific behaviour by the time Braun developed his theory, and he himself 
was no exception: thousands of pine cones were collected by him and his 
colleague, Carl Schimper, and meticulously sorted out and classified. And 
yet, Braun is steered by another drive than collector's mania or labelling neu-
rosis: he wants to unravel the hidden principle behind natural order as this 
becomes visible in the arrangement of leaves, seeds, petals and scales along 
a stem. 
What Braun finds is fascinating, but much more fascinating is to know 
what he is looking for. Braun was, in his own words, chasing the »joyful pre-
sumption of a law founded deeply in the life of the plants« [freudige Ahnung 
eines tief im Leben der Pflanze gegründeten Gesetzes).11 To this end, the exact de-
scription and classification of the outer appearance of the cones was not 
enough. In looking for his hidden law, Braun believed he was following na-
ture herself. And when he found the constitutive spiral, the row that dictat-
ed the position of all the scales, he welcomed this »miraculous regularity of 
order« (wunderbare Gesetzmässigkeit der 
Anordnung) with an almost religious 
respect: »In this last, One Row, dawn-
ing upon our expectation, we behold 
the true goal of our hope, the One 
Ground of phyllotaxis, on which all 
mult i tude and variety of rows must ^ 
rest.«12 
Braun's drawing, within a circle, /, 
of a bo t tom view of the pine cone 
shows one layer of this rational order. 
It is almost reminiscent of the picture 
in Zarlino's book: a rounded way of 
thinking that always comes back to its 
point of departure. 
11 Vergleichende Untersuchung, p. 3. 
12 »In dieser uns in der Erwartung vorschwebenden letzten, Einen Reihe erblicken wir 
das wahre Ziel unserer Hoffnung, den Einen Grund der Blattstellung, auf dem alle 
Vielheit und Vielartigkeit der Reihen beruhen muss.« Vergleichende Untersuchung, p. 22. 
97 
Albert van der Schoot 
There is an intriguing tension between unity and variety in Braun's 
conception of natural order, comparable to the way Zarlino deals with the 
perfection of consonants and their necessary differentiation in musical com-
position. The unity that is firmly established in the overall ruling of the fun-
damental spiral serves as a condition to bring out a multi tude of differenc-
es - differences by which the several species of cones can be distinguished 
and labelled. Braun's aim is a classification in the line of Linnaeus, arrived 
at by means of empirical observation, but his regulative conception is that 
of an overall rational order. In other words: Braun treats divergences as if 
they were musical intervals according to a traditional system of temperament, 
and he does so on the basis of a deeply rooted inner conviction that this is 
how nature behaves. Braun's phyllotaxis reflects an order of just intonation. 
It is to this preconception that the Bravais brothers oppose. There is 
no discrete classification of different divergences; when trying to attribute 
one of Braun's rational labels to a specific plant, the choice between, say, 
8{21 or 13j34 often seems quite arbitrary. None of Braun's alleged obser-
vations is as precise as the exactitude of the rational measure suggests. The 
brothers carefullyjustify this statement with a number of illustrations. What 
they object to is in fact not so much the validity of Braun's equally careful 
observations, but the very status of the starting point which led these obser-
vations to result in the conclusions that Braun presented. That starting point 
is the concept of orthostichy, which, to continue the metaphor I have jus t in-
troduced, in Braun's system of just botanical intonation fulfils the role of 
the octave, the point of reference for all the other intervals. The strong im-
pact of Braun's conception becomes clear when we read that Carl Friedrich 
Naumann considered the orthostichy as »the real essence« (das eigentliche 
Wesen) and parastichies as »a mere phenomenon of phyllotaxis« (ein blosses 
Phänomen der Blattstellung) The alternative which the Bravais b ro thers 
present comes down to granting identical rights to primordia in the same 
way tones have identical rights in equal temperature - with the proviso that 
in the case of the plants, this equality is granted by nature. 
Apart f rom carefully explaining their own theory, the Bravais brothers 
make a stand against Braun's position in a separate article.14 The tone of this 
article is (as opposed to Galilei's tone towards Zarlino) mild and respectful; 
Braun and Schimper are given ample credit for their research, and the 
opposition against the notion of orthostichy is very carefully presented. Braun 
13 C.F. Naumann, Über den Quincunx als Grundgesetz der Blattstellung vieler Pflanzen, 
Dresden/Leipzig 1845, Vorwort. 
14 Attached to the German translation of their work: L. & A. Bravais, Uber die geometrische 
Anordnung der Blätter und der Blüthenstände, Breslau 1839. 
98 
Rational Order in Tone Scales and Cone Scales 
is less attentive in his reply to the brothers in a later book.15 In order to coun-
terdict the French criticism, Braun tries to find a theoretical peg f rom an 
area where rational order had come to be understood and generally accept-
ed: crystallography. As this happened to be Auguste Bravais's field of exper-
tise, and as he had even been one of the pioneers in establishing which class-
es of crystals were morphologically possible, Braun seems to beat his oppo-
nent at his own game when he claims that wiping out the differences between 
the several rational divergences would amount to saying that all crystal forms 
are not really different because they all have the sphere as their limit.16 
This argument sounds stronger than it is. Crystal forms are different 
for constructive reasons; as opposed to phyllotaxis, each specific form is the 
result of a different chemical build-up that is discretely established f rom the 
beginning. Whatever possibilities there are, the sphere is no t among them. 
But it is an excellent illustration of Braun's way of thinking. He wants to see 
his covering law as a regulative principle, no t as a generalisation of empiri-
cal data. Transcendent unity must appear to the senses as phenomena l vari-
ety. The realm of truth is not to be found in experience, but in the mind: 
»All truth is mental«, says Braun; »all facts become recognized truths only 
when we can mentally construct them«.17 
It is almost touching to read how Nees von Esenbeck, the author of the 
introduction to the German translation of the Bravais writings, tries to unite 
the contribution of both parties in one encompassing reconciliation: hav-
ing made clear that it was his compatriots Braun and Schimper who led the 
way and who took care of the essential discoveries, he compliments the Bra-
vais brothers for their mathematical fine-tuning of the issue. The discovery 
of the »essentially irrational proportion« (das wesentlich irrationale Verhältniss) 
involved in the divergences, leads in his eye to the »ideal infinity of the fun-
damental spiral« (die ideale Unendlichkeit der Grundwendel). And he contin-
ues: »both these significant results are redeeming features not only for the 
metamorphosis of plants, but indeed for the philosophical contemplat ion 
of the organized world. It confirms the conviction that even the originally 
rational arrangements of leaves are subjected to the fundamenta l law of ir-
rational [phyllotaxis], and are recognized as mere multiples of them«.18 
15 A. Braun, Betrachtungen über die Erscheinung der Verjüngung in der Natur; Leipzig 1851. 
16 Betrachtungen, p. 126. 
17 A. Braun, »Dr. Carl Schimper's »Vorträge über die Möglichkeit eines wissenschaftlichen 
Verständnisses der Blattstellung«, in Flora, Jg. 18,1. Band, 1835, p. 146. 
18 There seems to be a word lacking in the German text; maybe the dash after irrationalen 
in the manuscript was meant to repeat Blattstellungen: »diese beiden bedeutsamen 
Resultate sind Lichtpuncte nicht allein für die Pflanzenmetamorphose, sondern für 
die philosophische Betrachtung der organisirten Welt überhaupt. Man sieht mit 
99 
Albert van der Schoot 
This is a surprising point of view. It combines the mathematical con-
clusion of the Bravais brothers concerning phyllotaxis with Braun's philo-
sophical idealism concerning the fundamental order that prevails in nature 
— and yet manages to squeeze in the idea that these arrangements are »orig-
inally rational«. 
It is not very difficult to round off empirical data concerning musical 
intervals or botanical primordia in such a way that the rationality hypothe-
sis is confirmed. Both tone scales and cone scales come very close indeed. 
But this rationality comes about as a result of human evaluation. Whether, 
in the end, nature does or doesn«t show rational order, depends - not on 
the nature of nature, but on the nature of our conception of natural order. 
verstärkter Ueberzeugung, wie selbst die ursprünglich rationalen Blattstellungen der 
Pflanzen sich dem Grundgesetze der irrationalen - unterordnen, und als blosse 
Vielfache derselben erkannt werden (....).« L. 8c A. Bravais, Uber die geometrische 
Anordnung der Blätter und der Btüthenstände, Breslau 1839, pp. V/VI. 
100