And yet it is heard volume 1 (2014) tito m tonietti

Science Networks
Historical Studies

Tito M. Tonietti

And Yet
It Is Heard
Musical, Multilingual and
Multicultural History
of the Mathematical Sciences —
Volume 1


Science Networks. Historical Studies


Science Networks. Historical Studies
Founded by Erwin Hiebert and Hans Wußing
Volume 46

Edited by Eberhard Knobloch, Helge Kragh and Volker Remmert

Editorial Board:
K. Andersen, Amsterdam
H.J.M. Bos, Amsterdam
U. Bottazzini, Roma
J.Z. Buchwald, Pasadena
K. Chemla, Paris
S.S. Demidov, Moskva
M. Folkerts, München
P. Galison, Cambridge, Mass.

I. Grattan-Guinness, London
J. Gray, Milton Keynes
R. Halleux, Liége

S. Hildebrandt, Bonn
D. Kormos Buchwald, Pasadena
Ch. Meinel, Regensburg
J. Peiffer, Paris
W. Purkert, Bonn
D. Rowe, Mainz
A.I. Sabra, Cambridge, Mass.
Ch. Sasaki, Tokyo
R.H. Stuewer, Minneapolis
V.P. Vizgin, Moskva


Tito M. Tonietti

And Yet It Is Heard
Musical, Multilingual and
Multicultural History of the
Mathematical Sciences — Volume 1


Tito M. Tonietti
Dipartimento di matematica
University of Pisa
Pisa
Italy

ISBN 978-3-0348-0671-8
ISBN 978-3-0348-0672-5 (eBook)
DOI 10.1007/978-3-0348-0672-5
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The history of the sciences is a grand fugue,
in which the voices of various peoples chime

in, each in their turn.
It is as if an eternal harmony conversed
with itself as it may have done in the bosom
of God, before the creation of the world
Wolfgang Goethe



Foreword

Musica nihil aliud est, quam omnium ordinem scire.
Music is nothing but to know the order of all things.
Trismegistus in Asclepius, cited by Athanasius Kircher, Musurgia universalis,
Rome 1650, vol. II, title page
Tito Tonietti has certainly written a very ambitious, extraordinary book in many
respects. Its subtitle precisely describes his scientific aims and objectives. His
goal here is to present a musical, multilingual, and multicultural history of the
mathematical sciences, since ancient times up to the twentieth century. To the best
of my knowledge, this is the first serious, comprehensive attempt to do justice to the
essential role music played in the development of these sciences.
This musical aspect is usually ignored or dramatically underestimated in descriptions of the evolution of sciences. Tonietti stresses this issue continually. He states
his conviction at the very beginning of the book: Music was one of the primeval
mathematical models for natural sciences in the West. “By means of music, it is
easier to understand how many and what kinds of obstacles the Greek and Roman
natural philosophers had created between mathematical sciences and the world of
senses.”
Yet, also in China, it is possible to narrate the mathematical sciences by means
of music, as Tonietti demonstrates in Chap. 3. Even in India, certain ideas would
seem to connect music with mathematics. Narrating history through music remains
his principle and style when he speaks about the Arabic culture.

Tonietti emphasizes throughout the role of languages and the existence of cultural
differences and various scientific traditions, thus explicitly extending the famous
Sapir-Whorf hypothesis to the mathematical sciences. He emphatically rejects
Eurocentric prejudices and pleads for the acceptance of cultural variety. Every
culture generates its own science so that there are independent inventions in different
contexts. For him, even the texts of mathematicians acquire sense only if they are set
in their context: “The Indian brahwana and the Greek philosophers developed their
mathematical cultures in a relative autonomy, maintaining their own characteristics.”

vii


viii

Foreword

To mention another of Tonietti’s examples: The Greek and Latin scientific
cultures, the Chinese scientific culture cannot be reduced to some general characteristics. Chinese books offered different proofs from those of Euclid. He draws a
crucial conclusion: Such differences should not be transformed into inferiority or
exclusion.
For the Chinese, as well as for the Indians, the Pythagorean distinction between
integers – or ratios between them – and other, especially irrational numbers does
not seem to make sense. The Chinese mathematical theory of music was invented
through solid pipes.
Tonietti does not conceal another matter of fact: In his perspective of history,
harmony is not only the daughter of Venus, but also of a father like Mars. For
good reasons he dedicates a long chapter to Kepler’s world harmony, which
indeed deserves more attention. He disagrees with the many modern historians of
science who transformed Kepler’s diversity into inferiority “with the aggravating
circumstances of those intolerable nationalistic veins from which we particularly

desire to stay at a good distance.”
Tonietti’s original approach enables him to gain many essential new insights: The
true achievements of Aristoxenus, Vincenzo Galilei, Stevin (equable temperament),
Lucretius’s contributions to the history of science overlooked up to now, the reasons
the prohibition of irrational numbers was eclipsed during the seventeenth century,
and the understanding of the reappearance of mathematics as the language essential
to express the new science in this century, to mention some of them. Or, as he puts
it: “The question has become rather how to interpret the musical language of the
spheres and not whether it came from God.”
Tonietti emphatically refuses corruptions, discriminations, distortions, simplifications, anachronisms, nationalisms of authors, and cultures trying to show that
“even the mathematical sciences are neither neutral nor universal nor eternal and
depend on the historical and cultural contexts that invent them.” He places music
in the foreground, he has not written a history of music with just hints to acoustic
theories.
In spite of all his efforts and the more than thousand pages of his book, Tonietti
calls his attempt a modest proposal, a beginning. It is certainly a provocative book
that is worth diligently studying and continuing even if not every modern scholar
will accept all of its statements and conclusions.
Berlin, Germany
February 2014

Eberhard Knobloch


Contents for Volume I

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1

In the Ancient World

2 Above All with the Greek Alphabet . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1 The Most Ancient of All the Quantitative Physical Laws . . . . . . . . . . . .
2.2 The Pythagoreans .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 Plato .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.5 Aristoxenus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.6 Claudius Ptolemaeus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.7 Archimedes and a Few Others . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.8 The Latin Lucretius .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.9 Texts and Contexts.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9
9
11
19
25
35
41
54
64
82

3 In Chinese Characters .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1 Music in China, Yuejing, Confucius .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.2 Tuning Reed-Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3 The Figure of the String .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4 Calculating in Nine Ways . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5 The Qi .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6 Rules, Relationships and Movements.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.1 Characters and Literary Discourse . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.2 A Living Organism on Earth .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.3 Rules, Models in Movement and Values . .. . . . . . . . . . . . . . . . . . . .
3.6.4 The Geometry of the Continuum in Language.. . . . . . . . . . . . . . .
3.7 Between Tao and Logos .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

97
97
100
111
117
124
140
141
143
147
151
153

4 In the Sanskrit of the Sacred Indian Texts . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 Roots in the Sacred Books . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.2 Rules and Proofs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3 Numbers and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

169

169
172
179
ix


x

Contents for Volume I

4.4
4.5
4.6

Looking Down from on High . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 187
Did a Mathematical Theory of Music Exist in India, or Not? . . . . . . . . 194
Between Indians and Arabs . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 203

5 Not Only in Arabic .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 Between the West and the East . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 The Theory of Music in Ibn Sina . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 Other Theories of Music . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4 Beyond the Greek Tradition .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.5 Did the Arabs Use Their Fractions and Roots
for the Theory of Music, or Not? . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.6 An Experimental Model Between Mathematical Theory
and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.7 Some Reasons Why . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

209

209
210
219
225

6 With the Latin Alphabet, Above All . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 Reliable Proofs of Transmission . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 Theory and Practice of Music: Severinus Boethius
and Guido D’Arezzo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 Facing the Indians and the Arabs: Leonardo da Pisa . . . . . . . . . . . . . . . . .
6.4 Constructing, Drawing, Calculating:
Leon Battista Alberti, Piero della Francesca,
Luca Pacioli, Leonardo da Vinci .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5 The Quadrivium Still Resisted: Francesco Maurolico,
the Jesuits and Girolamo Cardano . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.6 Variants of Pythagorean Orthodoxy: Gioseffo Zarlino,
Giovan Battista Benedetti . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.7 The Rebirth of Aristoxenus, or Vincenzio Galilei . . . . . . . . . . . . . . . . . . . .

257
257

232
238
244

259
271

276

291
303
309

A [From the] Suanfa tongzong [Compendium of Rules
for Calculating] by Cheng Dawei . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 329
B Al-qawl ‘ala ajnas alladhi bi-al-arba‘a [Discussion
on the Genera Contained in a Fourth] by Umar al-Khayyam.. . . . . . . . . . . . . . 335
C Musica [Music] by Francesco Maurolico . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.1 Rules to Compose Consonant Music . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.2 Rule of Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.3 Rule of Taking Away . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.4 The Calculation of Boethius for the Comparison of Intervals .. . . . . . .
C.5 Comment on the Calculation of Boethius . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.6 Guido’s Icosichord.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.7 MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

341
348
353
354
359
360
362
374

D The Chinese Characters .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 379
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 395

Contents for Volume II

7

Introduction to Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part II
8

1

In the World of the Scientific Revolution

Not Only in Latin, but also in Dutch, Chinese, Italian
and German .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1
Aristoxenus with Numbers, or Simon Stevin and Zhu Zaiyu . . . . . .
8.2
Reaping What Has Been Sown. Galileo Galilei,
the Jesuits and the Chinese . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3
Johannes Kepler: The Importance of Harmony .. . . . . . . . . . . . . . . . . . . .

18
37

Beyond Latin, French, English and German:
The Invention of Symbolism . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.1
From Marin Mersenne to Blaise Pascal . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9.2
René Descartes, Isaac Beeckman and John Wallis . . . . . . . . . . . . . . . . .
9.3
Constantijn and Christiaan Huygens . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

107
107
158
192

10 Between Latin, French, English and German:
The Language of Transcendence . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.1 Gottfried Wilhelm Leibniz .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.2 Sir Isaac Newton and Mr. Robert Hooke .. . . . . . .. . . . . . . . . . . . . . . . . . . .
10.3 Symbolism and Transcendence .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

227
227
265
291

9

11 Between Latin and French. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.1 Jean-Philippe Rameau, the Bernoullis and Leonhard Euler .. . . . . . .
11.2 Jean le Rond d’Alembert, Jean-Jacques Rousseau
and Denis Diderot .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.3 Counting, Singing and Listening: From Rameau to Mozart . . . . . . .

5

5

327
327
368
412

12 From French to German .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 431
12.1 From Music-Making to Acoustics: Luigi Giuseppe
Lagrange e Joseph Jean-Baptiste Fourier . . . . . . .. . . . . . . . . . . . . . . . . . . . 431

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Contents for Volume II

12.2
12.3
12.4
Part III

Too Much Noise, from Harmony to Harmonics:
Bernhard Riemann and Hermann von Helmholtz . . . . . . . . . . . . . . . . . . 439
Ludwig Boltzmann and Max Planck . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 466
Arnold Schönberg and Albert Einstein . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 483
It Is Not Even Heard

13 In the Language of the Venusians . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13.1 Black Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.2 Stones, Pieces of String and Songs . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.3 Dancing, Singing and Navigating . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

511
511
513
514

14 Come on, Apophis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 527
14.1 Gott mit uns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 527
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 535
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 577


Chapters 3, 6 and 8–12 have been respectively obtained by re-elaborating, completing or simplifying the following talks, articles and books:
Paper presented at Hong Kong in 2001, “The Mathematics of Music During the
16th Century: The Cases of Francesco Maurolico, Simon Stevin, Cheng Dawei,
Zhu Zaiyu”, Ziran kexueshi yanjiu [Studies in the History of Natural Sciences],
(Beijing), 2003, 22, n. 3, 223–244.
Le matematiche del Tao, Roma 2006, Aracne, pp. 266.
“Tra armonia e conflitto: da Kepler a Kauffman”, in La matematizzazione della
biologia, Urbino 1999, Quattro venti, 213–228.
“Disegnare la natura (I modelli matematici di Piero, Leonardo da Vinci e Galileo
Galilei, per tacer di Luca)”, Punti critici, 2004, n. 10/11, 73–102.
“The Mathematical Contributions of Francesco Maurolico to the Theory of
Music of the 16th Century (The Problems of a Manuscript)”, Centaurus, 48,
(2006), 149–200.
Paper presented at Naples in 1995, “Verso la matematica nelle scienze: armonia
e matematica nei modelli del cosmo tra seicento e settecento”, in La costruzione

dell’immagine scientifica del mondo, Marco Mamone Capria ed., Napoli 1999, La
Città del Sole, 155–219.
Paper presented at Perugia in 1996, “Newton, credeva nella musica delle sfere?”,
in La scienza e i vortici del dubbio, Lino Conti and Marco Mamone Capria eds.,
Napoli 1999, Edizioni scientifiche italiane, 127–135. Also “Does Newton’s Musical
Model of Gravitation Work?”, Centaurus, 42, (2000), 135–149.
Paper presented at Arcidosso in 1999, “Is Music Relevant for the History of
Science?”, in The Applications of Mathematics to the Sciences of Nature: Critical
Moments and Aspects, P. Cerrai, P. Freguglia, C. Pellegrino (eds.), New York 2002,
Kluwer, 281–291.
“Albert Einstein and Arnold Schoenberg Correspondence”, NTM - Naturwissenschaften Technik und Medizin, 5 (1997) H. 1, 1–22. Also Nuvole in silenzio
(Arnold Schoenberg svelato) , Pisa 2004, Edizioni Plus, ch. 58.
xiii


xiv

“Il pacifismo problematico di Albert Einstein”, in Armi ed intenzioni di guerra,
Pisa 2005, Edizioni Plus, 287–309.
Chapters 2, 4 and 5 are completely new. In the meantime, thanks to the
help of Michele Barontini, a part of Chap. 5 has become “Umar al-Khayyam’s
Contributions to the Arabic Mathematical Theory of Music”, Arabic Science and
Philosophy v. 20 (2010), pp. 255–279. The problems of Chap. 4 produced, in
collaboration with Giacomo Benedetti, “Sulle antiche teorie indiane della musica.
Un problema a confronto con altre culture”, Rivista di studi sudasiatici, v. 4 (2010),
pp. 75–109; also, “Toward a Cross-cultural History of Mathematics. Between the
Chinese, and the Arabic Mathematical Theories of Music: the Puzzle of the Indian
Case”, in History of the Mathematical Sciences II, eds. B.S. Yadav & S.L. Singh,
Cambridge 2010, Cambridge Scientific Publishers, 185–203.
In the meantime, a part of Chaps. 11 and 12 has been published as “Music

between Hearing and Counting (A Historical Case Chosen within Continuous LongLasting Conflict)”, in Mathematics and Computation in Music, Carlos Agon et al.
eds., Lectures Notes in Artificial Intelligence 6726, Berlin 2011, Springer Verlag,
285–296.
Appendix C is the translation of the edition for Maurolico’s Musica, edited by the
author for the relative Opera Mathematica in www.maurolico.unipi.it, subsequently
also Pisa-Roma, Fabrizio Serra editore, to be published, perhaps.


Chapter 1

Introduction

The history of the sciences can be (and has been) told in many ways. In general, however, treatments display systematic, recurring partialities. Many of the
characters who contributed to them also wrote about music, and sure, the best
approximation would be to say that all of them did. And yet the musical aspect,
though present on a relatively continuous basis during the evolution of sciences, is
usually ignored or underestimated. This omission would appear to be particularly
serious, seeing that music would enable us to represent in a better and more
characteristic way the main controversies at the basisp
of this history. For example,
the question of the so-called irrational numbers, like 2, may have a very simple,
direct musical representation.
This book will thus bring into full light some pages dealing with musical subjects,
that are scattered throughout the most famous scientific texts. The complementary
point of view is relatively more widespread, that is to say, the one that presents
the history of music as traversed also by the study of physical sound, for example
frequencies and harmonics. This happened because, for better or worse, science and
technology have succeeded in influencing the world we live in, unfortunately, more
than music, and thus they have also influenced music. At this point, it has become
necessary to recall that also music was capable, on the contrary, of playing a role

among the sciences and among scientists.
There is another not insignificant defect in the histories of the modern-day
sciences. Apart from, in the best of cases, a few brief mentions in the opening
chapters, the evolution of sciences seems to be taken place exclusively in Europe, or
to have reached its definitive climax in Europe. However, despite Euclid, Galileo
Galilei, Descartes, Kepler, Newton, Darwin and Einstein, it actually had other
important scenarios: China, India, the countries of the Arab world. The idea
that the sciences were practically an exclusively Western invention is due to a
Eurocentric prejudice. The reasons for this commonplace, which does not stand
up to careful historical examination, are manifold. They will emerge, if necessary,
in due time. But one of these, in view of its general character, deserves to be
discussed immediately. Scientific results, which are more often called discoveries

T.M. Tonietti, And Yet It Is Heard, Science Networks. Historical Studies 46,
DOI 10.1007/978-3-0348-0672-5__1, © Springer Basel 2014

1


2

1 Introduction

than inventions, are in a certain sense made independent of the social, cultural,
economic, political, national, linguistic and religious context. Deprived of all these
characteristics, which are those that can be observed in historical reality, that is
to say, in the environment where the inventors lived, the sciences are described
as elements of an ideal, unreal world, which may also be called the justification
context. This book is alien, not only to this philosophy of history, but also to
all others. Partly because it serves to arrive at the idea of a (rhetorical) scientific

progress and neutral sciences, for which the authors are not, after all, responsible.
Here, on the contrary, the sciences are shown to be rooted in the various cultures,
and to contain their values in some form, which is to be verified time by time.
Consequently, the contributions that come from countries outside Europe not only
appear to be significant, and not at all negligible, but their value lies above all
in the fact that they are characteristic, and different from Western contributions.
The language represents the deepest aspect of each culture, because it is through
language that each presents and cultivates its own system of values. Thus, the first
element, and one of the most important that we must underline, is which language
the scientific texts examined here were written in. This means that our multicultural
history of sciences necessarily also becomes a review of the various dominant
languages used in the different historical contexts. Just as the scientific community
generally expresses itself nowadays in English, in other periods, for several centuries
it had expressed itself in Greek or in Latin, and elsewhere in Arabic, in Sanskrit, or
in Chinese. Often the language used by a scholar to write his text was not his own
mother tongue, but that of the dominant culture of the area. For example, various
Persian scholars wrote in Arabic. The Swiss mathematician and physicist of the
eighteenth century, Leonhard Euler, who actually spoke German, has left us texts
written in Latin.
The attention dedicated here to cultural differences, in relationship to the
various scientific traditions, will also lead us to deal with the question of how the
characteristics of the languages influenced the relative inventions. Thus we shall
find arguments in a linear form, like the deductions from axioms, in a alphabetic
language like Greek, but also another kind of visual demonstration, expressed
in Chinese characters. An anthropologist and scholar of the hopi language like
Benjamin Lee Whorf (1897–1941) wrote “: : : linguistics is fundamental for the
theory of thought and, ultimately, for all human sciences”.1 Here the famous SapirWhorf hypothesis is even extended to the mathematical sciences.
Moved by the best intentions, other historians have taken great pains to recall the
great inventions of Arabs, Indians and Chinese. They have often presented them,
however, as contributions to a single universal science. Faith in this thus led them

to overlook cultural differences and consequently to deal with insoluble, absurd
questions of priority and transmission from one country to another.2 On the contrary,

1

Whorf 1970, p. 64.
A good example of how one can limit one’s studies to problems of transmission, completely
ignoring cultural differences and music, is offered by the great, in many ways fundamental classic,

2


1 Introduction

3

national pride animated the historians of countries unjustifiably ignored, leading
them to offer improbable, not to say incorrect, dates for the texts that they study.
Apart from the cases with sufficient documentation, the history of science has,
on the contrary, all to gain from the idea of independent inventions made in different
contexts. In general, every culture generates its own sciences. Among these, it then
becomes particularly interesting to make comparisons. However, it is advisable
to avoid constructing hierarchies, which inevitably depend on the values of the
historian making the judgements, but are extraneous to the people studied. A famous
anthropologist like Claude Lévi-Strauss complained, “: : : it seems that diversity of
cultures has rarely been seen by men as what it really is: a natural phenomenon,
resulting from direct or indirect relationships between societies; rather, it has been
seen as a sort of monstrosity, or scandal.”3
This does not mean renouncing the characteristics of sciences compared with
other human activities. Simply, they are not to be distinguished by making them

independent of the people who invent their rules, laws or procedures, thus transporting them into a mythical transcendent world (imagined, naturally, to be European),
or into the present epoch, with its specialisations of an academic kind.
While this book does not tell the story of the evolution of the sciences as if it took
place in an ideal world alien to history, it does not proceed, either, as if there were
never confrontations, unfortunately usually tragic, between the various cultures and
peoples. Even the idyllic islands of Polynesia saw the arrival, sooner or later, of
(war)ships that had set sail from Europe.
It is no desire of mine to deny that here in the West, the development of the sciences received a particularly fervid impulse, starting from the seventeenth century.
Nor do I wish to ignore their capacity to expand all over the world, establishing
themselves, for better or for worse, in the lifestyles of many populations.
But this does not constitute a criterion of superiority for Western sciences. Rather,
the historical events that have led up to this situation indicate as the ground for a
confrontation that of power and warfare. It is only on this basis that a hierarchic
scale can be imposed on different values, each of which is fruitful and effective
within its own culture, and each of which it is largely impossible to measure with
respect to the others. Briefly, when we are tempted to transform the characteristics
of Western sciences into an effective superiority, we need to realise that we are
implicitly accepting the criterion of war as the ultimate basis for the comparison.
As a result, this book reserves an equal consideration for extra-European sciences
as for those that flourished in Europe, for the same moral reasons that lead us to
repudiate the strength of arms and military success as a valid criterion to compare
different cultures that come into conflict.

Otto Neugebauer 1970. This German scholar typically considers only astronomy as the leading
science of the ancient world, and does not even remember that Ptolemy had also written a book
about music; see Sect. 2.6.
3
Lévi-Strauss 2002, p. 10.

4

1 Introduction

Actually, Western sciences penetrated into China, thanks mainly to the Jesuits,
precisely because they proved to be useful for the noble art of arms and war.
This was clearly spelled out in the Western books of science which had been
translated at that time into Chinese. And this, unfortunately, was to find practical
confirmation, both when the Ming empire was defeated by the Qing (also known as
the Manchu) empire, half-way through the seventeenth century, and two centuries
later, when the latter imperial dynasty was subdued by Western imperialistic and
colonialistic powers during the infamous Opium Wars.4 With the precise aim of
exposing the deepest roots of Eurocentric prejudice, on these occasions, the various
reasons connected with arms and warfare, which had influenced the evolution of the
sciences, were not ignored or covered up (as usually happened).
In Part I, dedicated to the ancient world, the Chap. 2 tells the story, in pages
dealing with music, of the Pythagorean schools, Euclid, Plato, Aristoxenus and
Ptolemy, and how that orthodoxy was created in the Western world, which was to
prohibit the use of irrational numbers. The consequences of this choice persisted for
2,000 years, and came to be the most important characteristics of Western sciences:
these included the typical dualisms of a geometry separated from numbers, and a
mathematics that transcended the world where we live. The dominant language in
that period was Greek. Nor can we overlook Lucretius, on the grounds that he was
outside the predominant line, like Aristoxenus.
In Chap. 3, the Chinese mathematical theories of music based on reed-pipes
reveal a scientific culture dominated, instead, by the idea of an energetic fluid,
called qi. Omnipresent and pervasive, it gave rise to a continuum, where it could
carry out its processes, where it could freely move geometric figures, and where
it could execute every calculation, including the extraction of roots. Accordingly,
the leading property of right-angled triangles was proved in a different way from

that of Euclid. Also the dualism between heaven and earth, with the transcendence
which was so important in the West, was lacking here. During the sixteenth and
seventeenth centuries, these two distant scientific cultures were to enter into contact
in a direct comparison. The relative texts were composed in Chinese characters.
In the Chap. 4, India comes on the scene, with its sacred texts written in Sanskrit.
Here, the need for a particular precision, motivated by the rituals for the construction
of altars, led to geometric reasoning. The fundamental property of right-angled
triangles was exploited, and it was explained how to calculate the area of a trapezoid
altar. Music, too, acquired great importance thanks to the rituals based on singing.
But, by a curious unsolved paradox, which marks the culture that invented our
modern numbers, their theory of music does not seem to demand exactness through
mathematics, but rather trusts its ears.
In Chap. 5, the Arabs appear, with these famous numbers brought from India,
and their books translated from, and inspired by, a Greek culture that had too long
been ignored in the mediaeval West. By now, scholars, even those from Persia, left
books usually written in Arabic. Their predominant musical theory was inspired by

4

Tonietti 2006a, pp. 175–179 e 197.


1 Introduction

5

that of the Greeks, above all by Pythagorean-Ptolemaic orthodoxy. Some of their
terms, such as “algebra” and “algorithm” were to change their meaning in time, and
to enter into the current modern usage of the scientific community.
In Chap. 6, we return to Europe, recently revitalised by Oriental cultures, whose

influence is increasingly cited, even more than that of Greece. Its lingua franca,
with universal claims, had become Latin. Here, the musical rhythms were now
represented on the lines and spaces of the stave. Variations on PythagoreanPtolemaic orthodoxy were appearing, and Vincenzio Galilei at last remembered
even the ancient rival school of Aristoxenus. Euclid still remained the general
reference model for mathematical sciences, he now began to be flanked by new
calculating procedures for algebraic equations, and the new Indo-Arabic numbers.
Appendix A contains a translation of the musical pages contained in the famous
Chinese manual of mathematics, Suanfa tongzong [Compendium of calculating
rules] written by Cheng Dawei in the sixteenth century, and discussed in Chap. 3.
This is followed, in Appendix B, by a translation of a short text about music by
Umar al-Khayyam, which is discussed in Chap. 5. Lastly, Appendix C contains
a translation of the manuscript entitled Musica, handed down to us among the
papers of Francesco Maurolico and presented in Chap. 6. In appendix D, the Chinese
characters scattered in the text are given.
In Part II, which is dedicated to the scientific revolution, Chap. 8 narrates the
evolution of the seventeenth century through the writings on music of Stevin, Zhu
Zaiyu, Galileo Galilei and Kepler. The German even included in the title of his most
important work his idea of harmony in the cosmos. The equable temperament for
instruments was now also represented by means of irrational numbers.
The Chap. 9 is taken up by Mersenne, Pascal, Descartes, Beeckman, Wallis,
Constantin and Christiaan Huygens, and their discussions about music, God, the
world, and natural phenomena. Together with Latin, which still dominated in
universities, national languages were increasingly used to communicate outside
traditional circles. We now find texts also in Flemish, Chinese, Italian, French,
German and English. Above all (as a consequence?), a new typically European
mathematical symbolism was adopted, as writing music on staves had been.
In Chap. 10, we discover that even Leibniz and Newton, not to mention Hooke,
had continued for a while to deal with music, and had ended up by preferring
the equable temperament, at least in practice. With them, mathematical symbolism
gained that (divine?) transcendence which was necessary to deal better with

infinities and infinitesimal calculations with numbers.
In Chap. 11, music enjoys its final season of excellence among the great scientists
of the eighteenth century. As happened at court and in diplomatic circles, French
became the language most widely used among scientists, though some of them still
insisted on Latin. Euler based his neo-Pythagorean theory on prime numbers. His
opponent, d’Alembert, at his ease among the musicians of Paris, preferred, on the
contrary, to follow his own ears. Both of them, however, were to come up against
the musician Jean-Philippe Rameau, while the Illuminists of the Encyclopédie also
took part in the discussion.


6

1 Introduction

In the following period, harmony was overshadowed by the din of the combustion
engine. Consequently, in Chap. 12, the ancient harmony became a not-so-central
part of acoustics, together with harmonics and Fourier’s mathematics. Bernhard
Riemann, Helmholtz and Planck vied in explaining to us the sensitivity that our
ears were guided by. Finally, the correspondence that passed between the musician
Schoenberg and the famous physicist Albert Einstein shows us the (great?) nature
of the period between the end of the nineteenth and the twentieth centuries. Their
language had become German. With the pianoforte, all music now followed the
equable temperament.
In Chap. 13 of Part III, only the caustic language of the Venusians would succeed
in expressing the impossible dream of finding harmony in the age of warfare and
violence. For this reason, we also need to remember the forgotten, destroyed cultures
of Africa, pre-Columbian America and Oceania.
In the fourteenth and last chapter, with all the knowledge acquired by the
mathematical sciences today, we speculate whether the asteroid Apophis and

nuclear bombs will allow us to continue to enjoy music (and life).


Part I

In the Ancient World


Chapter 2

Above All with the Greek Alphabet

2.1 The Most Ancient of All the Quantitative Physical Laws
I would like to begin with an argument which may be stated most clearly and most
forcefully as follows:
Music was one of the primeval mathematical models for natural sciences in the West.

The other model described the movement of the stars in the sky, and a close
relationship was postulated between the two: the music of the spheres.
This argument is suggested to us by one of the most ancient events of which trace
still remains. It is so ancient that it has become legendary, and has been lost behind
the scenes of sands in the desert. A relationship exists between the length of a taut
string, which produces sounds when it is plucked and made to vibrate, and the way
in which those sounds are perceived by the ear. The relationship was established in a
precise mathematical form, that of proportionality, which was destined to dominate
the ancient world in general. Given the same tension, thickness and material, the
longer the string, the deeper or lower the sound perceived will be; the more it is
shortened, the less deep the sound perceived: the length of the string and the depth
of the sound are directly proportional. If the former increases, the latter increases
as well; if the former decreases, the latter does as well. Or else, the sound could

be described as more or less acute, or high. In this case, the length of the string
generating it would be described as inversely proportional to the pitch. The shorter
the string, the higher the sound produced. None of the special symbols employed
in modern manuals were used to express this law, but just common language. If the
string is lengthened, the height of the sound is proportionally lowered.
Two thousand years were to pass until the appearance of the formulas to which
we are accustomed today. It was only after René Descartes (1596–1650) and
subsequently Marin Mersenne (1588–1648), that formulas were composed of the
kind
/

1
l

T.M. Tonietti, And Yet It Is Heard, Science Networks. Historical Studies 46,
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2 Above All with the Greek Alphabet

where the height was to be interpreted as the number of vibrations of the string in
time, that is to say, the frequency , and the length was to be measured as l.
The first volume will accompany us only as far as the threshold of this representation, that is to say, up to the affirmation of a mathematical symbolism increasingly
detached from the languages spoken and written by natural philosophers and
musicians, and this will be the starting-point for the second volume. Furthermore,
it is important to remember that fractions such as 1l or 32 were not used in ancient

times, but ratios were indicated by means of expressions like ‘3 to 2’, which I will
also write as 3:2. The ratio was thus generally fixed by two whole numbers. Whereas
a fraction is the number obtained by dividing them, when this is possible.
The same relationship between the length of the string and the height of the sound
would appear to have remained stable up to the present day, about 2,500 years later.
Is this the only natural mathematical law still considered valid? While others were
modified several times with the passing of the years? “: : : possibly the oldest of
all quantitative physical laws”, wrote Carl Boyer in his manual on the history of
mathematics.1 That “possibly” can probably be left out.
In Europe, a tradition was created, according to which it was the renowned
Pythagoras who was struck by the relationship between the depth of sounds and
the dimensions of vibrating bodies, when he went past a smithy where hammers of
different sizes were being used. However, the anecdote does not appear to be very
reliable, mainly because the above ratio regarded strings.
In any case, the sounds produced by instruments, that is to say, the musical notes
perceived by the ear, could now be classified and regulated. How? Strings of varying
lengths produced notes of different pitches, with which music could be made. But
Pythagoras and his followers sustained that not all notes were appropriate. In order
to obtain good music, it was necessary to choose the notes, following a certain
criterion. Which criterion? The lengths of the strings must stand in the respective
ratios 2:1, 3:2, 4:3. That is to say, a first note was created by a string of a certain
length, and then a second note was generated by another string twice as long, thus
obtaining a deeper sound of half the height. The two notes gave rise to an interval
called diapason. Nowadays we would say that if the first note were a do, the second
one would be another do, but deeper, and the interval is called an “octave”, and so it
is the do one octave lower. The same ratio of 2:1 is also valid if we take a string of
half the length: a new note twice as high is obtained, that is to say, the do one octave
higher. But musical notes were to be indicated in this kind of syllabic manner only
from Guido D’Arezzo on (early 1000s to about 1050).2
The other ratios produced other notes and other intervals. The ratio 3:2 generated

the interval of diapente (the fifth do – sol) and 4:3 the diatessaron (the fourth do –
fa). Thus the ratios established that what was important for music was not the single
isolated sound, but the relationship between the notes. In this way, harmony was
born, from the Greek word for ‘uniting, connecting, relationship’.

1
2

Boyer 1990, p. 65.
See Sect. 6.2.


2.2 The Pythagoreans

11

At this point, the history became even more interesting, and also relatively well
documented, because in the whole of the subsequent evolution of the sciences,
controversies were to develop continually regarding two main problems. What
notes was the octave to be divided into? Which of the relative intervals were
to be considered as consonant, that is to say ‘pleasurable’, and consequently
allowed in pieces of music, and which were dissonant? And why? The constant
presence of conflicting answers to these questions also allows us to classify sciences
immediately against the background of the different cultures: each of them dealt
with the problems in its own way, offering different solutions.
Anyway, seeing the surprising success of our original mathematical law model,
it was coupled here and there with other regularities that had been identified, and
was posited as an explanation for other phenomena. The most famous of these
was undoubtedly the movement of the planets and the stars; this gave rise to
the so-called music of the heavenly spheres, and connected with this, also the

therapeutic use of music in medicine. This original seal, this foundational aporia
remained visible for a long time. All, or almost all, of the characters that we are
accustomed to considering in the evolution of the mathematical sciences wrote
about these problems. Sometimes they made original contributions, other times
they repeated, with some personal variations, what they had learnt from tradition. It
might be named Pythagorean tradition, so called after the reference to its legendary
founder, to whom the original discovery was attributed, or the Platonic or neoPlatonic tradition. This was even to be contrasted with a rival tradition dating back to
Aristoxenus. In any case, many scholars felt an obligation to pay homage to tradition
in their commentaries, summaries, and sundry quotations, or in their actual theories.
In this second chapter, we shall review the Pythagoreans, and other characters
who harked back to their tradition, such as Euclid and Plato, but also significant
variations like that of Claudius Ptolemaeus (Ptolemy), or the different conception
of Aristoxenus. In Chaps. 6, 8–11, we shall see that the interest in the division of
the octave into a certain number of notes, and the interest in explaining consonances
passed unscathed, or almost so, through the epochal substitution (revolution?) of
the Ptolemaic astronomic system with the Copernican one during the seventeenth
century. It might be variously described as musical theory, or acoustics, or as the
music of mathematics, or the mathematics of music. All the same, it continued
without any interruption in the Europe of Galileo Galilei, Kepler, Descartes,
Leibniz, and Newton. It was not completely abandoned, even when, during the
eighteenth century, figures like d’Alembert and Euler felt the need to perfect the new
symbolic language chosen for the new sciences, and to address them in a general
systematic manner.

2.2 The Pythagoreans
Pythagoras, : : : constructed his own o Ã0 ˛ [wisdom] o
[learning] and IJÄo
Ã0 ˛ [art of deception].

˛Â Ã0 ˛

Heraclitus.