In these two volumes of a thousand pages, Tonietti gives a very personal selection of the history of mathematics, and in particular the pieces where music and mathematics meet. He has some strong views on certain aspects that are not always mainstream and these are strongly put forward. One of his pet peeves is that many of his colleagues approached the subject with an eurocentric bias. Another one is that mathematics and science in general is too often considered to be an abstract universal entity besides or above the socio-cultural soil in which they are rooted. That means for example that also the language spoken and or even more so, the lingua franca used by the scientists and philosophers has had its influence which often goes hand in hand with cultural and religious foundations. This is his contribution to prove the Sapir–Whorf hypothesis in the case of mathematical sciences. Every culture generates its own science, and qualifications like superior and inferior or questions of precedence are often absurd. And of course, probably the main reason for writing this book, is his conviction that the mutual influence of music on the development of mathematics and vice versa is grossly underestimated. In this context he brings several contributions to the forefront that were wrongfully neglected (Aristoxenus, Vincenzo Galilei, Simon Stevin, Kepler,...). Thus there are many provocative viewpoints that some historians will disagree with, yet his arguments are extensively documented. Note however that this is not really a book on the history of mathematics, and neither is it a history of music. The reader is supposed to be familiar with music theory and should have some background in mathematics too. The book is a long plea and an extensive argumentation to underpin the viewpoints of the author, like those just mentioned. There are practically no formulas in the text, and relatively few illustrations, but the number of citations is overwhelming. These are almost always given in the original language with translation in brackets. The Chinese citations are written in pinyin, but full Chines characters are added in an appendix. This illustrates the importance that Tonietti is attaching to the language, since indeed, the translation is always an approximation and often an interpretation of what the original text is meant to say.
Let's go quickly though some of the contents to illustrate what has been said above. Volume 1 contains Part I: The ancient world, and Volume 2 consists mainly of Part II: The scientific revolution, and a shorter Part III: It is not even heard.
Part I treads the ancient cultures united around their language used: The Greek, Chinese, Sanskrit, Arabic, and Latin. For the Greek, music was part of the quadrivium and hence coexisted at the same level as mathematics and astronomy in the schools of Pythagoras, Euclid, Plato and Ptolemy. The search for harmony in music was reflected in the music of the spheres, all based on an orthodoxy of commensurability, hence integers and rationals. The music theory was developed on the basis of length of strings. The often neglected unorthodox outsiders here are Aristoxenus (who does not restrict to rationals) and Lucretius (although the latter wrote in Latin, he is Greek in spirit) who get special attention.
The Chines on the other hand developed a theory of music studying the length of pipes (the lülü), bells, and chime stones. The cultural essense of qi is an energetic flow, a continuum which is apposite to the discrete orthodoxy of the Greek. Another difference is the lack of an equivalence for the verb "to be". This implies a different way of doing mathematics like for example the way in which they proved the Pythagoras theorem.
Indian rules and regulations stem from religion. Precise prescriptions of how to build an altar show mathematical knowledge. Of course there was music, mostly by singing mathras, but most curiously, there is no trace of a music theory left. Musicians had `to trust their ears'.
The Arabs are the saviors of the Greek culture. Most of what we know about the Greeks comes to us through them. The Syntaxis mathematica of Ptolemy came to us in Arabic as the Almagest: `the greatest' Greek collection of astronomical data. So they inherited the orthodoxy in music and mathematics from the Greek. They brought us our number system, but also terms like algorithm and algebra.
Meanwhile in Europe, Latin had conquered the scientific scenary. This brought about a clash between the people, like Fibonacci promoting the introduction of the new Indo-Arabic number system against the Roman numerals. The Greek orthodoxy prevails, with Euclid being the reference for mathematics. Music theory florishes (Beothius, Guido D'Arezzo, Maurolico, Cardano,...). Tonietti gives special attention to Vicenzo Galilei, the father of Galileo, who picked up some ideas of Aristoxenus again.
Besides the appendix with Chinese characters mentioned above, three other appendices are texts related to music translated from Chinese, Arabic, and Latin.
In Part II chapters are named again afer the main languages (mostly European) used to disseminate scientific results. The interplay between geometry, astronomy, and music becomes explicit in work by Stevin and Galileo, but most of all in Kepler's Harmonices Mundi Libri Quinque [Five books on the harmony of the world] in which he completed Ptolemy's APMONIKA and interwaves geometry, astronomy, music and geometry, reflecting the music of the spheres. Tonietti does not shy away from critique on colleagues who had different interpretations of Kepler's work. People started using national languages besides Latin in their writings and (perhaps because of that) mathematical symbolism increases like writing music on staves was adopted before. Transcendent symbolism was mixed with music, God, and natural phenomena in work of Mersenne, Descartes, Wallis, and Huygens (Constantijn and Christiaan). The latter was not only a musician and composer, he used the newly invented logarithms and Leibnitz's differential calculus in his music theory. All this, according to Tonietti, shows that the status of music should be reinstalled as an essential element that contributed to the development of mathematics. Also Leibnitz and Newton worked on music for some time, but of course their main contribution here is the mathematical symbolism that allowed to deal with the infinite and the infinitesimal. With the use of the twelfth root of two in the equable temperament, the Pythagorean-Plato orthodoxy was definitely finished. The music of the spheres had degenerated and became intense discussions about God and creation. Again Tonietti analyzes interpretations of other historians, philosophers, or theologians on these topics sometimes rebutting them with his own vision. While in the 18th century French became the prominent language in Europe, music theory received its last flares. The French composer Rameau wrote a treatise on harmony using a theoreical basis, but he was opposed by Euler who declared tones more pleasing when they could be represented more simply. He based his analysis on prime numbers, with reminiscences of Pythagoras. Another opponent was d'Alembert discussing the harmonics of the vibrating string and also the other illuminists compiling the Encyclopédie had their explanation for musical terms. Still musicians wrote of science and scientists wrote of music. Entering the 19th century, Lagrange and Fourier, and later Riemann entered the discussion about vibrations, while von Helmholtz also did the physical experiments to analyze sound. Max Planck not only wrote about music but even composed some and also Einstein loved playing the violin and had correspondence with the composer Schönberg.
Part III is very short. It gives a short round-up of things not discussed like Africa, Cental and South America, and more extensively the music and navigation skills of Polynesians. A last chapter briefly touches upon the science of acoustics, in which music is largely neglected. A quote from one of the final paragraphs that renders explicitly what Tonietti has allowed to emerge in his book:
The decision to move away from musicians and their music impoverished both natural philosophers, first of all, and then mathematicians and physicists. This influenced and facilitated the development of their research in those main directions which are known to everybody, but which continue to deserve to be criticised for their limitations and their (negative?) effects on our life.