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PDF | On Nov 1, , Serge Perrine and others published Mathematics and music. a diderot mathematical forum.

**Table of contents**

If, obviously, the broad use of computation in music involves some relation with logic, is there a residue in music rationality that is unreachable by mathematical or logical means? The Vienna component of the Diderot Forum was actually organized in conjunction with a workshop with a small number of invited speakers and a poster session. Altogether 34 participants of that event provided a written contribution for the Proceedings of that workshop, which have been published separately by the OECG the Austrian Computer Society , in December , and edited by H.

## Mathematical and Musical Harmony Essay

Feichtinger and M. Expanded versions of eight of those articles constitute a special issue on "Music and Mathematics" of the Journal of New Music Research Vo1.

The event ended with a participants concert given in the rooms of the old Bosendorfer piano factory. The main topics of that workshop where problems of the synthesis of musical sound such as physical modeling , analysis of musical sounds such as the transcription problem, time-frequency methods, or the quantitative analysis of instruments and musical interpretations , restoration and denoising of old recordings, instrument optimization and mathematical models for musical sound and rhythm.

At almost all levels of musical activities, especially if instruments are involved, some mathematical model can be seen in the background. Such a model may either explaining the kind of sound which is produced by the instrument as in the famous question: "can you hear the shape of a drum" , or is used as the basis for sound production successfully implemented in modern key-boards which are based on the principle of physical modeling. Of course time-frequency analysis sometimes appearing under the name of Gabor analysis is an important tool to analyze musical sound, by displaying the "harmonic contents" of a musical signal as a function of time, just like the musical score tells the performer at which instance in time she should produce which kind of harmony.

Nowadays we have CDs and DVDs as stable sources for the reproduction of sound on a digital basis, without being aware which kind of mathematical transformations or coding techniques are running in the background. For instance, only recently some new recordings of Caruso have been produced, with the orchestra part being "replaced" by a modern recording, while preserving resp.

Clearly, such things have become possible only due the an improved understanding of the mathematical nature of acoustic signals, and how different parts if it can be treated "independently". The present book collects the sixteen written contributions to this Forum, illustrating with its large variety of articles the rich and deep interactions that exists between Mathematics and Music in a broad and contemporary sense. From the historical perspectives of the first chapters to the modeling and computation of musical sounds, from examples of musical patterns to the cultural aspects, and from the mathematical formalization to the musical logic, this rare collection of paper presents a comprehensive list of topics relating fundamental mathematics, applications of mathematics and the relation of both to society.

The first article by the musicologist Manuel Pedro Ferreira, introduces the historical perspective of the role of proportions in Ancient and Medieval Music. One of the four divisions of the Quadrivium, together with Arithmetic, Geometry and Astronomy, Music was considered a Mathematical Science. From the Greek heritage and the Latin world, up to the late-medieval France, this contribution traces a particular mode of musical thought, based on proportional relationships, that influenced the aesthetics of the Ancients and had impact on the musical composition in the Middle Ages.

Eberhard Knobloch in his contribution on "The Sounding Algebra" shows the role of Combinatorics in the baroque conception of music. He also illustrates how Music was based on a rational foundation and how musicologists and composers of that period believed that beauty and harmony consist in order and their variety stems from composition, combination and arrangement of their parts. Referring to Lullism and its combinatorial art, it is interesting to see how Mersenne, Kircher, Leibniz and, in the 18th century, Euler considerably have contributed to the progress of Combinatorics by studying such mechanical ways of composing.

The paper by the mathematician Benedetto Scimemi shows the use of mechanical devices and numerical algorithms in the 18th century for the equal temperament of the musical scale. In ancient times, before the logarithms and the irrational numbers were theoretically established, music theorists and instruments makers used a number of mechanical devices, geometrical constructions and algebraic algorithms to produce acceptable approximations for the sequence of frequencies for the musical scales. This is exemplified in the Renaissance treatise by Zarlino and, in the "setecento" , the craftsmanship of the J.

Bach's contemporary Strahle and the theoretical works of Schroter and Tartini. In the article by Jean Dhombres on Lagrange, we discover how a "working mathematician" has contributed to our theoretical understanding of wind instruments and music. One of his first papers of , "Recherches sur la nature et la propagation du son", was written with an objective for the use of music, which was for Lagrange a technique to be explained and therefore a subject for scientific research.

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Another aspect of the relationship between Mathematics and Music was shown in the Robin Wilson talk in Lisbon, illustrated with several musical examples in a wide range of styles and musical scales. As we can read in the article "Musical Patterns" , many composers have used mathematical devices in their music, namely symmetries and mathematical transformations, such as canon, expansion, retrograde motion and inversion. Francois Nicolas, a contemporary music composer with a strong background in science, opens the contributions from the Paris conference by recalling three aspects of logic: a grammar, a tautology factory, a theory of consistency and identifies their partial resonances in the musical field: the syntax of musical language s , the coherence of large musical forms.

Nicolas shows through a series of historical compositional strategies that logic in music is mostly a dialectic one, and concludes that the strategy of each work must be thought within a specific inferential framework rather than a deviation from the broader formal system it inherits. The text by Marie-Jose Durand-Richard, an epistemologist and historian of science, recalls the movement of mathematization of logic occurring with the work of George Boole She traces the discussion, still alive, of the place and nature of meaning between the defenders of blind calculation and those of a subjacent ontology.

This debate has some resonance with its musical counterpart: is music a formal game or is it based on powerful perceptive cognitive schemes? More specific, Laurent Fichet, a musicologist, studies music analysis techniques in the 20th century.

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Several methods have been widely inspired by mathematical processes. Using those that seem more likely to give interesting results, he puts forward different analysis of the 2nd Sonata by Pierre Boulez, which seems to lend itself to a logical approach. The comparison between what these mathematical analyses lead to and what a more intuitive analysis might bring gives a balanced view of the links between musical logic and mathematical logic.

Shaw, K. Rauscher, K. Robinson, J. Journal of Aesthetic Education Nature Steele, S. Dalla Bella, I. Peretz et al. Taylor and B. Journal of College Reading and Learning Great Lives in History: Inventors and Inventions. Salem Press, Rodrigues, eds. Springer, , Hodges and R.

Wilson is a gold mine: canon, modified canon, retrograde motion, inversion and the twelve tone system. Supervised by Prof.

See Richard E. Fibonacci Quarterly Mathematics Magazine Cambridge University Press, , J. Robinson, tr. Here he questions Hans Josef Irmen, Also see Mario Livio.

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Random House, , Mozart: The Man, The Musician. Schirmer Books, , Bonavia and Gordon Jacob, notes. Mozart, Symphony 40 in G Minor.