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Music by the Numbers: From Pythagoras to Schoenberg

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How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music explicitly based on mathematical principles. Yet Eli Maor argues that music has influenced math at least as much as math has influenced music. Starting with Pythagoras, proceeding through the work of Schoenberg, and ending with contemporary string theory, Music by the Numbers tells a fascinating story of composers, scientists, inventors, and eccentrics who played a role in the age-old relationship between music, mathematics, and the sciences, especially physics and astronomy. Music by the Numbers explores key moments in this history, particularly how problems originating in music have inspired mathematicians for centuries. Perhaps the most famous of these problems is the vibrating string, which pitted some of the greatest mathematicians of the eighteenth century against each other in a debate that lasted more than fifty years and that eventually led to the development of post-calculus mathematics. Other highlights in the book include a comparison between meter in music and metric in geometry, complete with examples of rhythmic patterns from Bach to Stravinsky, and an exploration of a suggestive twentieth-century development: the nearly simultaneous emergence of Einstein's theory of relativity and Schoenberg's twelve-tone system.Weaving these compelling historical episodes with Maor's personal reflections as a mathematician and lover of classical music, Music by the Numbers will delight anyone who loves mathematics and music.


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How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music explicitly based on mathematical principles. Yet Eli Maor argues that music has influenced math at least as much as math has influenced music. Starting with Pythagoras, proceeding through the work of Schoenberg, and ending with contemporary string theory, Music by the Numbers tells a fascinating story of composers, scientists, inventors, and eccentrics who played a role in the age-old relationship between music, mathematics, and the sciences, especially physics and astronomy. Music by the Numbers explores key moments in this history, particularly how problems originating in music have inspired mathematicians for centuries. Perhaps the most famous of these problems is the vibrating string, which pitted some of the greatest mathematicians of the eighteenth century against each other in a debate that lasted more than fifty years and that eventually led to the development of post-calculus mathematics. Other highlights in the book include a comparison between meter in music and metric in geometry, complete with examples of rhythmic patterns from Bach to Stravinsky, and an exploration of a suggestive twentieth-century development: the nearly simultaneous emergence of Einstein's theory of relativity and Schoenberg's twelve-tone system.Weaving these compelling historical episodes with Maor's personal reflections as a mathematician and lover of classical music, Music by the Numbers will delight anyone who loves mathematics and music.

30 review for Music by the Numbers: From Pythagoras to Schoenberg

  1. 5 out of 5

    Jake

    Unclear what the book wished to be based on its contents but I found it to be a pretty superficial overview of the relationship between math and music. There is, somewhere out there, a hypothetically beautiful book relaying a history of how the two subjects elucidated one anther's harmonies. This though - was not that book. While there was a vague construct of a history - it was at most superficial. I sadly must extend my view that this issue of superficiality exists within the texts philosophic Unclear what the book wished to be based on its contents but I found it to be a pretty superficial overview of the relationship between math and music. There is, somewhere out there, a hypothetically beautiful book relaying a history of how the two subjects elucidated one anther's harmonies. This though - was not that book. While there was a vague construct of a history - it was at most superficial. I sadly must extend my view that this issue of superficiality exists within the texts philosophical elaborations, math, and exposition on music. Beyond that, it came off as a quite tangential text. I expect though that the author is a pretty smart guy, and as such I will move on to his other writings. But, I am sad to say that I did not enjoy this book. He ended a book with a quip on string theory. At least -in his eyes - ol Pythagoras had real (verifiable*) strings to play with was silly. (Oy) Perhaps this will make a good intro book on the relationship between the subjects of math and music. Or rather to simply there exists one at all. All and all I was unimpressed. Recommended for : -Those curious about the relationship between number and music *Shout out to popper

  2. 5 out of 5

    Fernando del Alamo

    Este es un librito escrito por un apasionado tanto de la física como de la música, intentando explicar la relación de una con la otra. No es un libro fácil de leer, pues el nivel no es bajo. Explica cómo salieron las notas, los estilos de los diferentes autores en la historia, los principios físicos y matemáticos (Fourier, por supuesto), etc. También habla del oído absoluto, cosa que sorprende, pues comenta que obras en diferentes tonos no suenan igual para los que lo tienen. Muy interesante, per Este es un librito escrito por un apasionado tanto de la física como de la música, intentando explicar la relación de una con la otra. No es un libro fácil de leer, pues el nivel no es bajo. Explica cómo salieron las notas, los estilos de los diferentes autores en la historia, los principios físicos y matemáticos (Fourier, por supuesto), etc. También habla del oído absoluto, cosa que sorprende, pues comenta que obras en diferentes tonos no suenan igual para los que lo tienen. Muy interesante, pero sólo lo recomendaré si sabes algo de solfeo y tienes claro qué son las series de Fourier. Si no, te pierdes.

  3. 4 out of 5

    Rachel Pollock

    Interesting academic (yet readable) book on the crossovers of music, mathematics, physics, acoustics, history, the Enlightenment, etc. You don’t have to be well-versed in, say, advanced calculus to follow along, but if geometry and algebra are daunting, give this a pass. Me, I enjoyed it.

  4. 4 out of 5

    Svalbard

    Da parte di un professore israeliano di storia della matematica, un libretto agile e comprensibile, oltre che assai bene informato - e soprattutto molto “umile”, nel senso che non si prefigge di dimostrare nessuna tesi - sui legami tra musica, fisica e matematica e la loro storia nel corso del tempo. Si parla ovviamente di tutti gli scienziati e i musicisti che si sono dati da fare ad indagare la questione - per primo Pitagora, poi molti altri - dei principi fisici del suono, del sistema tempera Da parte di un professore israeliano di storia della matematica, un libretto agile e comprensibile, oltre che assai bene informato - e soprattutto molto “umile”, nel senso che non si prefigge di dimostrare nessuna tesi - sui legami tra musica, fisica e matematica e la loro storia nel corso del tempo. Si parla ovviamente di tutti gli scienziati e i musicisti che si sono dati da fare ad indagare la questione - per primo Pitagora, poi molti altri - dei principi fisici del suono, del sistema temperato, dei presupposti acustici della scala diatonica, eccetera. Molto interessante il capitolo dedicato a Schoenberg e ad Einstein, alla corrispondenza delle intenzioni del primo di scardinare il sistema tonale togliendo tutti i punti di riferimento delle tonalità, mentre il secondo faceva lo stesso con la fisica newtoniana - ovviamente non è che ci fosse un’intenzione o una complicità, ma la coincidenza degli eventi dice molto sul momento storico e sulla percezione di sé dell’essere umano nell’universo. E anche il fatto che da un lato si continui ad ascoltare musica tonale - lo stesso Schoenberg avrebbe detto, in tarda età, che la tonalità di do maggiore ha ancora molto da dire, e le sue composizioni ancora oggi più ascoltate sono o quelle giovanili, ancora intrise di orchestralità wagneriana, o quelle tardive, non più dodecafoniche - e dall’altro a ragionare in termini newtoniani, significa che se queste sono le leggi della “nostra parte di universo”, come diceva Battiato, in qualche modo ci tocca seguirle. Inoltre l’autore esprime parecchi dubbi in merito al fatto se sia vero che le tonalità - do maggiore, re maggiore, ecc. siano necessarie e se sia vero che tonalità diverse possano esprimere emozioni diverse, come piaceva pensare ai romantici, soprattutto considerato che i portatori di “orecchio assoluto” sono molto rari. (Personalmente ritengo che il dubbio di Maor sia più che legittimo, ma per quello che capisco di musica - lui ammette di capirne poco - ritengo che la diversa espressività delle diverse tonalità, se esiste, dipende dall’uso che se ne è fatto nel corso del tempo, e del diverso comportamento degli strumenti in funzione dell’altezza assoluta del loro suono, come peraltro lui stesso ipotizza. Questo ha evidentemente creato dei “luoghi comuni espressivi” che probabilmente associamo a determinate composizioni e non ad altre; e in qualche modo ciò viene percepito anche senza conoscere nettamente la tonalità della composizione. Inoltre ogni tonalità tende ad essere modulata più agevolmente in altre tonalità più o meno vicine o lontane e non in altre, ed essendo la modulazione qualcosa di più complesso e riconoscibile (anche inconsciamente, per chi non è un addetto ai lavori) del semplice suono privo di riferimenti “relativi”, in qualche modo “colora” e “connota” l’impianto tonale di partenza. Ovviamente da questo ad affermare che una tonalità sia “olimpica e serena” e l’altra “tragica” ce ne corre. Sebbene ci siano parecchie formule ed equazioni, il libro è accessibile anche a chi non ha conoscenze matematiche particolarmente approfondite. Ho solo trovato un clamoroso errore a pagina 152, dove si dice che le trombe sono intonate in si diesis. Se fosse così, esse suonerebbero in… do maggiore, proprio per via del sistema temperato nella spiegazione del quale l’autore si è dilungato, giustamente, per molte pagine, e pertanto non sarebbero strumenti traspositori. Sicuramente intendeva si bemolle, anche perché afferma che hanno due alterazioni in chiave, e la tonalità, del tutto teorica, di si diesis di alterazioni ne avrebbe ben di più. Ma penso che sia un errore di stampa o di traduzione.

  5. 4 out of 5

    Daniel

    Relevant to my interests in many ways. Much was review for me, having studied basic music theory formally & informally + having a renewed interest in applied maths more recently. But much was also new & illuminating. The musical maths of the eponymous bookends (Pythagoras & Schoenberg) were already particularly well-worn territory, but much of the middle stuff was not: * 18th Century physical / musical / mathematical string debates of math / stats luminaries: Bernoulli, Euler, D'Alembert & Lagran Relevant to my interests in many ways. Much was review for me, having studied basic music theory formally & informally + having a renewed interest in applied maths more recently. But much was also new & illuminating. The musical maths of the eponymous bookends (Pythagoras & Schoenberg) were already particularly well-worn territory, but much of the middle stuff was not: * 18th Century physical / musical / mathematical string debates of math / stats luminaries: Bernoulli, Euler, D'Alembert & Lagrange * The ear's ability to perform Fourier transformations, decomposing complex tonalities into their constituent fundamentals (to the shame of the eye, which can only see blended color frequencies but never dissect them -- e.g. we see "green", not "blue & yellow simultaneously") * Schoenberg as the 'Einstein of music', unmooring the discipline from its stable referents & into the relative sea (recalling an earlier reading of mine) It's a brief survey, necessarily shallower, narrower & less complete than a proper study, but a pithy & substantive foray nonetheless. Also good looking.

  6. 4 out of 5

    Edoardo Casali

    Eccellente e scorrevolissimo libro particolarmente indicato a chi fosse interessato alla storia degli intervalli musicali (ossia dei rapporti matematici che li mettono in relazione fra di loro), alla storia e alla teoria del temperamento equabile e alla relazione del pitagorismo con la teoria musicale. Interessante anche il capitolo biografico su Schönberg e Einstein. L'edizione italiana presenta alcuni errori, trascurabili. Eccellente e scorrevolissimo libro particolarmente indicato a chi fosse interessato alla storia degli intervalli musicali (ossia dei rapporti matematici che li mettono in relazione fra di loro), alla storia e alla teoria del temperamento equabile e alla relazione del pitagorismo con la teoria musicale. Interessante anche il capitolo biografico su Schönberg e Einstein. L'edizione italiana presenta alcuni errori, trascurabili.

  7. 4 out of 5

    Helen

    MUSIC GRAPHS ARE SO PRETTY THE BOOK IS SO PRETTY MATHS BEAUTIFUL

  8. 4 out of 5

    Steven Abra

    This might be interesting to people with strong math background and significant interest but little knowledge about classical music, however, reading it from a musician's perspective, I found the musical substance of this to be mostly pretty unrewarding. (And the mathematical parallels largely went above my head.) The musical examples were mostly pretty pedestrian, or oversimplified, or just not that accurate. (For instance - repeatedly citing Mahler as an important progressive figure in the tra This might be interesting to people with strong math background and significant interest but little knowledge about classical music, however, reading it from a musician's perspective, I found the musical substance of this to be mostly pretty unrewarding. (And the mathematical parallels largely went above my head.) The musical examples were mostly pretty pedestrian, or oversimplified, or just not that accurate. (For instance - repeatedly citing Mahler as an important progressive figure in the transition away from traditional tonal harmony, on the same footing at Wagner or Schoenberg, but completely ignoring Débussy's (incomparably more important than Mahler's) role in that change.) It's probably a great source of conversation topics for academic mathematicians at parties, but not so great for people who don't fit in that niche.

  9. 5 out of 5

    Andrea

    Really good book. Maor explores the relationships between Mathematics and sounds/music with a lot of backgrounds. It's pretty satisfying especially if you can handle some musical instruments (I actually play guitar, bass, piano and I'm moderately expert in music theory) but, to be perfectly honest, there are plenty of free YouTube videos that do the same thing this book does in a more interesting way. Nevertheless I still think this little book deserve five stars. Once your pc is shut off, you s Really good book. Maor explores the relationships between Mathematics and sounds/music with a lot of backgrounds. It's pretty satisfying especially if you can handle some musical instruments (I actually play guitar, bass, piano and I'm moderately expert in music theory) but, to be perfectly honest, there are plenty of free YouTube videos that do the same thing this book does in a more interesting way. Nevertheless I still think this little book deserve five stars. Once your pc is shut off, you still can open the book. And grab a guitar and understand how math rules!

  10. 4 out of 5

    Devero

    Un libro non proprio semplice, specialmente se uno la musica la ha, poco, studiata solo alle elementari e medie. Sarebbe piaciuto maggiormente al mio cugino musicologo e organista. Di certo solleva qualche riflessione sull'insegnamento della musica in Italia, anche se non si parla mai dell'Italia, e sul valore che la musica ha nella formazione della mente. Io ho apprezzato soprattutto l'ultimo capitolo, con il parallelismo tra Einstein e Schoemberg. Inoltre è un bene che si viva nell'età di Youtub Un libro non proprio semplice, specialmente se uno la musica la ha, poco, studiata solo alle elementari e medie. Sarebbe piaciuto maggiormente al mio cugino musicologo e organista. Di certo solleva qualche riflessione sull'insegnamento della musica in Italia, anche se non si parla mai dell'Italia, e sul valore che la musica ha nella formazione della mente. Io ho apprezzato soprattutto l'ultimo capitolo, con il parallelismo tra Einstein e Schoemberg. Inoltre è un bene che si viva nell'età di Youtube: molte, se non tutte, le musiche e i pezzi citati come esempio si possono comodamente ascoltare mentre si legge il capitolo in cui se ne parla.

  11. 4 out of 5

    Sebastian Kluge

    Not very engagingly written unfortunately, it's basically a 150-page summary of the main math-music-linkages in this interdisciplinary history. I wish Maor would've added more personal elements in his book, like in the last chapter, where he describes his experience when looking for Schoenberg CDs in his library and only finding one compilation in the catalogue. Imagine you are looking for books on Picasso or James Joyce and only find a single book in your library's catalogue! Absolutely unimagi Not very engagingly written unfortunately, it's basically a 150-page summary of the main math-music-linkages in this interdisciplinary history. I wish Maor would've added more personal elements in his book, like in the last chapter, where he describes his experience when looking for Schoenberg CDs in his library and only finding one compilation in the catalogue. Imagine you are looking for books on Picasso or James Joyce and only find a single book in your library's catalogue! Absolutely unimaginable! Yet here lies the story of how 20th century modern music has been culturally neglected by culture itself.

  12. 4 out of 5

    Adrián M.

    El propósito del libro me atrajo sobremanera: conectar la historia de las matemáticas, de la física y de la música, y dilucidar las posibles influencias entre las tres disciplinas. Desgraciadamente, todos los eventos, descubrimientos y creaciones se tratan de forma muy superficial, por lo que la obra se acerca más a una simple y llana enumeración que a una explicación detallada de los episodios que va mencionando.

  13. 5 out of 5

    Luis González Ricardo

    Atractiva monografía que discute los paralelismos e interacciones entre música, matemática y física. El centro del libro es relacionar los aportes de A. Einstein a la física y los de A. Schoenberg a la música. Se requiere unos conocimientos mínimos de matemática para disfrutar el texto, pero no tanto de música, pues sería ventajoso conocer los rudimentos de la lectura de partituras.

  14. 4 out of 5

    Tomáš

    This book was both easy and difficult to read. I have somewhat of a mathematical/physics background, and this book has shed a lot of light on topics I thought I understood before. It also brushes up on both mathematics and music history. I would highly recommend this book to anyone interested in mathematics, but I'd suggest having at least basic knowledge of musical notation and terminology. This book was both easy and difficult to read. I have somewhat of a mathematical/physics background, and this book has shed a lot of light on topics I thought I understood before. It also brushes up on both mathematics and music history. I would highly recommend this book to anyone interested in mathematics, but I'd suggest having at least basic knowledge of musical notation and terminology.

  15. 4 out of 5

    Jo

    This is an interesting exploration of the history of the mathematics of music. It does need quite a lot of mathematical knowledge to be able to understand - the musical elements are much better explained for a non-musician than the mathematical elements are for a non-mathematician!

  16. 4 out of 5

    Chris Vig

    This book will be exceptionally interesting a specific cross-section of readers who are interested in both music and mathematics/engineering. That said, like the author somewhat self-consciously notes, there’s probably too much math for musicians and too much music for mathematicians.

  17. 4 out of 5

    Carlos Vargas

    I mainly pick this book for the cover, which arise from a current research. While the mathematics are easy to understand, as well as the history of some of the greatest mathemathicians, the musical concepts are not for an easy reading if you are not a musician (such is my case).

  18. 4 out of 5

    Alexander, Roi du Campgrain

    This book is unsure of what it wants to be.

  19. 4 out of 5

    Andrej Badilla Solano

    Por momentos el autor se concentra más en la matemática que en la música.

  20. 4 out of 5

    Lan

    A combination of music history and math history. It is interesting to end the book with the juxtaposition of Einstein's theory of relativity and Schoenberg's twelve-tone music. A combination of music history and math history. It is interesting to end the book with the juxtaposition of Einstein's theory of relativity and Schoenberg's twelve-tone music.

  21. 5 out of 5

    Jesus Saenz

    Interesting parallels between the developments of physics and music. Posses interesting questions about the psychological aspects of tonality.

  22. 5 out of 5

    Richard

    a valuable survey.

  23. 5 out of 5

    Arend

    Diverting, interesting at times, but I had been hoping for something more substantial or insightful.

  24. 5 out of 5

    Candy_K

    Attempt to say everything but ended up with nothing deeper.

  25. 4 out of 5

    Francesco Marcolini

    Probabilmente uno dei migliori libri su questo argomento. L'autore dimostra una straordinaria competenza sia in campo musicale che in campo fisico/matematico. Probabilmente uno dei migliori libri su questo argomento. L'autore dimostra una straordinaria competenza sia in campo musicale che in campo fisico/matematico.

  26. 4 out of 5

    Yee-Ning

    It's ironic that I'm giving this a 3 star because I think the author knows exactly what people would complain about--if you're mathematically minded, this has not enough math. If you're musically minded, this has too much math. I was trying to explore more of the mathematical side of things and, while this gives a nice jumping off point to intrigue those who may not have considered the ties between music and math, it did not delve into enough of the mathematics for my want. Again, I probably shou It's ironic that I'm giving this a 3 star because I think the author knows exactly what people would complain about--if you're mathematically minded, this has not enough math. If you're musically minded, this has too much math. I was trying to explore more of the mathematical side of things and, while this gives a nice jumping off point to intrigue those who may not have considered the ties between music and math, it did not delve into enough of the mathematics for my want. Again, I probably should change this to something more generous because maybe I am not quite the target audience, but oh well. A good survey if you wonder about the historical interplay between music and math.

  27. 4 out of 5

    Maurizio Codogno

    Si dice sempre che la musica è matematica. Ma è proprio vero? In questo libro Eli Maor prova a dare una risposta vedendo quello che è successo con i matematici che si occuparono di musica, a partire da Pitagora per arrivare a Schönberg (o Schoenberg, come preferì farsi chiamare dopo che ottenne la cittadinanza statunitense). Spero di non fare uno spoiler se vi dico che la risposta è negativa: i matematici hanno trovato tante regole matematiche che i musicisti hanno bellamente ignorato. Non è poi Si dice sempre che la musica è matematica. Ma è proprio vero? In questo libro Eli Maor prova a dare una risposta vedendo quello che è successo con i matematici che si occuparono di musica, a partire da Pitagora per arrivare a Schönberg (o Schoenberg, come preferì farsi chiamare dopo che ottenne la cittadinanza statunitense). Spero di non fare uno spoiler se vi dico che la risposta è negativa: i matematici hanno trovato tante regole matematiche che i musicisti hanno bellamente ignorato. Non è poi così strano: non tutte le strutture matematiche si applicano allo stesso modo, e soprattutto l'orecchio vuole anche una metastruttura, il che può spiegare perché la musica dodecafonica - che Maor ritiene "locale", con connessioni temporali limitate - non ha mai preso davvero piede nemmeno tra i musicisti. Ma anche Bach ha avuto un periodo di un secolo di oblio proprio perché "troppo matematico"... insomma bisogna trovare un equilibrio che dipende anche dal tempo. Purtroppo la parte più strettamente musicale è molto meno approfondita: mi sono per esempio stupito della mancanza di un capitolo sui vari temperamenti che sono solo stati accennati (e con l'usuale errore di pensare che il Clavicembalo ben temperato fosse pensato per il temperamento equabile e non per uno dei Werckmeister). Un'ultima nota per chi musicista non è: i si diesis delle pagine 152-154 sono in realtà bemolli.

  28. 4 out of 5

    Ele Ran

  29. 4 out of 5

    Fiona

  30. 4 out of 5

    William Thomas

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