European Mathematical Society - 97A30
https://euro-math-soc.eu/msc-full/97a30
enMusic by the Numbers From Pythagoras to Schoenberg
https://euro-math-soc.eu/review/music-numbers-pythagoras-schoenberg
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Music and mathematics have a long joint history. Music theory was part of the Greek quadrivium, and it has been designed and revised by mathematicians including Pythagoras, Simon Stevin, Kepler, etc. Many well known mathematicians were also skilled practitioners of some instrument (Einstein loved his violin, Feynman enjoyed playing the bongos, and Smullyan gave piano recitals,...). Of course several books were written on the subject already. For example D.J. Benson: <em>Music, A mathematical offering</em> (2007) or the monumental two-volumes historical survey by T.M. Tonietti <a href="/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols" target="_blank"><em>And yet it is heard</em></a> (2014). But also G.E. Roberts <em>Music and Mathematics</em> (2016); G. Loy <em>Musimathics: The Mathematical Foundations of Music</em> (2011); D. Wright <em>Mathematics and Music</em> (2009); N. Harkleroad <em>The Math Behind the Music</em> (2006). And the collection of papers J. Fauvel, R. Flood, R. Wilson (eds.) <a href="/review/music-and-mathematics-pythagoras-fractals-0" target="_blank"><em>Music and Mathematics</em></a> (2006), G. Assayag, H.G. Feichtinger (eds.) <em>Mathematics and Music</em> (2002). This is to name just a few. A simple internet search will give many more results.</p>
<p>
Maor is a writer of several popular mathematics books, and, although not a practitioner, he is a lover of music. In this relatively short booklet he draws a parallel between the history of mathematics and the history of music theory. It is again a book on popular mathematics for which no extra mathematics outside secondary school education is needed. However some familiarity with terms from music theory is advised, even though most of these concepts are explained. Maor selects some topics of (historical) interest and sketches evolutions both of mathematical history and of the historical approaches to music theory. Besides the obvious and obligatory topics, and a personal selection of the historical periods, there are also a number of side tracks added as curious anecdotes.</p>
<p>
Maor describes some pillars of the historical bridge that is spanning the wide gap of the eventful evolution of music and math since Pythagoras till our times. The opening chapter is describing the pillar on which that bridge is resting on our side of history. The early 20th century is the scenery where Hilbert challenges the mathematicians with his his list of problems. Solving some of them eventually leads to a crisis in the foundations of mathematics. Physics moves forward to a new era leaving Newtonian mechanics and entering an age of relativity theory. The rigid world of Laplace, acting as a clockwork, becomes a quantum world governed by probabilities. Likewise music changed its face. The fixed tonality, the reference frame, that had been the standard for ages was left and Mahler and Berlioz made this all relative, culminating in Schoenberg's twelve-tone system. This introduction sets the scene where the book will eventually lead to in some grand finale. But first we need to wade through the historical evolution to appreciate the meaning of these revolutionary ideas.</p>
<p>
Maor's guided tour starts at the other pillar of the history bridge at 500 BCE with a (physical) string theory by Pythagoras, defining a scale by introducing an octave, a fifth, and a fourth, which are logarithmic scales long before John Napier conceived logarithms. The Greek vision of a physical world dominated by integers was accepted during many centuries to follow and Galileo and Kepler were still Pythagoreans in this respect adhering to the music of the spheres.</p>
<p>
The Enlightenment was a first breach with the past. Galileo's father Vincenzo Galilei discovered that the pitch of the vibrating string was proportional to the square root of the tension of the string. Galileo in his <em>Dialogues</em> on the `New Sciences' was the first to have the word `frequency' in his book and Mersenne was the first to measure it. Although better known for his prime numbers, he was the first to write a book on vibrating strings: his <em>Harmonie Universelle</em> (1636). Even less known is Joseph Sauveur (1653-1716) who coined the term `acoustics' and who discretized the differential equation describing the vibrating string by considering it as an oscillating string of beads. Of course a true differential equations needs calculus that was being invented by Newton and Leibniz in those days and they have quickly conquered science in many aspects through the work of the Bernoullis (Jacob, Johann, Daniel), Euler, D'Alembert, and Lagrange. The differential equations of a vibrating string was related to music theory and harmonics, but it was only Fourier who finally discovered that almost any periodic function can be written as a sum of sine functions of different frequencies and this defines the acoustic spectrum and generalizes the idea of standing waves or the natural harmonics or overtones of instruments. These were further explored in the acoustic theory in books written by Helmholz in Germany and Rayleigh in Britain.</p>
<p>
The physics being established, Maor returns to music theory. The history of how to subdivide the octave has caused much confusion and disagreement, and has not only defined musical temperament but also heated the temperaments of the protagonists. As a transition to a discussion on rhythm, meter and metric, Maor introduces the tuning fork and the metronome as musical gadgets. When composers started using variable meters, a parallel is drawn with the local metric on Riemannian manifolds, just like Einstein used a local reference system for his relativistic observations. This idea is extended to other disciplines using reference systems such as cartography and the relativistic use of perspective in visual arts as explored in the work of Escher's and Dali.</p>
<p>
That brings Maor back to the nearby pillar of his narrative tension in a chapter where Schoenberg, a contemporary of Einstein, develops his relativistic music in the form of a strict twelve-tone system. However, while Einstein's theory has practical applications still used today, Schoenberg's experiment was less successful and he didn't have many followers. Maor closes the circle completely with some remarks on string theory in current theoretical physics, which of course links up with the strings studied by Pythagoras.</p>
<p>
Most interesting are also some of Maor's excursions on the side (there are five) about the musical nomenclature, the slinky (a periodic mechanical gadget in the form of a spiral that can `walk' down the stairs), some musical items worth an entry in the Guinness Book of Records, the poorly understood intrinsic rules that govern the change of the tonic to different keys, and the <em>Bernoulli</em> (an instrument invented by Mike Stirling with 12 radial strings equally tempered as like on a Bernoulli spiral and that actually looks like a spiral harp).</p>
<p>
Maor is an experienced story teller. His mixture of musical, mathematical, and physical history, enriched with personal experiences and some unexpected links and bridges are nice reading for anybody with a slight interest in music and science. No mathematical training required. Leisure reading. Do not expect deep analysis or high brow theoretical expositions. Just enjoy and let yourself be surprised.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Maor gives a selection of historical parallels that can be drawn between the evolution of mathematics and music theory. From the strings of Pythagoras to the string theory of theoretical physics. His main message is that at some point mathematics and physics have abandoned an overall reference system and accepted local reference frames (think of relativity theory and geometry). At about the same time something similar happened in music theory when keys were no longer maintained over a long time but they became local which has resulted in atonality and Schoenberg's twelve-tone theory.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-17690-1 (hbk); 978-1-400-88989-1 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">24.95 USD (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">176</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/11250.html" title="Link to web page">https://press.princeton.edu/titles/11250.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A30</a></li></ul></span>Tue, 29 May 2018 06:34:03 +0000Adhemar Bultheel48509 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/music-numbers-pythagoras-schoenberg#commentsThe Magic Garden of George B and Other Logic Puzzles
https://euro-math-soc.eu/review/magic-garden-george-b-and-other-logic-puzzles
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book was published in 2007 by <em>Polymetrica</em>, an Italian open access book publisher. However, since it does not seem to exist anymore, it is now made available as a World Scientific publication in 2015.</p>
<p>
In fact it contains two books. The first is a collection of logic and mathematical puzzles of the type that made Smullyan popular among his fans since he published his first collection <em>What is the name of this book?</em> in 1978. The promotion by Martin Gardner made it an instant success and later a dozen similar collections have appeared. Smullyan is a mathematician with a PhD from Princeton (1959) that he prepared under Alonzo Church on GĂ¶delian incompleteness theorems in formal systems. But he is also a gifted piano player, he worked as a magician and he published on Taoism, philosophy and religion. His recreational books may not be as diverse as Martin Gardner's but he has an outstanding ability to bring his logical puzzles and riddles in many variations and in a most entertaining way. Some are easy, but some are quite challenging, even for experienced puzzlers.</p>
<p>
The first book in this collection has 12 chapters where first, if needed, a general setting is explained. For example: Teresa, Thelma, Leila, and Lenore are four sisters the first two always tell the Truth, the other two always Lie. Then there follows a list of problems to solve. For example: If you meet one of them, but don't know which, what statement should she make to convince you that she is Lenore? Each chapter ends with the solutions and the arguments used. Most puzzles are logic, but some of them involve some algebra with integers or even some probability. Truthful to his nature, Smullyan cannot refrain himself from occasionally including a joke here and there, but there are not too many.</p>
<p>
The second book is an introduction to Boolean algebra (the George B of the title is of course George Boole). It starts out in the previous recreational style, but it soon mover to a more formal treatment. So this part is somewhat more demanding for the layman with proper formulas and truth tables, and even definitions and theorems. It starts with a first 'grand problem'. In George's magic garden grow flowers that are either red or blue for a whole day, but they may change their color from day to day. If you pick any three flowers on any day, then if the first two are both blue, the third one is red and if the first two are red, the third is blue. Moreover, for any two distinct flowers, there is at least one day on which their color will differ. If you know that there are between 200 and 500 flowers, how many are there exactly? The puzzle is not trivial, but there is exactly one solution. The problem is solved but only after some exercises with set theory, unions, intersections, complements, etc. which allow to tear it down in several solvable subproblems. Propositional logic and formal Boolean algebra's are the next mathematical fortresses to conquer. The ultimate 'grand problem' is to prove that one has a Boolean algebra once a set of axioms is satisfied.</p>
<p>
Hence what started as some fun problems, that in principle are solvable by anyone, the second book takes a turn towards a much more formal approach and mathematical ingredients which gives a proper introduction to propositional logic and Boolean algebra. The hope is of course that readers attracted by the first part will engage also into the second book if they want to learn a systematic approach to solve the logical problems. I doubt that they will reach the end unless they are strongly motivated since eventually indeed all the 'fun' elements have evaporated and turned into mathematics which, unfortunately is considered the opposite of fun by many. So this is a most remarkable attempt to join the extremes in one volume and hopefully some will be convinced. Of course for the professional, there is not much new in the mathematics, but he or she will certainly enjoy the puzzles. For the student who has to master the mathematics, this may turn out to be a very enjoyable way to get acquainted with the subject, guided by Smullyan, who is the grand master of this kind of entertainment. They will not be deceived. As an introduction to formal logic, his other book <em>Logical labyrinths</em> (A K Peters, 2007) is better and more elaborated. It does not stop after the propositional logic but also includes a further treatment of first order logic.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of logical puzzles combined with an introduction to propositional logic and Boole algebra. It is a book from 2007 that is now made available again.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/raymond-smullyan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Raymond Smullyan</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4678-55-1 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 19.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">180</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/9568" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9568</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03g05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03G05</a></li></ul></span>Tue, 02 Jun 2015 14:55:34 +0000Adhemar Bultheel46250 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/magic-garden-george-b-and-other-logic-puzzles#comments