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The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Chandrasekhar, S. Shakespeare, Newton and Beethoven or patterns of creativity. Current Sci. 70 (1996), no. 9, 810--822. SC: 01A99, MR: 1 387 202.
Discusses the creative lives of Shakespeare, Newton, and Beethoven. The example of Newton contrasts with the other two, particularly in how old they were when they did their most creative work. While the best work of poets is often later in life, G. H. Hardy tells us that the best work of mathematicians is generally when they are young. Chandrasekhar gives the additional examples of the mathematicians or scientists James Clerk Maxwell, George Gabriel Stokes, and Albert Einstein. Lord Rayleigh's example is different, and gives us a possible explanation of the differences we've seen. In the words of J. J. Thomson, "There are some great men of science whose charm consists in having said the first word on a subject, in having introduced some new idea which has proved fruitful; there are others whose charm consists perhaps in having said the last word on the subject, and who have reduced the subject to logical consistency and clearness. I think by temperament Lord Rayleigh belonged to the second group." Chandrasekhar then discusses the importance of beauty to mathematics and science, and concludes with statements of scientists and poets on one or the other of the two disciplines (some comments are more favorable than others). Closely related topics: Creativity, Shakespeare, Isaac Newton (1642-1727), and Beethoven.
Grattan-Guinness, I. Mozart 18, Beethoven 32: hidden shadows of integers in classical music. History of mathematics, 29--47, Academic Press, San Diego, CA, 1996. SC: 01A99 (00A69), MR: 97a:01075.
Discusses number symbolism in the works of Mozart and Beethoven. With Mozart, discusses in particular Die Zauberflöte and the last three symphonies (and particularly the Symphony in g of 1788). There is also some evidence that Mozart used gematria. Literary sources also attest to Mozart's interest in numerology. With Beethoven, focuses primarily on Piano Sonata op. 111 (no. 32), the Diabelli Variations, and the Missa Solemnis. The choice of opus numbers themselves appear to show an interest in numerology. The author suggests that some knowledge of the history and conventions of numerology would be useful before reading this article. The author's own article in the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences may be useful in this regard. The author also suggests some avenues for future research. Closely related topics: Music, Numerology, Gematria, Wolfgang Amadeus Mozart (1756-1791), and Beethoven.
Grattan-Guinness, I. Some numerological features of Beethoven's output. Ann. of Sci. 51 (1994), no. 2, 103--135. SC: 01A99 (00A69), MR: 1 278 119.
The author discusses possible occurrences of number symbolism in Beethoven's compositions. A large number of examples are used to buttress his arguments, and some prior familiarity with Beethoven's work might be useful. In some cases, numbers occur as the number of measures or notes of a them or motif, and in other cases in Beethoven's choice of opus numbers. (In contrast with the common practice of the time, Beethoven chose his opus numbers himself, and the numbers chosen could at times be seriously at variance with the order of composition.) The author's conclusions have been controversial, partly because Beethoven has often been regarded as being quite poor at arithmetic. The author discusses this objection and aspects of methodology in some detail. Closely related topics: Numerology, Music, and Beethoven.
Scriba, Christoph J. Mathematics and music. (Danish) Normat 38 (1990), no. 1, 3--17, 52. SC: 01A99 (00A69), MR: 91i:01154.
The author discusses the relationship between mathematics and music from Pythagorean through modern times. His story begins in in Pythagorean times, and as he explains, the notes of the musical scale were then determined by the ratio of a perfect fifth, i.e. 3:2. Twelve intervals of a fifth are roughly equal to seven octaves, but are in reality slightly more than seven octaves, the discrepancy being the "Pythagorean comma" of 312:219, or roughly 74:73. Whole steps in the scale were in the ratio 9:8, and half steps were in the ratio 256:243. Thus two half steps were slightly less than one whole step. In fact, Philolaus noted that one whole note is equal to two half notes plus a Pythagorean comma. Archytas showed that intervals like the octave 2:1, fifth 3:2, fourth 4:3, and whole tone 9:8, or any other interval in the ratio (n+1):n cannot in fact be divided with rational numbers into two equal intervals. However, he noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5. 5:4 can be thought of as a major third, and 6:5 can be though of as a minor third. So the ratio 3:2 is divided as 6:5:4. Similarly, the fourth of 4:3 can be divided into the product of 7:6 and 8:7, so the ratio 4:3 is divided as 8:7:6. The interval 7:6 can be though of as a shrunken minor third and 8:7 can be though of as an enlarged whole tone. Scriba suggests that the germs of the idea of making this division lie with the Babylonians.In the Renaissance, the musical scale was modified to take some of these ideas into account through the work of theoreticians like Ludovico Fogliano and Giusseppe Zarlino. For example, the ratio for the notes E:C and A:F were changed from the Pythagorean 81:64 (two whole tones) to the ratio 5:4. B moved to stay a whole tone of 9:8 above A. Thus the half tones F:E and c:B were now in the ratio 16:15 rather than the Pythagorean 256:243. The whole tones C:D, F:G, and A:B remained in the ratio 9:8, but the whole tones D:E and G:A were now in the ratio 10:9. (It was roughly in the same time interval that intervals of a third began to be considered consonant.) Sharps and flats did not coincide: C sharp and D flat were for example different notes. However, it wasn't long before there were efforts to make a scale of 12 uniform steps. The first to attempt to do so was Galileo Galilei's father, Vincenso Galilei. He tried to make each step of size 18:17, though that of course led to problems. It was Simon Stevin who first had the idea of making uniform steps of size 21/12.
Later on, some mathematicians even began to question the division of the scale into 12 tones, with the idea that a division into a different number of notes might lead to a more perfect representation of the intervals. For example Christiaan Huygens defined a 31-tone system of temperament in his Lettre touchant le cycle harmonique. One source even suggests that this has "led indirectly to a tradition of 31-tone music in the Netherlands in this century". Leonhard Euler's efforts involved an attempt to reconcile the ideal "octave" 2:1 with the ideal "fifth" 3:2. He analyzed the problem by using a continued fraction representation of the ratio log 2:log 3/2. The convergent 12/7 corresponds to the popular division of 7 octaves into a circle of 12 fifths. Other convergents include 17/12, 29/17, 41/24, and 53/31. In the last case, for example, 31 octaves would be divided into 53 fifths. These didn't answer the question of what kind of equally tempered scale best reconciles the intervals of an octave, fifth, and third (2:1, 3:2, and 5:4) simultaneously. This may or may not influence the course of music, but Scriba shows how an algorithm by the Norwegian mathematician Viggo Brun (1885-1978) gives an answer. If the best answers are written in terms of the number of steps in the three intervals, the best approximations are (2,1,1), (3,2,1), (5,3,2), (7,4,2), (12,7,4), (19,11,6), (31,18,10), (34,20,11), (53,31,17), (87,51,28), .... The triple (12,7,4) is the common case with 12 semitones in an octave, 7 in a "major fifth", and 4 in a "major third". As Scriba explains, the case of the 31 tone scale has been especially important historically. In fact, Scriba tells us that it was back in the middle of the 1600s that Nicolas Vicentino described a "archicembalo" with six manuals with the octave divided into 31 parts; as mentioned above, Huygens clarified this. Moreover, Scriba tells us that Zarlino and Salinas shortly thereafter discussed the division of the octave into 19 equal parts. There is apparently an organ built according to the principles of the Dutch physicist D. Fokker (1887-1972) that also divides the octave into 31 parts (it is now in the Teylers Museum in Haarlem). Along a different line, Euler tried to design a mathematical system to quantify the dissonance of chords, but it apparently did not work very well.
The next part of the article discusses some of the work of Wolfgang Graesers (1906-1928), who tried to do a mathematical study of Bach's Art of the Fugue (this was published under the name Bachs "Kunst der Fuge" (German) in the Bach-Jahrbuch 1924, pages 1-104). Here, group theoretic notions reflect the kinds of transformations, such as inversion, that can be used in a fugue. A background in music theory may be useful in understanding Graesers's work.