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MATHEMATICS and MUSIC Cheryl Periton Links Composers, Their Music, the Fibonacci Sequence, and the Golden Ratio

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MATHEMATICS and MUSIC Cheryl Periton Links Composers, Their Music, the Fibonacci Sequence, and the Golden Ratio © ATM 2013 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. MATHEMATICS AND MUSIC Cheryl Periton links composers, their music, the Fibonacci Sequence, and the Golden ratio hen thinking about cross-curricular Intrinsically there is however a philosophical themes how many of us consider, or difference between what we regard as just sound W are even really aware, of the links and what we regard as music. between mathematics and music? Current But, how do we view mathematics? initiatives focus on STEM yet using art or music Many people regard mathematics as rational and could be a way of hooking in some of those abstract; the manipulation of numbers and the students for whom the word “mathematics” performing of calculations; an objective subject results in a rolling of the eyes and a “shutdown rather than one linked to those emotions and mode” ensuing. feelings of music. In this article I would like to give an introduction Leibniz (1646 – 1716) said that; to one aspect of the links between what many of our students would seem to be two distinct and “Music is the pleasure the human soul completely unrelated subjects. In doing so it is experiences from counting without being important therefore to initially define what we mean aware that it is counting”. by mathematics, and what we mean by music. Above the door of Plato’s academy were the To many people music is a powerful form of non- words “Let no one ignorant of mathematics enter verbal communication that reaches past human here”. This together with his earlier quote about intellect directly into the soul. It engages directly music clearly links the two subjects. with our emotions; it is subjective in that we do not It is not just mathematicians that see these all respond in the same way to a piece of music. links. The 20th century Russian composer Igor This was the view propounded by Plato Stravinsky said; (428 -348 BCE) who said that “Musical form is close to mathematics – not perhaps to mathematics itself, but certainly “Musical training is a more potent instrument to something like mathematical thinking and than any other,because rhythm and harmony relationship”. find their way into the inward places of soul, on which they mightily fasten, imparting As the Golden Ratio (phi) is considered as grace, and making the soul of him who is being an exemplification of beauty in art and rightly educated graceful...” architecture, I am going to consider the ways in which the Fibonacci Sequence and Golden Ratio Others would describe it as sound that is can also be applied to music. organized in a meaningful way with rhythm, The Fibonacci sequence of numbers is melody and harmony. This definition would, for 1, 1, 2, 3 , 5, 8, 13 in which subsequent terms some, exclude rap music, which has rhythm but are formed by adding the previous two, and if not melody and harmony, a rhythmic rendition of one finds the ratio of terms one gets poetry maybe? 1, , , , , , , ……, with these ratios http://www.youtube.com/watch?v=lb4AEBWZx9g tending to a limit of 0.61803398…. , which is phi, or the work of composers such as John Cage the Golden Ratio. who also has a work entitled 4’33 of silence. Researchers such as Olsen suggest that http://www.youtube.com/watch?v=Pbgr74yNM7M Fibonacci and phi are often found in the timing of musical compositions. As an example the climax If we take the definition of music to encompass of songs is often found at roughly the phi point rhythm, melody and harmony, then a subsidiary (61.8%) of the song, as opposed to the middle question needs to be asked as to what is sound, or end of the song so in a 32 bar song this would and why is it not music? occur in the 20th bar. In lay terms sound can be defined as being a In the Hallelujah Chorus from Handel’s Messiah, form of energy that is perceived by our ears, this which lasts 94 bars, one of the most important energy being produced when air is set in motion points is the “King of Kings” which happens in by any means whatsoever. The mathematics behind this motion and the effects upon the bars 57 to 58, that is after about of the whole. sounds we hear could form a cross-curricular Similarly, after of the first 57 bars we have the investigation linked to the STEM agenda. entrance of the theme “The kingdom of glory”, Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY 24 Mathematics Teaching 233 Journal of the Association of Teachers of Mathematics © ATM 2013 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. MATHEMATICS AND MUSIC making another essential point. A third example Numbers or Fibonacci sequence for example, being the placement of “and he shall reign.” Doctor Steel http://www.youtube.com/watch?v=76RrdwElnTU http://www.youtube.com/watch?v=DGlr9iNBejA Fibonacci intervals, counting in semitones form In addition to composers using the Fibonacci the basis of the 3rd movement of Bela Bartok’s sequence (in) directly in their compositions “Sonata for Two Pianos and Percussion”. the sequence can also be seen in musical instruments as musical scales are based on Fibonacci numbers. There are 13 notes in the span of any note through its octave – 8 white keys and 5 black, with the black keys being arranged as 2 and 3. http://www.youtube.com/watch?v=8Lb0hgEsO3M Whilst the opening xylophone passage in the 3rd movement of Bartok’s “Music for Strings, Percussion and Celesta” uses Fibonacci rhythms such as : 1: 1: 2 : 3; 5 : 8 : 5 : 3 : 2 : 1: 1 http://www.youtube.com/watch?v=Bd-2Yfhy-LE In almost all of Mozart’s piano sonatas the relation between the exposition, development, and recapitulation conforms to the golden proportions. The first section of the sonata is the development and representation of the theme, the second is a visitation of the theme in different variations with the first section shorter than the second, with the division being in the golden ratio. Debussy used the Fibonacci numbers in the A scale is composed of 8 notes, of which the organization of the sections in the music of 3rd and 5th create the basic foundation of all the “Image, Reflections in Water”, in which the chords. The dominant note in the scale is the 5th sequence of keys is marked out by the intervals note of the major scale, which is also the 13th of 34, 21, 13 and 8. all the notes that comprise the octave. http://www.youtube.com/watch?v=p93fAq-qTol And, it is not only the piano that the geometric Other composers such as the Polish composer relationship that is the Golden Ratio can be Krzsystof Meyer structured work using Fibonacci seen. numbers such as “Trio for clarinet cello and piano.” Fibonacci and phi are used in the design of It is not only classical composers for whom the violins. Fibonacci numbers have significance. The American rock group “Tool” wrote a song entitled “Lateralus” that lasted 9 mins and 24 seconds which relies heavily on the Fibonacci sequence, featuring the sequence symbolically in the verses of the song. The syllables in the first verse count 1,1, 2,3,5,8,5,3,13,8,5,3. Missing bits of 2,1,1,2,3,5,8 are filled in during the second verse. The time signatures of the chorus change from to to . The drummer, Danny Carey is quoted as saying that the piece was originally entitled 9-8-7 for the time signatures, this is fit- ting as 987 is the 16th number in the Fibonacci sequence! http://www.youtube.com/watch?v=wS7CZIJVxFY Others have recorded songs entitled Fibonacci Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY March 2013 www.atm.org.uk 25 © ATM 2013 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. MATHEMATICS AND MUSIC http://www.youtube.com/watch?v=rEa8q7YUxVo So… what conclusions can we draw? Music and mathematics are inextricably linked. The Golden Section, seen in art and architecture as giving an impression of beauty and harmony, is equally applicable to musical composition. The mathematics of this underpins something that touches our emotions and reaches the very core of our humanity. .618 Mathematics is not just a tool, it is beautiful, it is 1 : a language with its own syntactic structure that Fibonacci and phi are used in the design of gives us a way of describing the world around us. violins. What better way could there be of enthusing our students than by allowing them to consider The guitar also contains the golden section. Fibonacci numbers in this way? Students could create their own songs and compositions either by using Fibonacci rhythms in the way the music is played, or in the way that the song is sung. Cheryl Periton, Senior Lecturer at Canterbury Christ Church University Use the red QR code to download the article onto your handheld device so that the web-sites can be accessed as you read. Yorgos Kertsopoulos interpreting on a classical guitar respecting “The Kertsopoulos mathematical model of the guitar”: Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY 26 Mathematics Teaching 233 Journal of the Association of Teachers of Mathematics The attached document has been downloaded or otherwise acquired from the website of the Association of Teachers of Mathematics (ATM) at www.atm.org.uk Legitimate uses of this document include printing of one copy for personal use, reasonable duplication for academic and educational purposes.
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