Bach and Mathematics
Date: 04/16/97 at 01:01:38 From: Mary Kane McAuslan Subject: Mathematics and music My son is in an advanced math program at his school (he's 10) and is trying to do a project for a math fair showing relationships between math and music. He has one good reference book that shows Fibonacci numbers in an octave on a piano keyboard. He is also looking at geometric transformations, such as translations and inversions, that occur in music as well as showing how music is divided into beats and measures and how musical notes are like fractions of measures. I have been trying to find more references for him but have been unsuccessful and thought your site might be able to lead us to some good resources. If you have any ideas, we would appreciate hearing from you. Thanks
Date: 04/16/97 at 03:48:16 From: Doctor Mitteldorf Subject: Re: Mathematics and music Dear Mary, This is quite an ambitious project already for a 10-year-old. Something you might find interesting is the story of Bach and the Well-tempered Klavier. Before Bach's time, the tuning on the piano made a perfect 5th exactly equal to a ratio of 3/2. A major third was 5/4 and a minor third 6/5.Bach noticed that this meant that some degrees of the scale were slightly larger than others, so when you wanted to play a piece in D on your harpsichord, you had to tune it differently from if you played in C. He proposed a fix: make equal intervals between each pair of adjacent notes on the piano. The trick is that it's not equal intervals in number of beats per second - rather it's equal _ratios_ between notes or equal _logarithmic_ intervals. The octave was always a factor of 2, before Bach's time and after. Since there are 12 intervals beteen a C and a C an octave above, the way to make all intervals equal is to make them all the twelfth root of 2. Now it happens that the twelfth root of two, multiplied together 7 times, makes 1.498, so that a perfect fifth is still very nearly 3/2. The major third and minor third are a little further from the "harmonic" values. Maybe working out these values would make a good addition to your son's project. Certainly understanding the concept of "equal logarithmic intervals" presents a challenge that many adults can't rise to! -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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