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/