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### Bach and Mathematics

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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
```

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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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High School History/Biography
High School Physics/Chemistry
Middle School History/Biography

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