Ratios Between Keys on a Piano

Date: 11/05/97 at 19:59:18
From: Michael M. Wiseman
Subject: Music: Formula to figure mathematical relations between keys 
on a 
piano

I know that there is a formula to describe a ratio from one key to 
another but I can't find it. I tried to calculate it using trial and 
error but I could only get it precise to 10 digits. My  daughter wants 
to explain the formula for a science project.

Thanks

Date: 11/06/97 at 14:00:57
From: Doctor Tom
Subject: Re: Music: Formula to figure mathematical relations between 
keys on 
a piano

Hi Michael,

The answer, of course, is, "it's complicated."

There is an easy answer that isn't quite right but is pretty close.  
Each time you go up an octave, the frequency has to double, and there 
are 12 half-steps to go up an octave:

  A -> A sharp -> B -> C -> C sharp, etc.

Twelve equal multiplicative steps have to amount to a factor of 2, so 
the 12th root of 2 is the answer. My calculator gives it as 
approximately:  1.05946309435929526455

So if A is 440 cycles/second, A sharp should be 440*1.059... cycles 
per second, and so on.

But musicians also like to talk about "thirds" and "fifths" and stuff 
like that. These correspond, physically, to the ratio of the 
frequencies of a stretched wire and of a wire 1/3 or 1/5 as long.  
These are close to the appropriate products of 1.059... by itself, but 
not exact.

So pianos are usually tuned to a "well-tempered" scale that's a sort 
of compromise between the mathematical ratios and the Pythagorean 
ratios that would give exact thirds and fifths. The well-tempered 
scale is purported to sound "good," but I don't have a terribly good 
musical ear, so I'm not one to judge.

In the old days, without high-tech tone generators that could make any 
frequency desired, piano tuners had a tuning fork (usually at A, or 
440 cycles/second) from which they'd get an exact tuning of one key.  
Then they'd work back and forth, starting with pure thirds and fifths, 
but using an approximation procedure that gradually zeroed in on a 
well-tempered scale. It used to be quite an art.

See if your daughter can dig up an old book on piano tuning - it's
really eye-opening.

Good luck.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/