Math and the Piano

Date: 09/27/98 at 13:17:54
From: Questkid
Subject: Pianos and math

I am doing a school project about math - about how math is important in 
my life - and I chose my piano. Do you have any ideas on how math is 
related to that?

For example, fingering (but please be specific).

Date: 09/28/98 at 11:23:37
From: Doctor Rick
Subject: Re: Pianos and math

Hi. I can think of lots of connections between math and music in 
general.

One that jumps out at you when you look at sheet music is the time
signature. Another is the tempo, sometimes given as a metronome rate. 
Note values are fractions (quarter, sixteenth), so a dotted note means 
multiply the value by 1.5. These things aren't very complicated, but 
they are right out there in front.

A little less visible but even more important is the matter of pitch. 
A pitch is created by a vibration. In the case of your piano, middle A
vibrates a string 440 times a second. Physicists call this a frequency 
of 440 hertz, or 440 cycles per second.

If you go up an octave, you double the frequency. Other intervals are 
other ratios. Going up a fifth (from A to E) multiplies the frequency 
by 3/2, so E is (about) 660 hertz. The Greeks discovered that two 
strings played together sounded pleasant if the lengths of the strings 
were in ratios of small whole numbers: 2:1 (octave), 3:2 (fifth), 4:3 
(fourth), 5:4 (third).

Keyboard instruments like your piano made things more complicated. Each 
piano key participates in a number of different chords, and it turns 
out that the key would have to be tuned slightly differently for each 
chord. Around the time of Bach, there was a lot of debate about this, 
and it was decided to go with a compromise called "equal tempering" 
that has each note a little bit off from what it should be, so that 
each chord will sound okay, though none is perfect.

Here's some math. The interval from A to E (a fifth), in equal 
tempering, isn't exactly 3/2 = 1.5, but 2^(7/12) = 1.4983. (The "^" 
represents an exponent.) The interval of a fourth isn't 4/3 = 1.3333, 
but 2^(5/12) = 1.3348. The interval of a third isn't 5/4 = 1.25, but 
2^(4/12) = 1.2599. I won't say exactly what equal tempering is, but 
you might be able to figure it out from what I said, depending on your 
math level. Equal tempering is related to a "logarithmic scale." You 
can look that up. 

Dave Rusin's Mathematical Atlas website makes other connections between 
music and math:

   http://www.math.niu.edu/~rusin/uses-math/music/   

Jim Campbell's page on "The Equal Tempered Scale and Some Peculiarities 
of Piano Tuning" goes into great detail about something that you depend 
on as you play that has a lot of math in it: piano tuning.

   http://www.izzy.net/~jc/Temper.html   

I hope these ideas help you. You may not need to know the math behind 
music, but it is truly important to you. Music sounds good to us 
because of its mathematical patterns - rhythms and pitches - and math 
has been used over the years to make music sound better. Math and 
music belong together.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/