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### Math and the Piano

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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).
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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).

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
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/
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