Math and Music: Harmonic Series

Date: 01/09/98 at 01:04:56
From: DBPanterA
Subject: Math and Music

I am currently writing a paper showing the mathematical properties 
within music. I know about the different scales, as well as 
Pythagoras' contributions. If you could supply me with
information on how math relates to music, it would greatly be 
appreciated.

Sincerely, 

Dietrich Bowar
St. Louis Park High School
St. Louis Park, Minnesota

Date: 01/09/98 at 04:45:59
From: Doctor Mitteldorf
Subject: Re: Math and Music

Dear DB,

First, an answer already in our archives:

Dr. Melissa says:

   First off, I suggest that you get your hands on this book:

   _Emblems of Mind: The Inner Life of Music and Mathematics_,
   by Edward Rothstein.

   It is full of interesting information, is fairly easy to 
   understand without assuming too much math knowledge, and (if I 
   recall correctly) has a nice bibliography that you could use to 
   track down more info.

   For material on the Web, I did a search of the Math Forum site 
   and came up with several items. This one looks the most promising:
-------------------------
   Fractal Music Project

   http://www-ks.rus.uni-stuttgart.de:80/people/schulz/fmusic/   

   New field of music research. "Fractal music is a result of a 
   recursive process where an algorithm is applied multiple times to 
   process its previous output. In wider perspective all musical 
   forms, both in micro and macro level can be modelled with this 
   process. Fractals provide extremely interesting musical results." 
   Mailing list; Fractal Jazz; papers on fractal music and related 
   topics; software; short examples of fractal music.
   -------------------------

   You might also find something useful in "Leonardo," a journal 
   published by the International Society of the Arts, Sciences and  
   Technology. It's on the Web at:

   http://mitpress.mit.edu/Leonardo/home.html   

   Hope this helps you out, and good luck with your project!
----------------------------------------------------------------------

Here are some more interesting facts to start you off:

If you move to the right along the piano, each note is higher than 
the next. But instead of equal steps as you move from one note to the 
next, the notes are actually in EQUAL RATIOS from one to the next.  
In fact, the ratio of the pitches of the C and C sharp notes is 
1.05946.  The ratio of a D to a C sharp is exactly the same, and the 
ratio from a D to a D sharp is again identical.

This number, 1.05946, is calculated in such a way that twelve such 
steps (which carries you a full octave from one C to the next) is 
exactly a factor of 2. This is an innovation that J.S. Bach came 
up with - tuning the piano in exactly this fashion. It's called 
"well-tempered tuning."

A series of numbers in which each one is bigger (or smaller) than the 
last by a constant factor is called a "harmonic series," precisely 
because such series were first studied in the context of music.
 
So if you take a string and pluck it, you get a pitch. If you divide
the string in half and pluck each half, you get a pitch that's exactly
twice as high, and the sound is an octave higher.

It just so happens that 7 of the 12 steps, each of which is a factor 
of 1.05946, gets you to a number that is very close to 1.5, or 3/2.  
This is the sound called a "perfect fifth," the sound of a C and a G 
on the piano.

You can get the sound of a fifth from a violin string by dividing the 
string in thirds with your finger - the sound that is produced is a 
fifth + an octave above the fundamental.

Furthermore, the ratio produced by four piano steps is close to 1.25, 
or 5/4. This means that the notes CEG, which are a basic "chord" on 
the piano, are in the ratio 6:5:4. All such chords, whether they start 
on C or any other note, have the same ratio 6:5:4. 
 
I hope you continue to read about this subject, and become an expert!

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