The Math Behind Music: Pitches, Scales, Geometry

Date: 12/03/97 at 21:30:06
From: Anonymous
Subject: Music

I have a project for math class about how music is connected to math. 
I understand about rhythm and frequency, but I am having trouble 
finding the rest of the information that I need. It would be very helpful 
if you could lend me a hand and tell me where to look. I would really 
appreciate it.

Date: 05/18/98 at 20:18:19
From: Doctor Santu
Subject: Re: Music

Angelica, we're late but in case you were really interested, we've put 
together a fairly extensive reply.

There are a number of connections between music, physics, and math. 
Some of these are easy to describe, while the others are probably more
intelligible only if you have some training in music.

The first set of facts is related to the physics of music, namely, the 
relation between frequencies and pitches, frequency-ratios and 
intervals.

Pitches, frequencies, ratios, and intervals
===========================================
"Pitch" simply means how high a note is. For instance, when I was a 
kid, our school had a tuning fork that vibrated 261.63 times a second,
and it was engraved with the words "middle C." We had a set of 12
tuning forks, and each one was engraved with its note:

   C       =  261.63
   C Sharp =  277.18
   D       =  293.66
   E Flat  =  311.13
   E       =  329.63
   F       =  349.23
   F Sharp =  369.99
   G       =  392.00
   A Flat  =  415.30
   A       =  440.00
   B Flat  =  466.16
   B       =  493.88 
   C'      =  523.25

I'm using the "=" sign carelessly here; I mean "corresponds to." The
note on the left corresponds to the number of vibrations per second on 
the right. We say that the frequency of A is 440 vibrations per 
second, or we say that the frequency of A is 440 Hertz, or sometimes 
we just say A equals 440, and most people know what is meant.

Most people can tell when two notes are an octave apart. Mathematically, 
the frequency of the higher note is exactly double that of the lower 
note. Since octaves of any note in music are denoted by the same 
letter-name, it follows that the frequencies 440, 880, 1760, 220, 110, 
55 are all A's. The 55 will be a really low A, almost the lowest on a 
piano.

In the time of the ancient Greeks, Pythagoras (the famous geometer) 
was fascinated by music. He founded a musical system, all the notes of 
which were simple fractions of each other. For instance, you could 
pick any note you liked to be C; then all the octaves of C would be 
double, four times, eight times the frequency of C, and so on, just as 
they are today. But the fifth note (what we call G) was 3/2 the 
frequency of C. F was 4/3. Some of these proportions are:

   D       =  9/8 of C
   E Flat  =  6/5 of C
   E       =  5/4 of C
   F       =  4/3 of C
   G       =  3/2 of C
   A       =  5/3 of C
   B       = 15/8 of C
   Upper C =  1/2 of C

If you got a computer to generate these frequencies, you would find 
that the notes are almost perfect, but just a shade off to our 
modern ears. However, violinists (and all those who play unfretted 
stringed instruments) generally play these Pythagorean intervals, 
since they sound much sweeter together in combination. Sometimes these 
intervals (distances between notes) are called "Just intervals," and 
some experts say that Just intervals are a little different from 
Pythagorean intervals, but most of them are the same, as far as I 
know.

The theory of overtones and sound quality is very interesting. I 
recommend this study to all mathematicians. Every note in nature is a 
loud basic note, with additional faint notes above it. These higher 
notes, called overtones, cannot be heard individually by ordinary 
people, but it's those overtones that make the difference between the 
sound of a flute, say, and a violin. Different instruments playing the 
same note also play the same overtones, but emphasize or de-emphasize 
them differently; some overtones are stronger in a violin than in a 
flute, for example. It is by putting in carefully engineered overtones 
that a synthesizer imitates instruments.

The overtones for the note C are:

   C', G', C'', E'', G'', [Bb]'', C''', D''', E''', [F#]''', G''', 
   [?]''', [Bb]''', B''', C'''' . . .

(I have put apostrophes to indicate higher octaves. C' is the C above 
middle-C, and so on. The G'' here is a Pythagorean, or "Just" G'', and 
so with the D'''s and the E'''s and so on. The notes in brackets are 
not part of our musical scales at all, so the note [Bb] is only very 
approximately our B Flat.)

If you write this as a vertical list, numbering as you go, an 
interesting fact emerges:

    0  C 
    1  C'
    2  C''
    3  G''
    4  C''
    5  E''
    6  G''
    7  [Bb]''
    8  C'''
    9  D'''
   10  E'''
   11  [F#]'''
   12  G'''
   13  [?]'''
   14  [Bb]'''
   15  B'''
   16  C''''
   ... .....

If you select the overtones that are multiples of 3, you get the 
overtone series for the note G. If you take the overtones that are 
multiples of 5, you get the overtone series for the note E, and so on. 
What we have here is a sort of fractal, a kind of self-similarity. 
Fractals are mathematical diagrams or objects that contain, within 
themselves, scale models of themselves. Examples are Heighway's Dragon 
and Sierpinski's Gasket. (See _The Science of Fractal Images_ by 
Hans-Otto Peitgen and others.)

The Equal-Tempered Scale
========================
As Doctor Jeremiah points out, around the early 1700s, a new idea came 
up. Why not tune pianos and harpsichords in such a way that the ratio 
of a note to the next note on the keyboard was exactly the same for 
every pair of neighbors? The pitches would be different, but the ratio 
of one pitch to the next would be fixed. What did this ratio have to 
be? Let's call it s, for semitone ratio.

One thing that nobody would give up, of course, was that the ratio of 
a note to its octave had to be 2.

Now:

   C#, the note next to C, has to be C*s.
   D, the note next to C#, has to be (C#)*s = C*s*s = C*s^2
   D#, the next note, has to be D*s = .... = C*s*s*s = C*s^3
   ...

In this way, we have this equality:

   C' = C*s*s*s*s*s*s*s*s*s*s*s*s = C*s^12

In other words, s must be a number such that s^12 = 2, so s must be 
the 12th root of 2, which is 1.05946309.

This is the origin of the famous "equal-tempered scale." There is a 
lot of evidence that this was the scale that J. S. Bach called the 
Well-Tempered Scale, which he celebrated by writing 48 preludes and 48 
fugues, four for each note of the scale, two in the major key, two in 
the minor (which you music maniacs will understand, I'm sure). On the 
other hand, there were improved-Just-Intonation scales that were 
called Well-Tempered Scales also, and to go into this question will 
get me in trouble with music experts.

Connections with Geometry
=========================
At a higher level, there are a number of connections. The idea of 
affine transformation is present in fugues. A fugue (pronounced 
"fyoog") is a piece of music with several "voices." The fugue contains 
a stretch of melody called its "subject," and the main idea is that 
this "subject" is repeated all over the fugue. In fact, the fugue 
usually simply drips with the subject. Suppose we consider the fugue 
to be a plane. This is easy when you think of music written on paper.  
Time goes from left to right, and the notes go from high to low, top 
to bottom. Then most of the occurrences of the subject will be just 
time-translations: left-to-right. Sometimes the fugues are moved to 
different pitches, often at the octave, sometimes at other intervals; 
so that's a frequency-translation (up-down). Then, sometimes, the 
subject is stretched out, doubled in length. This is a scaling 
composed with a translation. In extreme cases, the subject is turned 
upside-down, called inversion, and occasionally reversed, 
back-to-front, called retrograde motion.

The first three bars of a Bach fugue can be found at:



I have colored three portions of it. The portion in purple is the 
subject. Even if you don't know music notation, you can sort of 
imagine the melody from the rise and fall of the notes. The portion 
colored blue is the same melody, but inverted. The tune falls where it 
used to rise, rises where it used to fall - it's simply upside-down.  
The portion colored red is also upside down, but slowed down to 
half-speed, so that it lasts twice as long. The shapes of the notes 
reflect that: the longest notes are open, the shorter notes are 
filled-up, even shorter notes are attached to bars; the more bars, the 
shorter the note.

If you would like to hear this Bach piece, go to this URL

http://www.prs.net/bach.html#1000   

a page maintained by Mr. Peter Schwab. Look down the page for the
work called The Art of the Fugue, and click on Contrapunctus 7. Your 
computer has to be able to play MIDI files in order to hear this. 
If that's not possible, your local library just might have a recording 
of _The Art of the Fugue_ by Johann Sebastian Bach; play Contrapunctus 7. 
Another really pretty fugue is Contrapunctus 1. Listen to that and 
think of your friend Dr. Santu!

To really see the most interesting connections between music and math, 
you have to learn more music and more math! This is nice, because if 
you like either one of these, there's a lot of incentive to try and 
learn more of the other. Remember, many mathematicians were musicians, 
including Helmholtz, Poincare, and Sir James Jeans.

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